Electromagnetic Integral Equations: Insights in Conditioning and Preconditioning

Abstract

Integral equation formulations are a competitive strategy in computational electromagnetics but, lamentably, are often plagued by ill-conditioning and by related numerical instabilities that can jeopardize their effectiveness in several real case scenarios. Luckily, however, it is possible to leverage effective preconditioning and regularization strategies that can cure a large majority of these problems. Not surprisingly, integral equation preconditioning is currently a quite active field of research. To give the reader a propositive overview of the state of the art, this paper will review and discuss the main advancements in the field of integral equation preconditioning in electromagnetics summarizing strengths and weaknesses of each technique. The contribution will guide the reader through the choices of the right preconditioner for a given application scenario. This will be complemented by new analyses and discussions which will provide a further and more intuitive understanding of the ill-conditioning of the electric field (EFIE), magnetic field (MFIE), and combined field integral equation (CFIE) and of the associated remedies.

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Electromagnetic Integral Equations:
Insights in Conditioning and Preconditioning
Simon B. Adrian, Member, IEEE, Alexandre Dély, Davide Consoli, Student Member, IEEE,
Adrien Merlini, Member, IEEE, and Francesco P. Andriulli, Senior Member, IEEE
Integral equation formulations are a competitive strategy in computational electromagnetics but, lamentably, are often plagued by
ill-conditioning and by related numerical instabilities that can jeopardize their effectiveness in several real case scenarios. Luckily,
however, it is possible to leverage effective preconditioning and regularization strategies that can cure a large majority of these
problems. Not surprisingly, integral equation preconditioning is currently a quite active field of research. To give the reader a
propositive overview of the state of the art, this paper will review and discuss the main advancements in the field of integral
equation preconditioning in electromagnetics summarizing strengths and weaknesses of each technique. The contribution will guide
the reader through the choices of the right preconditioner for a given application scenario. This will be complemented by new
analyses and discussions which will provide a further and more intuitive understanding of the ill-conditioning of the electric field
(EFIE), magnetic field (MFIE), and combined field integral equation (CFIE) and of the associated remedies.
Index Terms—Integral Equations, Boundary Element Method, Computational Electromagnetic, Preconditioning, EFIE, MFIE.
I. Introduction
Integral equation formulations, solved by the boundary
element method (BEM), have become a well established tool
to solve scattering and radiation problems in electromagnet-
ics [1]–[4]. What makes these schemes so suitable for electro-
magnetic analyses is that, differently from approaches based
on differential equations such as the finite element method
(FEM) or the finite-difference time-domain method (FDTD),
they naturally incorporate radiation conditions without the
need for artificial absorbing boundary conditions, they only set
unknowns on boundary surfaces (two-dimensional manifolds)
instead of discretizing the entire volume, and they are mostly
free from numerical dispersion. On the other hand, linear sys-
tem matrices arising from differential equations schemes are
sparse [5], while those arising in BEM are, in general, dense.
This drawback, however, can be overcome if a fast method
such as the multilevel fast multipole method (MLFMM) [6],
the multilevel matrix decomposition algorithm (MLMDA) [7]
and later equivalents [8]–[12] are used at high frequency or
the adaptive cross approximation (ACA)/H-matrix methods
and related schemes [13]–[18] are used at lower frequencies.
These schemes are often capable of performing matrix-vector
This work was supported in part by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No 724846, project 321), by the Italian Ministry
of University and Research within the Program PRIN2017, EMVISION-
ING, Grantno. 2017HZJXSZ, CUP:E64I190025300, by the Italian Ministry
of University and Research within the Program FARE, CELER, Grantno.
R187PMFXA4, by the Région Bretagne and the Conseil Départemental du
Finistère under the project “TONNERRE”, by the ANR Labex CominLabs
under the project “CYCLE”, and by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) – SFB 1270/1–299150580.
S. B. Adrian is with Universität Rostock, Rostock, Germany (e-mail:
simon.adrian@uni-ro).
A. Dély is with the Politecnico di Torino, Turin, Italy (e-mail: alexan-
dre.dely@polito.it).
D. Consoli is with the Polytechnic University of Turin, Turin, Italy (e-
mail: davide.consoli@polito.it).
A. Merlini is with IMT Atlantique, Brest, France (e-mail: adrien.mer-
lini@imt-atlantique.fr).
F. P. Andriulli is with the Politecnico di Torino of Turin, Turin, Italy
(e-mail: francesco.andriulli@polito.it).
products in O(𝑁log 𝑁)or even 𝑂(𝑁)complexity, where 𝑁
denotes the number of unknowns (the linear system matrix
dimension). Thus the complexity to obtain the BEM solution
of the electromagnetic problem is, when an iterative solver
is used, O(𝑁iter𝑁log 𝑁)(or 𝑂(𝑁iter 𝑁)in the low-frequency
regime), where 𝑁iter is the number of iterations.
The number of iterations 𝑁iter is generally correlated with
the condition number of the linear system matrix, that is,
the ratio between the largest and smallest singular values of
the matrix [19]. This number is often a function of 𝑁and,
when the BEM formulation is set in the frequency domain,
of the wavenumber 𝑘. This can potentially result in a solution
complexity greater, and sometimes much greater, than 𝑂(𝑁2),
something that would severely jeopardize the other advantages
of using BEM approaches.
For this reason it is of paramount importance to address
and solve all sources of ill-conditioning for integral equations
and, not surprisingly, this has been the target of substantial
research in the last decade that this work will analyze, review,
and summarize.
For surface integral equations (SIEs) that model scatter-
ing or radiation problems for perfect electrical conductors
(PEC) geometries, we can typically distinguish the following
sources of ill-conditioning: i) the low-frequency breakdown, ii)
the h-refinement (dense-discretization) breakdown, iii) high-
frequency issues (including internal resonances and the high-
frequency breakdown), and iv) the lack of linear independence
in the basis elements (including lack of orthogonality and mesh
irregularities).
Some of the first methods explicitly addressing electromag-
netic integral equation ill-conditioning date back to the 1980s,
when the focus was on the low-frequency breakdown [20]
and on the problem of interior resonances [21]. Since then,
a plethora of schemes and strategies addressing one or more
of the issues i)–iv) have been presented and some of these
strategies are still the topic of intense research. In the past, a
few review articles have appeared that dealt with aspects of
stabilizing ill-conditioned electromagnetic integral equations.

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Most recently, Antoine and Darbas [22] presented an extensive
review on operator preconditioning with a focus on high-
frequency issues. A few years ago, Ylä-Oijala et al [23]
discussed issues in finding a stable and accurate integral
equation formulation and they addressed certain open issues
in preconditioning, and Carpentieri discussed preconditioning
strategies with a focus on large-scale problems [24], [25].
Finally, although for space limitation this paper will focus
on the electric field integral equation (EFIE) and magnetic
field integral equation (MFIE) operators (which are the fun-
damental building blocks for several other formulations), the
reader should not that a substantial amount of literature and
quite effective preconditioned methods have been presented
for modelling penetrable bodies both homogeneous and in-
homogeneous [26]–[29]. The reader should also be aware
that domain decomposition schemes can play a fundamen-
tal role in managing and solving electromagnetic problems
containing even severely ill-conditioned operators [30], [31].
These approaches, however, are per se a discipline within
Computational Electromagnetics and any brief treatment out-
side of a dedicated review would inevitably be insufficient
and partial. Moreover, domain decomposition algorithms are
not competing with the strategies discussed here but, often
times, complementary [32]. For these reasons, we will not
treat domain decomposition strategies in this review, but
rather refer the interested reader to the excellent contributions
in literature [33]–[35]. Similarly, discontinuous Galerkin and
related methods for handling non-conformal meshes will not
be treated here, as extensive additional treatments would be
required; the reader can refer to [36], [37] and references
therein for specific discussions on this family of methods.
The purpose of this article is two-fold: on the one hand,
we review and discuss the strategies that have been devised
in the past to overcome the sources of ill-conditioning i)-
iv) summarizing strengths and weaknesses, guiding the reader
through the choices of the right preconditioner for a given
application scenario. On the other hand, we complement the
overview with new results that contribute to better character-
izing the ill-conditioning of the EFIE and MFIE. Finally, we
will complement our discussions with a spectral analysis of
the formulations on the sphere, which will provide a further
and more intuitive understanding of the ill-conditioning of the
EFIE, MFIE, and combined field integral equation (CFIE) and
of the associated potential remedies. In contrast to [22], our
focus will include low-frequency effects and wideband stable
formulations as well Calderón and quasi-Helmholtz projection
strategies. Moreover, whenever appropriate, we will provide
implementational considerations and details that will enable
the reader to dodge all practical challenges that are usually
faced when engineering the most effective preconditioning
schemes.
This paper is organized as follows: Section II introduces
the background material and sets up the notation, Section III
reviews the connection of the spectrum of matrices and the
role of condition number in the solutions of the associated
linear systems. Section IV focuses on low-frequency scenarios
analyzing their main challenges and solution strategies. Sec-
tion V presents problems and solutions associated with highly
refined meshes, while Section VI focuses on scenarios in the
high-frequency regime. Section VII considers the low of mesh
and basis functions quality on the overall conditioning and
Section VIII presents the conclusions and final considerations.
II. Notation and Background
We are interested in solving the electromagnetic scattering
problem where a time-harmonic, electromagnetic wave (𝒆i,𝒉i)
in a space with permittivity 𝜀and permeability 𝜇impinges on
a connected domain 𝛺−⊂R3with PEC boundary 𝛤≔𝜕 𝛺−
resulting in the scattered wave (𝒆s,𝒉s). The total electric 𝒆B
𝒆i+𝒆sand magnetic 𝒉B𝒉i+𝒉sfields satisfy Maxwell’s
equations
∇ × 𝒆(𝒓)=+i𝑘𝒉(𝒓),for all 𝒓∈𝛺+,(1)
∇ × 𝒉(𝒓)=−i𝑘𝒆(𝒓),for all 𝒓∈𝛺+,(2)
where 𝛺+≔𝛺−
c,𝑘≔𝜔√𝜀𝜇 is the wave number, 𝜔
the angular frequency, and 𝒆,𝒉must satisfy the boundary
conditions for PEC boundaries
ˆ
𝒏×𝒆=0,for all 𝒓∈𝛤 , (3)
ˆ
𝒏×𝒉=𝒋𝛤,for all 𝒓∈𝛤 , (4)
where 𝒋𝛤is the induced electric surface current density. In
addition, 𝒆sand 𝒉smust satisfy the Silver-Müller radiation
condition [38], [39]
lim
𝑟→∞𝒉s×𝒓−𝑟𝒆s=0.(5)
We assumed (and suppressed) a time dependency of e−i𝜔𝑡 and
normalized 𝒉with the wave impedance 𝜂≔𝜇/𝜀.
To find (𝒆s,𝒉s), we can solve the EFIE
T𝑘𝒋𝛤=−ˆ
𝒏×𝒆i(6)
for 𝒋𝛤, where ˆ
𝒏is the surface normal vector directed into 𝛺+
and
T𝑘≔i𝑘T
A,𝑘 +1/(i𝑘)T
Φ,𝑘 (7)
is the EFIE operator composed of the vector potential operator
(T
A,𝑘 𝒋𝛤)(𝒓)=ˆ
𝒏×𝛤
𝐺𝑘(𝒓,𝒓0)𝒋𝛤(𝒓0)d𝑆(𝒓0)(8)
and the scalar potential operator
(T
Φ,𝑘 𝒋𝛤)(𝒓)=−ˆ
𝒏×grad𝛤𝛤
𝐺𝑘(𝒓,𝒓0)div𝛤𝒋𝛤(𝒓0)d𝑆(𝒓0),
(9)
where
𝐺𝑘(𝒓,𝒓0)=ei𝑘|𝒓−𝒓0|
4π|𝒓−𝒓0|(10)
is the free-space Green’s function. A definition of the surface
differential operators grad𝛤and div𝛤can be found in [40,
Appendix 3] or [41, Chapter 2]. Once 𝒋𝛤is obtained, 𝒆s,𝒉s
can be computed using the free-space radiation operators.
Alternatively, one can solve the MFIE for the exterior
scattering problem
ˆ
𝒏×𝒉i=M+
𝑘𝒋𝛤≔+I/2+K𝑘𝒋𝛤,(11)

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𝒗+
𝑛
𝒗−
𝑛
𝒓+
𝑛
𝒓−
𝑛
𝑐+
𝑛
𝑐−
𝑛
𝒆𝑛
Fig. 1. The vector field of an RWG function. The vector 𝒆𝑛denotes the
directed edge, 𝑐+
𝑛and 𝑐−
𝑛denote the domains of the cells, 𝑣+
𝑛and 𝑣−
𝑛denote
vertices on the edge 𝒆𝑛, and 𝒓+
𝑛and 𝒓−
𝑛are the vertices opposite to the edge
𝒆𝑛.
where Iis identity operator, M+
𝑘is the MFIE operator for the
exterior scattering problem, and
K𝑘𝒋𝛤(𝒓)B−ˆ
𝒏×𝛤∇𝐺𝑘(𝒓,𝒓0) × 𝒋𝛤d𝑆(𝒓0).(12)
The MFIE operator for the interior scattering problem is
M−
𝑘≔−I/2+K𝑘and will be used later in the construction
of preconditioners.
The EFIE and the MFIE have non-unique solutions for
resonance frequencies. A classical remedy is the use of the
CFIE [21]
−𝛼T𝑘𝒋𝛤+ (1−𝛼)ˆ
𝒏×M+
𝑘𝒋𝛤
=𝛼ˆ
𝒏×𝒆i+ (1−𝛼)ˆ
𝒏׈
𝒏×𝒉i(13)
which is uniquely solvable for all frequencies.
For the discretization of the EFIE, we employ Rao-Wilton-
Glisson (RWG) basis functions 𝒇𝑛∈𝑋𝒇which are here—in
contrast to their original definition in [42]—not normalized
with the edge length, that is,
𝒇𝑛=











𝒓−𝒓+
𝑛
2𝐴𝑐+
𝑛
for 𝒓∈𝑐+
𝑛,
𝒓−
𝑛−𝒓
2𝐴𝑐−
𝑛
for 𝒓∈𝑐−
𝑛
(14)
using the convention depicted in Figure 1.
Following a Petrov-Galerkin approach, we obtain the system
of equations
T𝑘j=i𝑘TA,𝑘 +1/(i𝑘)TΦ, 𝑘 j=−ei(15)
that can be solved to obtain an approximation of the solution
in the form 𝒋𝛤≈𝑛[j]𝑛𝒇𝑛and where
TA,𝑘 𝑛 𝑚 Bhˆ
𝒏×𝒇𝑛,T
A,𝑘 𝒇𝑚i𝛤,(16)
TΦ,𝑘 𝑛 𝑚 Bhˆ
𝒏×𝒇𝑛,T
Φ,𝑘 𝒇𝑚i𝛤,(17)
ei𝑛
Bhˆ
𝒏×𝒇𝑛,ˆ
𝒏×𝒆ii𝛤,(18)
with
h𝒇,𝒈i𝛤B𝛤
𝒇(𝒓) · 𝒈(𝒓)d𝑆(𝒓).(19)
Even though we are testing with ˆ
𝒏×𝒇𝑛, the resulting system
matrix T𝑘is the one from [42] (up to the fact that the
RWG functions we are using are not normalized), because
our definition of the EFIE operator includes an ˆ
𝒏×term (in
contrast to [42]).
For the discretization of the MFIE, functions dual to the
RWGs must be used for testing [43]. Historically, the first
dual basis functions for surface currents where introduced by
Chen and Wilton for a discretization of the Poggio-Miller-
Chang-Harrington-Wu-Tsai (PMCHWT) equation [44]. Later
and independently, Buffa and Christiansen introduced the
Buffa-Christiansen (BC) functions [45], which differ from the
Chen-Wilton (CW) functions in that the charge on the dual
cells is not constant. Figure 2 shows a visualization of a BC
function. In our implementation, we are using BC functions
and denote them as 𝒇∈𝑋
𝒇, where the tilde indicates that
the function is defined on the dual mesh. The analysis is,
however, applicable to CW functions as well, and thus, we
will mostly speak of “dual functions” to stress the generality
of our analysis. For a definition of the BC functions as well
as implementation details, we refer the reader to [46]. For the
discretization of the MFIE, we obtain
M+
𝑘j≔1/2Gˆ
𝒏×𝒇,𝒇+K𝑘j=hi,(20)
where
[K𝑘]𝑛𝑚 Bhˆ
𝒏×𝒇𝑛,K𝑘𝒇𝑚i𝛤,(21)
hi𝑛
Bhˆ
𝒏×𝒇𝑛,ˆ
𝒏×𝒉ii𝛤.(22)
and where the Gram matrix for any two function spaces 𝑋𝑓
and 𝑋𝑔is defined as
G𝑓 ,𝑔 𝑚𝑛 ≔h𝑓𝑚, 𝑔𝑛i𝛤,(23)
with 𝑓𝑚∈𝑋𝑓and 𝑔𝑛∈𝑋𝑔.
For the discretization of the CFIE, we have
C𝑘j≔−𝛼T𝑘+ (1−𝛼)G𝒇,𝒇G−1
ˆ
𝒏×
𝒇,𝒇M+
𝑘j
=𝛼ei+ (1−𝛼)G𝒇,𝒇G−1
ˆ
𝒏×
𝒇,𝒇hi(24)
with the combination parameter 0< 𝛼 < 1.
III. Condition Numbers, Iterative Solvers, and
Computational Complexity
To solve the linear system of equations arising from bound-
ary element discretizations, such as (15), one can resort either
to (fast) direct or to iterative solvers. For direct solvers, the time
to obtain a solution is independent from the right-hand side,
whereas for iterative solvers, the right-hand side as well as the
spectral properties of the system matrix influence the solution
time. Standard direct solvers such as Gaussian elimination
have a cubic complexity, which renders them unattractive for
large linear systems. Recent progress in the development of
fast direct solvers has improved the overall computational
cost [47]–[50].

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𝒗+
𝑛
𝒗−
𝑛
𝒓+
𝑛
𝒓−
𝑛
𝑐+
𝑛
𝑐−
𝑛
𝒆𝑛
Fig. 2. The vector field of a BC function.
Iterative solvers, on the other hand, start from an initial
guess of the solution, x(0), and compute a sequence of ap-
proximate solutions, where the following element of such a
sequence is based on the previously computed one, until a
desired accuracy is achieved. Formally, given a linear system
of equations
Ax =b,(25)
an iterative solver should stop when Ax(𝑖)−b/kbk< 𝜀,
where 𝜀 > 0is the solver tolerance and x(𝑖)the approximate
solution after the 𝑖th iteration. Whether an iterative solver
will converge or not, depends on the chosen solver and the
properties of A, as we will discuss in the following.
To assess the overall complexity in 𝑁for obtaining an
approximation of xwithin the tolerance 𝜀, a relation between
𝑁iter and 𝑁is needed. One way to obtain such a relationship
is via the condition number of the matrix, which is defined as
cond A=A2A−12
=𝜎max(A)
𝜎min(A),(26)
where kk2is the spectral norm, and 𝜎max/min denotes the
maximal and minimal singular value.
In the case of the conjugate gradient (CG) method, which
requires Ato be Hermitian and positive definite, there is an
upper bound on the error e(𝑖)≔x(𝑖)−xgiven by [51]
e(𝑖)A≤2√cond A−1
√cond A+1𝑖e(0)A,(27)
where kkAis the energy norm defined by kxkA≔x†Ax 1/2
and x†denotes the conjugate transpose of x. If the objective
is to reduce the relative error e(𝑖)/e(0)below 𝜀and by
considering limits for cond A1, one notes [51] that
𝑖≤1
2√cond Aln 2
𝜀(28)
iterations are at most needed (assuming an exact arithmetic).
If the condition number grows linearly in 𝑁, as observed for
the EFIE when the mesh is uniformly refined, this implies that
the complexity is at most O(𝑁1.5log 𝑁).
One could argue that this is an overly simplified picture
of the situation; indeed, the CG method is not applicable to
standard frequency domain integral equations as the resulting
system matrices are neither Hermitian nor positive definite.
One strategy to still obtain a bound on the number of iterations
is to use the CG method on the normal equation
A†Ax =A†b.(29)
The price for this, however, is that the condition number of the
resulting system matrix is (cond A)2and thus this approach
is, for the standard formulations, of little practical value. In
addition, round-off errors due to finite precision can lead to a
non-converging solver—despite the theory dictating that CG
should converge in at most 𝑁steps [52], [53]. Thus, the
condition number bound is relevant in practice often only in
the case that cond Ais small.
The problem with other popular Krylov methods such as
the generalized minimal residual (GMRES) or the conjugate
gradient squared (CGS) method is that, for general matrices,
no bound on the number of iterations in terms of the condition
number alone is available. In fact, even if two matrices have
the same condition number, their convergence behavior can
significantly differ: the distribution of the eigenvalues in the
complex plane impacts the convergence behavior as well [22].
Typically, a better convergence can be observed if all the
eigenvalues are located on either the real and or imaginary
axis and are either strictly positive or negative (if they are
on the imaginary axis, then positive or negative with respect
to Im (𝜆𝑖)). We will see in the following that, under certain
conditions, for low-frequency electromagnetic problems it is
possible to cluster the eigenvalues on the real axis and that
the condition number becomes a good indicator of the con-
vergence behavior. Moreover, some preconditioning strategies,
such as the refinement-free Calderón preconditioner which
will be discussed in Section V-A2, give rise to a Hermitian,
positive-definite system, and thus the CG and the associated
convergence theory is applicable.
For frequency-independent problems, it is customary to call
a formulation well-conditioned if cond Ais asymptotically
bounded by a constant 𝐶, which is independent from the
average edge length ℎof the mesh. For dynamic problems,
however, we also need to study the condition number as a
function of the frequency 𝑓≔𝜔/(2π), and one must specify
if a formulation is well-conditioned with respect to ℎ, to
𝑓, to both, or only in a particular regime, for example, for
frequencies where the corresponding wavelength is larger then
the diameter of 𝛤.
The classical remedy to overcome ill-conditioning and thus
improve the convergence behavior of iterative solvers is to
use a preconditioning strategy. Such a strategy results, in the
general case, in a linear system
PLAPRy=PLb,(30)
where x=PRyand the matrices should be chosen such that,
if possible,
cond (PLAPR)≤𝐶 , (31)

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where 𝐶is a constant both independent of ℎand 𝑓(in which
case the preconditioner is optimal). Normally, the matrix-
matrix products in (30) are not formed explicitly and, to be an
efficient preconditioner, the cost of a matrix-vector product
should not jeopardize the lead complexity set by the fast
method. In practice, to obtain an optimal preconditioner, the
nature of the underlying operators must be taken into account.
Thus, in the following sections, we will analyze the spectral
properties of the (discretized) EFIE, MFIE, and CFIE operator,
discuss the causes of their ill-conditioning as well as potential
remedies.
IV. Low-Frequency Scenarios
The low-frequency breakdown of the EFIE, that is, the
growth of the condition number when the frequency 𝒇de-
creases, was one of the first sources of ill-conditioning of the
EFIE to be studied. From a physical point of view, several
problems at low-frequency are rooted in the decoupling of the
electric and the magnetic field in the static limit: magnetostatic
loop currents excite the magnetic field and electrostatic charges
excite the electric field [20]. Both the EFIE and the MFIE
suffer from computational challenges at low-frequencies. As
we will see in this section, the EFIE suffers from conditioning
issues when the frequencies decreases and so does, albeit
for different reasons, the MFIE when applied to non-simply
connected geometries (i.e., geometries containing handles like
the torus illustrated in Figure 4, for example). The condition
number growth is, however, only one of the possible problems:
finite machine precision and inaccuracies due to numerical
integration that result in catastrophic round-off errors are
also plaguing the otherwise low-frequency well-conditioned
integral equations such as the MFIE on simply-connected
geometries. Together, these issues make the two formulations
increasingly inaccurate as the frequency decreases, which is
attested by the low-frequency radar cross sections illustrated
in Figure 3 that show wildly inaccurate results for the standard
formulations.
The low-frequency analysis of electromagnetic integral
equations benefits from the use of Helmholtz and quasi-
Helmholtz decompositions that we will summarize here for
the sake of completeness and understanding. The well-known
Helmholtz decomposition theorem states that any vector field
can be decomposed into a solenoidal, irrotational, and a
harmonic vector field, which in the case of a tangential surface
vector field such as 𝒋𝛤leads to [41, p. 251]
𝒋𝛤=curl𝛤𝛷+grad𝛤𝛹+𝑯(32)
where 𝛷and 𝛹are sufficiently smooth scalar functions,
curl𝛤𝛷≔grad𝛤𝛷׈
𝒏, and div𝛤𝑯=curl𝛤𝑯=0;
here, curl𝛤is the adjoint operator of curl𝛤, that is, we have
hcurl𝛤𝑓 , 𝒈i𝛤=h𝑓 , curl𝛤𝒈i𝛤(see [41, see (2.5.194)]). The
space of harmonic functions 𝐻𝑯(𝛤)is finite dimensional with
dim 𝐻𝑯(𝛤)=2𝑔on a closed surface, where 𝑔is the genus of
𝛤. The Helmholtz subspaces are all mutually orthogonal with
respect to the 𝑳2(𝛤)-inner product.
When 𝒋𝛤is a linear combination of div- but not curl-
conforming functions (e.g., RWG and BC functions), only
00.511.522.53
−1,400
−1,200
−1,000
Angle [rad]
Radar Cross Section [dBsm]
Mie series EFIE P-EFIE
Loop-star EFIE MFIE
Fig. 3. Radar cross sections calculated, with different formulations, for the
sphere of unit radius discretized with an average edge length of 0.15 m, and
excited by a plane wave of unit polarization along ˆ
𝒙and propagation along
ˆ
𝒛oscillating at 𝑓=10−20 Hz. The “EFIE” and “MFIE” labels refer to the
standard formulations (15) and (20), while the “Loop-star EFIE” and “P-
EFIE” refer to the EFIE stabilized with the loop-star (61) and quasi-Helmholtz
projectors (72), respectively.
Fig. 4. Illustration of a torus and the corresponding toroidal (in blue) and
poloidal (in orange) loops.
a quasi-Helmholtz decomposition is possible, where 𝒋𝛤is
decomposed into a solenoidal, a non-solenoidal, and a quasi-
harmonic current density. It is not possible to obtain irrota-
tional or harmonic current densities, since the curl of div-
conforming (but not curl-conforming) functions such as the
RWGs (or their dual counterparts) is, in general, not existing as
a classical derivative; therefore, it is termed quasi-Helmholtz
decomposition. Next we introduce the quasi-Helmholtz de-
compositions for primal (i.e., RWGs) and dual (i.e., BCs)
functions that we will use for our analysis in the next section.
Just as the Helmholtz decomposition (32) decomposes the
continuous solution 𝒋𝛤, a quasi-Helmholtz decomposition
decomposes the discrete solution jas
𝑁

𝑛=1[j]𝑛𝒇𝑛=
𝑁V

𝑛=1[j𝜦]𝑛𝜦𝑛+
𝑁C

𝑛=1[j𝜮]𝑛𝜮𝑛+
2𝑔

𝑛=1[j𝑯]𝑛𝑯𝑛,
(33)
where 𝜦𝑛∈𝑋𝜦are solenoidal loop functions, 𝜮𝑛∈𝑋𝜮
are non-solenoidal star functions, and 𝑯𝑛∈𝑋𝑯are quasi-

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harmonic global loops [54] and where j𝜦,j𝜮, and j𝑯are
the vectors containing the associated expansion coefficients;
moreover, 𝑁Vis the number of vertices and 𝑁Cis the number
of cells of the mesh.
We highlight some of the properties which we are going
to use throughout this article. First, and most importantly, the
functions 𝜦𝑛,𝑯𝑛, and 𝜮𝑛can be represented in terms of
RWG functions [54]. Thus the expansion coefficients are linked
by linear transformation matrices Λ,H, and Σ. For the loop
transformation matrix, we have
[Λ]𝑖 𝑗 =







1for 𝒗𝑗=𝒗−
𝑖
−1for 𝒗𝑗=𝒗+
𝑖
0otherwise,
(34)
where 𝒗𝑗is the 𝑗th vertex of the mesh (inner vertex if 𝛤is
open), and for the star transformation matrix
[Σ]𝑖 𝑗 =







1for 𝑐𝑗=𝑐+
𝑖
−1for 𝑐𝑗=𝑐−
𝑖
0otherwise,
(35)
where 𝑐𝑗is the 𝑗th cell of the mesh, following the conventions
depicted in Figure 1. With the definition of these matrices, the
quasi-Helmholtz decomposition in (33) can be equivalently
written as
j=Λj𝜦

=𝑗sol
+Hj𝑯

=jqhar
+Σj𝜮

=𝑗nsol
=jsol +jqhar +jnsol .(36)
The linear combinations of RWGs implied by the coefficient
vectors jsol,jnsol, and jqhar are solenoidal, non-solenoidal, and
quasi-harmonic current densities. These decompositions are
not unique: if we were to use, for example, the loop-tree
quasi-Helmholtz decomposition, we would obtain different
coefficient vectors jsol,jnsol, and jqhar . The decomposition is,
however, unique with respect to the loop-star space, that is,
when the linear dependency of loop and of star functions
(see [55] and references therein) is not resolved by arbitrarily
eliminating a loop and a star function; what sets the loop-star
basis apart from other quasi-Helmholtz decompositions is the
symmetry with respect to dual basis functions. A symmetry
that we are now going to further highlight.
First, we give to Λand Σa meaning that goes beyond
merely interpreting them as basis transformation matrices.
The matrices Λand Σare edge-node and edge-cell incidence
matrices of the graph defined by the mesh and they are
orthogonal, that is, ΣTΛ=0. It follows that jsol and jnsol are
𝒍2-orthogonal, that is, jT
nsoljsol =0. We find this noteworthy
for two reasons: i) the loop 𝜦𝑖and star functions 𝜮𝑗are, in
general, not 𝑳2-orthogonal (after all, 𝜮𝑗is not irrotational);
ii) the 𝒍2-orthogonality is not true for other quasi-Helmholtz
decompositions such as the loop-tree basis. In light of this
consideration, the matrices Λand Σcould be interpreted as the
graph curl (Λ) and graph gradient (Σ) of the standard mesh,
an interpretation that further increases the correspondence with
the continuous decomposition (32).
For global loops 𝑯𝑛, no such simple graph-based definition
exists. Indeed, they are, in general, not uniquely defined and
must be constructed from a search of holes and handles. For
any global loop basis so obtained, we have ΣTH=0; however,
ΛTH=0is, in general, not true. This property can be enforced
by constructing Has the right nullspace of Λ ΣT. Such a
construction is possible, for example, via a full singular value
decomposition (SVD), or, via more computationally efficient
randomized projections [56]. However, the computational cost
is higher, in general, compared with using a global loop-
finding algorithm, in particular, since Hwill be a dense matrix.
A similar decomposition can be obtained for dual functions
𝑁

𝑛=1[m]𝑛𝒇𝑛=
𝑁V

𝑛=1m
𝜦𝑛
𝜦𝑛+
𝑁C

𝑛=1m
𝜮𝑛
𝜮𝑛+
2𝑔

𝑛=1m
𝑯𝑛
𝑯𝑛,
(37)
where, in contrast to the RWG case, 
𝜦𝑛are non-solenoidal
dual star and 
𝜮𝑛are solenoidal dual loop functions. In matrix
notation, we have
j=Σm
𝜮

msol
+
Hm 
𝑯

mqhar
+Λm
𝜦

mnsol
=msol +mqhar +mnsol .(38)
Note that the same matrices Σand Λare present both in
the decomposition of RWG functions and in the one of dual
functions. However, while for RWGs the transformation matrix
Λdescribes solenoidal functions and the transformation matrix
Σdescribes non-solenoidal functions, the opposite is true for
the dual functions: it is Σthat describes solenoidal functions,
while Λdescribes non-solenoidal. Thus on the dual mesh, Λ
acts as graph gradient and Σas a graph curl. This is consistent
with the definition of dual functions: dual basis functions can
be interpreted as a div-conforming “rotation” by 90°of the
primal functions (note that the functions ˆ
𝒏×𝒇𝑖are a rotation
by 90°, which is not div-confirming); given that curl𝛤𝛷≔
∇𝛤𝛷׈
𝒏, it is consistent that the roles of Λand Σas graph
counterparts to continuous differential surface operators are
swapped on the dual mesh with respect to the primal mesh.
Regarding the quasi-harmonic functions, it must be empha-
sized that we cannot identify 
H=H. This equality is only true
if His the nullspace of Λ ΣT, a condition, which evidently
leads to the aforementioned unique definition of H. Even
though the construction of Has the nullspace of Λ ΣTis
cumbersome—and by introducing quasi-Helmholtz projectors
in the following, we will sidestep it—it suggests that these
global loops are capturing the analytic harmonic Helmholtz
subspace better than arbitrarily chosen global loops.
A. Electric Field Integral Equation
To put into light the low-frequency challenges that plague
the EFIE, its behavior on both the solenoidal and the non-
solenoidal subspaces must be analyzed. The following de-
velopments focus on geometries that do not contain global
loops, however the results can be immediately extended to the
general case by considering that—in the case of the EFIE—
global and local loops have similar properties. While they have
practical limitations, loop-star bases are a convenient tool to
perform this analysis. The loop-star transformed EFIE matrix

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TLS
𝑘BΛ ΣTT𝑘Λ Σcan be represented in block
matrix form as
TLS
𝑘=ΛTT𝑘Λ ΛTT𝑘Σ
ΣTT𝑘Λ ΣTT𝑘Σ,(39)
and the corresponding matrix equation now reads TLS
𝑘𝒋LS =
Λ ΣT𝒆i, where 𝒋=Λ Σ𝒋LS . In these definitions, the
Λand Σmatrices refer to the full-rank transformation matrices
in which linearly dependent columns have been removed:
for each connected component of 𝛤one star basis function
(column of Σ) must always be removed and one loop basis
function must be removed (column of Λ) if the component is
closed [57].
To evidence the different low-frequency behaviors of the
EFIE matrix on the solenoidal and non-solenoidal subspaces,
the properties ΛTTΦ, 𝑘 =0and TΦ,𝑘 Λ=0, which follow di-
rectly from the divergence-free nature of solenoidal functions,
must be enforced. In addition, the behavior of the matrix terms
must be derived by performing a Taylor series expansion of
the Green’s function in both T
A,𝑘 and T
Φ,𝑘 for 𝑘→0. For
instance,
hˆ
𝒏×𝜮𝑚,T
A,𝑘 𝜦𝑛i𝛤=
𝑘→0𝛤𝛤
𝜮𝑚(𝒓) · 𝜦𝑛(𝒓0)
4π
1
𝑅−𝑘2𝑅
2−i𝑘3𝑅2
6+ O(𝑘4)d𝑆(𝒓0)d𝑆(𝒓),(40)
where 𝑅=|𝒓−𝒓0|and where we have used
𝛤i𝑘𝜦𝑛(𝒓0)d𝑆(𝒓0)=0. We can deduce that, in general,
Re hˆ
𝒏×𝜮𝑚,T
A,𝑘 𝜦𝑛i𝛤=
𝑘→0O(1),(41)
Im hˆ
𝒏×𝜮𝑚,T
A,𝑘 𝜦𝑛i𝛤=
𝑘→0O(𝑘3).(42)
This process can be repeated for both T
A,𝑘 and T
Φ,𝑘 when
both expansion and testing functions are non-solenoidal and
when at least one of the two is solenoidal. In summary,
Re hˆ
𝒏×𝜮𝑚,T𝑘𝜮𝑛i𝛤=
𝑘→0O(𝑘2),(43)
Im hˆ
𝒏×𝜮𝑚,T𝑘𝜮𝑛i𝛤=
𝑘→0O(𝑘−1),(44)
Re hˆ
𝒏×𝜮𝑚,T𝑘𝜦𝑛i𝛤=
𝑘→0O(𝑘4),(45)
Im hˆ
𝒏×𝜮𝑚,T𝑘𝜦𝑛i𝛤=
𝑘→0O(𝑘).(46)
By symmetry, both hˆ
𝒏×𝜦𝑚,T𝑘𝜮𝑛i𝛤and hˆ
𝒏×𝜦𝑚,T𝑘𝜦𝑛i𝛤
have the same low-frequency behavior as hˆ
𝒏×𝜮𝑚,T𝑘𝜦𝑛i𝛤.
The scaling of the behavior of the block matrix is now
straightforward to obtain
Re TLS
𝑘=
𝑘→0O(𝑘4) O(𝑘4)
O(𝑘4) O(𝑘2),(47)
Im TLS
𝑘=
𝑘→0O(𝑘) O(𝑘)
O(𝑘) O(𝑘−1),(48)
and the dominant behavior of TLS
𝑘is that of its imaginary part.
These results can be used to demonstrate the issues plaguing
the EFIE at low frequencies, starting with its ill-conditioning.
Consider the block diagonal matrix D𝑘=diag 𝑘−1/2𝑘1/2
in which the block dimensions are consistent with that of the
loop star decomposition matrix. Clearly,
D𝑘TLS
𝑘D𝑘=
𝑘→0O(1) O(𝑘)
O(𝑘) O(1),(49)
is a well-conditioned matrix, in the sense that
lim𝑘→0cond D𝑘TLS
𝑘D𝑘≕𝛾is finite. It then follows
that
cond TLS
𝑘=cond D−1
𝑘D𝑘TLS
𝑘D𝑘D−1
𝑘
≤(cond D𝑘)2cond D𝑘TLS
𝑘D𝑘
(50)
and thus lim𝑘→0cond TLS
𝑘=O(𝑘−2). A lower bound for the
condition number of interest can be obtained through the
application of the Gershgorin disk theorem after diagonal-
ization of the bottom right block of TLS
𝑘, which proves that
cond TLS
𝑘≥𝜎min𝑘−2where 𝜎min is the smallest singular
value of (ΣTTΦ,0Σ). Considering these results and that the
loop-star transformation matrix is invertible and frequency
independent, we conclude that cond (T𝑘)∼𝑘−2when 𝑘→0.
The second source of instability of the EFIE at low frequen-
cies is the loss of significant digits in the right-hand side ei,
solution j, or radiated fields. To see this effect, the behavior of
the right-hand side of the EFIE must be considered. Here we
will restrict our developments to the plane-wave excitation,
but similar results can be obtained for other problems [58].
Following the same procedure as for the matrix elements, we
can determine the behavior of the loop and star right-hand side
elements
Re hˆ
𝒏×𝜦𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘→0O(𝑘2),(51)
Im hˆ
𝒏×𝜦𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘→0O(𝑘),(52)
Re hˆ
𝒏×𝜮𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘→0O(1),(53)
Im hˆ
𝒏×𝜮𝑚,ˆ
𝒏×𝒆i
PWi𝛤=
𝑘→0O(𝑘),(54)
where 𝒆i
PW is the electric field of the incident plane-wave.
It is crucial to remember that when the standard EFIE—
with no treatment—is solved numerically in finite precision
floating point arithmetic, the real parts (resp. imaginary parts)
of the loop and star components of the right-hand side are
stored in the same floating point number. In particular, the
real part of the solenoidal component that behaves as O(𝑘2)
is summed with an asymptotically much larger non-solenoidal
component behaving as O(1). In the context of finite precision
arithmetic, the dynamic range of the floating point number will
be imposed by the larger of the two components, meaning that
the floating point number will become increasingly incapable
of storing accurately the smaller one. This loss of significant
digits will worsen until the solenoidal component has com-
pletely vanished from the numerical value. This phenomenon
is not necessarily damageable per se, but can lead to drastic
losses in solution accuracy. In the particular case of the plane-
wave excitation, we will study the effect of this loss of accuracy

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on the dominant parts of the solution. Using the well-known
relations on block matrix inverses [59], one can show that
Re TLS
𝑘−1
=
𝑘→0O(𝑘2) O(𝑘4)
O(𝑘4) O(𝑘4),(55)
Im TLS
𝑘−1
=
𝑘→0O(𝑘−1) O(𝑘)
O(𝑘) O(𝑘),(56)
which, in combination with the right-hand side results yields
the behavior of the solution coefficients
Re (j𝜦)=
𝑘→0O(1),(57)
Im (j𝜦)=
𝑘→0O(𝑘),(58)
Re (j𝜮)=
𝑘→0O(𝑘2),(59)
Im (j𝜮)=
𝑘→0O(𝑘),(60)
which are indeed the behavior predicted by physics [60]. Note
that the inaccurate right-hand side component will only have a
significant contribution to the imaginary part of the solenoidal
component of the solution, which is non-dominant. As such,
although the error of the current could be low, the error of the
charge or field could be quite high.
Finally, the reader should note that to numerically observe
these results—and successfully implement the remedies that
we will see later on—the vanishing of all relevant integrals
must be explicitly enforced in some way, because floating
point arithmetic and numerical integration are not capable of
obtaining an exact zero in their computation and will saturate
at machine precision, in the best case scenarios. Indeed, had
they not been enforced, the solenoidal and non-solenoidal parts
of the solution would have had the same behavior and, as such,
would not yield a solution behaving as predicted by physics.
1) Loop-Star/Tree Approaches
Historically, the loop-star and loop-tree decompositions
have been used to cure the low-frequency breakdown of the
EFIE [20], [54] and as such are well-known and studied [55].
The fundamental curing mechanism of these approaches is
to decompose the EFIE system using a RWG-to-loop-star or
RWG-to-loop-tree mapping and isolate the solenoidal and non-
solenoidal parts of the system. This separation allows for a
diagonal preconditioning of the decomposed matrix to cure
its ill-conditioning (as was done in Section IV-A). In addition,
this separation makes it possible to enforce that the required
integrals and matrix products vanish and cures the loss of
significant digits that plagues the EFIE, since the loop and star
contributions of each entity are stored in separate floating point
numbers. In the case of the loop-star approach, the stabilized
matrix system is
D𝑘TLS
𝑘D𝑘jDLS =D𝑘Λ ΣTei,(61)
where j=Λ ΣD𝑘jDLS, following the notations of Sec-
tion IV-A. Once the intermediate solution jDLS has been ob-
tained, it must be handled with particular care. If, for instance,
the quantity of interest is the field radiated by the solution,
the radiation operators must be applied separately on the
solenoidal and non-solenoidal parts of the solution that can be
retrieved as jsol =Λ0D𝑘jDLS and jsol =0ΣD𝑘jDLS,
because additional vanishing integrals must be enforced in the
scattering operators when applied to solenoidal functions. In
addition, any explicit computation of jwould be subject to a
numerical loss of significance and would further compromise
the accuracy of the fields.
The key difference between loop-tree and loop-star tech-
niques is that, in the former, the quasi-Helmholtz decompo-
sition leverages a tree basis in place of the star basis, as
indicated by their names. To define this tree basis consider
the connectivity graph joining the centroids of all adjacent
triangle cells of the mesh. To each edge of this graph cor-
responds a unique RWG function. Then, given a spanning
tree of this graph, a tree basis can be defined as the subset
𝜽𝑗of the RWG functions whose corresponding edge in the
connectivity graph is included the spanning tree [54], [61].
The rationale behind the technique is that, by construction,
such a basis will not be capable of representing any loop
function. Clearly, the construction of this basis in not unique,
since it depends on the choice of spanning tree. In practice, the
loop-tree approach results in a matrix system similar to (61),
in which the RWG-to-loop-star mapping Λ Σis replaced
by an RWG-to-loop-tree mapping Λ Θand TLS
𝑘becomes
TLT
𝑘BΛ ΘTT𝑘Λ Θwhere
[Θ]𝑖 𝑗 =1if 𝒇𝑖=𝜽𝑗
0otherwise, (62)
is the general term of the RWG-to-tree transformation matrix.
The resulting preconditioned equation is
D𝑘TLT
𝑘D𝑘jDLT =D𝑘Λ ΘTei,(63)
where j=Λ ΘD𝑘jDLT.
At first glance, the computational overhead of the two meth-
ods seems low, since Λ,Σ,Θ, and D𝑘are sparse matrices.
However, while both methods adequately address the low-
frequency breakdown of the EFIE, in the sense that they yield
the correct solution (Figure 3) and prevent the conditioning
of the system to grow unbounded as the frequency decreases
(Figure 5), they cause the conditioning of the system matrix
to artificially worsen because the loop-star and loop-tree bases
are ill-conditioned [62]. This has led to the development of a
permutated loop-star and loop-tree bases to reduce the number
of iterations required to solve the preconditioned system using
iterative solvers [61]. In general, the loop-tree preconditioned
EFIE was observed to converge faster than the loop-star
preconditioned [63], which can be explained by the fact that
Λand Σcan be interpreted as the discretizations of the graph
curl and graph gradient [55], [62], that are ill-conditioned
derivative operators. While a rigorous proof of the effect of
this ill-conditioning on the preconditioned EFIE matrix is
out of the scope of this review, pseudo-differential operator
theory can be used to show that the differential strength of
the loop-star transformation operators is sufficiently high not
to be compensated by that of the vector potential. To illustrate
this adverse effect, the conditioning of the system matrices
has been obtained numerically and is presented in Figure 6.
Clearly, the standard EFIE matrix shows a condition number
growing as the frequency decreases. However, at moderate

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10−44 10−33 10−22 10−11 1001011
100
103
106
Frequency 𝑓[Hz]
Condition number
EFIE MFIE P-EFIE Loop-star EFIE
Fig. 5. Comparison of the conditioning of the system matrices for several
formulations on a sphere of radius 1 m discretized with an average edge length
of 0.15 m, for varying frequency.
105106107
100
102
104
106
108
Frequency 𝑓[Hz]
Condition number
EFIE Loop-star EFIE Loop-tree EFIE
P-EFIE P-CMP-EFIE
Fig. 6. Comparison of the conditioning of the loop-star, loop-tree, and
projector-based preconditioned EFIE matrices on a spheres of radius 1 m
discretized with an average edge length of 0.3 m (solid lines) and 0.2 m (dotted
lines) as a function of the frequency. The labels “Loop-tree EFIE” and “P-
CMP-EFIE” refer to the EFIE stabilized with the loop-tree approach (63) and
the Calderón EFIE stabilized with quasi-Helmholtz projectors (128).
frequencies, the conditioning of the of the loop-star and loop-
tree preconditioned matrices is significantly higher than that
of the original matrix.
2) Quasi-Helmholtz Projectors
From the previous sections it is clear that although the loop-
star/tree decompositions are helpful in analyzing the reasons
behind of the low-frequency breakdown and that historically
provided a cure for it, they still give rise to high condition
numbers since they introduce an ill-conditioning related to
the mesh discretization. Moreover, for non-simply connected
geometries, loop-star decompositions require a search for the
mesh global cycles, an operation that can be computationally
cumbersome.
A family of strategies to overcome the drawbacks of loop-
star/tree decompositions while still curing the low-frequency
breakdown is the one based on quasi-Helmholtz projec-
tors [62], [64]. Quasi-Helmholtz projectors can decompose the
current and the operators into solenoidal and non-solenoidal
components (just like a loop-star/tree decomposition does) but,
being projectors, have a flat spectrum that, differently from
loop-star/tree decompositions, do not alter the spectral slopes
of the original operators and thus do not introduce further
ill-conditioning.
Starting from the quasi-Helmholtz decomposition (36)
j=Σj𝜮+Λj𝜦+Hj𝑯,(64)
the quasi-Helmholtz projector for the non-solenoidal part is
the operator that maps jinto Σj𝜮. Since
ΣTj=ΣTΣj𝜮,(65)
the looked for projector is
PΣBΣ(ΣTΣ)+ΣT,(66)
where +denotes the Moore–Penrose pseudoinverse. The pro-
jector for the solenoidal plus harmonic components can be
obtained out of complementarity as
PΛH≔I−PΣ.(67)
The same reasoning for dual functions leads to the dual
definitions of the projector
PΛBΛ(ΛTΛ)+ΛT(68)
which is the non-solenoidal projector for dual functions. The
solenoidal plus harmonic projector for dual functions is, again,
obtained by complementarity as
PΣH≔I−PΛ.(69)
It is important to note that, even though the projectors pre-
sented so far include a pseudo-inverse in their definition, they
can be applied to arbitrary vectors in quasi-linear complexity
by leveraging algebraic multigrid preconditioning [62], [65],
[66] and, as such, are fully compatible with standard fast
solvers.
Quasi-Helmholtz projectors can be used to cure the dif-
ferent deleterious effects of the low-frequency breakdown by
isolating the solenoidal and non-solenoidal parts of the system
matrix, unknowns, and right-hand side and rescaling them
appropriately. Thus they are an alternative to loop-star/tree de-
compositions that presents several advantages when compared
to these schemes. Quasi-Helmholtz projectors have been used
to cure the low-frequency breakdowns of several formulations,
however for the sake of readability and conciseness, we will
only detail their application to the standard EFIE where it
is more straightforward, but will point to relevant papers
describing their applications to other well-known formulations.
Preconditioning the original system (15) with matrices of the
form
PB𝛼PΛH+𝛽PΣ,(70)
where, following a frequency analysis similar to the one used
for loop-star/tree decompositions, an optimal coefficient choice
can be found to be 𝛼∝𝑘−1
2and 𝛽∝𝑘1
2, that is,
P𝑘B𝐶/𝑘PΛH+i/√𝐶𝑘 PΣ,(71)

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0 100 200 300 400 500
10−35
10−24
10−13
10−2
Coefficient index
Current density A m−1
EFIE Loop-star EFIE P-EFIE MFIE
Fig. 7. Comparison of the solenoidal part of the surface current density
induced on a sphere of radius 1 m discretized with an average edge length of
0.3 m at 𝑓=10−20 Hz, computedwith different formulations.
resulting in a new system of equations
P𝑘T𝑘P𝑘y=P𝑘ei,(72)
where P𝑘y=j. The constant 𝐶can be obtained by maximizing
the components of the solution current that are recovered [60]
and by enforcing an equal contribution of the vector and scalar
potential components; this results in
𝐶≔kTΦ,𝑘 k
kPΛHTA,𝑘 PΛHk.(73)
The analysis of the conditioning effect of the projector can
mimic the strategy used for the loop-star decomposition. In
particular, the EFIE preconditioned with the projectors has a
frequency-independent limit
lim
𝑘→0P𝑘T𝑘P𝑘→i𝐶PΛHTA,0PΛH+i/𝐶TΦ,0,(74)
where we used that PΣTΦ,𝑘 PΣ=TΦ, 𝑘 and PΛTΦ,𝑘 =TΦ, 𝑘 PΛ=
0and thus
lim
𝑘→0cond (P𝑘T𝑘P𝑘)=𝛾 , (75)
where 𝛾is a frequency independent constant. This approach
can be proved to simultaneously solve the problem of catas-
trophic round-off errors in both the current and the right-
hand side of the EFIE [60]. Finally, the use of the projectors
has clear advantages in terms of conditioning with respect
to the use of loop-star or related decompositions that can be
seen in Figure 6. The impact on current and right-hand side
cancellation effects can be observed in Figures 7 and 8.
3) Other Strategies for the EFIE Low-Frequency Regular-
ization
From previous sections it is clear that the main drawbacks
of loop-star/tree decompositions reside in their constant-in-
frequency, but still high, condition number and also in the
need to be enriched with global loop functions [67], [68].
Both of these drawbacks can be overcome by the use of
quasi-Helmholtz projectors, as explained above, but other
schemes can alternatively be used as effective cures for one
0 100 200 300 400 500
10−64
10−43
10−22
Coefficient index
Current density A m−1
EFIE Loop-star EFIE P-EFIE MFIE
Fig. 8. Comparison of the non-solenoidal part of the surface current density
induced on a sphere of radius 1 m discretized with an average edge length of
0.3 m at 𝑓=10−20 Hz, computed with different formulations.
or both of the drawbacks above. By using a rearranged non-
solenoidal basis, for example, the conditioning of a loop-
star or a loop-tree preconditioned EFIE could be further
improved [61]. Moreover, to avoid the construction of global
loops on multiply-connected geometries, formulations have
been presented that consider the saddlepoint formulation of the
EFIE [69], [70], where the charge is introduced as unknown,
in addition to the current in the RWG basis. The most notable
are the current-charge formulation [71] and the augmented
EFIE [72]. However, these formulations are, in general, not
free from round-off errors in the current or the right-hand side
so that, for example, perturbation methods need to be used [58]
for further stabilization. An alternative to the perturbation
method is the augmented EFIE with normally constrained
magnetic field and static charge extraction, which includes
a boundary integral equation for the normal component of
magnetic field [73]. A disadvantage of current-charge formula-
tions is the introduction of an additional unknown, the charge;
hence, methods have been presented to save memory by
leveraging nodal functions [74]. An entirely different approach
is used in [75], where a closed-form expression of the inverse
of the EFIE system matrix is derived based on eigenvectors
and eigenvalues of the generalized eigenvalue problem.
Another class of strategies forfeits the EFIE approaches;
instead, they are based on potential formulations [76], [77].
These formulations are low-frequency stable on simply- and
multiply-connected without the need for searching global
loops. The potential-based approaches [76], [77] are also
dense-discretization stable. This property is shared with hi-
erarchical basis and Calderón-type preconditioners. With a
few exceptions [78], hierarchical basis preconditioners are
based on explicit quasi-Helmholtz decomposition [79]–[84],
since it then suffices to find a hierarchical basis for scalar-
valued functions. While they yield an overall improved con-
dition number with respect to classical loop-star and loop-
tree approaches, they require the search for global loops on
multiply-connected geometries; a suitable combination with

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quasi-Helmholtz projectors has been shown to alleviate the
need for this search [85]. Calderón-type preconditioners will
be discussed in the next section in greater detail. At this point,
we are content to say that standard Calderón preconditioned
EFIEs have a spectral behavior similar to that of the MFIE
operator: thus certain low-frequency issues that plague the
MFIE (and which we discuss in the next subsection) persist
in the Calderón preconditioned EFIE. Initially remedies relied
on combining Calderón preconditioners with loop-star precon-
ditioners. However, the Gram matrix becomes ill-conditioned
and global loops must be explicitly recovered [86], [87]. As
we will show in Section V, this can be avoided by using quasi-
Helmholtz projectors [64].
B. Handling of the right-hand side and field computation
As we have already mentioned, a well-conditioned dis-
cretization alone is not sufficient to accurately compute j: the
right-hand side suffers typically from numerical inaccuracies
due to finite integration precision and from round-off errors.
The main reason for this is that the quasi-Helmholtz compo-
nents scale differently in frequency. As an example, for the
case of the plane-wave excitation, the asymptotic behavior is
noted in (51)–(54).
Strategies have been presented in the past to yield stable
discretizations of the right-hand side [20], [61], which work
with arbitrary right-hand side excitations. For the plane-wave
excitation, a simple solution is to not only compute eas in
(18), but also an eextracted, where the static contribution is
extracted. We obtain this by replacing ei𝑘ˆ
𝒌·𝒓with ei𝑘ˆ
𝒌·𝒓−1,
where ˆ
𝒌denotes the direction of propagation. Then
𝐶/𝑘PΛHeextracted +i√𝐶𝑘 PΣe(76)
is a stable discretization of the preconditioned right-hand side.
To obtain a stable discretization for small arguments of the
exponential function, the subtraction in ei𝑘ˆ
𝒌·𝒓−1should be
replaced by a Taylor series, where the static part is omitted.
Similarly, the far-field cannot be computed by simply eval-
uating 𝛤
𝑁

𝑛=1[j]𝑛𝒇𝑛(𝒓0)e−i𝑘ˆ
𝒓·𝑟0d𝑆(𝒓0),(77)
where j=P𝑘yfrom (72), ˆ
𝒓=𝒓/|𝒓|. On the one hand,
by computing the unknown vector of the unpreconditioned
formulation j=P𝑘y, the different asymptotic behavior
in 𝑘of the quasi-Helmholtz components of jas denoted
in (57)–(60) would lead to a loss of the solenoidal/quasi-
harmonic components in the static limit due to finite machine
precision. Thus for the field computation, one should keep
the unpreconditioned components of jseparately, that is,
jsol-qhar =𝐶/𝑘PΛHyand jnsol =i√𝐶𝑘 PΣy. On the other
hand, it has been pointed out that also the far-field computation
suffers from round-off errors [88]. To avoid these, we compute
the far-field in two steps: we compute the contribution of jnsol
to the far-field by evaluating
𝑬far
nsol (𝒓)=
𝑁

𝑛=1[jnsol]𝑛𝛤
𝒇𝑛(𝒓0)e−i𝑘ˆ
𝒓·𝒓0d𝑆(𝒓0)(78)
and the contribution of jsol-qhar by
𝑬far
sol-qhar (𝒓)=
𝑁

𝑛=1jsol-qhar𝑛𝛤
𝒇𝑛(𝒓0)e−i𝑘ˆ
𝒓·𝒓0−1d𝑆(𝒓0),
(79)
where a Taylor-series expansion should be used for small
arguments of the exponential; then
𝑬far (𝒓)=𝑬far
nsol (𝒓) + 𝑬far
sol-qhar (𝒓)(80)
Also for the near-field computation, the separation in jsol-qhar
and in jnsol must be maintained, the static contribution removed
from the Green’s function, and, in addition, the divergence of
the scalar potential explicitly enforced by omitting it.
C. Magnetic Field Integral Equation
The MFIE has, other than in the Green’s function kernel,
no explicit dependency on 𝑘and should thus be expected to
remain well-conditioned in frequency for 𝑘→0. Indeed, for
simply-connected geometries 𝛤, we have cond(M𝑘)=O(1)
when 𝑘→0. In the case of multiply-connected geometries,
the MFIE operator exhibits a nullspace associated with the
toroidal (for the exterior MFIE) or poloidal loops (for the
interior MFIE) in the static limit [89]–[91]. This leads to an ill-
conditioned system matrix [91]; in the following, we are going
to show that cond (M𝑘)≥𝐶/𝑘2for some constant 𝐶∈R+.
To prove this result on the condition number, we will
consider the low-frequency behavior of block matrices that
result from a discretization of the MFIE with a loop-star
basis. For the analysis, we must however distinguish two types
of harmonic functions, the poloidal and the toroidal loops.
If 𝛺−has genus 𝑔, then the space 𝐻𝑯(𝛺−)defined by the
harmonic functions in 𝛺−and the space 𝐻𝑯(𝛺+)defined by
the harmonic functions in 𝛺+have both dimension 𝑔. The
space defined by 𝐻ˆ
𝑯P(𝛤)≔ˆ
𝒏×𝐻𝑯(𝛺−) |𝛤are the poloidal
loops and the space defined by 𝐻ˆ
𝑯T(𝛤)≔ˆ
𝒏×𝐻𝑯(𝛺+) |𝛤
are the toroidal loops [92], [93]. 𝐻ˆ
𝑯P(𝛤)has been show [89]
to be the nullspace of M−
0and 𝐻ˆ
𝑯T(𝛤)the nullspace of M+
0
and that ˆ
𝒏׈
𝑯T𝑛∈𝑯ˆ
𝑯P(𝛤)and ˆ
𝒏׈
𝑯P𝑛∈𝑯ˆ
𝑯T(𝛤).
We need to address how quasi-harmonic functions formed
from primal (RWG) or dual (CW/BC) functions are related to
harmonic functions. On the one hand, neither with RWG nor
with BC functions we can find linear combinations that are in
𝐻ˆ
𝑯P(𝛤)or in 𝐻ˆ
𝑯T(𝛤), a consequence of the fact that these
functions are not curl-conforming, as mentioned in Section IV.
On the other hand, quasi-harmonic functions 𝑯𝑛(and dual
quasi-harmonic functions 
𝑯𝑛, respectively) are associated with
the holes and handles of the geometry. They are not, unlike the
locally defined loop functions 𝜦𝑛, derived from a continuous
scalar potential on 𝛤. Together with the fact that 𝑯𝑛and

𝑯𝑛are solenoidal but not irrotational (since the RWG/BC
functions are not curl-conforming), (32) implies that quasi-
harmonic loops are linear combinations of solenoidal and har-
monic functions (i.e., quasi-harmonic functions are harmonic
functions with solenoidal perturbation). Clearly, any quasi-
harmonic basis {𝑯𝑛}2𝑔
𝑛=1can be rearranged into two bases,
where one basis {𝑯T𝑛}𝑔
𝑛=1is orthogonal to poloidal loops and
the other basis {𝑯P𝑛}𝑔
𝑛=1is orthogonal to toroidal loops. In the

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following, ˆ
𝑯T𝑛∈𝐻ˆ
𝑯T(𝛤)denote the harmonic toroidal and
ˆ
𝑯P𝑛∈𝐻ˆ
𝑯P(𝛤)denote the harmonic poloidal basis functions
on 𝛤, while 𝑯T𝑛,𝑯P𝑛∈𝑋𝒇and 
𝑯T𝑛,
𝑯P𝑛∈𝑋
𝒇are their
quasi-harmonic counterparts.
In [91], scalings were reported of the blocks of the system
matrix of M+
𝑘in terms of a quasi-Helmholtz decomposition

𝜦 𝜮 𝑯T𝑯P
ˆ
𝒏×
𝜮O(𝑘2) O(1) O(𝑘2) O(𝑘2)
ˆ
𝒏×
𝜦O(1) O(1) O(1) O(1)
ˆ
𝒏×
𝑯PO(𝑘2) O(1) O(𝑘2) O(𝑘2)
ˆ
𝒏×
𝑯TO(𝑘2) O(1) O(𝑘2) O(1)

.(81)
An analogous result can be obtained for M−
𝑘with exchanged
roles for poloidal and toroidal loops. To observe such a
frequency behavior, it is necessary that the testing functions
are curl-conforming. Indeed, for the historical MFIE tested
with RWG functions, the scalings are not observed [94].
To derive the asymptotic behavior for 𝑘→0of the block
matrices, we start by considering the Taylor series of the
Green’s function kernel of the MFIE
∇𝐺(𝒓,𝒓0)=𝑹
4π𝑅3(i𝑘 𝑅 −1)ei𝑘 𝑅 (82)
=𝑹
4π𝑅3−1+1
2(i𝑘)2𝑅2+1
3(i𝑘)3𝑅3+. . .,(83)
where 𝑅=|𝒓−𝒓0|and 𝑹=𝒓−𝒓0. An O(𝑘2)-scaling is
observed for a block matrix if the contribution due to the
static term in the Taylor series vanishes. For 𝜮𝑛as expansion
function or ˆ
𝒏×
𝜦as testing function, the static contribution
does not vanish and thus we conclude that the scalings of the
blocks in the second column and the second row of (81) are
constant in 𝑘.
We now consider the static MFIE, which models the mag-
netostatic problem, where 𝒋𝛤is either solenoidal or harmonic.
In this case, we have
hˆ
𝒏×
𝜮𝑚,M±
0𝜦𝑛i𝛤=0,(84)
hˆ
𝒏×
𝜮𝑚,M±
0𝑯T𝑛i𝛤=0,(85)
hˆ
𝒏×
𝜮𝑚,M±
0𝑯P𝑛i𝛤=0.(86)
This can be seen by considering that a solenoidal function 𝜦
has a corresponding scalar potential
𝜦=curl𝛤𝛷 . (87)
Furthermore, we have the equality
hˆ
𝒏×𝜦,M±
0𝒋𝛤i𝛤=−h𝜦,ˆ
𝒏×M±
0𝒋𝛤i𝛤.(88)
Inserting (87) for the testing function in the right-hand side
of (88) and using the fact that curl𝛤is the adjoint operator of
curl𝛤, we obtain
hˆ
𝒏×𝜦,M±
0𝒋𝛤i𝛤=h𝛷, curl𝛤ˆ
𝒏× (M±
0𝒋𝛤)i𝛤
=−h𝛷, curl𝛤𝒉∓
Ti𝛤,(89)
where 𝒉∓
T≔ˆ
𝒏×(M±
0𝒋𝛤)is the (rotated) tangential component
of the magnetic field
𝒉(𝒓)=curl 𝛤
𝐺𝑘(𝒓,𝒓0)𝒋𝛤(𝒓0)d𝑆(𝒓0)for 𝒓∈𝛺−,(90)
that is, 𝒉−
Tis the (rotated) tangential component of 𝒉when 𝛤is
approached from within 𝛺−, and 𝒉+
Twhen 𝛤is approach from
within 𝛺+. We recall that (90) is obtained by finding a vector
potential 𝒂such that 𝒉=curl 𝒂and noting that curl curl 𝒂=𝒋𝛤
under the assumption that div 𝒋𝛤=0[40, see Chapter 6.1].
From
curl𝛤𝒗±
T(𝒓)=lim
𝛺±3𝒓0→𝒓curl 𝒗(𝒓0)·ˆ
𝒏(𝒓0).(91)
and from [40, (6.17)]
curl 𝒗={curl 𝒗}+ˆ
𝒏×𝒗+−𝒗−δ𝛤,(92)
where curly braces {} mean that this part is evaluated only in
𝛺±and δ𝛤is the surface Dirac delta function, together with
(90), we have
curl𝛤𝒉∓
T=curl 𝒉·ˆ
𝒏=0.(93)
Next we will establish that hˆ
𝒏×𝑯P𝑚,M±
0𝜦𝑛i𝛤=0and
hˆ
𝒏×𝑯T𝑚,M±
0𝜦𝑛i𝛤=0. First, note that the exterior MFIE
operator has the mapping properties that [89]
M+
0ˆ
𝑯T𝑛=0(94)
and
M+
0ˆ
𝑯P𝑛=ˆ
𝑯P𝑛,(95)
while for the interior MFIE operator, we have the mappings
M−
0ˆ
𝑯P𝑛=0(96)
and
M−
0ˆ
𝑯T𝑛=ˆ
𝑯T𝑛.(97)
Furthermore, for any two surface functions 𝒇,𝒈, we have [89,
Section 5]
h𝒈,K𝒇i𝛤=h−ˆ
𝒏×K𝒈׈
𝒏,𝒇i𝛤
=h−K𝒈׈
𝒏,𝒇׈
𝒏i𝛤(98)
Then we find
hˆ
𝒏׈
𝑯T𝑛,M+
0𝜦i𝛤=hM−
0ˆ
𝑯T𝑛,𝜦׈
𝒏i𝛤
(97)
=0,(99)
hˆ
𝒏׈
𝑯P𝑛,M+
0𝜦i𝛤=hM−
0ˆ
𝑯P𝑛,𝜦׈
𝒏i𝛤
(96)
=0,(100)
hˆ
𝒏׈
𝑯T𝑛,M−
0𝜦i𝛤=hM+
0ˆ
𝑯T𝑛,𝜦׈
𝒏i𝛤
(94)
=0,(101)
hˆ
𝒏׈
𝑯P𝑛,M−
0𝜦i𝛤=hM+
0ˆ
𝑯P𝑛,𝜦׈
𝒏i𝛤
(95)
=0,(102)
where we used (94)-(97) and the orthogonality of harmonic
and irrotational functions ˆ
𝒏×𝜦. Now consider that 
𝑯P𝑛=
ˆ
𝑯P𝑛+
𝜮P𝑛and 
𝑯T𝑛=ˆ
𝑯T𝑛+
𝜮T𝑛, where 
𝜮P/T𝑛is the respective
perturbation. Thus by taking into account (84), we have
hˆ
𝒏×
𝑯P𝑚,M±
0𝜦𝑛i𝛤=0,(103)
hˆ
𝒏×
𝑯T𝑚,M±
0𝜦𝑛i𝛤=0.(104)
From now on, we will only consider the harmonic functions
ˆ
𝑯T𝑛and ˆ
𝑯P𝑛instead of their quasi-harmonic counterpart
since, as we have seen, the solenoidal pertubation will always
vanish. Then for M+
0, we have
hˆ
𝒏׈
𝑯T𝑚,M+
0ˆ
𝑯T𝑛i𝛤=hˆ
𝒏׈
𝑯P𝑚,M+
0ˆ
𝑯T𝑛i𝛤
(94)
=0,(105)
hˆ
𝒏׈
𝑯T𝑚,M−
0ˆ
𝑯P𝑛i𝛤=hˆ
𝒏׈
𝑯P𝑚,M−
0ˆ
𝑯P𝑛i𝛤
(96)
=0(106)

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due to the nullspace. Finally, we have
hˆ
𝒏׈
𝑯P𝑚,M+
0ˆ
𝑯P𝑛i𝛤
(95)
=hˆ
𝒏׈
𝑯P𝑚,ˆ
𝑯P𝑛i𝛤=0,(107)
as ˆ
𝒏׈
𝑯P𝑚∈𝐻ˆ
𝑯T(𝛤)and this space is orthogonal to 𝐻ˆ
𝑯P(𝛤).
Likewise, we can conclude that there is at least one 𝑚such
that
hˆ
𝒏׈
𝑯T𝑚,M+
0ˆ
𝑯P𝑛i𝛤≠0.(108)
Analogously, we obtain
hˆ
𝒏׈
𝑯P𝑚,M−
0ˆ
𝑯T𝑛i𝛤≠0.(109)
This has established the scalings of the different block matrices
composing M±
𝑘.
To prove the ill-conditioning in 𝑘, we consider the static
limit 𝑘→0, where j𝜮=0and
kj𝛬k=j𝑯T=j𝑯P=O(1),
following the argumentation in [95]. Forming the matrix-vector
product, we find
M+
𝑘
j𝜦
0
j𝑯T
j𝑯P

=
O(𝑘2)
O(1)
O(𝑘2)
O(1)

.(110)
Thus for 𝑘→0, we have the reduced system

M+
0,ˆ
𝒏×
𝜦,𝜦M+
0,ˆ
𝒏×
𝜦,𝑯T
M+
0,ˆ
𝒏×
𝜦,𝑯P
  M+
0,ˆ
𝒏×
𝑯T,𝑯P
  
𝑁
𝜦+𝑁
𝑯T×𝑁𝜦+𝑁𝑯T+𝑁𝑯P

j𝜦
j𝑯T
j𝑯P

𝑁𝜦+𝑁𝑯T+𝑁𝑯P×1
=O(1)
O(1)
  
𝑁
𝜦+𝑁𝑯P×1
,(111)
where 𝑁𝑿is the dimensionality of the respective finite element
space of the basis functions 𝑿𝑛. Since 𝑁
𝜦=𝑁𝜦, this implies
that M+
0has an 𝑁𝑯P=𝑔-dimensional nullspace.
To obtain a lower bound on the condition number depen-
dency of M±
𝑘on 𝑘, we consider that for all j±
0∈null M±
0,
we have M±
𝑘j±
0=O(𝑘2)for 𝑘→0. Thus cond(M±
𝑘)&𝑘2.
1) Quasi-Helmholtz Projectors
It is not possible to cure the low-frequency breakdown of
the MFIE on multiply-connected geometries with projectors.
The reason is that only a part of the harmonic Helmholtz
subspace is affected. What, however, is more critical for the
MFIE in low-frequency scenarios than the ill-conditioning,
is that round-off errors and finite precision of integration
rules lead to non-vanishing static components. Equation (81)
shows that several block matrix entries should vanish when
𝑘→0, for example, we should observe PΣM±
0PΛ=0;
this is not observed in practice due to errors, for example,
in numerical integration. To obtain accurate solutions even
for 𝑘→0, the contribution of the static kernel K0must
be removed. On simply-connected geometries, it is sufficient
to explicitly set the term PΣM±
0PΛto zero. On multiply-
connected geometries, this is not possible due to the different
scaling of the block matrices in (81) associated with poloidal
and toroidal loops (i.e., it could be cured with projectors on
the poloidal and toroidal quasi-Helmholtz subspace. These are,
however, numerically expensive to construct).
Quasi-Helmholtz projectors can be used to ensure the van-
ishing of the static components that ought to be, according to
(81), zero, but are not due to numerical integration errors. This
is obtained by a combination of the interior and the exterior
MFIE operator together with quasi-Helmholtz projectors as
outlined in [96], that is,

P𝑘
M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇M+
𝑘−PΣHM−
0G−1
ˆ
𝒏×
𝒇,𝒇M+
0PΛH
  
remove static kernel contribution

P𝑘y
=
P𝑘M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇hi,(112)
where 
P𝑘=1/√𝑘PΣH+i√𝑘PΛ(113)
is the counterpart of P𝑘on the dual mesh. The formulation
in (112) symmetries the behavior on the quasi-harmonic
Helmholtz subspace and ensures that the MFIE can be solved
accurately for 𝑘→0since no catastrophic round-off errors
occur in the right-hand side and j, and since the numerical
integration error that leads to a nonphysical contribution of the
static kernel is removed. Moreover, as we will see in the next
section, it can be directly combined with an EFIE resulting in
a stable CFIE formulation.
It should be noted that for a stable implementation of (112),
one should, instead of removing the static kernel contribution
by a subtraction, build the system matrix without this contri-
bution in the first place. To this end, we define the operator
K0
𝑘𝒋𝛤(𝒓)B−ˆ
𝒏×𝛤∇ei𝑘|𝒓−𝒓0|−1
4π|𝒓−𝒓0|×𝒋𝛤d𝑆(𝒓0).(114)
and its discretization as
K0
𝑘𝑛𝑚 Bhˆ
𝒏×𝒇𝑛,K0
𝑘𝒇𝑚i𝛤,(115)
where we note that for small 𝑘|𝒓−𝒓0|a Taylor-series should
be used to remove the static contribution. Then the overall
system matrix is
1/√𝑘PΣH(−K0
𝑘)G−1
ˆ
𝒏×
𝒇,𝒇M+
𝑘+M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇K0
𝑘(1/√𝑘PΛH)
+1/√𝑘PΣHM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇M+
𝑘(i√𝑘PΣ)
+i√𝑘PΛM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇M+
𝑘(1/√𝑘PΛH)
+i√𝑘PΛM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇M+
𝑘(i√𝑘PΣ),(116)
which evidently does not contain the static kernel contribution
PΣHM−
0G−1
ˆ
𝒏×
𝒇,𝒇M+
0PΛH.

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D. Handling of the right-hand side
Similarly to the case of EFIE, the right-hand side requires
special treatment. First, we note

P𝑘M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇hi
=
P𝑘M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΣHhi
extracted +
P𝑘M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΛhi,(117)
where hi
extracted is computed analogously to ei
extracted. Again,
the equality is only true in exact arithmetic; the first term
on the right-hand side needs no further treatment as the
asymptotically constant blocks of K−
𝑘will dominate. The
second term, however, needs to be further stabilized. We note

P𝑘M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΛhi=
P𝑘M−
𝑘(PΛ+PΣH)G−1
ˆ
𝒏×
𝒇,𝒇PΛhi
=
P𝑘M−
𝑘PΛG−1
ˆ
𝒏×
𝒇,𝒇PΛhi,(118)
where we used PΣHG−1
ˆ
𝒏×
𝒇,𝒇PΛ=0from Lemma 1. Using (81),
we have

P𝑘M−
𝑘PΛG−1
ˆ
𝒏×
𝒇,𝒇PΛhi
=1/√𝑘PΣH(−K0
𝑘)PΛG−1
ˆ
𝒏×
𝒇,𝒇PΛhi+i√𝑘PΛM−
𝑘PΛG−1
ˆ
𝒏×
𝒇,𝒇PΛhi
=1/√𝑘PΣH(−K0
𝑘)G−1
ˆ
𝒏×
𝒇,𝒇PΛhi+i√𝑘PΛM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΛhi.
(119)
Summarizing, we obtain for the right-hand side

P𝑘M−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΣHhi
extracted
+1/√𝑘PΣH(−K0
𝑘)G−1
ˆ
𝒏×
𝒇,𝒇PΛhi
+i√𝑘PΛM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΛhi.(120)
which can be rearranged into
1/√𝑘PΣHM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΣHhi
extracted
+1/√𝑘PΣH(−K0
𝑘)G−1
ˆ
𝒏×
𝒇,𝒇PΛhi+
i√𝑘PΛM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇hi(121)
using
i√𝑘PΛM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΣHhi
extracted
i√𝑘PΛM−
𝑘G−1
ˆ
𝒏×
𝒇,𝒇PΛhi.(122)
1) Other Low-Frequency Regularizations of the MFIE
When it comes to the low-frequency regularization of the
MFIE, the body of literature falls shorter compared with the
EFIE. In light of the discussion of the previous section, it
is evident that a correct low-frequency behavior can only be
expected when the MFIE operator is tested with dual basis
functions [43], even though improvements can be expected
for the standard MFIE by using, for example, a perturbation
method [94]. Instead of quasi-Helmholtz projectors, explicit
quasi-Helmholtz decompositions can be used to ensure the
vanishing of the static components of the discretized MFIE.
Standard bases such as the loop-star worsen, however, the
conditioning in ℎ[62] since the identity operator is turned into
a discretized Laplace-Beltrami operator. This can be avoided
with hierarchical bases as denoted in [97] in the context of the
CFIE, or by using quasi-Helmholtz projectors as outlined in
Section IV-C1.
A remaining issue is the ill-conditioning due to the quasi-
static nullspace of either the toroidal or the poloidal loops of
the exterior or the interior MFIE operator, respectively. Strate-
gies that have been presented in the past typically introduce
an extra condition based on vector potential considerations
and on an explicit detection of global loops to remove the
ill-conditioning [98], [99].
To obtain a wideband stable solver, an extension of the pre-
conditioning schemes to the CFIE are necessary. For Calderón
preconditioning, we will discuss this in greater detail in the
next section, but a few words should be said on the other
techniques. As mentioned earlier, explicit quasi-Helmholtz
decompositions can be readily extended to the CFIE, however,
some such as the loop-star basis will worsen the condition-
ing of the CFIE. Hierarchical basis preconditioners can be
extended without the worsening [97]. An augmented CFIE has
been presented [100], however, it does not appear to resolve
numerical round-off losses.
In concluding this section on the EFIE and the MFIE low-
frequency behavior, a word should be said regarding fast
methods. Typically, a fast method is employed to accelerate
the matrix-vector products. On of the most popular choices,
the MLFMM, is not low-frequency stable [6]. Even if a
low-frequency stable method is chosen, such as the ACA,
catastrophic round-off errors can appear in the EFIE due to
the different scaling of the vector and scalar potential part
in frequency. This can either be resolved by simply storing
those contributions separately [101]. More memory efficient
approaches have been presented [102] at the price of a more
complicated implementation.
V. Dense-Discretization Scenarios
The dense-discretization breakdown is sometimes confused
with the low-frequency breakdown. They are, however, two
distinct phenomena: the dense-discretization breakdown im-
plies that the condition number grows when ℎ→0.
The condition number of the EFIE is known to grow as
cond (T)=O(1/ℎ2).(123)
Unlike the low-frequency breakdown, it is more difficult to
prove this statement rigorously. One possible line of argumen-
tation is based on pseudo-differential operator theory [103].
Another line of argumentation follows from a functional
analytic point of view, where the link between the coefficient
space C𝑁and the domain and range of the respective operator
is considered [104]. Based on inverse inequalities, the norm
of a Sobolev space 𝐻𝑚is bounded by the norm of 𝐻𝑙, where
𝑙 < 𝑚. In the case of the EFIE operator, this is approach
needs a non-trivial extension as the 𝑯−1/2
div -space necessitates
a Helmholtz decomposition.
To avoid the introduction of either Sobolev space or pseudo-
differential operator theory [105], one can also study the

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1.522.533.544.5 5
101
102
103
104
1/ℎm−1
Singular value
𝜎max for 𝑓=1×108Hz 𝜎min for 𝑓=1×108Hz
𝜎max for 𝑓=5×107Hz 𝜎min for 𝑓=5×107Hz
Fig. 9. The figures shows 𝜎max (T𝑘)and 𝜎min (T𝑘)as a function of the
maximum spectral index 1/ℎfor a sphere of unit radius.
singular value spectrum. This is typically only possible for
a few canonical geometries and thus the results are not a
rigorous proof. They do provide, however, a more intuitive
understanding of the underlying physics. In the appendix, we
have included a discussion of the singular value spectrum of
the EFIE on a sphere and its link to the breakdowns.
From Appendix A-B, it is evident that T
Ahas singular
values that accumulate at zero with a rate of ℎ, while T
Φ
has singular values that grow to infinity with a rate of 1/ℎ
resulting in an overall condition number scaling proportional to
1/ℎ2. The discretized operator inherits the spectral properties
of the analytic EFIE operator, though the effect of the basis
function should be removed by considering the spectrum of
G−1
𝒇,𝒇T. Figure 9 shows the spectrum of the EFIE operator
discretized on a sphere, confirming that indeed the condition
number scales as 1/ℎ2.
For the MFIE, we observe that the singular values are
bounded from above and below. Indeed, second-kind integral
equations such as the MFIE are known to result in well-
conditioned system matrices if the expansion and testing
functions are 𝑳2-stable (i.e., that the Gram matrix is well-
conditioned) [106].
For the CFIE, the conditioning improves to
cond (C𝑘)=O(1/ℎ),(124)
due to the MFIE part of the CFIE: the identity of the
MFIE introduces a lower bound on the singular values, which
annihilates the ill-conditioning due to T
A. Hence, only T
Φ
contributes to the growing condition number.
A. Calderón Preconditioning
The Calderón identity
T𝑘◦T𝑘=M−
𝑘◦M+
𝑘=−I
4+K2
𝑘(125)
suggests that the EFIE operator can be turned into a second-
kind integral operator, which, as other second-kind integral
equation like the MFIE, is well-conditioned in ℎ. The stable
discretization of (125) is not trivial however (see, for example,
the pioneering works [103], [107]–[109]) and research for a
stable discretization lead to discovery of an alternative to CW
functions in the form of the BC functions [45] that leveraged
the seminal work in operator preconditioning of Steinbach and
Wendland [110].
1) Calderón preconditioning with BC functions
The Calderón multiplicative preconditioner (CMP) is an
effective and well-conditioned scheme for intricate geome-
tries [46]. With the notation of this paper the CMP-EFIE can
be expressed as
T𝑘G−1
ˆ
𝒏×𝒇,
𝒇T𝑘j=−
T𝑘G−1
ˆ
𝒏×𝒇,
𝒇ei,(126)
where 
T𝑘𝑛𝑚
Bhˆ
𝒏×𝒇𝑛,T𝑘𝒇𝑚i𝛤.(127)
The preconditioning strategies has been used extensively and
successfully in several application scenarios with effective
improvements [111]–[113].
However, when the frequency is extremely low (when 𝑘2
normalized by the norm ratio of the scalar and vector potentials
reaches 10−16 in double precision), the standard CMP will
break down: in fact, from Section IV, we can expect that (126)
inherits the low-frequency issues that the MFIE is suffering
from on multiply-connected geometries due to the Calderón
identity −I/4+K2
𝑘=(I/2+K𝑘)(−I/2+K𝑘). To overcome
the low-frequency breakdown due to the harmonic Helmholtz
subspace and due to catastrophic round-off errors, a quasi-
Helmholtz decomposition must be introduced.
In [64], quasi-Helmholtz projectors are leveraged yielding

P𝑘
T𝑘
P𝑘G−1
ˆ
𝒏×𝒇,
𝒇P𝑘T𝑘P𝑘y=−
P𝑘
T𝑘
P𝑘G−1
ˆ
𝒏×𝒇,
𝒇P𝑘ei.(128)
The projectors ensure that the overall system matrix is well-
conditioned also on multiply-connected geometries. Figure 10
shows that the condition number of the preconditioned system
matrix in (128) is asymptotically bounded. A similar result
holds even on the more challenging NASA almond benchmark
[114] (Figure 11).
2) Refinement-Free Calderón Preconditioning
The use of BC functions increases, unfortunately, the nu-
merical costs due to their larger support compared with
RWG functions (though specific implementations can help
offset the overhead [115], [116]). Moreover, no extension to
geometries with junctions is available. Fortunately, a Calderón
preconditioned EFIE has been obtained without the use of BC
functions by indirectly using the Laplace-Beltrami operator as
preconditioner. The formulation reads [117]
P†
oT†
𝑘PmT Poi=−P†
oT†
𝑘Pmei(129)
where the outer matrix Pois
Po≔PΛH/𝛼+iPgΣ/𝛽(130)
with
PgΣ≔ΣΣTΣ+G−1

𝜆, 𝑝 ΣT,(131)
where G
𝜆, 𝑝 is the Gram matrix of piecewise linear dual basis
functions as defined in [45] and patch basis functions with

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0510 15 20
100
102
104
106
108
1/ℎm−1
Condition number
EFIE LS-EFIE P-EFIE P-CMP-EFIE
RF-CMP-EFIE MFIE P-CMP-MFIE CMP-EFIE
Fig. 10. Condition number of the system matrices as a function of 1/ℎ, which
is proportional to the maximum spectral index, for a cube of side 1 m and
a frequency of 107Hz. The labels “CMP-EFIE”, “CFIE”, “RF-CMP-EFIE”,
“P-CMP-MFIE”, and “P-CMP-CFIE” refer to the standard Calderón EFIE
(126), the conforming CFIE, the refinement-free Calderón preconditioned
EFIE (129), and the Calderón preconditioned MFIE and CFIE with projectors
(112) and (142).
10 20 30 40 50 60 70
101
104
107
1/ℎm−1
Condition number
EFIE LS-EFIE P-EFIE P-CMP-EFIE
RF-CMP-EFIE MFIE P-CMP-MFIE CMP-EFIE
Fig. 11. Conditioning of the different formulations on the NASA almond —
scaled to be of 1.1 m in diameter — and a simulating frequency of 5×107Hz.
patch height 1/𝐴𝑖, where 𝐴𝑖is the area of cell 𝑖of the mesh;
furthermore, the middle matrix is
Pm≔PmΛ+PmΣ(132)
with the definitions
PmΛ≔ΛG−1
𝜆𝜆ΛT/𝛼2+PΛH/𝛾 , (133)
PmΣ≔ΣΣTΣ+G−1
𝑝 𝑝 ΣTΣ+Σ/𝛽2,(134)
where G𝜆𝜆 and G𝑝 𝑝 are the Gram matrices of piecewise linear
and piecewise constant basis functions on the primal mesh (for
22.533.54
·108
101
102
103
Frequency 𝑓[Hz]
Condition number
EFIE P-EFIE P-CMP-EFIE MFIE
P-CMP-MFIE CFIE P-CMP-CFIE Yu-CMP-CFIE
Fig. 12. Condition number of the system matrices of several formulations,
for a sphere of radius 1 m and average edge length of 0.15 m as a function of
the frequency 𝑓. The label “Yu-CMP-CFIE” refers to the standard Calderón
CFIE with modified Yukawa wavenumber (141).
a detailed definition, we refer the reader to [117]), and the
scaling coefficients are
𝛼=4
PΛHT†
A,𝑘 ΛG −1
𝜆𝜆ΛTTA, 𝑘 PΛH2,(135)
𝛽=4
PT
gΣT†
Φ,𝑘 ΣΣTΣ+G−1
𝑝 𝑝 ΣTΣ+ΣTTΦ, 𝑘 PgΣ2,
(136)
𝛾=(PΛH/𝛼)T†
A,𝑘 PΛHTA, 𝑘 (PΛH/𝛼)2,(137)
where the norms can be estimated with the power iteration
method for computing the largest singular value. We note that
for 𝑘→0we have
𝛼∝√𝑘 , (138)
𝛽∝1/√𝑘 , (139)
𝛾∝𝑘 , (140)
which are similar scalings as encounter for the quasi-
Helmholtz projectors. A key property of this formulation is
that the resulting system matrix is Hermitian, positive definite.
This allows the use of CG, which in exact arithmetic guaranties
convergence, and has a lower computational cost with respect
to CGS or GMRES [53].
Figure 13 shows the number of iterations for the refinement-
free preconditioned compared with (128), a standard EFIE
and a loop-tree preconditioned EFIE. It displays that the
number of iterations become bounded independently from
ℎ, this confirming the dense-discretization stability of the
formulation.
3) Alternative Calderón-type Strategies
Many alternative Calderón-identity based preconditioning
strategies have been presented, most prominently the pioneer-
ing works by Adams and Brown [118] and by Christiansen and
Nédélec [103], [119]. Often the focus was not only obtaining
a well-conditioned EFIE, but rather a well-conditioned CFIE

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101
103
105
1/ℎm−1
Condition number
EFIE LS-EFIE P-EFIE P-CMP-EFIE
RF-CMP-EFIE MFIE P-CMP-MFIE CMP-EFIE
Fig. 13. Number of iterations needed by GMRES(50) to converge for tolerance
10−4when solving the linear systems arising from different formulations
discretized on a cube, for a frequency of 107Hz. For the RF-CMP from
(129) also the iterations needed by CG are shown (since for this formulation
the system matrix is Hermitian, positive definite).
or combined source integral equation (CSIE) [120], [121]
with the pioneering works by Contopaganos et al [108] and
Adams [122]. These schemes do not use dual basis functions,
but instead require ad-hoc implementations of modified oper-
ators or use Nyström discretizations.
Another class of preconditioners is based on Dirichlet-to-
Neumann operators and on-surface radiation conditions [109],
[123]–[125]. As will be discussed in the next section, some
of these methods have been extended to handle the high-
frequency breakdown. Naturally, these extension are all ca-
pable of curing the dense-discretization breakdown. In these
schemes, low-frequency issues have not always been addressed
in the original papers, but we believe they could be extended
to handle those as well.
To stabilize the standard Calderón preconditioner in the
static limit, it has been proposed to discretize the EFIE with
a loop-star basis first [86], [87], which implies ill-conditioned
Gram matrices and the search for global loops. A perturbation
method based approach was also suggested [126].
Another strategy employs a discretization with the 𝑯div -
inner product [127]; combined with a Calderón preconditioner,
that cures both the low-frequency and the dense-discretization
breakdown; however, the scheme has not yet been extended to
multiply-connected geometries.
For open problems, the Calderón preconditioner in (128)
yields only a logarithmic bound in ℎ. Recently, a formulation
has been proposed that yields an ℎ-independent bound by
leveraging an analytic transformation for the open disc and
Lipschitz transformations of the geometry [128].
B. Quasi-Optimal Hierarchical Preconditioners
One alternative to Calderón preconditioning for the EFIE are
hierarchical basis approaches. Hierarchical bases (also referred
to as multiresolution or wavelet methods) have been pioneered
in the context of scalar integral equations; they can not only be
used as preconditioner, but also to compress the resulting sys-
tem matrix [129]–[140]. Both the compressibility of the system
matrix [78], [141], [142], as well as the preconditioning effect
has been demonstrated for the EFIE system matrix [79], [80],
[83], [142], [143]. For unstructured meshes, a hierarchically
preconditioned EFIE could initially only be obtained for the
TΦ,𝑘 -part of T𝑘[81]. Recently, a scheme based on generalized
primal and dual Haar prewavelets has been presented, where
the condition grows only logarithmically in 𝑁, and which—
in contrast to other hierarchical bases—extends naturally to
unstructured meshes [84].
The logarithmic growth renders hierarchical basis precon-
ditioners quasi-optimal compared with Calderón schemes,
where the condition number can be asymptotically bounded
by a constant. This disadvantage can, in practice, often be
compensated by the fact that unlike Calderón schemes, no
second multiplication with the EFIE matrix is required. As
mentioned in Section IV-A3, hierarchical basis preconditioners
can be applied to the CFIE [97] and the search for global
loops can be avoided by a suitable combination with quasi-
Helmholtz projectors [85].
C. Alternative Strategies
From the alternative strategies that remedy the low-
frequency breakdown mentioned in Section IV-A3, only a few
are dense-discretization stable without further changes. The
most prominent example is the decoupled potential integral
equation [76]. The augmented EFIE, on the other hand, can
be combined with a Calderón preconditioner [144].
As an alternative to operator preconditioning based strate-
gies, algebraic preconditioners have been proposed. In gen-
eral, this class of preconditioners will not provide asymptotic
bounds on the condition number in ℎ. Very often those
methods have been designed with electrically large problems
in mind and they typically employ the MLFMM: wideband
stability is typically not a construction criterion.
The idea of algebraic preconditioners is to obtain an ap-
proximation Bof the system matrix A, where B−1can be
computed explicitly (a direct inverse is obtained) or implicitly
(a linear system has to be solved at each iteration step) fast.1
A detailed review of algebraic preconditioners for electromag-
netic integral equations is provided in [25]. Here, we review
only the most well-known techniques.
A typical example of explicit preconditioners are sparse
approximations [145]–[148]. A challenge is that if a fast
method such as the MLFMM is used, the full system matrix
is not available. It is customary to use, for example, the near-
field interactions of the MLFMM to construct a precondi-
tioner [148]. While this matrix is indeed sparse, the goal is to
obtain a sparse inverse and, in general, the inverse of a sparse
matrix is not sparse. This sparse matrix is typically obtained as
solution to the minimization problem kI−ANBkF, where k·kF
is the Frobenius norm and ANrefers to the near-interaction ma-
trices [149]. A filtering of ANis often necessary to reduce the
1Operator preconditioning, in this logic, strives to find Bas an approxi-
mation of A−1.

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computational cost [150] and the use of macro basis functions
has also been proposed to reduce the problem size [151]. Since
the methods are based on the near-interactions, it is a local in
nature, instead, to consider global interactions, [152] proposes
to use interpolative decompositions, while [149] employs an
embedded iterative scheme. Recently, hybrid parallelizations
for clusters have been presented [153]. Also, the sparse approx-
imate preconditioners can be combined with other methods
such as a two-grid spectral preconditioner [154].
Some of the most prominent among the implicit methods
are based on the incomplete LU decomposition [155]–[158].
The costs for obtaining the incomplete LU decomposition
are often lower than the the costs of constructing a sparse
approximate inverse. However, highly indefinite systems pose
a significant challenge as they can give rise to ill-conditioned
triangular factors [25]. As an alternative, GMRES-based near-
zone preconditioning has been proposed [159].
VI. High-Frequency Issues
For resonant frequencies, both the EFIE and the MFIE
have nullspaces associated with the interior resonances and
are thus ill-conditioned. To overcome the problem of interior
resonances, a CFIE or CSIE formulation is typically em-
ployed [21], [160].
A. Calderón-Yukawa Combined Field Integral Equation
Calderón preconditioning can also be extended to a com-
bined field framework. A trivial combination of the Calderón
preconditioned EFIE (126) and the MFIE from (20) would
lead to an overall operator suffering from interior resonances:
the identity (125) implies that the Calderón preconditioned
EFIE and the MFIE share part of their nullspaces. To obtain
a wideband stable CFIE, the EFIE from (126) should be com-
bined with the MFIE from (20) with one important adjustment:
the wavenumber 𝑘in 
T𝑘needs to be made a complex number.
One possibility for a complex 𝑘is to use the Yukawa potential
resulting in
−𝛼
Ti𝑘G−1
ˆ
𝒏×𝒇,
𝒇T𝑘+ (1−𝛼)M+
𝑘j
=𝛼
Ti𝑘G−1
ˆ
𝒏×𝒇,
𝒇ei+ (1−𝛼)hi.(141)
Originally, this has been proposed in [108] and later em-
ployed in the Calderón multiplicative preconditioner for the
CFIE [161]; Section V-A3 includes a discussion of Calderón-
type preconditioners for the CFIE and CSIE.
This standard approach is, however, not stable until arbitrar-
ily low frequencies, because of the numerical issues detailed in
Section IV. A stable formulation can be obtained by combining
the Calderón preconditioned EFIE (128) and the stabilized
MFIE from (112). Similar to the preconditioning EFIE matrix,
050 100 150 200 250 300
10−18
10−12
10−6
100
Spectral Index 𝑖
Singular value 𝜎𝑖
CMP-EFIE P-CMP-EFIE P-CMP-MFIE
P-CMP-CFIE
Fig. 14. Low frequency (10−12 Hz) spectra of several formulations putting
into evidence their magnetostatic nullspace, or lack thereof. The matrices have
been obtained by discretizing a torus with a square the cross section of 0.2 m
sides and with a minimum inner diameter of 2 m. The average edge length of
the mesh is of 0.2 m and the spectra have been normalized to make the figure
more readable. The combination parameter 𝜉=iwas used here.
a complex wavenumber should be used for preconditioning
MFIE. This results in the overall formulation

P𝑘
Ti𝑘
P𝑘G−1
ˆ
𝒏×𝒇,
𝒇P𝑘T𝑘P𝑘
+𝜉M−
i𝑘G−1
ˆ
𝒏×
𝒇,𝒇M+
𝑘−PΣHM−
0G−1
ˆ
𝒏×
𝒇,𝒇M+
0PΛHP𝑘y
=−
P𝑘
Ti𝑘
P𝑘G−1
ˆ
𝒏×𝒇,
𝒇P𝑘ei−𝜉M−
i𝑘G−1
ˆ
𝒏×
𝒇,𝒇hi,(142)
with a suitably chosen 𝜉and in high-frequency scenarios, the
scaling of the projectors must be set to a unitary value [96].
Figure 12 shows that the CFIE formulation is free from interior
resonances and Figure 13 the number of iterations as a function
of the spectral index 1/ℎ. For the presented results, we have
weighted the EFIE and MFIE part equally. Finally, this com-
bined formulation has the advantage of not exhibiting a static
nullspace. This is because, unlike the standard the Calderón
EFIE (126), the Calderón EFIE stabilized with projectors (128)
has no static nullspace, which can additively cure that of the
MFIE and results in equation (142) that is free from spurious
resonances, as illustrated in Figure 14.
While both Calderón-Yukawa CFIEs (141) and (142) are
free from the dense-discretization breakdown (i.e., when all
parameters are kept constant expect from ℎ, the condition num-
ber can be asymptotically bounded for ℎ→0), the condition
number cannot be bounded independently from 𝑘. This effect
has been designated as the high-frequency breakdown.
B. High-Frequency Breakdown of the Calderón-Yukawa
CFIE
The high-frequency breakdown corresponds to the regime
in which the wavenumber 𝑘increases and the mesh parameter
ℎdecreases proportionally to the inverse of the wavenumber.

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Concretely, it corresponds to increasing the frequency while
keeping the number of basis functions per wavelength constant
(e.g., ten subdivisions per wavelength is a common choice).
In other words, we are interested in the high-frequency regime
which is characterized by ℎ𝑘 =const while 𝑘→ ∞.
It can be shown that in general both the standard CFIE
and the Calderón-Yukawa CFIE have singular values that
are unbounded in 𝑘. As, however, the problem is that the
singular values of the analytic operator are unbounded in 𝑘, the
condition number of the Calderón-Yukawa CFIE grows when
𝑘→ ∞. On spherical geometries, it has been proved that
the condition number grows as O(𝑘2/3)[162]. As was seen
in the previous sections, the spectrum of the electromagnetic
boundary integral operators can be separated into parts: one for
the solenoidal functions and the other for the non-solenoidal
functions. The O(𝑘2/3)growth of the condition number is
due to a maximum singular value in the solenoidal part of
the spectrum that grows as O(𝑘1/3)and a minimum singular
value in the non-solenoidal part of the spectrum that decreases
as O(𝑘−1/3). This effect is related to the high-frequency
breakdown in the scalar Helmholtz equation in acoustics,
where both the Dirichlet problem solved with a standard CFIE
and the Neumann problem solved with a Calderón-Yukawa
CFIE have condition numbers that grow as O(𝑘1/3)on a
sphere [163], [164]. A more detailed analysis of the high
frequency breakdown on the sphere is given in the appendix.
C. Remedies
The preconditioners to cure the high-frequency breakdown
that are found in the literature are based on the use of a
modified wavenumber 𝑘m. This modified wavenumber 𝑘mis
the original wavenumber 𝑘plus a shift in the complex plane.
This modified wavenumber has the form [165]
𝑘m=𝑘+i𝑐𝑘 1/3𝑎−2/3,(143)
where 𝑐is a dimensionless constant and 𝑎is a length that de-
pends on the geometry. For example, on spherical geometries,
𝑎is the radius of the sphere and 𝑐is often chosen to be 𝑐≈0.4,
which is actually based on optimizations for the conditioning
of the acoustic operators [166], but also give good results in
the electromagnetic case. A heuristic from [162] is to choose
as the maximum of the absolute value of the mean curvature
of the object boundary 𝐻=1/𝑎.
Among the solutions found in the literature, [162] proposes
a Calderón-like preconditioning with a modified wavenumber
𝑘min the preconditioner of the EFIE operator, then an un-
changed MFIE is added in a CFIE fashion. The work [167]
also uses a Calderón-like preconditioning of the MFIE with
the same 𝑘m. The work in [123] uses the inverse square root of
the vector Helmholtz operator with the modified wavenumber
as a preconditioner for the non-solenoidal part of the equation.
In [168], two different symmetric formulations for the EFIE are
proposed that are high-frequency stable (in addition to being
low-frequency and dense-discretization stable): one based on
a loop-star decomposition with a scalar Helmholtz operator
(with 𝑘m) and its inverse to precondition separately the loop
and star components, and the other based on quasi-Helmholtz
projectors to precondition independently the solenoidal and
the non-solenoidal part with the vector Helmholtz operator
and its inverse. Essentially, in the two cases, the EFIE is
squared with the Helmholtz operator in the middle, which
avoids the discretization of the EFIE with dual basis functions
similar to the refinement-free Calderón preconditioner from
Section V-A2.
VII. A Note on the Role of the Basis Function Gram
Matrix on Conditioning
Given that several of the techniques described above aim to
obtain a preconditioned equation which is spectrally equivalent
to an identity, it follows that the final condition number is
related to the one of the (potentially mixed) Gram matrix
between the rightmost expansion and leftmost testing basis
functions used. In the special case in which these two bases
coincide, the Gram matrix of the basis will be the one dictating
ultimately the conditioning behavior. There are two main
causes of ill-conditioning for a Gram matrix associated with
a specific basis function set: (i) the mesh quality is low or
(ii) the basis functions linear dependency increases with one
of the relevant parameters (mesh size and polynomial order,
for example). Below we will briefly discuss these two cases
together with related solution approaches.
A. Ill-shaped mesh elements
Simply speaking, a mesh is ill-shaped if it has narrow
triangles (see Figure 15 for a mesh with narrow triangles;
for a more nuanced discussion on measures for the quality
of meshes, see [169]). Ill-shaped meshes lead to higher con-
dition numbers of the discretized operator compared with a
discretization based on a well-shaped mesh, that is, where
the angles of each triangle have roughly the same size. To
quantify the impact of ill-conditioning due to the mesh, one
can consider the condition number of discretized I, that
is, G𝒇,𝒇: the condition number will be a baseline for any
other integral operator discretized with RWG basis functions.
Clearly, this problem is not specific to surface integral equation
methods, volume integral equation methods or finite element
methods also suffer from this issue.
This kind of Gram matrix ill-conditioning can be cured by
the optimization of the geometry or the application of mesh
smoothers, a strategy to avoid this cumbersome (often manual)
work has been presented by Stephenanson and Lee [170].
Based on a study of the SVD of the EFIE system matrix,
degrees of freedom associated with the ill-shaped triangles
are eliminated. An alternative strategy can be the use of
discontinuous Galerkin (DG) strategies. In fact, one source
of ill-shaped meshes stems from geometries that necessitate
a finer discretization of certain parts of the geometry to
either capture important details or to avoid a work intensive
cleanup of the geometry. With DG individual parts of the
surface are meshed independently from each other. Weak
conditions are used to enforce current continuity [37]. Lastly,
one can entirely avoid this issue by discretizing directly on
the spline-based geometry, that is, leveraging methods from

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Fig. 15. Example of ill-shaped triangles.
isogeometric analysis. For the electric field integral equation
such approaches have recently been presented [171]–[174].
Improvements of the Calderón preconditioning in (128) can
be expected by placing an additional Gram matrix resulting in
G−T
ˆ
𝒏×𝒇,
𝒇
P𝑘
T𝑘
P𝑘G−1
ˆ
𝒏×𝒇,
𝒇P𝑘T𝑘P𝑘y
=−G−T
ˆ
𝒏×𝒇,
𝒇
P𝑘
T𝑘
P𝑘G−1
ˆ
𝒏×𝒇,
𝒇P𝑘ei,(144)
where G−T
ˆ
𝒏×
𝒇,𝒇
=GT
ˆ
𝒏×
𝒇,𝒇−1
. It ensures that the condition
number can be bounded independently from ℎand the bases
used for discretization. For a well-conditioned bases such
as RWG and BC functions, this is not necessary. However,
it typically improves the condition number, in particular, if
the mesh contains ill-shaped triangles or if the discretization
is non-homogeneous which occurs, for instance, in the case
of meshes resulting from adaptive refinement. The cost for
inverting G−T
ˆ
𝒏×
𝒇,𝒇are often outweighed by a saving in the
number of iterations required to solve the preconditioned
system since the cost of a matrix-vector product with Tor

Tare often more expensive than inverting G−T
ˆ
𝒏×
𝒇,𝒇iteratively.
B. Basis Functions Linear Dependency
The choice of basis functions affects the conditioning of the
Gram matrix and thus of the discretized operator (especially
when the latter is of the second kind, as it often happens for
preconditioned formulations). A particularly important case
of Gram matrix ill-conditioning though is the one found
in higher-order basis functions. The most popular types of
higher-order bases for electromagnetic integral equations are
interpolatory and hierarchical higher-order bases [175]–[186].
Hierarchical higher-order basis functions should not be
confused with hierarchical bases used for curing the dense-
discretization breakdown, which we discussed in Section V-B.
Typical hierarchical bases for curing the dense-discretization
breakdown of the EFIE are of the same polynomial order as
first order RWG bases. While the use of such hierarchical
basis functions leads with an increasing order 𝑝to a faster
convergence with respect to the analytic solution, it also
increases the condition number. For interpolatory bases, an
exponential growth of the condition number in 𝑝has been
reported, in contrast to a polynomial growth for hierarchical
bases [181, p. 187].
Strategies to overcome the 𝑝-breakdown rely on construct-
ing orthogonal higher-order basis functions. In fact, hierarchi-
cal higher-order basis functions enforce orthogonality between
functions on the same level [178], [179]. Further improvements
have been presented for wire-, quadrilateral-, and brick-type
elements [187]–[189].
For some of these hierarchical higher-order basis functions,
Calderón-type preconditioners have been presented [190]. As
a consequence from operator preconditioning theory [110],
[191], it is possible to obtain a higher-order discretization of
the EFIE, where the condition number is bounded indepen-
dently from ℎand 𝑝. It must be noted though that while this
is true for the overall condition number, the (inverse) Gram
matrices that appear in the Calderón preconditioner remain
ill-conditioned in 𝑝.
VIII. Conclusion
This paper has presented an overview of the state of the
art in electromagnetic integral equation preconditioning. Dif-
ferent spectral regularization strategies have been discussed,
further analyzed, and framed in the overall context of solving
challenging large computational electromagnetics problems.
Bibliographic reviews have been alternated with theoretical
treatments and corroborating numerical results to show the
practical impacts of all different strategies.
Appendix A
Continuous Interpretations
In this section, 𝛤is a sphere of radius 𝑎centered at the
origin. The scalar spherical harmonics (SH) are noted 𝑌𝑙𝑚 with
𝑙≥0and |𝑚|< 𝑙. The vector spherical harmonics (VSH) are
defined as
𝒀𝑙𝑚 (ˆ
𝒓)=ˆ
𝒓𝑌𝑙𝑚 (ˆ
𝒓),(145)
𝑿𝑙𝑚 (ˆ
𝒓)=𝑎
i𝑙(𝑙+1)ˆ
𝒓× ∇𝑌𝑙𝑚(ˆ
𝒓),(146)
𝑼𝑙𝑚 (ˆ
𝒓)=−𝑎
i𝑙(𝑙+1)∇𝑌𝑙𝑚(ˆ
𝒓),(147)
They form an orthonormal basis for vector fields on 𝛤with
respect to the weighted inner product
h𝒖,𝒗i=𝛤
𝒖·𝒗d𝑆(𝒓)
𝑎2=2π
𝜑=0π
𝜃=0
𝒖·𝒗sin 𝜃d𝜃d𝜑 , (148)
Note that 𝑿𝑙 𝑚 and 𝑼𝑙𝑚 are tangential, while 𝒀𝑙𝑚 are radial
vector fields. Therefore, only 𝑿𝑙 𝑚 and 𝑼𝑙𝑚 are used in the
analysis of the surface integral operators. By construction, this
basis is Helmholtz decomposed: 𝑿𝑙 𝑚 are solenoidal and 𝑼𝑙𝑚
are irrotational.
In the following, J𝑙is the Riccati-Bessel function and H𝑙is
the Riccati-Hankel function of first kind. They are related to
the Bessel function 𝐽𝑙and the Hankel function of first kind 𝐻𝑙
by J𝑙(𝑧)=π𝑧/2 J𝑙+1/2(𝑧)and H𝑙(𝑧)=π𝑧/2𝐻𝑙+1/2(𝑧). Also,
0denotes their derivative with respect to their argument. We
thus obtain the analytic expressions of the EFIE operator [108],
[192], [193]
T𝑘𝑿𝑙𝑚 =−J𝑙(𝑘 𝑎)H𝑙(𝑘 𝑎)𝑼𝑙 𝑚 ,(149)
T𝑘𝑼𝑙𝑚 =J0
𝑙(𝑘𝑎)H0
𝑙(𝑘𝑎)𝑿𝑙𝑚 ,(150)
the exterior MFIE operator
(I/2+K𝑘)𝑿𝑙𝑚 =iJ0
𝑙(𝑘𝑎)H𝑙(𝑘𝑎)𝑿𝑙𝑚 ,(151)
(I/2+K𝑘)𝑼𝑙𝑚 =−iJ𝑙(𝑘 𝑎)H0
𝑙(𝑘𝑎)𝑼𝑙𝑚 ,(152)

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and the interior MFIE operator
(−I/2+K𝑘)𝑿𝑙 𝑚 =iJ𝑙(𝑘𝑎)H0
𝑙(𝑘𝑎)𝑿𝑙𝑚 ,(153)
(−I/2+K𝑘)𝑼𝑙 𝑚 =−iJ0
𝑙(𝑘𝑎)H𝑙(𝑘𝑎)𝑼𝑙𝑚 .(154)
The tangential vector fields 𝑿𝑙𝑚 and 𝑼𝑙 𝑚 are eigenvectors
of the MFIE operator, and they are singular vectors of the
EFIE operators. Assume we discretize the operator Twith
the VSH, then it is clear that the singular values associated
to the right singular vectors 𝑿𝑙 𝑚 are 𝜎𝑙, 𝑿=|J𝑙(𝑘𝑎)H𝑙(𝑘𝑎)|,
and the singular values associated to the right singular vectors
𝑼𝑙𝑚 are 𝜎𝑙,𝑼=|J0
𝑙(𝑘𝑎)H0
𝑙(𝑘𝑎)|. By studying the behavior of
these singular values in different regimes, we can obtain the
asymptotic behavior of the condition number of the discretized
EFIE. Since ∇ · 𝑿𝑙𝑚 =0, the study of the operators applied to
𝑿𝑙𝑚 gives the behavior of the operators applied to solenoidal
functions. Whereas the study of the operators applied to
𝑼𝑙𝑚 gives the behavior of the operators applied to non-
solenoidal functions. The following asymptotic expressions
can be obtained easily from the asymptotic behavior of the
special functions in different regimes [194].
A. Low-Frequency Breakdown and other Issues
1) Low-Frequency Breakdown
We study the condition number of the discretized EFIE for
𝑘→0, where we keep the number of basis functions constant.
In this case, it corresponds to considering only the orders 𝑙less
than some constant 𝑙max. Thus in the asymptotic expansion 𝑙
is fixed and independent of 𝑘. In the following, we will use
the asymptotic behavior of the Riccati-Bessel, Riccati-Hankel,
and their derivatives for small arguments given by
J𝑙(𝑘𝑎) ∼
𝑘→0(𝑘𝑎)𝑙+1
(2𝑙+1)!! (155)
J0
𝑙(𝑘𝑎) ∼
𝑘→0(𝑙+1)(𝑘 𝑎)𝑙
(2𝑙+1)!! (156)
H𝑙(𝑘𝑎) ∼
𝑘→0−i(2𝑙−1)!!
(𝑘𝑎)𝑙(157)
H0
𝑙(𝑘𝑎) ∼
𝑘→0
i𝑙(2𝑙−1)!!
(𝑘𝑎)𝑙+1(158)
where !! is the double factorial (or semifactorial) that is defined
by the recurrence relation 𝑛!! =(𝑛−2)!!𝑛and 1!! =0!! =1.
Therefore, as 𝑘→0, the singular values scale as
𝜎𝑙,𝑿=|J𝑙(𝑘 𝑎)H𝑙(𝑘 𝑎)| =O(𝑘),(159)
𝜎𝑙,𝑼=|J0
𝑙(𝑘𝑎)H0
𝑙(𝑘𝑎)| =O(𝑘−1).(160)
As a consequence, the condition number grows as O(𝑘2). The
singular values of the EFIE operator on a sphere have been
represented on Figure 16.
A way to solve this problem is by scaling the solenoidal
part by 𝑘−1and by scaling the non-solenoidal part by 𝑘. In
the VSH basis, it is simply a diagonal preconditioner, but in
practical applications one has to use some techniques such
as a loop-star decomposition or quasi-Helmholtz projectors to
decompose the two subspaces before rescaling them. Another
way to solve the problem is by Calderón preconditioning. In
10−1100101
10−2
10−1
100
101
𝑘 𝑎 [rad]
Singular value
𝜎𝑙,𝑼, 𝑙 =1𝜎𝑙,𝑼, 𝑙 =2𝜎𝑙,𝑼, 𝑙 =3O ( ( 𝑘 𝑎)−1)
𝜎𝑙,𝑿, 𝑙 =1𝜎𝑙 ,𝑿, 𝑙 =2𝜎𝑙, 𝑿, 𝑙 =3O ( ( 𝑘 𝑎)1)
Fig. 16. Singular values of the EFIE associated to the first three harmonic
orders (𝑙=1,2,3) as functions of the frequency. The low frequency behavior
is given by a growth of the maximum singular value as (𝑘 𝑎)−1, and the
decrease of minimum singular value as (𝑘 𝑎), which results in a condition
number that grows proportionally to (𝑘 𝑎)−2as 𝑘 𝑎 →0. In higher frequencies
there are the first resonant frequencies (at 𝑘 𝑎 ≈2.74,3.87,etc.) around which
the condition number is unbounded.
this case, the main operator is (T𝑘)2, whose singular values
are
|J𝑙(𝑘𝑎)H𝑙(𝑘𝑎)J0
𝑙(𝑘𝑎)H0
𝑙(𝑘𝑎)| =O(1)(161)
as 𝑘→0. As a result, the condition number of the Calderón
EFIE remains bounded as 𝑘→0.
2) Low-Frequency Round-Off Errors in the Excitation
We consider the plane wave 𝒇pw (𝒓)=𝑒0ei𝑘ˆ
𝒛·𝒓ˆ
𝒙. Its tangen-
tial component expands as [195]
h𝑿𝑙,±1,𝒇pw i=2πi𝑙𝑒02𝑙+1
4π
J𝑙(𝑘𝑎)
𝑘𝑎 ,(162)
h𝑼𝑙,±1,𝒇pwi=±2πi𝑙𝑒02𝑙+1
4π
J0
𝑙(𝑘𝑎)
𝑘𝑎 .(163)
For 𝑘→0, the dominant terms of the solenoidal and non-
solenoidal parts are
h𝑿1,±1,𝒇pwi=O(𝑘𝑎),(164)
h𝑼1,±1,𝒇pwi=O(1).(165)
When the RWG basis functions are used for the discretization,
the two components are summed together; due to finite ma-
chine precision and the associated numerical round-off errors,
the solenoidal part is lost.
3) Low-Frequency Round-off Errors in the Solution
Consider the EFIE T𝑘𝒋𝛤=−ˆ
𝒏×𝒆iwhen the excitation is
the plane wave 𝒆i(𝒓)=𝑒0ei𝑘ˆ
𝒛·𝒓ˆ
𝒙. From (149) and (162), the
solution 𝒋𝛤has the following expansion
h𝑿𝑙,±1,𝒋𝛤i=2πi𝑙
H𝑙(𝑘𝑎)𝑘 𝑎 2𝑙+1
4π𝑒0,(166)
h𝑼𝑙,±1,𝒋𝛤i=±2πi𝑙
H0
𝑙(𝑘𝑎)𝑘 𝑎 2𝑙+1
4π𝑒0.(167)

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For 𝑘→0, the dominant terms of the solenoidal and non-
solenoidal parts are respectively
h𝑿1,±1,𝒋𝛤i=O(1),(168)
h𝑼1,±1,𝒋𝛤i=O(𝑘𝑎).(169)
Again, similarly to the round-off errors in the excitation, the
solution has two Helmholtz components that scale differently
with the frequency. Therefore, in a numerical solver, unless
some care is taken by treating the two components separately,
the non-solenoidal part will get canceled.
B. Dense-Discretization Breakdown
We study the condition number of the discretized EFIE
when ℎdecreases and the frequency remains constant. A
decrease in ℎis equivalent to an increase of the maximum
spacial frequency of the functions that are discretized. In the
case of the VSH, their spacial frequency increases with their
order 𝑙. The number of VSH whose order is less or equal
to 𝑙grows quadratically with 𝑙, due to their multiplicity on
the order 𝑚(|𝑚| ≤ 𝑙). When we use local basis functions
such as RWG to discretize the operators, the number of
unknowns also grows quadratically with 𝑎/ℎ. Therefore, to
study the dense-discretization breakdown, we can use the
correspondence 𝑙=O(𝑎/ℎ). In the following, we will use
the asymptotic behavior of the Riccati-Bessel, Riccati-Hankel,
and their derivatives for large order given by
J𝑙(𝑘𝑎) ∼
𝑙→+∞
1
√2e e𝑘𝑎
2𝑙+1𝑙+1
,(170)
J0
𝑙(𝑘𝑎) ∼
𝑙→+∞ e
2
𝑙+1
2𝑙+1e𝑘𝑎
2𝑙+1𝑙
,(171)
H𝑙(𝑘𝑎) ∼
𝑙→+∞ −i2
e2𝑙+1
e𝑘𝑎 𝑙
,(172)
H0
𝑙(𝑘𝑎) ∼
𝑙→+∞
i𝑙√2e
2𝑙+12𝑙+1
e𝑘𝑎 𝑙+1
.(173)
Therefore, as 𝑙→ +∞, the singular values scale as
𝜎𝑙,𝑿=|J𝑙(𝑘 𝑎)H𝑙(𝑘 𝑎)| =O(𝑙−1)=O(ℎ),(174)
𝜎𝑙,𝑼=|J0
𝑙(𝑘𝑎)H0
𝑙(𝑘𝑎)| =O(𝑙)=O(ℎ−1).(175)
As a consequence, the condition number grows as O(ℎ−2). The
spectrum of the EFIE have been represented on Figure 17 for
different frequencies to illustrate this growth of the condition
number as the discretization density increases.
This problem can be solved using Calderón preconditioning:
as 𝑙→ +∞ (or equivalently as ℎ→0), the singular values of
(T𝑘)2are
|J𝑙(𝑘𝑎)H𝑙(𝑘𝑎)J0
𝑙(𝑘𝑎)H0
𝑙(𝑘𝑎)| =𝜎𝑙, 𝑿𝜎𝑙,𝑼=O(1).(176)
As a result the condition number of the Calderón EFIE remains
bounded as ℎ→0.
C. High-Frequency Resonances
We study the condition number for 𝑘𝑎 1. Here the point
to notice is that 𝑙is kept constant as 𝑘𝑎 → +∞, and
J𝑙(𝑘𝑎)=sin(𝑘 𝑎 −𝑙π/2) + O((𝑘 𝑎)−1),(177)
J0
𝑙(𝑘𝑎)=cos(𝑘 𝑎 −𝑙π/2) + O((𝑘 𝑎)−1).(178)
So the Riccati-Bessel function and its derivative have zeros,
and as a consequence, there are an infinite number of fre-
quencies such that one of the 𝜎𝑙,𝑿or 𝜎𝑙,𝑼is 0. Therefore
the condition number of the EFIE is unbounded around these
resonant frequencies, which makes the EFIE unsolvable in
practice without further treatment. The MFIE suffers from the
same problem.
The resonant frequencies can be read on Figure 16 with
the first one appearing at 𝑘 𝑎 ≈2.74. Also, it is clear from
Figure 16 that the higher 𝑘 𝑎 is, the more common the resonant
frequencies are. The result of the resonances on the spectrum
is clear on Figure 17 where there is a finite number of singular
values that are unbounded from below at a fixed frequency.
The common way to solve this problem is to discretize the
CFIE which combines the EFIE (6) scaled by 𝛼and the MFIE
(11) on which is applied (1−𝛼)ˆ
𝒏×, where 0< 𝛼 < 1is a
dimensionless constant. The CFIE is
𝛼ˆ
𝒏×𝒆i+ (1−𝛼)ˆ
𝒏׈
𝒏×𝒉i=
−𝛼T𝑘+ (1−𝛼)ˆ
𝒏×I/2+K𝑘𝒋𝛤.(179)
The singular values of this CFIE operator associated to the
right singular vector 𝑿𝑙 𝑚 and 𝑼𝑙𝑚 are, respectively,
|𝛼J𝑙(𝑘𝑎) + i(1−𝛼)J0
𝑙(𝑘𝑎)|| H𝑙(𝑘𝑎)|,(180)
|𝛼J0
𝑙(𝑘𝑎) − i(1−𝛼)J𝑙(𝑘 𝑎)||H0
𝑙(𝑘𝑎)|,(181)
which are never 0 since the zeros of J𝑙and J0
𝑙never coincide
(due to the fact that the zeros of the Bessel functions are simple
with an exception at 𝑘𝑎 =0which we do not consider). In
other words, the CFIE can always be inverted.
Note that the CFIE only fixes the problem of the resonances
that make the MFIE operator and EFIE operator non-invertible
for some frequencies. Around these resonant frequencies the
condition number is unbounded. However, the CFIE still
suffer from the dense-mesh breakdown. The Calderón-Yukawa
CFIE is dense-mesh stable, low-frequency stable and free
from resonances. But it still suffers from the high-frequency
breakdown which is an asymptotic growing of the condition
number as 𝑘→ ∞ while keeping 𝑘 ℎ constant. It is discussed
in the next section.
D. High-Frequency Breakdown
In this section we study the asymptotic behavior of the
singular values of the Calderón-Yukawa CFIE operator [96]
−Ti𝑘T𝑘+ (I/2−Ki𝑘)(I/2+K𝑘).(182)
Recall that the CFIE is used to avoid the interior resonances,
while Calderón preconditioning avoids the dense-discretization

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100101102
10−3
10−2
10−1
100
101
102
Order 𝑙
Singular value
𝜎𝑙,𝑼, 𝑘 𝑎 =1𝜎𝑙,𝑼, 𝑘 𝑎 =5𝜎𝑙,𝑼, 𝑘 𝑎 =10
𝜎𝑙,𝑿, 𝑘 𝑎 =1𝜎𝑙,𝑿, 𝑘 𝑎 =5𝜎𝑙, 𝑿, 𝑘 𝑎 =10
𝑙=O (𝑎/ℎ)𝑙−1=O ( ℎ/𝑎)
Fig. 17. Singular values of the operator of the EFIE on a sphere for three
different frequencies. The maximum singular value grows as ℎ−1and the
minimum singular value decreases as ℎ, resulting in a condition number
growth proportional to ℎ−2.
breakdown. The Yukawa part denotes the use of pure imagi-
nary wavenumbers in the preconditioning operators so that the
preconditioners do not reintroduce interior resonances.
We show in this section that the maximum singular value,
which is associated with the solenoidal functions, scales at
least proportionally to 𝑘1/3, while the minimum singular value,
which is associated with the non-solenoidal functions, scales
at most as O(𝑘−1/3). The result is a growth of the condition
number faster than a constant times 𝑘2/3where numerical
evidences on the spherical harmonics show that this limit
is in practice attained. The singular values of the Calderón-
Yukawa CFIE have been represented on Figure 18 as well
as the asymptotic bounds that are derived in the following
sections.
a) Minimum singular value: The singular value associ-
ated with the nonsolenoidal VSH 𝑼𝑙𝑚 are
𝜎𝑙,𝑼=|H𝑙(i𝑘𝑎)H0
𝑙(𝑘𝑎)(J0
𝑙(i𝑘𝑎)J𝑙(𝑘𝑎) + J𝑙(i𝑘 𝑎)J0
𝑙(𝑘𝑎))| .
(183)
It is useful to treat the order 𝑙as a real variable instead
of an integer. We obtain an upper bound for the minimum
singular values by, choosing as 𝑙the value for which J𝑙(𝑘𝑎)=
π𝑘𝑎/2 J𝑙+1/2(𝑘𝑎)=0, that is, for which 𝑘𝑎 is a zero of J𝑙+1/2
(Jis the Bessel function and Jis the Ricatti-Bessel function).
For the first zero of J𝑙+1/2, the asymptotic expansion is given
by [194, eq. 10.21.40]
𝑘𝑎 ∼𝑙+𝛼𝑙1/3+1/2,(184)
or, equivalently, when
𝑙∼𝑘𝑎 −𝛼(𝑘𝑎)1/3−1/2(185)
with 𝛼=−2−1/3𝑎1≈1.856, where 𝑎1≈ −2.34 is the first
zero of Ai, which denotes the Airy function of the first kind.
Using the uniform asymptotic expansion of the derivatives of
the Bessel and Hankel functions for large orders [194, eq.
10.20.7-9], we find that the upper bound for the minimum
singular value is asymptotically
πAi0(𝑎1)|Ai0(𝑎1) − i Bi0(𝑎1)|
27/6(𝑘𝑎)−1/3≈0.69(𝑘 𝑎)−1/3,
(186)
where Bi denotes the Airy function of the second kind.
b) Maximum singular value: The singular values associ-
ated with the solenoidal VSH 𝑿𝑙𝑚 are
𝜎𝑙,𝑿=|H0
𝑙(i𝑘𝑎)H𝑙(𝑘𝑎)(J0
𝑙(i𝑘𝑎)J𝑙(𝑘𝑎) + J𝑙(i𝑘 𝑎)J0
𝑙(𝑘𝑎))| .
Similarly to the previous case, there is another constant 𝛼such
that the index of a lower bound for the maximum singular value
is asymptotically
𝑙∼𝑘𝑎 −𝛼(𝑘𝑎)1/3−1/2.(187)
Using the asymptotic expansion of the Bessel functions in the
transition region [194, eq. 10.19.8-9] we find that the dominant
term in 𝜎𝑙, 𝑿is proportional to Ai(−21/3𝛼)|Ai(e−iπ/321/3𝛼)|.
So 𝛼has to maximize this quantity, and a derivative of it
shows that 𝛼is solution of
Ai0(𝑧)(2 Ai (𝑧)2+Bi(𝑧)2) + Ai(𝑧)Bi(𝑧)Bi0(𝑧)=0,(188)
where 𝑧=−21/3𝛼. Numerically, we obtain 𝛼≈0.623, and the
lower bound for the maximum singular value is asymptotically
πAi(𝑧)| Ai(𝑧) − i Bi (𝑧)|
25/6(𝑘𝑎)1/3≈0.524(𝑘 𝑎)1/3.(189)
c) High-frequency stabilization: Notice that the
Calderón-EFIE (and the MFIE, respectively) does, in some
sense, not suffer from the high-frequency breakdown since
the singular values of its operator are bounded from above
by a constant independent from 𝑘, but it is unstable due
to the interior resonances. In contrast, the Calderón-Yukawa
CFIE is free from interior resonances, but it suffers from the
high-frequency breakdown as the maximum singular values
of its operator is unbounded in 𝑘. To solve this issue one
can to construct a Calderón-CFIE, but instead of using the
wavenumber 𝑘in the preconditioner, one uses a modified
wavenumber 𝑘mthat is shifted in the complex plane to avoid
reintroducing the resonances resulting in [167]
−T𝑘mT𝑘+ (I/2−K𝑘𝑚)(I/2+K𝑘).(190)
This modified wavenumber for the sphere has the form [123],
[162]
𝑘m=𝑘+i𝑐𝑘 1/3𝑎−2/3(191)
where 𝑐∈Ris a constant. A common choice that gives
a good condition number is 𝑐≈0.4. The singular values
of this operator are similar to those of the Calderón-Yukawa
CFIE with the pure imaginary wavenumber i𝑘replaced by 𝑘𝑚,
namely
𝜎𝑙,𝑼=|H𝑙(𝑘m𝑎)H0
𝑙(𝑘𝑎)(J0
𝑙(𝑘m𝑎)J𝑙(𝑘𝑎) + J𝑙(𝑘m𝑎)J0
𝑙(𝑘𝑎))|
(192)
𝜎𝑙,𝑿=|H0
𝑙(𝑘m𝑎)H𝑙(𝑘𝑎)(J0
𝑙(𝑘m𝑎)J𝑙(𝑘𝑎) + J𝑙(𝑘m𝑎)J0
𝑙(𝑘𝑎))| .
(193)

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These singular values have been plot in Figure 19 for increas-
ing frequencies. It is evident that the condition number is
bounded.
Appendix B
Inverse Gram matrix properties
Lemma 1. For the inverse mixed Gram matrix Gˆ
𝒏×
𝒇,𝒇, we
have
PΣHG−1
ˆ
𝒏×
𝒇,𝒇PΛ=0(194)
PΣG−1
ˆ
𝒏×
𝒇,𝒇PΛH=0(195)
Proof. First, we define the transformation matrix Q=
Λ H Σ , where the global loops are constructed such that
ΛTH=0. Furthermore, without loss of generality, we assume
that loop and star functions have been eliminated such that
they are linearly independent. In this case, we can denote the
projectors as
PΛH=Λ(ΛTΛ)−1ΛT+H(HTH)−1HT,(196)
PΣH=Σ(ΣTΣ)−1ΣT+H(HTH)−1HT.(197)
To reveal how G−1
ˆ
𝒏×
𝒇,𝒇acts on the different quasi-
Helmholtz subspaces, we derive an explicit expression for
Q−TQTGQQ−1−1
=QQTGQ−1
QT, where in the follow-
ing we abbreviate Gˆ
𝒏×
𝒇,𝒇=G. First, we recall the block
structure of
QTGQ =
ΛTGΛ ΛTG H ΛTGΣ
HTGΛ HTGH HTGΣ
ΣTGΛ Σ TGH ΣTGΣ
=
ΛTGΛ ΛTG H ΛTGΣ
0HTGH HTGΣ
0 0 ΣTG Σ
,(198)
which follows from the orthogonality of solenoidal (i.e., 𝜦)
and irrotational functions (i.e., ˆ
𝒏×
𝜮)/ harmonic functions
with irrotational perturbation (i.e., ˆ
𝒏×
𝑯).
Using the Schur complement, we have
A B
0D−1
=A−1−A−1BD−1
0D−1(199)
Recursively application of (199) to (198) yields (200) shown
at the top of this page; explicitly computing QQTGQ−1
QT
results in (201). Evidently, we have
HTG−1Λ=0,(202)
ΣTG−1Λ=0,(203)
ΣTG−1H=0,(204)
and considering the definition of projectors, the lemma fol-
lows. 
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QTGQ−1
=

(ΛTGΛ)−1−(ΛTGΛ)−1ΛTGH(HTGH)−1(ΛTGΛ)−1ΛTGH (HTGH)−1HTGΣ (ΣTGΣ)−1− (ΛTGΛ)−1ΛTGΣ (ΣTGΣ)−1
0(HTGH)−1−(HTG H)−1HTGΣ(ΣTGΣ)−1
0 0 (ΣTGΣ )−1
(200)
QQTGQ−1
QT=Λ(ΛTGΛ)−1ΛT−Λ(ΛTG Λ)−1ΛTGH (HTGH )−1HT+Λ(ΛTGΛ)−1ΛTGH(HTG H)−1HTGΣ(ΣTGΣ)−1ΣT
−Λ(ΛTGΛ)−1ΛTG Σ(ΣTGΣ)−1ΣT+H(HTGH)−1HT−H(HTG H)−1HTGΣ(ΣTGΣ)−1ΣT+Σ(ΣTG Σ)−1ΣT(201)
100101102
10−1
100
Order 𝑙
Singular value
𝜎𝑙,𝑼, 𝑘 𝑎 =50 𝜎𝑙,𝑼, 𝑘 𝑎 =100 𝜎𝑙,𝑼, 𝑘 𝑎 =200
𝜎𝑙,𝑿, 𝑘 𝑎 =50 𝜎𝑙,𝑿, 𝑘 𝑎 =100 𝜎𝑙, 𝑿, 𝑘 𝑎 =200
O ( ( 𝑘𝑎 )1/3) O ( ( 𝑘𝑎 )−1/3)
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Simon B. Adrian (S’09–M’19) Simon Adrian re-
ceived the Bachelor of Science (B.Sc.) degree in
Electrical Engineering and Information Technology
from the Technical University of Munich (TUM),
Munich, Germany, in 2009, the Master of Science
(M.S.) degree in Electrical and Computer Engi-
neering from the Georgia Institute of Technology,
Atlanta, GA, USA, in 2010, the Diplom-Ingenieur
(Dipl.-Ing.) degree in Electrical Engineering and
Information Technology from the TUM in 2012,
and the Doktor-Ingenieur (Dr.-Ing) degree from the
TUM and the École nationale supérieure Mines-Télécom Atlantique Bretagne
Pays de la Loire (IMT Atlantique), Brest, France, in 2018.
From 2012 to 2019, he was research assistant at the TUM. In 2018, he was
Visiting Professor with the Politecnico di Torino, Turin, Italy. From 2019 to
2020, he was a Senior Engineer with Infineon AG, Neubiberg, Germany. Since
2020, he is Assistant Professor with the University of Rostock. His research
interest is in computational electromagnetics with a focus on integral equation
solvers. In particular, he is interested in preconditioning techniques, low-
frequency stable formulations, and fast solvers. Areas of application include
antenna modeling and bio-electromagnetic problems.
Dr. Adrian has been a member of the IEEE Antennas and Propagation
Society Education Committee since 2016, where he has been chair of the
Student Activities Subcommittee since 2021. He has been member of the
Union Radio-Scientifique Internationale (URSI) since 2018. He has been an
Associate Editor of the IEEE Transactions on Antennas and Propagation since
2020 and of the IEEE Antennas and Propagation Magazine since 2021. He is
laureate of an IEEE Antennas and Propagation Society Pre-Doctoral Research
Award and an IEEE Antennas and Propagation Society Doctoral Research
Award. He was awarded with the Kurt-Fischer-Preis 2012 for his diploma
thesis. His contribution to the student paper competition at IEEE APS/URSI
Symposium in 2013 was selected as an honorable mention paper. He received
the Second Prize in the Third International URSI Student Prize Paper
Competition of the URSI General Assembly and Scientific Symposium 2014,
the Best Young-Scientist-Paper Award of the Kleinheubacher Tagung 2014,
the Third Prize Student Paper Award of the IEEE APS/URSI Symposium
2015, the First Prize EMTS 2016 Young Scientist Best Paper Award of the
URSI International Symposium on Electromagnetic Theory 2016, the Third
Prize Student Paper Award of the IEEE APS/URSI Symposium 2018, and the
Dr.-Georg-Spinner-Hochfrequenz-Preis for the doctoral thesis in 2018.
Alexandre Dély received the M.Sc. Eng. degree
from the École Nationale Supérieure des Télé-
communications de Bretagne (Télécom Bretagne),
France, in 2015. He received the Ph.D. degree from
the École Nationale Supérieure Mines-Télécom At-
lantique (IMT Atlantique), France, and from the Uni-
versity of Nottingham, United Kingdom, in 2019.
He is currently working at Politecnico di Torino,
Turin, Italy. His research focuses on preconditioned
and fast solution of boundary element methods, fre-
quency domain and time domain integral equations.
Davide Consoli (S’19) Davide Consoli received
the MSc (Laurea) degree from the Politecnico di
Torino, Italy, in 2018 and he is currently working
towards his PhD in the same institute. His research
interests are in computational electromagnetics with
a focus on fast and preconditioned integral equation
formulations.
Adrien Merlini (S’16–M’19) received the M.Sc.
Eng. degree from the École Nationale Supérieure
des Télécommunications de Bretagne (Télécom Bre-
tagne), France, in 2015 and received the Ph.D.
degree from the École Nationale Supérieure Mines-
Télécom Atlantique (IMT Atlantique), France, in
2019.
From 2018 to 2019, he was a visiting Ph.D. sudent
at the Politecnico di Torino, Italy, which he then
joined as a Research Associate. Since 2019, he has
been an Associate Professor with the Microwave
Department, IMT Atlantique. His research interests include preconditioning
and acceleration of integral equation solvers for electromagnetic simulations
and their application in brain imaging.
Dr. Merlini received a Young Scientist Award at the URSI GASS 2020
meeting. In addition, he has co-authored a conference paper recipient of an
honorable mention at the URSI/IEEE-APS 2020. He is a member of IEEE-
HKN, the IEEE Antennas and Propagation Society, URSI France, and of the
Lab-STICC laboratory.

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Francesco P. Andriulli (S’05–M’09–SM’11) re-
ceived the Laurea in electrical engineering from the
Politecnico di Torino, Italy, in 2004, the MSc in
electrical engineering and computer science from the
University of Illinois at Chicago in 2004, and the
PhD in electrical engineering from the University of
Michigan at Ann Arbor in 2008. From 2008 to 2010
he was a Research Associate with the Politecnico
di Torino. From 2010 to 2017 he was an Associate
Professor (2010-2014) and then Full Professor with
the École Nationale Supérieure Mines-Télécom At-
lantique (IMT Atlantique, previously ENST Bretagne), Brest, France. Since
2017 he has been a Full Professor with the Politecnico di Torino, Turin,
Italy. His research interests are in computational electromagnetics with focus
on frequency- and time-domain integral equation solvers, well-conditioned
formulations, fast solvers, low-frequency electromagnetic analyses, and mod-
eling techniques for antennas, wireless components, microwave circuits, and
biomedical applications with a special focus on Brain Imaging.
Dr. Andriulli was the recipient of the best student paper award at the
2007 URSI North American Radio Science Meeting. He received the first
place prize of the student paper context of the 2008 IEEE Antennas and
Propagation Society International Symposium. He was the recipient of the
2009 RMTG Award for junior researchers and was awarded two URSI Young
Scientist Awards at the International Symposium on Electromagnetic Theory
in 2010 and 2013 where he was also awarded the second prize in the best
paper contest. He also received the 2015 ICEAA IEEE-APWC Best Paper
Award. In addition, he co-authored with his students and collaborators other
three first prize conference papers (EMTS 2016, URSI-DE Meeting 2014,
ICEAA 2009), a second prize conference paper (URSI GASS 2014), a third
prize conference paper (IEEE–APS 2018), three honorable mention conference
papers (ICEAA 2011, URSI/IEEE–APS 2013, URSI/IEEE–APS 20) and other
three finalist conference papers (URSI/IEEE-APS 2012, URSI/IEEE-APS
2007, URSI/IEEE-APS 2006). Moreover, he received the 2014 IEEE AP-S
Donald G. Dudley Jr. Undergraduate Teaching Award, the triennium 2014-
2016 URSI Issac Koga Gold Medal, and the 2015 L. B. Felsen Award for
Excellence in Electrodynamics.
Dr. Andriulli is a member of Eta Kappa Nu, Tau Beta Pi, Phi Kappa Phi,
and of the International Union of Radio Science (URSI). He is the Editor
in Chief of the IEEE Antennas and Propagation Magazine, he serves as a
Track Editor for the IEEE Transactions on Antennas and Propagation, and
as an Associate Editor for IEEE Access, URSI Radio Science Letters and
IET-MAP. He is the PI of the ERC Consolidator Grant “321”.