On the Eigenvalues of the Volume Integral Operator of Electromagnetic Scattering.
The volume integral equation of electromagnetic scattering can be used to compute the scattering by inhomogeneous or anisotropic scatterers. In this paper we compute the spectrum of the scattering integral operator for a sphere and the eigenvalues of the coefficient matrices that arise from the discretization of the integral equation. For the case of a spherical scatterer, the eigenvalues lie mostly on a line in the complex plane, with some eigenvalues lying below the line. We show how the spectrum of the integral operator can be related to the well-posedness of a modified scattering problem. The eigenvalues lying below the line segment arise from resonances in the analytical series solution of scattering by a sphere. The eigenvalues on the line are due to the branch cut of the square root in the definition of the refractive index. We try to use this information to predict the performance of iterative methods.  For a normal matrix the initial guess and the eigenvalues of the coefficient matrix determine the rate of convergence of iterative solvers. We show that when the scatterer is a small sphere, the convergence rate for the nonnormal coefficient matrices can be estimated but this estimate is no longer valid for large spheres.
Introduction
. The convergence of iterative solvers for systems of linear equations is closely related
to the eigenvalue distribution of the coefficient matrix. To be able to predict the convergence of
iterative solvers one therefore needs to look at how the matrix and its eigenvalues arise from a discretization
of a physical problem.
In this article we examine the eigenvalues of the coefficient matrix arising from a discretization of
a volume integral equation of electromagnetic scattering and the behavior of iterative solvers for this
problem. In the early stages of this research project we noticed that in the case of a spherical scatterer,
most of the eigenvalues of the matrix lie on a line in the complex plane. The line segment can also be
seen for other types of scatterers.
The coefficient matrix arises from a discretization of a physical problem. Thus, the eigenvalues of
the coefficient matrix should resemble the spectrum of the corresponding integral operator in a function
space. The spectrum of an operator is the infinite-dimensional counterpart of the eigenvalues of a matrix.
In the case of a spherical scatterer, we show that it is possible to relate the spectrum of the scattering
integral operator to the behavior of the analytic solution of scattering by a sphere. The main result is
that the eigenvalues of the coefficient matrix are related either to resonances in the analytic solution or
to the branch cut of the complex square root used in the definition of the refractive index.
An analysis of the eigenvalues of the surface integral operator in electromagnetic scattering has been
carried out by Hsiao and Kleinman [12], who studied the mapping properties of integral operators in an
appropriate Sobolev space. Colton and Kress have also studied these weakly singular surface integral
operators [3]. In our case, we look at the spectrum of a strongly singular volume integral operator. We
have not studied the mapping properties of this operator.
The paper is organized as follows: In x 2 we give the volume integral equation formalism and discuss
its discretization. In x 3 we compute the eigenvalues of the coefficient matrix. Section 4 shows how the
spectrum of the integral operator can be determined with the help of the analytic solution. Numerical
Parts of this work have been carried out at the Center for Scientific Computing, Finland, and at the Helsinki University
of Technology. This work was supported by the Jenny and Antti Wihuri foundation and by the Cultural Foundation of
Finland.
y CERFACS, 42 av G. Coriolis, 31057 Toulouse Cedex, France (Jussi.Rahola@cerfacs.fr).
J. RAHOLA
experiments are presented that show how well the eigenvalues of the coefficient matrix can approximate
the spectrum of the integral operator. In x 5 we show how the eigenvalue distribution can be used in the
prediction of the convergence of iterative solvers. In x 6 we conclude and point out some areas of future
research.
2. Volume integral equation for electromagnetic scattering. The volume integral equation
of electromagnetic scattering is employed especially for inhomogeneous and anisotropic scatterers. For
homogeneous scatterers it also offers some advantages over the surface integral equation, namely a simple
description of the scatterer with the help of cubic cells and the use of the fast Fourier transform in the
computation of the matrix-vector products. On the other hand, the volume integral equation uses many
more unknowns than the surface integral formulation for a given computational volume.
We will study the scattering of monochromatic electromagnetic radiation of frequency f , wavelength
wave number In our presentation we assume
that the electric field is time-harmonic and thus its time-dependence is of the form exp(\Gammai!t). The
scattering material is described by its complex refractive index defined by
~ffl, where the complex
permittivity ~ ffl is given by ~ is the (real) permittivity and oe is the conductivity. We
have assumed that the magnetic permeability of the material is the same as that of vacuum. The volume
integral equation is typically used for dielectric or weakly conducting objects.
The volume integral equation of electromagnetic scattering is given by [9, 15, 16]
Z
where V is the volume of space the scatterer occupies, E(r) is the electric field inside the object, E inc (r)
is the incident field and G is the dyadic Green's function given by
where
The Green's function also has an explicit representation by
ae \Gammaae 2
ae
The scattering integral equation can be discretized in various ways. The simplest discretization uses
cubic cells and assumes that the electric field is constant inside each cube. By requiring that the integral
equation (2.1) be satisfied at the centers r i of the N cubes (point-matching or collocation technique) and
by using simple one-point integration, we end up with the equation [14, 15, 22]
is the volume of the computational cube, M is given by
ii
e ikb(3=4-) 1=3
EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 3
b is the length of the side of the computational cube and T ij is given by
ae 3
ae 3
The factor M arises from the analytical integration of the self-term, using a sphere whose volume is equal
to the volume of the cube [15].
Note that all the physical dimensions of the problem are of the form kr. This means that the
scattering problem depends only on the ratio of the size of the object to the wavelength. In the rest of
the paper we will give the dimensions of the scattering object in the form
The systems of linear equations (2.5) has a dense complex symmetric coefficient matrix. Various
ways of solving the linear system with iterative solvers has been studied by Rahola [17, 23, 22] who chose
the complex symmetric version of QMR [7] as the iterative solver. The matrix-vector products can be
computed with the help of the fast Fourier transform if the cells sit on a regular lattice. For the FFT,
the computational grid must be enlarged with ghost cells to a cube [10].
3. Eigenvalues of the coefficient matrix. In this section we show some examples of the eigen-values
of the coefficient matrices. All the eigenvalues were computed with the eigenvalue routines in the
library [1].

Figure

3.1 shows the eigenvalues of a coefficient matrix for a sphere of radius refractive
computational cells, respectively, are used to discretize the
sphere. Note how most of the eigenvalues lie on a line in the complex plane. When the discretization is
refined, the line segment becomes "denser" but there remains a small number of eigenvalues off the line.
An interesting feature of these coefficient matrices is that when the discretization is refined, the
number of iterations needed for iterative solvers to converge remains constant even when the linear
system is not preconditioned [21, 22]. This is of course the optimal situation for iterative solvers. In
practical calculations, when the number of computational points is increased, the size of computational
cells is kept constant and the physical size of the object is increased. In this case the number of iterations
naturally grows with the size of the problem.
4. Spectrum of the integral operator. In this section we will explain what is meant by the
spectrum of a linear operator, how to compute points in the spectrum of the scattering integral operator
and how they correspond to the eigenvalues of the coefficient matrix of the discretized problem.
The spectrum of a linear operator T is the set of points z in the complex plane for which the operator
does not have an inverse operator that is a bounded linear operator defined everywhere. Here 1
stands for the identity operator.
A matrix is the prototype of a finite-dimensional linear operator. The spectrum of a matrix is
exactly the set of its eigenvalues, that is the point spectrum. In the rest of this paper, we reserve the
word 'spectrum' only for the infinite-dimensional integral operator and talk only about eigenvalues of
matrices. For an eigenvalue -, there exists an eigenvector x such that thus the matrix
singular and not invertible.
For an arbitrary linear operator, there is no general way of obtaining its spectrum. Each case must
be analyzed separately with different analytical tools. We will consider the case of the scattering integral
equation (2.1). We will write it in the form
where K is the integral operator defined by
Z
4 J. RAHOLA
Fig. 3.1. Eigenvalues of the coefficient matrix for a spherical scatterer of radius refractive index
0:05i. The sphere is discretized with 136 computational cells (upper) and 455 computational cells (lower).
We do not know of any direct way of computing the spectrum of the scattering integral operator
K. However, with the following observation, we can relate the spectrum to the well-posedness of some
related scattering problems.
To find the spectrum of the operator K we need to find the complex points z for which the operator
does not have a well-defined inverse. For a homogeneous scatterer the integral operator (4.2)
has the form
Z
The crucial observation is the following: the operator (z1 \Gamma K) is a scaled version of the operator
z
which is equivalent to the scattering integral equation (2.1) by the same homogeneous object but with a
different refractive index m 0 defined by
z
EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 5
Thus, to find the points z in the spectrum for a scatterer with a given shape and refractive index m,
it suffices to find all possible refractive indices m 0 for which the scattering problem is not well defined.
After this, the points in the spectrum are recovered from
In all our figures, we actually plot the points in the spectrum of the operator
correspond to the eigenvalues of the matrix.
4.1. Scattering by a sphere. In this section we recall the analytic solution of scattering by a
sphere. This solution will be used to find the points in the spectrum of the integral operator. The
solution uses the spherical vector wave functions M lm and N lm [2, 5] that are defined by
l ('; OE));
l ('; OE) are the spherical harmonics and f l (kr) stands for the spherical Bessel or Hankel function.
The functions M lm (r) and N lm (r) are both solutions to the vector Helmholtz equation r \Theta r \Theta
The scattering by a sphere can be computed analytically using the so-called Mie theory [2, 25]. The
incoming electric field E inc , the electric field inside the object E 1 and the scattered electric field E s are
all expanded in terms of the spherical vector wave functions:
l
m=\Gammal
l M (1)
l N (1)
lm (kr)
l
m=\Gammal
l M (1)
l N (1)
l
m=\Gammal
l M (3)
l N (3)
lm (kr)
In these formulas k is the free-space wave number and k is the wave number inside the
scatterer. The notation M (1)
lm refers to the spherical vector wave functions that have a radial dependency
given by the spherical Bessel function j n (kr) while for M (3)
lm the radial dependency is given by the spherical
Hankel function h (1)
(kr). The magnetic field can easily be computed from these expansions.
One then requires that the tangential components of the electric and magnetic field are continuous
across the boundary of the sphere at radius kr = x. Now we can integrate the expansions and the vector
spherical harmonics over the full solid angle and the orthogonality of the vector wave functions shows
that the single mode M (1)
lm (r) in the incoming field excites the corresponding modes M (1)
lm (r) and M (3)
lm (r)
in the internal and scattered fields, respectively. This means that the coefficients of the internal and
scattered fields are related to the coefficients ff m
l and fi m
l of the incoming field by i m
l
l ,
l , and
l . At the radius kr = x, the boundary conditions and the expansions for
the electric and magnetic fields give four linear equations for the coefficients a l , b l , c l , and d l . The Mie
coefficients giving the field inside the object are [2]
l
l (x)[xj l (x)] 0
l (mx)[xh (1)
l
l (x)[mxj l (mx)] 0
l
l (x)[xj l (x)] 0
l
l (x)[mxj l (mx)] 0
6 J. RAHOLA
4.2. Resonances in the Mie series. To find points in the spectrum of the scattering integral
operator, we now need to find those values of the refractive index m for which the Mie solution is not
well-behaved. One such possibility is that the denominator in Equation (4.11) or (4.12) becomes zero.
The case when a denominator of a Mie coefficient becomes zero or close to zero is called the Mie
resonance. It means that a single spherical harmonic mode is greatly amplified and efficiently suppresses
all the other modes. The resonances in Mie scattering have been studied quite extensively, see, e.g.,
[4, 8, 13, 26]. In these studies both the refractive index m of the scatterer and the size of the scatterer,
which is measured by the size parameter kr = 2-r=-, can obtain complex values.
The imaginary part of the refractive index determines how the object absorbs electromagnetic energy.
A zero imaginary part means no absorption, a positive imaginary part is used for absorbing materials,
while a negative imaginary part signifies a somewhat unphysical case when the objects are generating
electromagnetic energy.
If the size parameter is real, the denominators of Mie coefficients can become zero only if the refractive
index has a negative imaginary part. However, in some cases the resonance appears with a refractive
index that has a very small negative imaginary part. In this case the proximity of the resonance is visible
in the scattering calculations also if we set the imaginary part of the resonant refractive index to zero.
Likewise, for a object with a refractive index with zero or positive imaginary part, the resonance can
appear only if the size parameter becomes complex. Sometimes the resonant size parameter has a very
small imaginary part, making the resonance visible at the corresponding real size parameter.
In this study we have a fixed the size parameter and try to find all the values for the complex refractive
index that satisfy the resonance conditions exactly. We must stress that this is purely a mathematical
trick to find the points in the spectrum of the integral equation and thus the refractive indices have no
physical meaning here.
To find the resonance points, we find the approximate locations of the zeros of denominators for Mie
coefficients c l and d l for all orders l by visual inspection of the absolute value of the denominators when m
varies in the complex plane. After this, we feed these values as initial guesses to the root-finding routine
DZANLY in the IMSL library that uses Muller's method [20].

Figure

4.1 shows some of the resonant complex refractive indices m for a sphere of radius kr = 1.
Note how the resonance positions gather along the positive real axis and at around \Gammai. We did not
compute the resonant refractive indices with large positive real parts, because all these will be mapped
to very small z-values in Equation (4.5). If the refractive index m 0 is resonant, so is \Gammam 0 , but these will
be mapped to the same z-point.
The preceding analysis indicates also that the coefficient matrix should become singular when we try
to compute scattering by a sphere with a resonant refractive index. For a resonant refractive index, the
mapping (4.5) indicates that 1 belongs to the spectrum of the integral operator K and thus there is a
zero eigenvalue in the coefficient matrix. In addition to this, the matrix should become badly conditioned
in the vicinity of such refractive indices and thus the convergence of iterative solvers should slow down.
This has indeed been observed in another study [11].
4.3. Branch cut of the square root. Now we turn our attention to the line segment in the
eigenvalue plots of the matrix and ask in what other way besides a resonance can the Mie solution fail.
To this end we will study the definition of the refractive index in the whole complex plane.
Recall that the refractive index is defined by
~ffl, where ~ ffl is the complex permittivity. The
square root y of a complex number z is defined as the solution of the equation y z. But this equation
has two solutions. We will define the square root as being the solution that lies in the right half-plane,
i.e., with non-negative real part. If the complex number z is given by re iOE , where \Gamma- ! OE -,
then the square root can be uniquely defined as
re iOE=2 . This is the main branch corresponding to
positive square roots for positive real numbers.
There is a discontinuity associated with this definition of the square root. When z approaches the
negative real axis from above,
z approaches the positive imaginary axis. However, when z approaches the
EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 7
-1.4
-1.2
-0.4
Fig. 4.1. Some of the complex refractive indices (m) giving rise to a resonance in the Mie series for a sphere of radius
negative real axis from below, p
z approaches the negative imaginary axis. Thus there is a discontinuity
in p
z as z crosses the negative imaginary axis. The negative imaginary axis is called the branch cut of
the complex square root.
We recall that in the Mie series solution, inside the scatterer the radial dependency of the solution
is given by the spherical Bessel function where the wave number inside the object is given by
mk. Thus in the scattering problem, when the complex permittivity ~
ffl sits on the negative real axis,
the refractive index being purely imaginary, there is an ambiguity in the Mie solution due to the branch
cut. Using mapping (4.5) these refractive indices imply that the corresponding points z belong to the
spectrum of the integral operator and that these points form a line segment in the complex plane.
The analysis of the branch cut of the refractive index and the mapping (4.4) are valid for any
homogeneous scatterer, not just the sphere. Thus it is clear that the position and size of the line segment
in the spectrum only depends on the refractive index, but not on the shape or size of the object. On the
other hand, the part of the spectrum corresponding to the resonances does depend on the shape and size
of the scatterer.
4.4. Numerical experiments. In Figure 4.2 we show the eigenvalues of the coefficient matrix for
two different discretizations of a sphere with kr = 1. We also show the points in the spectrum of the
found with the help of the resonances of the Mie series and the branch cut of the square
root. If m 0 is a refractive index for which the scattering problem is not well defined, the corresponding
points in the spectrum of operator are recovered from
Most of the eigenvalues lying outside the line can be explained by the resonance points and when the
discretization is refined, the agreement gets better. The endpoint of the line segment corresponding to
the branch cut, i.e., scattering with purely imaginary refractive indices, is also shown. These refractive
indices are of type iy, where y is real. When y approaches infinity, the points z in the spectrum
approach 1. When y goes to zero, the points in the spectrum approach the point m 2 , which is marked in
the plots with a large circle.
In

Figure

4.3 we show the same information as in Figure 4.2 but with computational cells)
and cells), respectively. These figures show that when the size of the sphere is increased,
8 J. RAHOLA
Fig. 4.2. The eigenvalues of the coefficient matrix (small black dots) and some points in the spectrum of the integral
operator due to the resonances (small circles). The points in the spectrum of the integral operator that correspond to
scattering by objects with purely imaginary refractive indices extend from the point one to the point marked with a large
circle. The eigenvalues were computed from a discretization with 280 computational cells (upper) and 1000 computational
cells (lower). The radius of the sphere is given by kr = 1.
the number of eigenvalues off the line also increases.

Figure

4.4 shows the eigenvalues for an anisotropic spherical scatterer with and the refractive
indices in the direction of the coordinate axes being m
Three line segments are visible in the eigenvalue plots. The segments start from 1, and they correspond
to scattering with purely imaginary refractive indices in the mapping (4.13), where the refractive index
m is replaced by any of the above indices.
The fact that the scattering integral operator has a continuous spectrum implies that the operator
cannot be compact. However, in a related situation, there is a theorem by Colton and Kress [3, Section 9]
that states that the scattering integral operator is a compact operator if the refractive index is a smooth
function of r.
In the case of scattering by a homogeneous sphere, the refractive index is not smooth but instead has
a discontinuity at the surface of the scatterer. Corresponding to the case analyzed by Colton and Kress
EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 9
-1.4
-1.2
-0.4
-0.20.2
Real(l)
Real(l)
kr=5, m=1.4+0.05*i
Fig. 4.3. Same as Figure 4.2 but with size kr = 3 and with 480 computational cells (upper) and with size
with 1064 computational cells (lower).
we also computed the eigenvalues of the coefficient matrices of the integral equation when the refractive
index varies smoothly. In this case, the line segment was still visible in the eigenvalue plots, but the
eigenvalues no longer resided uniformly along the line but converged towards 1. This is the behavior one
would expect of the discretization of the identity operator plus a compact operator.
5. Convergence estimates for iterative solvers. In this section we study the possibility of
estimating the speed of convergence of iterative solvers from the knowledge of the eigenvalues of the
coefficient matrix. In general, the convergence of iterative methods is dictated by the distribution of the
eigenvalues, the conditioning of the eigenvalues and the right-hand side vector.
We shall now give a basic convergence results for a Krylov-subspace method, i.e., iterative methods
based on only the information given by successive matrix-vector products of the matrix A. We shall
denote the linear system by b, the current iterate by xn and the residual by r
and r 0 being the initial guess and initial residual, respectively.
Iterative Krylov-subspace methods produce iterates xn such that the corresponding residual r n is
J. RAHOLA
Real(l)
Fig. 4.4. The eigenvalues of an anisotropic sphere with refractive indices in the direction of the coordinate
axes being 0:3i. The sphere is discretized with 1064 computational
cells. For each of the three indices, the endpoint of the line segments corresponding to purely imaginary refractive indices
in the mapping (4.13) are shown as large circles.
given by r is a polynomial of degree n with pn 1. The task of iterative
methods is to construct polynomials pn such that the norm of the residual decreases rapidly when n
increases.
Suppose that the coefficient matrix A is diagonalizable and thus has distinct eigenvectors v i ,
. Denote by V the matrix whose columns are v i and by   the diagonal matrix of the corresponding
eigenvalues - i . The matrix A can be decomposed as . Given this decomposition we can
estimate the norm of the matrix polynomial pn (A) in terms of the polynomial evaluated at the eigenvalues:
where the condition number of the eigenvalue matrix is given by -(V
In our experiments we will use two iterative methods: the generalized minimal residual method
(GMRES) [24] without restarts and the complex symmetric version of the quasi-minimal residual method
(QMR) [7]. GMRES minimizes the residual of the current iterate among all the possible iterates in
a Krylov subspace, a subspace generated by successive multiplications by the initial residual with the
coefficient matrix. QMR has only an approximate minimization property and thus converges slower
than GMRES, but its iterates are much cheaper to compute. QMR is the iterative solver used in our
production code.
For the iterative method GMRES, we have the following theorem [18]:
kr nk
pn (z)
pn (0)=1
In other words, the iterative method GMRES finds a polynomial such that the maximum value of the
polynomial at the eigenvalues is minimized. In the rest of this paper, we assume that the eigenvectors
are well-conditioned and thus -(V ) is close to unity.
The question of the optimal convergence speed in iterative methods has been studied by, e.g., Nevanlinna
[19], who studied iteration of the problem g. We study the so-called linear phase of the
convergence of iterations. The fastest possible linear convergence of the residual is given by kr k
EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 11
where j is the optimal reduction factor. Nevanlinna shows by potential-theoretic arguments that this is
given by is the conformal map from the outside of the spectrum of K to the outside
of the unit disk. Here we have assumed that the spectrum of K is a simply connected region in the
complex plane.
Now we will try to estimate the convergence speed of iterative solvers assuming the spectrum is a line
segment in the complex plane, thus neglecting the few eigenvalues off the line. We are working with the
spectrum of the scattering integral operator K defined in (4.2), not the operator that corresponds to
the coefficient matrix. The line segment in the spectrum of K starts from zero, has a negative imaginary
part, makes and angle of ff with the negative real axis, and has a length of d. From the analysis in the
preceeding section it follows that
The conformal map that maps the outside of the line segment to the outside of the unit disk consists
of three parts: rotates the line segment to the negative imaginary axis
(i.e., to the segment [\Gammad; 0]), OE 2 translates this segment to the segment [\Gamma1; 1] and finally OE 3 maps the
outside of this segment to the outside of the unit disk. These maps are explicitly given by
d
Thus the optimal reduction factor is given by
where
d

Figure

5.1 shows the convergence behavior or QMR and GMRES together with the estimated optimal
convergence speed for a sphere with refractive index 0:05i and with size of kr
image) and (lower image). It can be seen that the convergence estimate and the actual behavior
of iterative methods are very close in the case kr = 1. Also, the convergence of QMR is very close to
GMRES. In the case of kr = 3 we notice that the observed convergence of iterative methods is much
slower than the predicted rate. This is due to the increased number of eigenvalues that lie off the line, as
can be seen from Figure 4.3. The convergence could be better estimated by drawing a polygon around
the spectrum and using the Schwarz-Cristoffel theorem [6] to compute the values of a conformal map
from this polygon to the outside of the unit disk, but this approach has not been pursued further.
6. Conclusions. We have studied the eigenvalues of the coefficient matrix arising from a discretization
of the volume integral equation of electromagnetic scattering. The eigenvalues of a coefficient matrix
for a spherical scatterer consist of a line segment plus some isolated points. We have studied how these
eigenvalues and the spectrum of the scattering integral operator are related. To find the spectrum of the
integral operator, we show that it is sufficient to find all the refractive indices for which the scattering
problem is not well defined. These indices are then mapped to the points in the spectrum of the operator.
We have shown that the isolated eigenvalues of the coefficient matrix correspond to exact resonances
in the analytical solution of scattering by a sphere. The line segment in the eigenvalue plots corresponds
to scattering by purely imaginary refractive indices, which is related to the branch cut of the square root
in the definition of the complex refractive index and the wave number inside the scatterer.
The knowledge of the eigenvalues of the integral operator can help us to understand the convergence
of iterative solution methods for the discretized scattering problem. For example, we have observed that
J. RAHOLA
Iteration number
Residual
norm
Iteration number
Residual
norm
Fig. 5.1. Convergence given by the optimal reduction factor (solid line), convergence of GMRES (dashed line) and of
QMR (dash-dotted line) for a sphere with and with refractive index
when the same object is discretized with increasing resolution, the number of unpreconditioned iterations
is practically constant. The convergence of iterative solvers depends on the eigenvalue distribution of the
coefficient matrix. Successively finer discretizations of the scattering integral equation produce coefficient
matrices which have approximately the same eigenvalue distribution and thus the same convergence
properties.
In contrast, when partial differential equations are discretized with increasingly finer meshes, the
convergence of unpreconditioned iterative solvers typically gets worse. This situation can arise if zero
belongs to the spectrum of the partial differential operator. When such a problem is discretized with
finer and finer meshes, some of the eigenvalues of the coefficient matrices move closer and closer to zero,
giving rise to poor convergence of iterative solvers. This problem can sometimes be remedied with the
help of preconditioners.
Finally, we have tried to predict the convergence speed of iterative solvers based on the knowledge
of the eigenvalues. In doing so, we have only used the location and length of the line segment in the
eigenvalue plots, thus neglecting the isolated eigenvalues that lie outside the line. This strategy gives
EIGENVALUES OF INTEGRAL OPERATOR IN SCATTERING 13
convergence estimates that are very close to the observed convergence for small scatterers. Once the
size of the sphere is increased, the number of eigenvalues lying off the line is increased and thus the
convergence estimate quickly becomes useless.
The analysis presented in this paper could be applied to other scattering geometries, such as spheroids,
for which analytical solutions exist. It would also be interesting to compute the eigenvalues arising from
more refined discretization schemes of the volume integral operator. The convergence analysis of iterative
solvers presented here could be augmented to account for the eigenvalues lying off the line segment in the
complex plane. However, for a given physical problem it is quite tedious to compute all the resonance
locations and thus this approach will probably not give a practical convergence analysis tool.

Acknowledgment

. I would like to thank Francis Collino for fruitful discussions on the subject.



--R


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--TR

--CTR
Matthys M. Botha, Solving the volume integral equations of electromagnetic scattering, Journal of Computational Physics, v.218 n.1, p.141-158, 10 October 2006
electromagnetic scattering;spectrum of linear operators;eigenvalues of matrices;iterative methods;integral equations