Laplacian Filtered Loop-Star Decompositions and Quasi-Helmholtz Laplacian Filters: Definitions, Analysis, and Efficient Algorithms

Abstract

Quasi-Helmholtz decompositions are fundamental tools in integral equation modeling of electromagnetic problems because of their ability of rescaling solenoidal and non-solenoidal components of solutions, operator matrices, and radiated fields. These tools are however incapable, per se, of modifying the refinement-dependent spectral behavior of the different operators and often need to be combined with other preconditioning strategies. This paper introduces the new concept of filtered quasi-Helmholtz decompositions proposing them in two incarnations: the filtered Loop-Star functions and the quasi-Helmholtz Laplacian filters. Because they are capable of manipulating large parts of the operators' spectra, new families of preconditioners and fast solvers can be derived from these new tools. A first application to the case of the frequency and h-refinement preconditioning of the electric field integral equation is presented together with numerical results showing the practical effectiveness of the newly proposed decompositions.

Full text

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Laplacian Filtered Loop-Star Decompositions and
Quasi-Helmholtz Laplacian Filters:
Definitions, Analysis, and Efficient Algorithms
Adrien Merlini, Member, IEEE, Clément Henry, Member, IEEE, Davide Consoli, Student Member, IEEE,
Lyes Rahmouni, Alexandre Dély, and Francesco P. Andriulli, Senior Member, IEEE
Abstract—Quasi-Helmholtz decompositions are fundamental
tools in integral equation modeling of electromagnetic problems
because of their ability of rescaling solenoidal and non-solenoidal
components of solutions, operator matrices, and radiated fields.
These tools are however incapable, per se, of modifying the
refinement-dependent spectral behavior of the different operators
and often need to be combined with other preconditioning strate-
gies. This paper introduces the new concept of filtered quasi-
Helmholtz decompositions proposing them in two incarnations:
the filtered Loop-Star functions and the quasi-Helmholtz Lapla-
cian filters. Because they are capable of manipulating large parts
of the operators’ spectra, new families of preconditioners and fast
solvers can be derived from these new tools. A first application
to the case of the frequency and h-refinement preconditioning
of the electric field integral equation is presented together with
numerical results showing the practical effectiveness of the newly
proposed decompositions.
Index Terms—Integral equations, quasi-Helmholtz decomposi-
tions, quasi-Helmholtz projectors, preconditioning, EFIE.
I. INTRODUCTION
INTEGRAL equation formulations are effective numerical
strategies for modeling radiation and scattering by perfectly
electrically conducting objects [1]–[3]. Their effectiveness
primarily derives from the fact that they only require the
scatterers’ surfaces to be discretized, automatically impose ra-
diation conditions and, thanks to the advent of fast algorithms
[4], give rise to linear-in-complexity approaches when solved
with iterative schemes—provided that the conditioning of the
linear system matrices resulting from their discretizations is
independent of the number of unknowns [5]. Among the well-
established formulations, the electric field integral equation
(EFIE) plays a crucial role, both in itself and within combined
field formulations [6]. The EFIE, lamentably, becomes ill-
conditioned when the frequency is low or the discretization
density high [7]. These phenomena—respectively known as
the low-frequency and h-refinement breakdowns—cause the
solution of the EFIE to become increasingly challenging to ob-
tain, as the number of iterations of the solution process grows
unbounded, which jeopardizes the possibility of achieving an
overall linear complexity.
D. Consoli, L. Rahmouni, A. Dély, and F. P. Andriulli are with the
Department of Electronics and Telecommunications, Politecnico di Torino,
10129 Torino, Italy; e-mail: name.surname@polito.it.
A. Merlini and C. Henry are with the Microwave department, IMT Atlan-
tique, 29238 Brest cedex 03, France; e-mail: name.surname@imt-atlantique.fr.
Manuscript received April 19, 2005.
Traditional approaches to tackle the low-frequency break-
down rely on standard quasi-Helmholtz decompositions such
as Loop-Star/Tree bases [8]–[12] that, despite curing the low-
frequency behavior, worsen the h-refinement ill-conditioning
of the EFIE [11] because of the derivative nature of the
change of basis [12]. A way to circumvent the issue is the
use of hierarchical strategies both on structured [13], [14]
and unstructured meshes [15]–[18]. These schemes, when
designed properly, can solve both the low-frequency and
the h-refinement problems but still rely on the construction
on an explicit, basis-based, quasi-Helmholtz decomposition
that requires the cumbersome detection of topological loops
whenever handles are present in the geometry. A popular
alternative strategy leverages Calderón identities to form a
second kind integral equation out of the EFIE. Calderón
approaches concurrently solve the low-frequency and the h-
refinement breakdowns without calling for an explicit quasi-
Helmholtz decomposition [19]–[26]. In their standard incar-
nations they do, however, require the use of a dual dis-
cretization and global loop handling, because global loops
reside in the static null-space of the Calderón operator. The
introduction of implicit quasi-Helmholtz decompositions via
the so called quasi-Helmholtz projectors [27], when combined
with Calderón approaches, led to the design of several well-
conditioned formulations, free from static nullspaces (see [7],
[27]–[29] and references therein) and, in some incarnations,
free from the need of performing a barycentric refinement [30].
Quasi-Helmholtz projectors have shown to be an effective and
efficiently computable tool for performing quasi-Helmholtz
decompositions, but, by themselves, they can only tackle
the low-frequency breakdown and must be combined with
Calderón-like strategies that involve multiple operators, to
obtain h-refinement spectral preconditioning effects. A set of
tools as versatile as the projectors that could also manipulate
the operator spectra beyond a simple rescaling would thus be
desirable.
This paper introduces such a new family of tools. The con-
tribution of this work is in fact threefold: (i) we will introduce
the concept of Laplacian-filtered Loop-Star decompositions, a
new quasi-Helmholtz decomposition approach that will allow
for a finer tuning of the operator spectrum with respect to
their standard Loop-Star counterparts. (ii) Just like standard
Loop-Star bases give rise to the quasi-Helmholtz projectors,
a suitable choice of projections on the filtered Loop-Star
spaces will give rise to a new family of mathematical objects,
arXiv:2211.07704v1 [math.NA] 14 Nov 2022

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the quasi-Helmholtz Laplacian filters, that will not require
handling basis functions and explicit decompositions, while
still providing the spectral tuning properties of (i). (iii) We
will obtain new frequency and h-refinement preconditioners
for the EFIE based on (i) and (ii) that represent a natural
first application of the newly proposed techniques. We believe,
however, that the applicability of the new spectral filters will
extend beyond EFIE preconditioning in further investigations.
The contribution will be further enriched by a section devoted
to efficient implementations of the newly defined tools that
will be obtained by leveraging strategies developed in the
context of polynomial preconditioning approaches and in
graph wavelet theory [31]–[34]. Numerical results will then
corroborate and confirm our theoretical considerations.
The paper is organized as follows: the background material
and the notation are presented in Section II, the new Laplacian
filtered Loop-Star decompositions and the quasi-Helmholtz
Laplacian filters are presented in Section III and Section IV,
respectively, along with their main properties. Various strate-
gies for computing the filters, in practical scenarios, are
detailed in Section V. Preconditioners tackling simultaneously
the low-frequency and h-refinement breakdowns of the EFIE
are then derived in Section VI. Finer details relating to the
implementation and computation of the filters and precon-
ditioners are then presented in Section VII. Illustrations of
the effectiveness of the schemes are provided in Section VIII,
before concluding in Section IX.
Preliminary results from this work were presented in the
conference contributions [35], [36].
II. NOTATI ON A ND BAC KG ROUND
Let Γbe a smooth surface modeling the boundary of a per-
fectly electrically conducting (PEC), closed scatterer enclosed
in a homogeneous background medium with permittivity and
permeability µ. The boundary Γcan be multiply connected
and contain holes. We denote by ˆ
n(r)the outward pointing
normal field at r. When illuminated by a time-harmonic
incident electric field Ei, a surface current density Jis
induced on Γthat satisfies the electric field integral equation
(EFIE)
TJ=TsJ+ThJ=−ˆ
n×Ei,(1)
where
TsJ=ˆ
n(r)×ikZΓ
eikkr−r0k
4πkr−r0kJ(r0)dS(r0),(2)
ThJ=−ˆ
n(r)×1
ik∇ZΓ
eikkr−r0k
4πkr−r0k∇0·J(r0)dS(r0),
(3)
and kis the wavenumber of the electromagnetic wave in
the background medium. Equation (1) can be solved nu-
merically by approximating Γwith triangular elements of
average edge length hand by approximating the current
density as J≈PN
n=1[j]nfnwith the Rao-Wilton-Glisson
basis functions {fn}n[37], in which Nis the number of
edges in the mesh, jis the vector of the coefficients of the
expansion, and fnis defined as
fn(r) = 






r−r+
n
2A+
n
if r∈c+
n
−r−r−
n
2A−
n
if r∈c−
n,
(4)
where the notation of Fig. 1 was employed and where A±
nis
the area of the cell c±
n.
The final step to obtain the discretized EFIE is to test (1)
with the rotated RWG functions {ˆ
n×fn}, which results in
the linear system
Tj= (Ts+Th)j=v,(5)
in which [Ts]mn =hˆ
n×fm,Ts(fn)i,[Th]mn =hˆ
n×
fm,Th(fn)i,[v]m=hˆ
n×fm,−ˆ
n×Eii, and ha,bi=
RΓa·bds. The EFIE can also be discretized on the dual mesh
using dual functions defined on the barycentric refinement.
Both Buffa-Christiansen [38] and Chen-Wilton [39] elements
can be used for this dual discretization. For the sake of brevity,
we will omit the explicit definitions of the dual elements
that will be denoted by {gn}nin the following; the reader
can refer to [7] and references therein for a more detailed
treatment. We will also need the definition of the standard
and dual Gram matrices whose entries are [G]mn =hfm,fni
and [G]mn =hgm,gni. While they are not required for
the discretization of the EFIE itself, we introduce the patch
and pyramids scalar basis functions sets {pn}and {λn},
respectively composed of NSand NLfunctions, that will
be required for some of the following developments. These
functions are defined as
pm(r) = (A−1
mif r∈cm,
0otherwise, (6)
and
λm(r) = 




1r=vm,
0r=vn, n 6=m ,
linear otherwise,
(7)
where {vn}nare the vertices of the mesh. The number of
these functions can be deduced from the mesh properties:
NSis the number of mesh triangles and NLis the number
of mesh vertices. The Gram matrices corresponding to these
bases are Gpfor the patch functions and Gλfor the pyramids
with [Gp]mn =hpm, pniand [Gλ]mn =hλm, λni. The dual
of these functions, living in the barycentric refinement of the
original mesh will also be required, and their definitions, omit-
ted here for conciseness, can be found in [30]. The NLdual
patches will be designated as {˜pn}n, the NSdual pyramids as
{˜
λn}n, and the corresponding gram matrices as G˜pand G˜
λ
with [G˜p]mn =h˜pm,˜pniand [G˜
λ]mn =D˜
λm,˜
λnE.
Because this contribution deals with discrete quasi-
Helmholtz decompositions we will recall some of their prop-
erties. The continuous solution Jcan be decomposed into a
solenoidal, irrotational, and (on non simply-connected mani-
folds) harmonic components
J=∇ × ˆ
nλ+∇sφ+h.(8)

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ei
v
−
i
v+
i
r
−
i
r+
ic
−
i
c+
i
Fig. 1: Convention used for the RWGs: each function is
defined on the two triangles c+
iand c−
ithat are formed
with their common edge eiand the vertices r+
iand r−
i,
respectively.
When Jis discretized by the approximate expansion in RWG
functions, a discrete counterpart of (8) holds for the coefficient
vector j
j=Λl+Σσ+Hh,(9)
where Λ∈RN×NLand Σ∈RN×NSare the Loop-to-RWG
and Star-to-RWG transformation matrices [9], [12], [40]–[42]
defined, following the convention of Fig. 1 and the definition
of the RWG in (4), as
[Λ]mn =




1if node nequals v+
m,
−1if node nequals v−
m,
0otherwise,
(10)
and
[Σ]mn =




1if the cell nequals c+
m,
−1if the cell nequals c−
m,
0otherwise.
(11)
With these definitions ΛTΛand ΣTΣare respectively the
vertices- and the cells-based graph Laplacians [12]. The
explicit use of the change of basis matrix Hwill not be
required and we omit here its explicit definition, for the sake
of conciseness, which could however be found in [7] and
references therein.
With the definitions above, the standard quasi-Helmholtz
projectors [12], [27] are defined as
PΣ=ΣΣTΣ+ΣT,
PΛH =I−PΣ(12)
for the primal ones,
PΛ=ΛΛTΛ+ΛT,
PΣH =I−PΛ(13)
for the dual ones, and
PH=PH=I−PΣ−PΛ(14)
for the projector to quasi-harmonic subspace, where +denotes
the Moore-Penrose pseudo-inverse.
III. LAPLACIAN FILTERED LOOP-STAR DECOMPOSITIONS
In this section, we will extend the notion of Loop-Star bases
by introducing the concept of filtered (generating) functions.
We will first treat graph-based decompositions (a direct gen-
eralization of the standard case) and we will then move on
to their Gram matrix normalized counterparts that will be
more effective in treating problems involving inhomogeneous
meshes.
A. The Standard Case
Consider the singular value decomposition (SVD) [43] of a
matrix X∈RN×Nx
X=UXSXVT
X(15)
where Xis a placeholder for either Σor Λ,UX∈RN×N,
VX∈RNx×Nx, and SX∈RN×Nx. The matrices UXand
VXare unitary and SXis a block diagonal matrix with the
singular values σX,i of Xas entries (in decreasing order).
Clearly the SVD of (XTX)is VXST
XSXVT
X, and, by defining
the diagonal matrix LX,n ∈RNx×Nx, with 1≤n≤Nx, such
that
[LX,n]ii =(σX,i if i > Nx−n ,
0otherwise, (16)
we define the filtered graph Laplacians
(XTX)n:=VXL2
X,nVT
X,(17)
from which we introduce the filtered Loop-to-RWG and fil-
tered Star-to-RWG matrices we propose in this work
Σn=ΣΣTΣ+ΣTΣn,(18)
Λn=ΛΛTΛ+ΛTΛn.(19)
These matrices contains the coefficients of sets of linearly
dependent filtered Loop-Star functions.
Properties: We now study some properties of the filtered
Loop-Star matrices. Because ΣTΛ=0[7], we have ∀n, m
ΣT
nΛm=ΣTΣnΣTΣ+ΣTΛΛTΛ+ΛTΛm
=0.(20)
Otherwise said, the new filtered Loop-Star functions are
coefficient-orthogonal (l2-orthogonal) like their non-filtered,
standard counterparts.
From the definition of LX,n in (16), it follows that
LX,nLX,m =L2
X,min{n,m}. Thus from (17)
(XTX)n(XTX)m=VXLX,nVT
XVXLX,mVT
X
=VXLX,nLX,m VT
X=VXLX,min{n,m}LX,min{n,m}VT
X
=VXLX,min{n,m}VT
XVXLX,min{n,m}VT
X(21)
= (XTX)2
min{n,m}.
We thus have
ΣT
mΣn=ΣTΣmΣTΣ+ΣTΣΣTΣ+ΣTΣn
=ΣTΣ+ΣTΣmΣTΣn
=ΣTΣ+ΣTΣmin{n,m}ΣTΣmin{n,m}(22)
=ΣT
min{n,m}Σmin{n,m}.
Similarly,
ΛT
mΛn=ΛT
min{n,m}Λmin{n,m}.(23)
Given integers such that m<n<p<q, the property
(Σm−Σn)T(Σp−Σq) =
ΣT
m(Σp−Σq)−ΣT
n(Σp−Σq) = 0,(24)

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holds and, similarly,
(Λm−Λn)T(Λp−Λq) = 0.(25)
Properties (24) and (25) show that non-intersecting differences
of filtered Star or Loop bases are mutually orthogonal (and
thus generate linearly independent spaces), a property that will
be useful to build invertible changes of basis, as will be shown
in Section VI-A.
B. Generalization for Non-homogeneously Meshed Geome-
tries
When the filtered Loop-Star decompositions are to be used
on geometries with non-homogenous discretizations, both the
standard discretizations of the EFIE and the graph Laplacian
matrices may lead to suboptimal performance and a proper
normalization with Gram matrices must be employed. In
this context, we define the normalized EFIE electromagnetic
operator matrices
˜
T=G−1/2TG−1/2,(26)
˜
Ts=G−1/2TsG−1/2,(27)
˜
Th=G−1/2ThG−1/2,(28)
and the normalized Loop and Star matrices
˜
Σ=G−1/2ΣG1/2
p,(29)
˜
Λ=G1/2ΛG−1/2
λ.(30)
Following the same strategy as in (18) and (19), the normalized
filtered Loop-Star matrices are consistently defined as
˜
Σn=˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Σn,(31)
˜
Λn=˜
Λ˜
ΛT˜
Λ+˜
ΛT˜
Λn.(32)
When dealing with dual Loop-Star decomposition matrices,
the normalization is different from that of the primal ones, and
the dually-normalized Loop and Star transformation matrices
are defined as
˜
Σ=G1/2ΣG−1/2
˜
λ,(33)
˜
Λ=G−1/2ΛG1/2
˜p,(34)
and the associated filtered decomposition matrices as
˜
Σn=˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Σn,(35)
˜
Λn=˜
Λ˜
ΛT˜
Λ+˜
ΛT˜
Λn.(36)
Properties: The primal and dual normalized Loop-Star
bases keep satisfying the orthogonality properties
˜
ΛT˜
Σ=G−1/2
λΛTG1/2G−1/2ΣG1/2
p=0,(37)
˜
ΣT˜
Λ=G1/2
˜pΣTG−1/2G1/2ΛG−1/2
˜
λ=0,(38)
because ΣTΛ=0. Moreover, because (21) holds, we obtain,
similarly to Section III-A, that
˜
ΣT
m˜
Σn=˜
ΣT
min{n,m}˜
Σmin{n,m},(39)
˜
ΛT
m˜
Λn=˜
ΛT
min{n,m}˜
Λmin{n,m},(40)
˜
ΛT
m˜
Λn=˜
ΛT
min{n,m}˜
Λmin{n,m},(41)
˜
ΣT
m˜
Σn=˜
ΣT
min{n,m}˜
Σmin{n,m}.(42)
Using these properties and the same reasoning as previously,
the counterparts of the properties of the non-normalized fil-
tered Loop-Star matrices can be obtained. In particular the
counterparts of (24) and (25) can be obtained by replacing
each matrix with its normalized (“tilde”) counterpart.
IV. QUASI-HELMHO LTZ LAPLACIAN FILT ERS
Although explicit quasi-Helmholtz decomposition bases are
useful in applications in which a direct access to the Helmholtz
components of the current is required, oftentimes, especially
when the main target is preconditioning and regularization,
implicit Helmholtz decompositions can be more efficient. An
implicit Helmholtz decomposition was obtained in [27], where
the concept of quasi-Helmholtz projector was introduced.
Following a similar philosophy, and leveraging the filtered
Loop-Star functions introduced above, we can now define
quasi-Helmholtz Laplacian filters.
A. The Standard Case
The idea behind the projectors was to obtain a basis-
free quasi-Helmholtz decomposition that would not worsen
the conditioning of the original equation. If we follow the
definitions (12) and (13) by replacing the standard Star basis
with the new filtered sets, we obtain
ΣnΣT
nΣn+ΣT
n=ΣΣTΣ+ΣTΣn
ΣTΣnΣTΣ+ΣTΣΣTΣ+ΣTΣn+
ΣTΣnΣTΣ+ΣT=ΣΣTΣ+
nΣT,(43)
and, similarly,
ΛnΛT
nΛn+ΛT
n=ΛΛTΛ+
nΛT.(44)
This justifies the following definitions of the new primal filters
PΣ
n=ΣΣTΣ+
nΣT,(45)
PΛH
n=ΛΛTΛ+
nΛT+I−PΣ−PΛ(46)
and dual filters
PΛ
n=ΛΛTΛ+
nΛT,(47)
PΣH
n=ΣΣTΣ+
nΣT+I−PΛ−PΣ.(48)
The reader should note that, in the special case of simply
connected geometries PΛH
n=ΛΛTΛ+
nΛTand PΣH
n=
ΣΣTΣ+
nΣTsince PΛ+PΣ=I. Moreover, by construc-
tion,
PΣ
NS=PΣ,(49)
PΛ
NL=PΛ,(50)

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and thus
PΛH
NL=PΛ+I−PΣ−PΛ=PΛH ,(51)
PΣH
NS=PΣ+I−PΛ−PΣ=PΣH ,(52)
which means that, with these definitions, the quasi-Helmholtz
Laplacian filters converge to the standard quasi-Helmholtz
projectors when the Laplacian is unfiltered (n=NX).
Properties: From these definitions, a few useful properties
of the quasi-Helmholtz Laplacian filters can be derived and
will be summarized here. First, the filters still behave as
projectors since
PΣ
nPΣ
n=ΣΣTΣ+
nΣTΣΣTΣ+
nΣT
=ΣΣTΣ+
nΣT=PΣ
n,
(53)
PΛ
nPΛ
n=ΛΛTΛ+
nΛTΛΛTΛ+
nΛT
=ΛΛTΛ+
nΛT=PΛ
n,
(54)
and, similarly,
PΛH
nPΛH
n=PΛH
n,(55)
PΣH
nPΣH
n=PΣH
n.(56)
Moreover, ∀m, n
PΣ
mPΛH
n=ΣΣTΣ+
nΣTΛΛTΛ+
nΛT
+ΣΣTΣ+
nΣTI−PΣ−PΛ=0,(57)
where the properties ΣTΛ=0and ΣTI−PΣ−PΛ=0
have been used. A similar property and proof hold for the dual
projectors
PΛ
mPΣH
n=0,∀m, n . (58)
For integers m<n<p<q, we have the following
orthogonality property
PΣ
m−PΣ
nPΣ
p−PΣ
q
=ΣΣTΣ+
mΣT−ΣΣTΣ+
nΣT
ΣΣTΣ+
pΣT−ΣΣTΣ+
qΣT
=ΣΣTΣ+
mΣT−ΣΣTΣ+
mΣT
+ΣΣTΣ+
nΣT−ΣΣTΣ+
nΣT=0,
(59)
where (21) has been used. In a similar way, one can prove
that
PΛ
m−PΛ
nPΛ
p−PΛ
q=0.(60)
Moreover, given that PΛH
n−PΛH
m=PΛ
n−PΛ
mand PΣH
n−
PΣH
m=PΣ
n−PΣ
m∀n, m—which can be deduced from (46)
and (48)—the remaining properties
PΛH
m−PΛH
nPΛH
p−PΛH
q=0,(61)
PΣH
m−PΣH
nPΣ H
p−PΣH
q=0(62)
follow. All the properties listed above, will be useful when
building invertible transforms, similarly to their basis-based
counterpart (24).
B. Generalization for Non-homogeneously Meshed Geome-
tries
The definitions of the normalized Loop and Star matrices
in (29) and (30) suggest the following definition for the
associated normalized quasi-Helmholtz projectors
˜
PΣ=˜
Σ˜
ΣT˜
Σ+˜
ΣT,(63)
˜
PΛ=˜
Λ˜
ΛT˜
Λ+˜
ΛT.(64)
Moreover, as is proved in Appendix A, the complementarity
property ˜
PΣ=I−˜
PΛ(65)
holds on simply connected geometries; on general geometries
and together with definitions (31) and (32), this justifies
the following definition for the normalized quasi-Helmholtz
Laplacian filters
˜
PΣ
n=˜
Σ˜
ΣT˜
Σ+
n
˜
ΣT,(66)
˜
PΛH
n=˜
Λ˜
ΛT˜
Λ+
n
˜
ΛT+I−˜
PΣ−˜
PΛ.(67)
By analogy, we can define the normalized dual quasi-
Helmholtz projectors as
˜
PΛ=˜
Λ˜
ΛT˜
Λ+˜
ΛT,(68)
˜
PΣ=˜
Σ˜
ΣT˜
Σ+˜
ΣT,(69)
with the property
˜
PΛ=I−˜
PΣ(70)
holding on simply connected geometries (see Appendix A for
the proof). Thus, dually to the primal case, we define, on
general geometries,
˜
PΛ
n=˜
Λ˜
ΛT˜
Λ+
n
˜
ΛT,(71)
˜
PΣH
n=˜
Σ˜
ΣT˜
Σ+
n
˜
ΣT+I−˜
PΛ−˜
PΣ.(72)
Properties: Since the primal and dual normalized Loop-Star
bases still satisfy the orthogonality properties (37) and (38) and
because of the properties (39)-(42), the same reasoning yields
all counterparts of the properties (53)-(62), after replacing each
matrix with its normalized (“tilde”) counterpart.
V. EFFICIENT FILTERING ALGORITHMS
The definitions of the filtered Loop and Star functions
and of the quasi-Helmholtz Laplacian filters in Sections III
and IV involve an SVD which, while ensuring a clear and
compact theoretical treatment, is in general computationally
inefficient. This section will be devoted to presenting algo-
rithms allowing for SVD-free matrix-vector products for the
filtered graph Laplacians ΣTΣnand ΛTΛn, which are
the two key operations on both approaches presented in the
previous section. Moreover, while our treatment will deal with
the graph matrices Σand Λ, it is intended that substantially
the same strategies can be applied when replacing those
matrices with their normalized counterparts ˜
Σand ˜
Λwith

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minor modifications. In fact the additional products with the
inverse square roots of (well-conditioned) Gram matrices can
be obtained efficiently by using matrix function strategies [44].
A. Power Method Filtering
For filters with a filtering index that is independent on the
total number of degrees of freedom, preconditioned inverse
power methods [43] yield the last singular vectors and singular
values of the ΣTΣand ΛTΛmatrices at the price of a
constant number of matrix-vector products. Given that the
matrices involved are sparse, the resulting method is linear
in complexity and the filtered projectors can be efficiently
obtained. These schemes are well known and we do not
provide extensive details here for the sake of brevity. We
just mention that a special care should be put into using this
schemes in the presence of degenerate spectra (arising from
symmetries for example); under these conditions, the scheme
presented in the following section should rather be preferred.
B. Butterworth Matrix Filters
An alternative strategy for the previous scenario, i.e. when a
filter is needed that has filtering index which is independent on
the total number of degrees of freedom, is provided by a matrix
function and filtering approach. Given a scalar (squared)
Butterworth filter of positive order mand cutoff parameter
xc>0, characterized by
fm,xc(x) = (1 + (x/xc)m)−1, x ≥0,(73)
the spectrum of a symmetric positive matrix A∈RN×N
composed of the set of singular values {σi(A)}ican be filtered
by generalizing fm,xcto matrix arguments and applying it to
A, yielding the filtered matrix
Afilt :=fm,xc(A)=(I+ (A/xc)m)−1,(74)
with singular values {fm,xc(σi(A))}i. The filtered matrix
ΣTΣncan now be expressed as
ΣTΣn= (ΣTΣ) lim
m→∞ fm,σn(ΣTΣ)ΣTΣ.(75)
The presence of high exponents in (75) may render its com-
putation unstable. Hence we propose to use the following
factorization formula that leverages the roots of unity
ΣTΣn=ΣTΣ
lim
m→∞
m
Y
k=1 ΣTΣ
σn(ΣTΣ)−e(2k+1)iπ/N I−1
.(76)
For practical purposes the infinite products in this expression
can be truncated at the desired precision. Regarding the value
of σn(ΣTΣ), an approximation can be obtained either with
ad-hoc heuristics or by the approximation σn(ΣTΣ)≈(Ns−
n)/kΣTΣ+k. Finally, when the filtering point is a constant
with respect to the number of unknowns, a multigrid approach
is effective in providing the inverse required by (76).
C. Filter Approximation via Chebyshev Polynomials
When the filtering index is proportional to the number of
unknowns, the computational burden of the two methods above
can become high. In this regime we can leverage the ideas
of polynomial preconditioning and graph wavelets [31]–[34]
and adopt a method based on a polynomial expansion of the
spectral filter.
Because we are interested in cases in which the filtering
index is proportional to the number of degrees of freedom
(for instance, n=NS/2) we can leverage a polynomial
approximation of fm,xcon the interval [0, σNS(ΣTΣ)]; a
natural basis for this approximation is that of the Chebyshev
polynomials {Tn(x)}n, defined by the recurrence relation
Tn(x) = 




1if n= 0
xif n= 1
2xTn−1(x)−Tn−2(x)otherwise.
(77)
The approximated filtered matrix now reads
ΣTΣn≈ −c0
2I+
nc
X
k=1
ckTkΣTΣ
σn(ΣTΣ),(78)
where the cnare the expansion coefficients of fm,σn(ΣTΣ)in
the basis of the first nc+1 Chebyshev polynomials. Algorithms
for their computation can be found, among others, in [45].
Because the cutoff frequency of this filter is proportional
to the number of unknowns and so is the domain size, the
order of the polynomial that is required to obtain a given
approximation of the Butterworth filter, will not need to be
changed with increasing discretizations. In other words, the fil-
ters obtained by following this approach will require the same
number of sparse matrix-vector multiplication for increasing
discretization when the filtering index will be proportional
to the number of degrees of freedom. It should be noted
that in the transition region between the filters described in
the previous two sections (constant filtering index) and the
scenario described here (filtering index will be proportional to
the number of degrees of freedom) the Chebyshev approach
decreases in efficiency and further treatments may be required
[34].
VI. A FI RS T APP LI CATI ON CA SE SCENARIO: LAPLACIAN
FILTE R BAS ED PRECONDITIONING
As a first application case scenario of the new filters intro-
duced here, we will develop two families of preconditioners
for the EFIE in (5). This equation is known to suffer from
ill-conditioning both for decreasing frequency and average
mesh length h(phenomena known as the low-frequency and
h-refinement breakdowns, respectively, see [7] and references
therein). In the following we will cure both breakdowns by
developing preconditioners based both on filtered functions
decompositions and on quasi-Helmholtz Laplacian filters.
The reader should note that in this Section and in the
subsequent ones, we will study the singular value spectrum
of potentially singular matrices. When dealing with such
matrices, the condition number will be defined as cond(A) =
kAkkA+k. Moreover, inverse powers of singular matrices in
the following will always denote the corresponding positive
power of the pseudoinverse of the matrix.

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A. Filtered Bases Approach
The primal and dual Laplacians can be used to precondition
the single layer and the hypersingular operator [1], [18], [46],
[47], thus VΛ, and VΣfollowed by a diagonal precondition-
ing are valid bases for regularizing the vector and scalar poten-
tial parts of the EFIE. In particular, for Th, this results from
the fact that an operator spectrally equivalent to the single layer
can be obtained from Th. In fact, noticing that Th=ΣRΣT
[48], where Ris the patch-function discretization of the single
layer operator, i.e. [R]mn =hpm,Spniwith
(Sp) (r):=ZΓ
eikkr−r0k
4πkr−r0kp(r0)dS(r0),(79)
and defining ˜
R:=G−1/2
pRG−1/2
p, we obtain ˜
Th=˜
Σ˜
R˜
ΣT.
The equivalence between ˜
ΣT˜
Σ+˜
ΣT˜
Th˜
Σ˜
ΣT˜
Σ+
and
˜
Rthus follows. To conclude the reasoning, we note that,
because
˜
ΣT˜
Σ1/4˜
R˜
ΣT˜
Σ1/4
=˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ˜
R˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ,(80)
is well conditioned for increasing discretization—as a conse-
quence of the results proven in [47], since ˜
ΣT˜
Σis a valid
discretization of a Laplacian matrix [49]—we have
cond ˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Th
˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ=O(1) , h →0.(81)
The reader should note that, since ˜
V˜
Σis unitary, we also have
cond ˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Th
˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ=
cond ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Th
˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4.(82)
Such an approach would require the computation of the matrix
˜
V˜
Σand ˜
S˜
Σwhich are prohibitively expensive to obtain. A key
observation, however, is that we do not need to use the entire
diagonal of ˜
S˜
Σ, but a logarithmic sampling of it will suffice.
In other words, define D˜
Σthe vector containing the entries of
the diagonal of ˜
ST
˜
Σ
˜
S˜
Σand define the block diagonal matrix
˜
D˜
Σ,α = diag D˜
ΣNS−NS,α+1 INrem
S,α ,
D˜
ΣNS−NS,α
α+1 INS,α
α
,...,D˜
ΣNSI1,(83)
where NS,α =αblogα(NS)c,Nrem
S,α =NS−
(1 −NS,α) (1 −α)−1, and Inis the identity matrix of
size n, or, more programmatically,
h˜
D˜
Σ,αiii =D˜
Σf˜
Σ(i),(84)
with f˜
Σ(i) = NS−αblogα(NS−i+1)c+ 1. Note that the
construction of this matrix only requires explicit knowledge of
logα(NS)terms of D˜
Σ. Few passages—omitted here—suffice
to show that
cond ˜
D1/4
˜
Σ,α ˜
VT
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Th˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
D1/4
˜
Σ,α
=O(α) = O(1) , h →0,(85)
which is reminiscent of hierarchical strategies (see [7] and
references therein). Because ˜
V˜
Σis unitary, we obtain equiv-
alently
cond ˜
V˜
Σ˜
D1/4
˜
Σ,α ˜
VT
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Th
˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
D1/4
˜
Σ,α ˜
VT
˜
Σ=O(α) = O(1) .(86)
This preconditioning strategy can be slightly altered to
leverage the filtered basis presented in Section III by introduc-
ing an additional Laplacian in (81) and adjusting the exponent
of ˜
ST
˜
Σ
˜
S˜
Σaccordingly. In particular, we have
˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ1/4˜
VT
˜
Σ=
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Σ˜
V˜
Σ˜
ST
˜
Σ˜
S˜
Σ−3/4˜
VT
˜
Σ,(87)
which, following the reasoning detailed above, means that
˜
B˜
Σ:=˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Σ˜
V˜
Σ˜
D−3/4
˜
Σ,α ˜
VT
˜
Σ(88)
is a valid left and right symmetric preconditioner for ˜
Th.
Finally, thanks to the properties introduced in Section III, we
have
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Σ˜
V˜
Σ˜
D−3/4
˜
Σ,α ˜
VT
˜
Σ=
NS,α
X
l=2 ˜
Σαl−1−˜
Σαl−1−1D˜
Σ−3/4
NS−αl−1+1
+˜
Σ−˜
ΣαNS,α −1D˜
Σ−3/4
NS−NS,α+1 =:˜
Σp,α (89)
and thus from (87) and (89) it follows that
cond ˜
ΣT
p,α ˜
Th˜
Σp,α=O(1) , h →0.(90)
A similar reasoning for ˜
Ts, following from the precondi-
tioning of the hypersingular operator, leads to
cond ˜
ST
˜
Λ˜
S˜
Λ−1/4˜
VT
˜
Λ˜
ΛT˜
Ts˜
Λ˜
V˜
Λ˜
ST
˜
Λ˜
S˜
Λ−1/4=O(1)
(91)
and
cond ˜
V˜
Λ˜
ST
˜
Λ˜
S˜
Λ−1/4˜
VT
˜
Λ˜
ΛT˜
Ts
˜
Λ˜
V˜
Λ˜
ST
˜
Λ˜
S˜
Λ−1/4˜
VT
˜
Λ=O(1) , h →0.(92)
From this, following a dual reasoning as the one of the
previous section, we obtain
cond ˜
ΛT
p,α ˜
Ts˜
Λp,α=O(1) , h →0.(93)

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where
˜
Λ˜
ΛT˜
Λ+˜
ΛT˜
Λ˜
V˜
Λ˜
D−1/4
˜
Λ,α ˜
VT
˜
Λ=
NL,α
X
l=2 ˜
Λαl−1−˜
Λαl−1−1D˜
Λ−1/4
NL−αl−1+1
+˜
Λ−˜
ΛαNL,α −1D˜
Λ−1/4
NL−NL,α+1 =:˜
Λp,α ,(94)
and
˜
D˜
Λ,α = diag D˜
ΛNL−NL,α+1 INrem
L,α ,
D˜
ΛNL−NL,α
α+1 INL,α
α
,...,D˜
ΛNLI1,(95)
with Nrem
L,α =NL−(1 −NL,α) (1 −α)−1,D˜
Λthe vector
containing the elements of the diagonal of ˜
ST
˜
Λ
˜
S˜
Λ, and NL,α =
αblogα(NL)c.
The previous preconditioners can then be combined to
obtain a complete regularization of the EFIE system, for both
low-frequency and h-refinement breakdowns, that reads
˜
WT˜
T˜
W˜
j=˜
WT˜
v,(96)
where ˜
v=G−1/2v,j=G−1/2˜
W˜
j,˜
W=
√c˜
Λ˜
Λp,α √c˜
Σ˜
Σp,α,c˜
Σ=k˜
ΣT
p,α ˜
Th˜
Σp,αk−1,c˜
Λ=
k˜
ΛT
p,α ˜
Ts˜
Λp,αk−1, and where we assume that the appropriate
number of columns have been removed from ˜
Σp,α and ˜
Λp,α
(e.g. 1 column must be removed from each for a simply con-
nected, closed scatterer) to account for the linear dependence
in the underlying Loop and Star bases [8], as is done in stan-
dard Loop-Star preconditioning. The reader should note that,
as in the case of standard Loop-Star functions, this operations
will create a small number of isolated singular values, that
however will not impact the convergence properties of the
preconditioned equation. This effect will not be present instead
in the scheme of next Section. The h-refinement regularization
effect of this preconditioner can be deduced from the previous
derivations for each of the potentials [50]. The low frequency
regularization, can be demonstrated following the same rea-
soning as for standard Loop-Star approaches [7], since the new
filtered bases retain the crucial properties that made Loop-Star
so adapted low-frequency regularization in the first place—
˜
ΛT
p,α ˜
Th=0,˜
Th˜
Λp,α =0, and ˜
ΛT
p,α ˜
Σp,α =0. Finally, we
have
cond ˜
WT˜
T˜
W=O(1) ,when h→0, k →0.(97)
B. Quasi-Helmholtz Filters Approach
In several application scenarios, an explicit quasi-Helmholtz
decomposition, such as the Loop-Star decomposition, is not
necessary, and quasi-Helmholtz projectors [7] could be used
instead. Similarly, instead of using filtered Loop-Star pre-
conditioning approaches, basis-free approaches, based on the
quasi-Helmholtz Laplacian filters, will often be more effective.
This Section will explore this approach that, as an additional
advantage, will also have the avoidance of the burden of
global-loop detection for multiply connected scatterers.
Following the same philosophy as in Section VI-A, we will
form preconditioners for the solenoidal part of ˜
Tsand for ˜
Th
that will then be combined into a full EFIE preconditioner. We
can transition from a basis-based Helmholtz decomposition
to a projector based Helmholtz decomposition by leveraging
the correspondences between ˜
Σand ˜
Λand their respective
projectors ˜
PΣand ˜
PΛ. In particular, because ˜
B˜
Σwas a valid
preconditioner for ˜
Th(equations (89) and (90)), ˜
B˜
Σ0,
once applied left and right to ˜
Thwill yield a block diag-
onal matrix which is well conditioned away from its large
nullspace. This, in turns, means that ˜
C˜
Σ0, with ˜
C˜
Σ=
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Σ˜
V˜
Σ˜
D−5/4
˜
Σ
˜
VT
˜
Σ
˜
V˜
Σ˜
D1/2
˜
Σ, will also yield
a well-conditioned (up to its nullspace) matrix. Finally, be-
cause multiplications by unitary matrices do not compromise
conditioning properties, we can form the preconditioner
˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
Σ˜
V˜
Σ˜
D−5/4
˜
Σ
˜
VT
˜
Σ˜
V˜
Σh˜
D1/2
˜
Σ0i˜
UT
˜
Σ=
˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
D−1/4
˜
Σ
˜
VT
˜
Σ˜
ΣT.(98)
This allows us to form the preconditioner ˜
Q˜
Σ
p,α, of additive
Schwarz type, based on quasi-Helmholtz filters
˜
Σ˜
ΣT˜
Σ+˜
V˜
Σ˜
D−1/4
˜
Σ
˜
VT
˜
Σ˜
ΣT=
NS,α
X
l=2 ˜
P˜
Σ
αl−1−˜
P˜
Σ
αl−1−1D˜
Σ−1/4
NS−αl−1+1
+˜
P˜
Σ−˜
P˜
Σ
αNS,α −1D˜
Σ−1/4
NS−NS,α+1 =:˜
Q˜
Σ
p,α (99)
for which
cond ˜
Q˜
Σ
p,α ˜
Th˜
Q˜
Σ
p,α=O(1) , h →0.(100)
Similarly, a preconditioner for the solenoidal part of ˜
Tsis
˜
Q˜
Λ
p,α :=
NL,α
X
l=2 ˜
P˜
Λ
αl−1−˜
P˜
Λ
αl−1−1D˜
Λ1/4
NL−αl−1+1
+˜
P˜
Λ−˜
P˜
Λ
αNL,α −1D˜
Λ1/4
NL−NL,α+1 (101)
for which
cond ˜
Q˜
Λ
p,α ˜
Ts˜
Q˜
Λ
p,α=O(1) , h →0.(102)
The full EFIE preconditioner is then an appropriate linear
combination of the solenoidal and non-solenoidal precondi-
tioners above to cure also the low-frequency breakdown. In
particular we define
˜
Q=pb˜
Λ˜
Q˜
Λ
p,α +ipb˜
Σ˜
Q˜
Σ
p,α +pb˜
H˜
PH,(103)
where ˜
PH=I−˜
PΣ−˜
PΛand
b˜
Λ=k˜
Q˜
Λ
p,α ˜
Ts˜
Q˜
Λ
p,αk−1,(104)
b˜
Σ=k˜
Q˜
Σ
p,α ˜
Th˜
Q˜
Σ
p,αk−1,(105)
b˜
H=k˜
P˜
H˜
Ts˜
PHk−1,(106)
account for the frequency-scaling of the operators and the
diameter of Γ. The preconditioned EFIE system is
˜
Q˜
T˜
Q˜
jqH =˜
Q˜
v,(107)
with j=G−1/2˜
Q˜
jqH.

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VII. IMP LE ME NTATION RELATED DETAI LS A ND F URT HE R
IM PROV EM EN TS
In addition to the efficient filtering algorithms presented in
Section V, obtaining a fast and efficient implementation of the
proposed preconditioning scheme based on filtered projectors
requires that particular attention be given to parts of their
implementation. First, all the terms of the form ThQΛ
p,α,
QΛ
p,αTh,PHTh, or ThPHmust be explicitly set to 0to
avoid numerical instabilities. Further treatments on the right
hand side and on the solution vector, are required to ensure
that the solution of the system remains accurate until arbi-
trarily low frequencies. These treatments are straightforward
generalization of those required for standard quasi-Helmholtz
preconditioning techniques that can be found in [7].
The condition numbers obtained when employing the
schemes introduced in Section VI-A and Section VI-B, while
stable, can be further brought down by slightly modifying
the preconditioners. The diagonal preconditioning based on
the theoretical Laplacian eigenvalues can be altered to instead
employ matrix norms; the new preconditioners then become
QΣ
p,α =
NS,α
X
l=2 PΣ
αl−1−PΣ
αl−1−1bl+
PΣ−PΣ
αNS,α −1bNS,α+1 ,(108)
where
bl=

PΣ
αl−1−PΣ
αl−1−1TThPΣ
αl−1−PΣ
αl−1−1


−1/2,
2≤l≤NS,α ,
(109)
bNS,α+1 =


PΣ−PΣ
αNS,α −1T
ThPΣ−PΣ
αNS,α −1



−1/2
.
(110)
The same modification can be performed for QΛ
p,α that be-
comes
QΛ
p,α =
NL,α
X
l=2 PΛ
αl−1−PΛ
αl−1−1dl+
PΛ−PΛ
αNL,α −1dNS,α+1 ,(111)
with
dl=

PΛ
αl−1−PΛ
αl−1−1TTsPΛ
αl−1−PΛ
αl−1−1


−1/2,
2≤l≤NL,α ,
(112)
dNS,α+1 =


PΛ−PΛ
αNL,α −1T
TsPΛ−PΛ
αNL,α −1



−1/2
.
(113)
To ensure that the overall complexity of the algorithm is not
increased, the values of {bl}land {dl}lcan be efficiently
computed using, for example, power methods. The reader
should note that the preconditioning approach delineated above
requires filter profiles with support both proportional to and
independent from the number of unknowns, which can be
100101102103
10−2
10−1
100
101
102
Singular value index
Singular value
PΛTsPΛΛT
p,αTsΛp,α ΛT
p,αΛp,α ξ1/2
PΛTsPΛΛT
p,αTsΛp,α ΛT
p,αΛp,α ξ−1/2
Fig. 2: Spectrum of the solenoidal part of the vector potential,
its preconditioner, and its preconditioned counterpart. These
spectra have been obtained for a smoothly-deformed sphere
with a maximum diameter of 7.17 m (see insert), a frequency
of 106Hz, and for two different average edge lengths 0.31 m
and 0.20 m. The spectra have been normalized so that their
first singular value is one, for readability. Perfect filters built
out of SVD have been used in these results.
efficiently obtained with the approaches described in Sec-
tion V. As said in the previous Section, filters in the transition
region could be less efficient to obtain, as the Chebyshev
approach decreases in efficiency away from the middle of
the spectrum [34]. All preconditioning real case scenarios
presented here, however, are not impacted by this fact as
shown in Section VIII.
VIII. NUMERICAL RES ULTS
All numerical results presented in this section have been
obtained with non-normalized matrices (Λ,Σ) to illustrate that
graph matrices are often enough for practical cases. Equally
good or superior performance, however, can be obtained by
using normalized matrices ( ˜
Λ,˜
Σ) instead. In the first set
of examples we have leveraged perfect filters obtained by
SVD before presenting results based on SVD-free approaches.
The filtered Loop-Star preconditioning approach presented
in Section VI-A leverages the spectral equivalences between
the appropriately scaled filtered bases and Tsand Th. To
numerically illustrate these equivalences, the spectra of these
operators and their preconditioned counterparts are illustrated
in Figures 2 and 3. These spectra correspond to a smoothly
deformed sphere (see Fig. 2 and 3), and the ordering of the
singular values is obtained by projection against the graph
Laplacians’ eigenvectors. The original spectrum of Tsand
Thshow the expected ξ−1/2and ξ1/2—with ξthe spectral
index—behaviors, predicted by pseudo-differential operator
theory. Given the construction of the preconditioners, it is
then not surprising that the preconditioned operators show a
spectrum bounded (and away from zero) with the expected
variations in the spectrum.
To illustrate that the preconditioning schemes based on
filtered bases do regularize the EFIE, the condition number of

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100101102103
10−2
10−1
100
101
102
Singular value index
Singular value
PΣThPΣΣT
p,αThΣp,α ΣT
p,αΣp,α ξ1/2
PΣThPΣΣT
p,αThΣp,α ΣT
p,αΣp,α ξ−1/2
Fig. 3: Spectrum of the non-solenoidal part of the scalar po-
tential, its preconditioner, and its preconditioned counterpart.
These spectra have been obtained for a smoothly-deformed
sphere with a maximum diameter of 7.17 m (see insert),
a frequency of 106Hz, and for two different average edge
lengths 0.31 m and 0.20 m. The spectra have been normalized
so that their first singular value is one, for readability. Perfect
filters built out of SVD have been used in these results.
the original and preconditioned schemes will be compared for
varying frequencies and discretizations. First, the conditioning
of a filtered Loop-Star preconditioned EFIE for the NASA
almond [51] is reported in Figure 4. The low frequency and
dense discretization breakdowns of the original equations are
apparent, while the preconditioned equation (corresponding to
(96)) shows a constant conditioning. This is in contrast with
the standard Loop-Star approach that does regularize the low
frequency conditioning breakdown, but actually worsens the
dense discretization behavior of the equation.
A similar study has been performed with the filtered projec-
tors schemes. In Figures 5 and 6 the spectra of the dominant
solenoidal and non-solenoidal parts of the EFIE operators are
displayed alongside their preconditioners. The precondition-
ing performance on the overall EFIE system is illustrated
in Figure 7 for a torus. The approach yields satisfactory
conditioning that remains stable in both low frequency and
dense discretization, which in turns shows that the scheme
can also handle multiply-connected geometries.
Finally, a conditioning study of the NASA almond is
reported in Figure 8 that has been obtained using Chebyshev-
interpolated filters (78) corresponding to Butterworth filters of
order 100, expanded into 200 Chebyshev polynomials. The
coefficients of the filters are obtained via the norm estimates
detailed in (108) and (111) and the cutting point of the filters
is determined using the approximate Laplacian spectrum de-
scribed bellow (76). The excellent stability of preconditioned
scheme for a structure such as the NASA almond showcases
the effectiveness of the scheme when using the fast techniques
presented in this paper.
101.1101.2101.3101.4101.5101.6
100
104
108
1012
h−1m−1
Condition number
EFIE 106Hz LS EFIE 106Hz fLS EFIE 106Hz
EFIE 104Hz LS EFIE 104Hz fLS EFIE 104Hz
Fig. 4: Condition number of the EFIE (5), Loop-Star EFIE, and
filtered Loop-Star EFIE (96) as a function of discretization for
several frequencies. The condition number has been obtained
after eliminating the isolated singular values, which have
minimal impact on the convergence, arising from the deletion
of one column from each of the preconditioning matrices. The
solid lines correspond to a simulating frequency of 106Hz
and the dotted lines to a frequency of 104Hz. The simulated
structure is the NASA almond re-scaled to be enclosed in a
bounding box of diameter 1.09 m. Perfect filters built out of
SVD have been used in these results.
100101102103
10−2
10−1
100
101
102
Singular value index
Singular value
PΛTsPΛQΛ
p,αTsQΛ
p,α QΛ
p,αQΛ
p,α ξ1/2
PΛTsPΛQΛ
p,αTsQΛ
p,α QΛ
p,αQΛ
p,α ξ−1/2
Fig. 5: Spectrum of the solenoidal part of the vector potential,
its preconditioner, and its preconditioned counterpart. These
spectra have been obtained for a smoothly-deformed sphere
with a maximum diameter of 7.17 m (see insert), a frequency
of 106Hz, and for two different average edge lengths 0.31 m
and 0.20 m. The spectra have been normalized so that their
first singular value is one, for readability. Perfect filters built
out of SVD have been used in these results.

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100101102103
10−2
10−1
100
101
102
Singular value index
Singular value
PΣThPΣQΣ
p,αThQΣ
p,α QΣ
p,αQΣ
p,α ξ1/2
PΣThPΣQΣ
p,αThQΣ
p,α QΣ
p,αQΣ
p,α ξ−1/2
Fig. 6: Spectrum of the non-solenoidal part of the scalar po-
tential, its preconditioner, and its preconditioned counterpart.
These spectra have been obtained for a smoothly-deformed
sphere with a maximum diameter of 7.17 m (see insert),
a frequency of 106Hz, and for two different average edge
lengths 0.31 m and 0.20 m. The spectra have been normalized
so that their first singular value is one, for readability. Perfect
filters built out of SVD have been used in these results.
100.95 101101.05 101.1101.15
100
104
108
1012
h−1m−1
Condition number
EFIE 106Hz qH EFIE 106Hz fqH EFIE 106Hz
EFIE 104Hz qH EFIE 104Hz fqH EFIE 104Hz
Fig. 7: Condition number of the EFIE (5), quasi-Helmholtz
(qH) projector EFIE, and filtered qH projector EFIE (107)
as a function of discretization for several frequencies. The
solid lines correspond to a simulating frequency of 106Hz
and the dotted lines to a frequency of 104Hz. The simulated
structure is a torus with inner radius 0.9 m and outer radius
1.1 m. Perfect filters built out of SVD have been used in these
results.
101.3101.4101.5101.6101.7
100
104
108
1012
h−1m−1
Condition number
EFIE 104Hz qH EFIE 104Hz fqH EFIE 104Hz
Fig. 8: Condition number of the EFIE (5), quasi-Helmholtz
(qH) projector EFIE, and filtered qH projector EFIE (107) as
a function of discretization for several frequencies. The simu-
lated structure is the NASA almond re-scaled to be enclosed
in a bounding box of diameter 1.09 m. The preconditioner
is built without using SVDs, but by leveraging Chebyshev-
interpolated filters (78) corresponding to Butterworth filters of
order 100, expanded into 200 Chebyshev polynomials.
IX. CONCLUSION
A new family of strategies has been introduced for perform-
ing filtered quasi-Helmholtz decompositions of electromag-
netic integral equations: the filtered Loop-Star decompositions
and the quasi-Helmholtz Laplacian filters. These new tools are
capable of manipulating large parts of the operators’ spectra
to obtain new families of preconditioners and fast solvers. A
first application to the case of frequency and h-refinement
preconditioning of the electric field integral equation has been
presented and numerical results have shown the practical
effectiveness of the newly proposed tools.
APPENDIX A
COMPLEMENTARITY OF THE PROJECTORS
In this appendix, we show that the properties ˜
PΛ+˜
PΣ=I
and ˜
PΛ+˜
PΣ=Ihold true on simply connected geometries.
To this end, we first prove that the normalized coefficients
˜
jof the RWG functions can be decomposed with ˜
Λand ˜
Σ,
similarly as in (9) where we assume that the proper number of
columns from the matrices have been removed as is standard
to ensure a full column rank, such that
˜
j=G−1
2j=˜
Λ˜
l+˜
Σ˜
s(114)
in which ˜
land ˜
sare the coefficient vectors of the normalized
Loop and Star parts in this decomposition. Since G,Gp, and
Gλare invertible matrices, we have rank(˜
Σ) = rank(Σ)and
rank(˜
Λ) = rank(Λ). Moreover, since ˜
ΛT˜
Σ=0, we also
obtain that ˜
Λand ˜
Σhave their column linearly independent,
which yields rank([˜
Λ˜
Σ]) = rank(˜
Λ) + rank(˜
Σ) = N, from
which the existence and (unicity) of (114) follows.
Subsequently, using (114), we can form a new set of
normalized projectors to retrieve ˜
Λ˜
land ˜
Σ˜
sseparately. The

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first step is to apply ˜
ΛTand ˜
ΣTto (114) to express ˜
jin the
two different bases
˜
ΛT˜
j=˜
ΛT˜
Λ˜
l,(115)
˜
ΣT˜
j=˜
ΣT˜
Σ˜
s,(116)
since ˜
ΛT˜
Σ=0and ˜
ΣT˜
Λ=0, given (20). Subsequently, we
express the coefficients of the normalized Loop and Star bases
as a function of ˜
j
˜
l=˜
ΛT˜
Λ+˜
ΛT˜
j,(117)
˜
s=˜
ΣT˜
Σ+˜
ΣT˜
j.(118)
Finally, we express ˜
Λ˜
land ˜
Σ˜
sin terms of ˜
jby applying ˜
Λ
and ˜
Σto (117) and (118)
˜
Λ˜
l=˜
Λ˜
ΛT˜
Λ+˜
ΛT˜
j=˜
PΛ˜
j,(119)
˜
Σ˜
s=˜
Σ˜
ΣT˜
Σ+˜
ΣT˜
j=˜
PΣ˜
j,(120)
and we obtain that ˜
PΛ+˜
PΣ=Iby leveraging (119), (120),
and (114). Following the same procedure, except that now ˜
Λ
and ˜
Σare employed in the initial decomposition, we can show
that the property ˜
PΛ+˜
PΣ=Ialso holds true.
ACKNOWLEDGMENT
This work has been funded in part by the European Research
Council (ERC) under the European Union’s Horizon 2020
research and innovation program (ERC project 321, grant
No.724846) and in part by the ANR Labex CominLabs under
the project “CYCLE”.
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PLACE
PHOTO
HERE
Michael Shell Biography text here.
John Doe Biography text here.
Jane Doe Biography text here.