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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=ΣΣTΣ+ΣT,

PΛH =I−PΣ(12)

for the primal ones,

PΛ=ΛΛTΛ+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=ΣΣTΣ+ΣTΣn,(18)

Λn=ΛΛTΛ+ΛTΛn.(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=ΣTΣnΣTΣ+ΣTΛΛTΛ+ΛTΛm

=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=ΣTΣmΣTΣ+ΣTΣΣTΣ+ΣTΣn

=ΣTΣ+ΣTΣmΣTΣn

=ΣTΣ+ΣTΣmin{n,m}ΣTΣmin{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˜

Σn,(31)

˜

Λn=˜

Λ˜

ΛT˜

Λ+˜

ΛT˜

Λn.(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˜

Σn,(35)

˜

Λn=˜

Λ˜

ΛT˜

Λ+˜

ΛT˜

Λn.(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

ΣnΣT

nΣn+ΣT

n=ΣΣTΣ+ΣTΣn

ΣTΣnΣTΣ+ΣTΣΣTΣ+ΣTΣn+

ΣTΣnΣTΣ+ΣT=ΣΣTΣ+

nΣT,(43)

and, similarly,

ΛnΛT

nΛn+ΛT

n=ΛΛTΛ+

nΛT.(44)

This justiﬁes the following deﬁnitions of the new primal ﬁlters

PΣ

n=ΣΣTΣ+

nΣT,(45)

PΛH

n=ΛΛTΛ+

nΛT+I−PΣ−PΛ(46)

and dual ﬁlters

PΛ

n=ΛΛTΛ+

nΛT,(47)

PΣH

n=ΣΣTΣ+

nΣT+I−PΛ−PΣ.(48)

The reader should note that, in the special case of simply

connected geometries PΛH

n=ΛΛTΛ+

nΛTand PΣH

n=

ΣΣTΣ+

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=ΣΣTΣ+

nΣTΣΣTΣ+

nΣT

=ΣΣTΣ+

nΣT=PΣ

n,

(53)

PΛ

nPΛ

n=ΛΛTΛ+

nΛTΛΛTΛ+

nΛT

=ΛΛTΛ+

nΛT=PΛ

n,

(54)

and, similarly,

PΛH

nPΛH

n=PΛH

n,(55)

PΣH

nPΣH

n=PΣH

n.(56)

Moreover, ∀m, n

PΣ

mPΛH

n=ΣΣTΣ+

nΣTΛΛTΛ+

nΛT

+ΣΣTΣ+

nΣTI−PΣ−PΛ=0,(57)

where the properties ΣTΛ=0and ΣTI−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

nPΣ

p−PΣ

q

=ΣΣTΣ+

mΣT−ΣΣTΣ+

nΣT

ΣΣTΣ+

pΣT−ΣΣTΣ+

qΣT

=ΣΣTΣ+

mΣT−ΣΣTΣ+

mΣT

+ΣΣTΣ+

nΣT−ΣΣTΣ+

nΣT=0,

(59)

where (21) has been used. In a similar way, one can prove

that

PΛ

m−PΛ

nPΛ

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

nPΛH

p−PΛH

q=0,(61)

PΣH

m−PΣH

nPΣ 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 ΣTΣnand ΛTΛn, 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

ΣTΣncan now be expressed as

ΣTΣn= (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

ΣTΣn=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)/kΣTΣ+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

ΣTΣn≈ −c0

2I+

nc

X

k=1

ckTkΣTΣ

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˜

Σ1/4˜

R˜

ΣT˜

Σ1/4

=˜

V˜

Σ˜

ST

˜

Σ˜

S˜

Σ1/4˜

VT

˜

Σ˜

R˜

V˜

Σ˜

ST

˜

Σ˜

S˜

Σ1/4˜

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˜

Σ1/4˜

VT

˜

Σ˜

ΣT˜

Σ+˜

ΣT˜

Th

˜

Σ˜

ΣT˜

Σ+˜

V˜

Σ˜

ST

˜

Σ˜

S˜

Σ1/4˜

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˜

Σ1/4˜

VT

˜

Σ˜

ΣT˜

Σ+˜

ΣT˜

Th

˜

Σ˜

ΣT˜

Σ+˜

V˜

Σ˜

ST

˜

Σ˜

S˜

Σ1/4˜

VT

˜

Σ=

cond ˜

ST

˜

Σ˜

S˜

Σ1/4˜

VT

˜

Σ˜

ΣT˜

Σ+˜

ΣT˜

Th

˜

Σ˜

ΣT˜

Σ+˜

V˜

Σ˜

ST

˜

Σ˜

S˜

Σ1/4.(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˜

ΣNS−NS,α+1 INrem

S,α ,

D˜

ΣNS−NS,α

α+1 INS,α

α

,...,D˜

ΣNSI1,(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˜

Σf˜

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˜

Σ1/4˜

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−1−1D˜

Σ−3/4

NS−αl−1+1

+˜

Σ−˜

ΣαNS,α −1D˜

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−1−1D˜

Λ−1/4

NL−αl−1+1

+˜

Λ−˜

ΛαNL,α −1D˜

Λ−1/4

NL−NL,α+1 =:˜

Λp,α ,(94)

and

˜

D˜

Λ,α = diag D˜

ΛNL−NL,α+1 INrem

L,α ,

D˜

ΛNL−NL,α

α+1 INL,α

α

,...,D˜

ΛNLI1,(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−1−1D˜

Σ−1/4

NS−αl−1+1

+˜

P˜

Σ−˜

P˜

Σ

αNS,α −1D˜

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−1−1D˜

Λ1/4

NL−αl−1+1

+˜

P˜

Λ−˜

P˜

Λ

αNL,α −1D˜

Λ1/4

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−1−1bl+

PΣ−PΣ

αNS,α −1bNS,α+1 ,(108)

where

bl=

PΣ

αl−1−PΣ

αl−1−1TThPΣ

αl−1−PΣ

αl−1−1

−1/2,

2≤l≤NS,α ,

(109)

bNS,α+1 =

PΣ−PΣ

αNS,α −1T

ThPΣ−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−1−1dl+

PΛ−PΛ

αNL,α −1dNS,α+1 ,(111)

with

dl=

PΛ

αl−1−PΛ

αl−1−1TTsPΛ

αl−1−PΛ

αl−1−1

−1/2,

2≤l≤NL,α ,

(112)

dNS,α+1 =

PΛ−PΛ

αNL,α −1T

TsPΛ−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−1m−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−1m−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−1m−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.