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Numerische Mathematik (2019) 142:577–609
https://doi.org/10.1007/s00211-019-01034-w
Numerische
Mathemat ik
Asymptotic convergence of spectral inverse iterations
for stochastic eigenvalue problems
Harri Hakula1
·Mikael Laaksonen1
Received: 10 August 2017 / Revised: 9 January 2019 / Published online: 19 March 2019
© The Author(s) 2019
Abstract
We consider and analyze applying a spectral inverse iteration algorithm and its sub-
space iteration variant for computing eigenpairs of an elliptic operator with random
coefficients. With these iterative algorithms the solution is sought from a finite dimen-
sional space formed as the tensor product of the approximation space for the underlying
stochastic function space, and the approximation space for the underlying spatial func-
tion space. Sparse polynomial approximation is employed to obtain the first one, while
classical finite elements are employed to obtain the latter. An error analysis is presented
for the asymptotic convergence of the spectral inverse iteration to the smallest eigen-
value and the associated eigenvector of the problem. A series of detailed numerical
experiments supports the conclusions of this analysis.
Mathematics Subject Classification 65C20 ·65N12 ·65N15 ·65N25 ·65N30
1 Introduction
During the recent years numerical solution of stochastic partial differential equations
(sPDE) has attracted a lot of attention and become a well-established field. However,
the field of stochastic eigenvalue problems (sEVP) and their numerical solution is
still in its infancy. It is natural that, after the source problem, more effort is put on
addressing the eigenvalue problem.
Harri Hakula: The work of this author was supported by the (FP7/2007–2013) ERC Grant Agreement No.
339380. Mikael Laaksonen: The work of this author was supported by the Magnus Ehrnrooth Foundation.
BHarri Hakula
harri.hakula@aalto.fi
Mikael Laaksonen
mikael.j.laaksonen@aalto.fi
1Department of Mathematics and Systems Analysis, Aalto University, Espoo, Finland
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578 H. Hakula, M. Laaksonen
A few different algorithms have recently been suggested for computing approximate
eigenpairs of sEVPs. As with sPDEs, the solution methods are typically divided into
intrusive and non-intrusive ones. A benchmark for non-intrusive methods is the sparse
collocation algorithm suggested and thoroughly analyzed by Andreev and Schwab
[1]. An attempt towards a Galerkin-based (intrusive) method was made by Verhoosel
et al. [20], though this method omits uniform normalization of the eigenmodes. Very
recently Meidani and Ghanem proposed a spectral power iteration, in which the eigen-
modes are normalized using a quadrature rule over the parameter space [16]. The
algorithm has been further developed and studied by Sousedík and Elman [19]. How-
ever, neither of the papers present a comprehensive error analysis for the method.
Inspired by the original method of Meidani and Ghanem we have suggested a purely
Galerkin-based spectral inverse iteration, in which normalization of the eigenmodes
is achieved via solution of a simple nonlinear system [11]. This method, and its gen-
eralization to a spectral subspace iteration, is the focus of the current paper. Although
the algorithms in [16,19] differ from ours in the way normalization is performed, the
basic principles are still the same and hence our results on convergence should apply
to these methods as well.
In this work we consider computing eigenpairs of an elliptic operator with random
coefficients. We assume a physical domain D⊂Rdand, in order to capture the
random dimension of the system, a parameter domain Γ⊂R∞with associated
measure ν. One may think of a parametrization that arises from Karhunen-Loève
representations of the random coefficients in the system, for instance. Discretization
in space is achieved by standard FEM and associated with a discretization parameter
h, whereas discretization in the random dimension is achieved using collections of
certain multivariate polynomials. These collections are represented by multi-index
sets Aof increasing cardinality #Aas →0.
In the current paper we present a step-by-step analysis that leads to the main result:
the asymptotic convergence of the spectral inverse iteration towards the exact eigenpair
(μ, u). In this context the eigenpair of interest is the ground state, i.e., the smallest
eigenvalue and the associated eigenfunction of the system. However, analogously to
the classical inverse iteration, the computation of other eigenpairs may be possible
by using a suitably chosen shift parameter λ∈R. We show that under sufficient
assumptions the iterates of the algorithm (μk,uk)for k=1,2,...obey
||u−uh,A,k||L2
ν(Γ )⊗L2(D)h1+l+(#A)−r+λk
1/2(1)
and
||μ−μh,A,k||L2
ν(Γ ) h2l+(#A)−r+λk
1/2,(2)
where l∈Nis the degree of polynomials used in the spatial discretization and r>0
depends on the properties of the region to which the solution, as a function of the
parameter vector, admits a complex-analytic extension. The quantity λ1/2reflects the
gap between the two smallest eigenvalues of the system and should be less than one.
The first term in the formulas (1) and (2) is justified by standard theory for Galerkin
approximation of eigenvalue problems, a simple consequence of which we have
recapped in Theorem 1. The second term can be deduced from Theorem 2, which
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Asymptotic convergence of spectral inverse iterations for… 579
bounds the Galerkin approximation errors by residuals of certain polynomial approx-
imations of the solution. Using best P-term polynomial approximations, we see that
these residuals are ultimately expected to decay at an algebraic rate r>0, see [5] and
[7]. Finally, the third term follows from Theorem 3, which states that asymptotically
the iterates of the spectral inverse iteration converge to a fixed point in geometric
fashion. Here the analogy to classical inverse iteration is evident. Each of these three
important steps that comprise the main result is separately verified through detailed
numerical examples.
A variant of our algorithm for spectral subspace iteration is also presented. No anal-
ysis of this algorithm is given, but the numerical experiments support the conclusion
that it converges towards the exact subspace of interest, and that the rate of conver-
gence is analogous to what we would expect from classical theory. This is despite
the fact that the individual eigenmodes, as defined by the pointwise order of magni-
tude of the eigenvalues, are not continuous functions over the parameter space due to
an eigenvalue crossing. To the authors’ knowledge such a scenario has not yet been
considered in the scientific literature.
The rest of the paper is organized as follows. Our model problem and its fundamental
properties are assessed in Sects. 2and 3. A detailed review of the discretization of the
spatial and stochastic approximation spaces is given in Sect. 4. Analysis of the spectral
inverse iteration, supported by thorough numerical experiments, is given in Sect. 5.
Finally, the algorithm of spectral subspace iteration and numerical experiments of its
convergence are presented in Sect. 6.
2 Problem statement
In this work we consider eigenvalue problems of elliptic operators with random coeffi-
cients. It is assumed that the random coefficients admit a parametrization with respect
to countably many independent and bounded random variables. As a model problem
we consider the eigenvalue problem of a diffusion operator with a random diffusion
coefficient. It will be evident, however, that our methods and analysis in fact cover a
much broader class of problems.
2.1 Model problem
Let (Ω, F,P)be a probability space, Ωbeing the set of outcomes, Faσ-algebra of
events, and Pa probability measure defined on Ω. We denote by L2
P(Ω) the space
of square integrable random variables on Ωand define for v∈L2
P(Ω) the expected
value
E[v]=Ω
v(ω) dP(w)
and variance Var[v]=E[(v −E[v])2].
Let D⊂Rdbe a bounded convex domain with a sufficiently smooth boundary
and assume a diffusion coefficient a:D×Ω→Rthat is a random field on D.
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580 H. Hakula, M. Laaksonen
The diffusion coefficient is assumed to be strictly uniformly positive and uniformly
bounded, i.e., for some positive constants amin and amax it holds that
Pω∈Ω:amin ≤ess inf
x∈Da(x,ω)≤ess sup
x∈D
a(x,ω)≤amax =1.(3)
We now formulate the model problem as: find functions μ:Ω→Rand u:D×Ω→
Rsuch that the equations
−∇ · (a(·,ω)∇u(·,ω)) =μ(ω)u(·,ω) in D
u(·,ω) =0on∂D(4)
hold P-almost surely. In order to make the solutions physically meaningful we also
impose a normalization condition ||u(·,ω)||L2(D)=1 that should hold P-almost
surely.
2.2 Parametrization of the random input
We make the assumption that the input random field admits a representation of the
form
a(x,ω) =a0(x)+∞
m=1
am(x)ym(ω), (5)
where {ym}∞
m=1are mutually independent and bounded random variables. For sim-
plicity, we assume here that each ymis uniformly distributed. Thus, after possible
rescaling, the dependence on ωis now parametrized by the vector y=(y1,y2,...) ∈
Γ:= [−1,1]∞. We denote by νthe underlying uniform product probability measure
and by L2
ν(Γ ) the corresponding weighted L2-space.
The usual convention is that the parametrization (5) results from a Karhunen-Loève
expansion, which gives a(x,ω) as a linear combination of the eigenfunctions of the
associated covariance operator. The distinguishing feature of the Karhunen-Loève
expansion compared to other linear expansions is that it minimizes the mean square
truncation error [9].
It is easy to see that a0∈L∞(D)and
ess inf
x∈Da0(x)> ∞
m=1||am||L∞(D)(6)
are sufficient conditions to ensure the assumption (3). In order to ensure analyticity of
the eigenpair (μ, u)with respect to the parameter vector y=(y1,y2,...) we assume
that ∞
m=1||am||p0
L∞(D)<∞(7)
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Asymptotic convergence of spectral inverse iterations for… 581
for some p0∈(0,1)and that for a certain level of smoothness s∈Nwe have
a0∈Ws,∞(D)and ∞
m=1||am||ps
Ws,∞(D)<∞(8)
for some ps∈(0,1). In particular, we consider the interesting case of algebraic
||am||L∞(D)≤Cm−ς,ς>1,m=1,2,...
decay of the coefficients in the series (5).
2.3 Parametric eigenvalue problem and its variational formulation
With the diffusion coefficient given by (5), the model problem (4) becomes an eigen-
value problem of the operator
(A(y)v)(x):= −∇ · (a(x,y)∇v(x)), x∈D,y∈Γ,
where
a(x,y)=a0(x)+∞
m=1
am(x)ym.
Thus, we obtain the parametric eigenvalue problem: find μ:Γ→Rand u:Γ→
H1
0(D)such that
A(y)u(y)=μ(y)u(y)∀y∈Γ. (9)
We denote by σ(A(y)) the set of eigenvalues of A(y)for y∈Γ.
For any fixed y∈Γthe problem (9) reduces to a single deterministic eigenvalue
problem. In variational form this is given by: find μ(y)∈Rand u(·,y)∈H1
0(D)
such that
b(y;u(·,y), v) =μ(y)u(·,y), v L2(D)∀v∈H1
0(D), (10)
where
b(y;u,v) := D
a(·,y)∇u·∇vdx.
Under assumption (6) the bilinear form b(y;u,v)is continuous and elliptic. Thus, as in
[1,11], we deduce that the problem (10) admits a countable number of real eigenvalues
and corresponding eigenfunctions that form an orthogonal basis of L2(D).
3 Analyticity of eigenmodes
A key issue in the analysis of parametric eigenvalue problems is that eigenvalues may
cross within the parameter space. Here we first disregard this possibility and recap
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582 H. Hakula, M. Laaksonen
the main results from [1] for simple eigenvalues that are sufficiently well separated
from the rest of the spectrum. In Sect. 6we briefly comment on the case of possibly
clustered eigenvalues and associated invariant subspaces.
We call an eigenvalue μof problem (9) strictly nondegenerate if
(i) μ(y)is simple as an eigenvalue of A(y)for all y∈Γand
(ii) the minimum spectral gap inf y∈Γdist(μ(y), σ ( A(y))\{μ(y)})is positive.
In the case of strictly nondegenerate eigenvalues, the eigenpair (μ, u)is in fact analytic
with respect to the parameter vector y.
Proposition 1 Consider a strictly nondegenerate eigenvalue μof the problem (9)and
the corresponding eigenfunction u normalized so that ||u(y)||L2(D)=1for all y ∈Γ.
For s ∈Nassume that a0∈Ws,∞(D)and the assumptions (6)–(8)hold for some
p0,ps∈(0,1). Given τ=(τ1,τ
2,...) ∈R∞
+define
E(τ ) := {z∈C∞|dist(zm,[−1,1])≤τm}.
Then there exists C1>0independent of m such that with C2>0arbitrary and τ
given by
τm=min C1||am||p0−1
L∞(D),C2||am||ps−1
Ws,∞(D),m=1,2,...
the eigenpair (μ, u)can be extended to a jointly complex-analytic function on E (τ )
with values in C×(Hs+1(D)∩H1
0(D)).
Proof This is analogous to Corollary 2 of Theorem 4 in [1].
It is well known that for elliptic operators on a connected domain Dthe smallest
eigenvalue is simple [12]. Thus, Proposition 1may at least be applied for the smallest
eigenvalue of problem (9).
4 Stochastic finite elements
Proposition 1, under sufficient assumptions, guarantees the existence of an analytic
eigenpair for problem (9). It now makes sense to look for the eigenvalue in the space
L2
ν(Γ ) and the eigenfunction in the space L2
ν(Γ ) ⊗H1
0(D). The space H1
0(D)may
be discretized by means of the traditional finite element method. For the discretiza-
tion of L2
ν(Γ ), we follow the usual convention in stochastic Galerkin methods and
construct a basis of orthogonal polynomials of the input random variables. Orthogo-
nal polynomials for various probability distributions exist and the use of these as the
approximation basis has been observed to yield optimal rates of convergence [18,21].
Here we consider uniformly distributed random variables which lead to the choice of
tensorized Legendre polynomials.
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Asymptotic convergence of spectral inverse iterations for… 583
4.1 Galerkin discretization in space
Let Vh⊂H1
0(D)denote a finite dimensional approximation space associated with
the discretization parameter h>0. We assume approximation estimates
inf
vh∈Vh||v−vh||L2(D)≤Chl+1||v||Hl+1(D)(11)
and
inf
vh∈Vh||v−vh||H1
0(D)≤Chl||v||Hl+1(D)(12)
that are standard for piecewise polynomials of degree l.
Fix y∈Γand let (μh,uh)be the solution to the variational equation
b(y;uh(·,y), vh)=μh(y)uh(·,y), vhL2(D)∀vh∈Vh,(13)
where b(y;·,·)is as in (10). Then we have the following bounds for the discretization
error.
Theorem 1 Assume (11)and (12).Fory ∈Γlet μ( y)be a simple eigenvalue
of (10)and μh(y)an eigenvalue of (13)such that limh→0μh(y)=μ( y). Let
u(·,y)∈H1+l(D)and uh(·,y)∈Vhdenote the associated eigenfunctions nor-
malized in L 2(D). Then there exists C >0such that
|μ(y)−μh(y)|≤Ch2l,(14)
and
||u(·,y)−uh(·,y)||L2(D)≤Ch1+l||u(·,y)||H1+l(D).(15)
as h →0.
Proof This follows from the theory of Galerkin approximation for variational eigen-
value problems. See Section 8 in [3] and Section 9 in [8].
Let Vh=span{ϕi}i∈Jwhere J:= {1,2,...,N}. Then (13) can be written as a
parametric matrix eigenvalue problem: find μh:Γ→Rand uh:Γ→RNsuch that
K(0)+∞
m=1
K(m)ymuh(y)=μh(y)Muh(y)∀y∈Γ, (16)
where uh(x,y)=i∈Jϕi(x)(uh)i(y). The coefficient matrices are given by
K(m)
ij =D
am∇ϕi·∇ϕjdx,m=0,1,...
and
Mij =D
ϕiϕjdx.
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584 H. Hakula, M. Laaksonen
For each fixed y∈Γthe problem (16) reduces to a positive-definite generalized
matrix eigenvalue problem.
4.2 Legendre chaos
Recall that y=(y1,y2,...) ∈Γis a vector of mutually independent uniform random
variables and νis the underlying constant product probablity measure. Now
E[v]=Γ
v(y)dν(y)(17)
whenever the integral is finite. We define (N∞
0)cto be the set of all multi-indices with
finite support, i.e.,
(N∞
0)c:= {α∈N∞
0|# supp(α) < ∞},
where supp(α) ={m∈N|αm= 0}. Given a multi-index α∈(N∞
0)cwe now define
the multivariate Legendre polynomial
Λα(y):=
m∈supp α
Lαm(ym),
where Lp(x)denotes the univariate Legendre polynomial of degree p. We will assume
the normalization E[Λ2
α]=1 for all α∈(N∞
0)c.
The system {Λα(y)|α∈(N∞
0)c}forms an orthonormal basis of L2
ν(Γ ). Therefore,
we may write any square integrable random variable vinaseries
v(y)=
α∈(N∞
0)c
vαΛα(y)(18)
with convergence in L2
ν(Γ ). The expansion coefficients are given by vα=E[vΛα].
Due to the orthogonality of the Legendre polynomials we have E[Λα]=δα0and
E[ΛαΛβ]=δαβ for all α, β ∈(N∞
0)c. Moreover, we denote
cαβγ := E[ΛαΛβΛγ], α,β,γ ∈(N∞
0)c
cmαβ := E[ymΛαΛβ],m∈N,α,β∈(N∞
0)c
c0αβ := δαβ ,α,β∈(N∞
0)c.
4.3 Sparse polynomial approximation in the parameter domain
We fix a finite set A⊂(N∞
0)cand employ the approximation space WA=
span{Λα}α∈A⊂L2
ν(Γ ).WeletPAand RAdenote the underlying projection and
residual operators so that v∈L2
ν(Γ ) is approximated by
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Asymptotic convergence of spectral inverse iterations for… 585
PA(v)(y)=
α∈A
vαΛα(y)
and the approximation error is given by RA(v) =v−PA(v). Since
||RA(v)||2
L2
ν(Γ ) =E⎡
⎣
α∈Ac
vαΛα2⎤
⎦=
α∈Ac
v2
α,(19)
where Ac={α∈(N∞
0)c|α/∈A}, we conclude that the choice of the multi-index
set Aultimately determines the accuracy of our expansion.
We proceed as in [5] and use best P-term approximations to prove convergence of
the approximation error.
Proposition 2 Let H be a Hilbert space. Assume that v:Γ→H admits a complex-
analytic extension in the region
E(τ ) := {z∈C∞|dist(zm,[−1,1])≤τm}
with
τm≥Cm,>1,m=1,2,... (20)
Given >0define
A:= ⎧
⎨
⎩α∈(N∞
0)c
m∈supp α
ηαm
m>⎫
⎬
⎭,
where
ηm:= τm+1+τ2
m−1
,m=1,2,...
Then
||RA(v)||L2
ν(Γ )⊗H≤||v||L∞(E(τ );H)(21)
and as →0we have
#A≤C(, r)−1/r(22)
for any 0<r<−1
2.
Proof Fix >0 and let P=#A. Set M=max{m∈N|∃α∈As.t. αm= 0}
and vM(z)=v(z1,...,zM,0,0,...) so that PA(v) =PA(vM). The norm of the
residual may now be separated into two parts in the following sense
||RA(v)||L2
ν(Γ )⊗H≤||v−vM||L2
ν(Γ )⊗H+||vM−PA(vM)||L2
ν(Γ )⊗H.(23)
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586 H. Hakula, M. Laaksonen
For the second term we may apply the proof of Proposition 3.1 in [6] and obtain
||vM−PA(vM)||L2
ν(Γ )⊗H≤C(, r)P−r||vM||L∞(E(τ );H).(24)
On the other hand, in order to bound the first term we note that
∞
m>M
(ηm−1)−1≤C∞
m>M
m−≤C∞
M
x−dx ≤C()M1−.(25)
Thus, by Lemmas 4.3. and 4.4 in [2], we obtain
||v−vM||L2
ν(Γ )⊗H≤C||v||L∞(E(τ );H)
∞
m>M
(ηm−1)−1≤C()P−r||v||L∞(E(τ );H)
(26)
for any M≥C() Pr/(−1). The claim follows from combining (24) and (26).
4.4 Stochastic Galerkin approximation of vectors and matrices
We now generalize the concept of sparse polynomial approximation to vector and
matrix valued functions. Assume that the dimensions of the approximation spaces
Vhand WAare Nand Prespectively. We denote by WN
A(or WN×N
A) the space of
functions v:Γ→RN(or A:Γ→RN×N) whose every component is in WA.
Whenever v∈WN
Aand α∈Awe set vαi=(vi)αand use vαto denote the vector
of coefficients {vαi}i∈J∈RN. Moreover, we associate any v∈WAwith the array
of coefficients ˆv:= {vα}α∈A∈RPand similarly any v∈WN
Awith the array of
coefficients ˆ
v:= {vαi}α∈A,i∈J∈RPN.
We denote by ·,·RN
Mthe inner product on RNinduced by the positive definite
matrix Mand by ||·||RN
Mthe associated norm. Furthermore, we let ||·||RPdenote the
standard norm on RPand || · ||RP⊗RN
Mdenote the tensorized norm on RPN given by
||ˆ
v||2
RP⊗RN
M:=
α∈A
i∈J
j∈J
vαiMijvαj.
Remark 1 Observe that if v∈WA⊗Vhis written as v(x,y)=i∈Jϕi(x)vi(y),
then ||v||2
L2
ν(Γ )⊗L2(D)=||v||2
L2
ν(Γ )⊗RN
M=||ˆ
v||2
RP⊗RN
M
.(27)
Let us consider the linear system defined by a parametric matrix A∈WN×N
A.The
Galerkin approximation of this system is: given f∈WN
Afind v∈WN
Asuch that
PA(Av)(y)=f(y)∀y∈Γ. (28)
We define moment matrices G(m)∈RP×Pfor m∈N0and G(α) ∈RP×Pfor α∈A
by setting [G(m)]αβ =cmαβ and [G(α)]βγ =cαβγ . Using this notation we may write
(28) as the fully discrete system: given ˆ
f∈RPN find ˆ
v∈RPN such that
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Asymptotic convergence of spectral inverse iterations for… 587
α∈A
G(α) ⊗Aαˆ
v=ˆ
f,(29)
where Aα=E[AΛα]∈RN×N. The existence of a solution, i.e. the invertibility of
the coefficient matrix, is guaranteed by the following lemma.
Lemma 1 If A∈WN×N
Ais a parametric matrix such that A(y)is positive-definite for
every y ∈Γ, then for any f∈WN
Athere exists a unique v∈WN
Asuch that (28)holds.
Furthermore,
||v||L2
ν(Γ )⊗RN
M≤sup
y∈Γ
λ−1(y)||f||L2
ν(Γ )⊗RN
M,(30)
where λ(y)is the smallest eigenvalue of A(y)for each y ∈Γ.
Proof Observe that the system (28) is equivalent to the variational form
E[vTAw]=E[fTw]∀w∈WN
A.(31)
The left hand side of (31) is a symmetric and elliptic bilinear form so the existence of a
unique solution is guaranteed by the Lax-Milgram Lemma. Moreover, the associated
coefficient matrix in (29) is positive definite.
Now let ˜
λ∈Rbe such that ˜
λ<inf y∈Γλ(y). The matrix A(y)−˜
λIN, where INis
the identity matrix, is positive definite for all y∈Γ. Thereby the eigenvalues of the
associated coefficient matrix should be positive. Let χbe an eigenvalue of (29), i.e.,
there exists w∈WN
Asuch that
PA(Aw)(y)=χw(y)∀y∈Γ. (32)
Then
PA((A−˜
λIN)w)(y)=PA(Aw)(y)−˜
λw(y)=(χ −˜
λ)w(y)∀y∈Γ(33)
and we deduce that χ>˜
λ. Equation (30) now follows from taking the limit ˜
λ→
inf y∈Γλ(y).
5 Spectral inverse iteration
In this section we introduce the algorithm of spectral inverse iteration, analyze its
asymptotic convergence, and present numerical examples to support our analysis. The
spectral inverse iteration, see [11], can be considered as an extension of the classical
inverse iteration to the case of parametric matrix eigenvalue problems. In the spectral
version each of the elementary operations is computed in Galerkin sense via projecting
to the sparse polynomial basis WA. Optimal convergence of the algorithm requires
that the eigenmode of interest, i.e., the smallest eigenvalue of the parametric matrix,
is strictly nondegenerate.
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588 H. Hakula, M. Laaksonen
5.1 Algorithm description
Fix a finite set of multi-indices A⊂(N∞
0)cand let P=#A. The spectral inverse
iteration for the system (16) is now defined in Algorithm 1. One should note that,
if the projections in the algorithm were precise, the algorithm would correspond to
performing classical inverse iteration pointwise over the parameter space Γ. We expect
the algorithm to converge to an approximation of the eigenvector corresponding to the
smallest eigenvalue of the system.
Algorithm 1 (Spectral inverse iteration) Fix t o l >0and let u(0)∈WN
Abe an initial
guess for the eigenvector. For k =1,2,...do
(1) Solve v∈WN
Afrom the linear equation
PA(Kv)=Mu(k−1).(34)
(2) Solve s ∈WAfrom the nonlinear equation
PA(s2)=PA||v||2
RN
M.(35)
(3) Solve u(k)∈WN
Afrom the linear equation
PAsu(k)=v.(36)
(4) Stop if ||u(k)−u(k−1)||L2
ν(Γ )⊗RN
M<tol and return u(k)as the approximate eigen-
vector.
Once the approximate eigenvector u(k)∈WN
Ahas been computed, the correspond-
ing eigenvalue μ(k)∈WAmay be evaluated from the Rayleigh quotient, as in [11],
or alternatively from the linear system
PA(sμ(k))=1.(37)
Lemma 1guarantees the invertibility of the linear system (34) and, assuming that
s(y)>0 for all y∈Γ, the invertibility of the systems (36) and (37). The nonlinear
system (35) may be solved using for instance Newton’s method.
Remark 2 For the computation of non-extremal eigenmodes, one may proceed as in
[11] and replace K(y)in (34) with (K(y)−λM), where λ∈Ris a suitably chosen
parameter. In this case we expect the algorithm to converge to an eigenpair for which
the eigenvalue is close to λ. Note, however, that now the existence of a unique solution
to (34) is not necessarily guaranteed by Lemma 1.
We try to write Algorithm 1in a computationally more convenient form. The pro-
jections in the algorithm can be computed explicitly using the notation introduced in
123
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Asymptotic convergence of spectral inverse iterations for… 589
Sect. 4. It is easy to verify that Eqs. (34)–(36) become
∞
m=0
β∈A
K(m)vβcmαβ =Mu(k−1)
α∀α∈A,(38)
β∈A
γ∈A
sβsγcαβγ =
β∈A
γ∈Avβ,vγRN
Mcαβγ ∀α∈A,(39)
β∈A
γ∈A
sβu(k)
γcαβγ =vα∀α∈A(40)
respectively. Given ˆs={sα}α∈A∈RPwe define matrices
Δ(ˆs):=
α∈A
G(α)sα,
K:=
M(A)
m=0
G(m)⊗K(m),
M:= IP⊗M,
S:=
M−1
K,
T(ˆs):= Δ( ˆs)⊗IN,
where M(A):= max{m∈N|∃α∈As.t. αm= 0}and IP∈RP×Pand IN∈RN×N
are identity matrices. We also define the nonlinear function F:RP×RPN →RP
via
Fα(ˆs,ˆ
v):= ˆs·G(α) ˆs−ˆ
v·(G(α) ⊗M)ˆ
v,α∈A
and let Fs:RP×RP→RPand Fv:RPN ×RPN →RPdenote the associated
bilinear forms given by Fs
α(ˆs,ˆ
t):= ˆs·G(α) ˆ
tand Fv
α(ˆ
v,ˆ
w):= ˆ
v·(G(α) ⊗M)ˆ
w.Now
Algorithm 1may be rewritten in the following form.
Algorithm 2 (Spectral inverse iteration in tensor form) Fix t o l >0and let ˆ
u(0)=
{u(0)
αi}α∈A,i∈J∈RPN be an initial guess for the eigenvector. For k =1,2,... do
(1) Solve ˆ
v={vαi}α∈A,i∈J∈RPN from the linear system
Kˆ
v=
Mˆ
u(k−1).(41)
(2) Solve ˆs={sα}α∈A∈RPfrom the nonlinear system
F(ˆs,ˆ
v)=0(42)
with the initial guess sα=||ˆ
v||RP⊗RN
Mδα0for α∈A.
123
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590 H. Hakula, M. Laaksonen
(3) Solve ˆ
u(k)={u(k)
αi}α∈A,i∈J∈RPN from the linear system
T(ˆs)ˆ
u(k)=ˆ
v.(43)
(4) Stop if ||ˆ
u(k)−ˆ
u(k−1)||RP⊗RN
M<tol and return ˆ
u(k)as the approximate eigen-
vector.
The approximate eigenvalue ˆμ(k)∈RPmay now be solved from the equation
Δ(ˆs)ˆμ(k)=ˆe1,(44)
where ˆe1={δα0}α∈A∈RP.
Remark 3 In [11] Newton’s method with the initial guess sα=||vα||RN
Mwas suggested
for the system of Eq. (42). Here the initial guess is somewhat different and corresponds
to s0=||v||L2
ν(Γ )⊗RN
M(and sα=0forα= 0).
In general it is not guaranteed that the Newton iteration for the system (42) converges
to a solution. The following proposition will give some insight to the conditions under
which this happens to be the case.
Proposition 3 Fix ˆ
v∈RPN and let ˆs(0)={s(0)
α}α∈A∈RPbe given by s(0)
α=
||ˆ
v||RP⊗RN
Mδα0for α∈A. Assume that there is a norm || · ||∗on RPand r >0such
that
||Fs(ˆs,ˆ
t)||∗≤CF||ˆs||∗||ˆ
t||∗
for all ˆs,ˆ
tinB(ˆs(0),r):= {ˆs∈RP|||ˆs−ˆs(0)||∗≤r}.If
||F(ˆs(0),ˆ
v)||∗<C−1
F||ˆ
v||2
RP⊗RN
M
then the Newton method for F (·,ˆ
v)=0with the initial guess ˆs(0)converges to a
unique solution in B(ˆs(0),r).
Proof This is a direct application of the Newton-Kantorovich theorem for the equation
F(·,ˆ
v)=0, see [13] (Theorem 6, 1.XVIII). Note that the first derivative (Jacobian) of
F(·,ˆ
v)at ˆs(0)is 2||ˆ
v||RP⊗RN
MIPand the second derivative is represented by the tensor
of coefficients 2cαβγ .
From Proposition 3we see that convergence of the Newton iteration is a conse-
quence of the boundedness of the function Fs, which again is ultimately determined
by the structure of the multi-index set A.
123
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Asymptotic convergence of spectral inverse iterations for… 591
5.2 Analysis of convergence
Due to a lack of general mathematical theory for multi-parametric eigenvalue problems
we rely on a slightly unconventional approach in analyzing our algorithm. First of all,
we restrict ourselves to asymptotic analysis since the underlying problem is nonlinear
and thus hard to analyze globally. Second, we will analyze the solutions pointwise
in the parameter space and deduce convergence theorems from classical eigenvalue
perturbation bounds.
5.2.1 Characterization of the dominant fixed point
The classical inverse iteration converges to the dominant eigenpair of the inverse
matrix. In a somewhat similar fashion the spectral inverse iteration tends to converge
to a certain fixed point, which we shall refer to as the dominant fixed point. Here we
will establish a connection between this dominant fixed point of the spectral inverse
iteration and the dominant eigenpair of the inverse of the parametric matrix under
consideration. This connection is obtained by considering the fixed point as a pointwise
perturbation of the eigenvalue problem of the parametric matrix.
If uA∈WN
Ais a fixed point of the Algorithm 1, then there exists a pair (s,v)∈
WA×WN
Asuch that uA=M−1PA(Kv)and
⎧
⎨
⎩
PA(sP
A(Kv))=Mv
PAs2−||v||2
RN
M=0.(45)
We call uAthe dominant fixed point if, whenever (˜s,˜
v)= (s,v)also solves the system
(45), then s(y)>˜s(y)for all y∈Γ. For any fixed y∈Γwe may write (45)as
⎧
⎨
⎩
s(y)K(y)v(y)=Mv(y)+s(y)RA(Kv)(y)+RA(sP
A(Kv))(y)
s2(y)=||v(y)||2
RN
M+RAs2−||v||2
RN
M(y). (46)
The following Lemma will be helpful in establishing a connection between the eigen-
pair of interest and the system (46).
Lemma 2 Denote by ||·|| the standard Euclidean norm on RN. Assume that S ∈RN×N
can be diagonalized as
S(xX)=(xX)λ10
0Λ,(47)
where λ1∈R,Λ=diag(λ2,...,λ
N)is real, and (xX)is orthogonal. Assume also
that λ1>λ
2≥... ≥λNand denote ˆ
λ:= λ1−λ2. Let ρ∈Rand r ∈RNbe such
that |ρ|≤1/2and ||r|| ≤ ˆ
λ/8. Then there exist κ≥1/2and π∈RN−1such that
(i) The pair (s,w)given by s =λ1−κ−1xTr and w=κx+Xπsolves the system
Sw=sw+r
||w||2=1+ρ. (48)
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592 H. Hakula, M. Laaksonen
(ii) If (˜s,˜w) = (s,w) also solves the system (48), then s >˜sorx
T˜w<0.
(iii) There exists C >0such that |κ−1|≤C(|ρ|+ˆ
λ−2||r||2)and ||π|| ≤ Cˆ
λ−1||r||.
Proof (i) Let s(κ) =λ1−κ−1xTr. For any κ≥1/2wehave|κ−1xTr|≤ˆ
λ/4so
that
min
2≤i≤N|λi−s(κ)|= min
2≤i≤N|λ1−λi−κ−1xTr|≥ˆ
λ−1
4ˆ
λ> 1
2ˆ
λ(49)
and
||(Λ −s(κ)I)−1|| ≤ 2ˆ
λ−1.(50)
The function
f(κ) =κ2+||(Λ −s(κ)I)−1XTr||2−1−ρ(51)
is strictly increasing for κ≥1/2 since
κ2f(κ) =2κ3+2xTr||(Λ −s(κ) I)−3
2XTr||2
≥2(κ3−(2ˆ
λ−1)3||r||3)
>2(κ3−2−3)≥0.(52)
One may also verify that f(1/2)<0 and f(2)>0. Thus, we may choose κ>1/2
such that f(κ ) =0. For w=κx+Xπwe obtain
Sw−sw=κSx +SXπ−κsx −sXπ=κ(λ1−s)x+X(Λ −sI)π (53)
so the equation Sw=sw+ris equivalent to
xT(Sw−sw−r)=κ(λ1−s)−xTr=0
XT(Sw−sw−r)=(Λ −sI)π −XTr=0.(54)
Choosing s=s(κ) and π=(Λ −sI)−1XTrwe see that both equations are
satisfied. Moreover
||w||2=κ2+||π||2=f(κ) +1+ρ=1+ρ. (55)
(ii) Suppose (˜s,˜w) also solves the system (48) and write ˜w=˜κx+X˜πfor some
˜κ∈Rand ˜π∈RN−1. In the nontrivial case we have ˜κ=xT˜w>0. Assume first
that 0 ≤˜κ≤1/2. We have
˜s=˜wTS˜w−˜wTr
|| ˜w||2=λ1˜κ2+˜πTΛ˜π−˜wTr
|| ˜w||2≤λ1˜κ2+λ2|| ˜π||2+||˜w||||r||
|| ˜w||2
=λ2+˜κ2
1+ρˆ
λ+||r||
(1+ρ)1
2
.(56)
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Asymptotic convergence of spectral inverse iterations for… 593
Since s≥λ1−κ−1||r||, we deduce that
s−˜s≥ˆ
λ−κ−1||r||− ˜κ2
1+ρˆ
λ−||r||
(1+ρ)1
2
>1−1
4−1
2−√2
8ˆ
λ>0.(57)
Now let ˜κ≥1/2. If (˜s,˜w) is to solve (48) then, as in part (i), we should have
˜κ(λ1−˜s)−xTr=0
(Λ −˜sI)˜π−XTr=0.(58)
From the first equation we obtain ˜s=λ1−˜κ−1xTr.Dueto|˜κ−1xTr|≤ˆ
λ/4the
matrix (Λ −˜sI)is invertible so the second equation gives ˜π=(Λ −˜sI)−1XTr.
Here ˜κ≥1/2 must be chosen so that f(˜κ) =0 and therefore (˜s,˜w) =(s,w).
(iii) From f(κ) =0 and κ≥1/2 we deduce that
|κ−1|≤(κ +1)−1(|ρ|+||(Λ −s(κ)I)−1XTr||2)≤|ρ|+4ˆ
λ−2||r||2(59)
and
||π|| = ||(Λ −s(κ)I)−1XTr|| ≤ 2ˆ
λ−1||r||.(60)
Thus, the claim follows.
Applying Lemma 2to the system (46) pointwise for y∈Γwe obtain the following
result.
Proposition 4 Let uA∈WN
Abe the dominant fixed point of Algorithm 1and denote
by (s,v)the associated pair in WA×WN
Athat solves (45). Let μA∈WAbe such that
PA(sμA)=1.For y ∈Γdenote by ˆ
λ(y)the gap between the two largest eigenvalues
of K−1(y)M. Assume that inf y∈Γs(y)>0and inf y∈Γˆ
λ(y)>0.For y ∈Γdefine
r(y):= K−1(y)RA(Kv)(y)+s−1(y)K−1(y)RA(sP
A(Kv))(y)
and
ρ(y):= s−2(y)RAs2−||v||2
RN
M(y).
If
r∗:= sup
y∈Γˆ
λ−1(y)||r(y)||RN
M<1
8
and
ρ∗:= sup
y∈Γ|ρ(y)|<1
2
123
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594 H. Hakula, M. Laaksonen
then there exists C >0such that
|μA(y)−μh(y)|≤Cmax{μ2
h(y), s−2(y)}||r(y)||RN
M+s−1(y)|RA(sμA)(y)|
(61)
and
||uA(y)−uh(y)||RN
M
≤C|ρ(y)|+ˆ
λ−1(y)||r(y)||RN
M+s−1(y)||M−1RA(sP
A(Kv))(y)||RN
M,(62)
where μh:Γ→Ris the smallest eigenvalue of M−1K(y)and uh:Γ→RNis the
corresponding eigenvector normalized in || · ||RN
M(and with appropriate sign).
Proof It is easy to see that the system (46) is equivalent to
M1
2K−1(y)M1
2w(y)=s(y)w(y)−M1
2r(y)
||w(y)||2
RN=1−ρ(y), (63)
where w(y)=s−1(y)M1
2v(y). By Lemma 2the solution with the pointwise largest
s(y)can be written as
s(y)=λ1(y)+κ−1(y)xT(y)M1
2r(y)
w(y)=κ(y)x(y)+X(y)π(y), (64)
where κ:Γ→[1/2,∞)and π:Γ→RN−1are such that
|κ(y)−1|2+||π(y)||2
RN≤C|ρ(y)|+ˆ
λ−2(y)||r(y)||2
RN
M2+Cˆ
λ−2(y)||r(y)||2
RN
M
,
λ1(y)=μ−1
h(y)is the pointwise largest eigenvalue of S(y)and x(y)=M1
2uh(y)
is the corresponding eigenvector. The matrix (x(y)X(y)) is orthonormal for every
y∈Γ. A Taylor expansion of s−1(y)yields
|s−1(y)−μh(y)|=C(μ−1
h(y)+ξ(y))−2||r(y)||RN
M
≤Cmax{μ2
h(y), s−2(y)}||r(y)||RN
M,(65)
where ξ(y)is such that 0 ≤ξ(y)≤κ−1(y)xT(y)M1
2r(y). Combining this with the
equation
μA(y)=s−1(y)+s−1(y)RA(sμA)(y)(66)
obtained from the condition PA(sμA)=1, we have altogether that
|μA(y)−μh(y)|≤Cmax{μ2
h(y), s−2(y)}||r(y)||RN
M+s−1(y)|RA(sμA)(y)|.(67)
123
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Asymptotic convergence of spectral inverse iterations for… 595
Furthermore,
M1
2uA(y)=M−1
2PA(Kv)(y)=s−1(y)M−1
2RA(sP
A(Kv))(y)+w(y)(68)
from which it follows that
||uA(y)−uh(y)||2
RN
M=||M1
2uA(y)−w(y)||2
RN+||w(y)−M1
2uh(y)||2
RN
=||s−1(y)M−1
2RA(sP
A(Kv))(y)||2
RN+||(κ(y)−1)x(y)+X(y)π(y)||2
RN
=s−1(y)||M−1RA(sP
A(Kv))(y)||2
RN
M+|κ(y)−1|2+||π(y)||2
RN
≤C|ρ(y)|+ˆ
λ−1(y)||r(y)||RN
M+s−1(y)||M−1RA(sP
A(Kv))(y)||RN
M2
.
(69)
This concludes the proof.
Remark 4 Note that we have not proven the existence of a dominant fixed point of
the Algorithm 1. The residuals rand ρin Proposition 4depend on the pair (s,v)∈
WA×WN
Aand hence Lemma 2by itself is not sufficient to guarantee the existence
of a dominant fixed point.
5.2.2 Convergence of the dominant fixed point to a parametric eigenpair
The next step in our analysis is to bound the error between the dominant fixed point
of Algorithm 1and the dominant eigenpair of the inverse of the parametric matrix. To
this end we will use the pointwise estimate obtained previously.
From Proposition 4we may easily deduce the following result.
Theorem 2 Let uA∈WN
Abe the dominant fixed point of Algorithm 1and denote by
(s,v)the associated pair in WA×WN
Athat solves (45). Let μA∈WAbe such that
PA(sμA)=1.For y ∈Γdenote by ˆ
λ(y)the gap between the two largest eigenvalues
of K−1(y)M. Assume that s∗:= inf y∈Γs(y)>0,ˆ
λ∗:= inf y∈Γˆ
λ(y)>0and that
the quantity
||RA(Kv)||L∞(Γ )⊗RN
M+||RA(sP
A(Kv))||L∞(Γ )⊗RN
M+RAs2−||v||2
RN
ML∞(Γ )
is small enough. Then there exists C >0such that
||μA−μh||L2
ν(Γ ) ≤C||RA(Kv)||L2
ν(Γ )⊗RN
M
+||RA(sP
A(Kv))||L2
ν(Γ )⊗RN
M+||RA(sμA)||L2
ν(Γ )(70)
123
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596 H. Hakula, M. Laaksonen
and
||uA−uh||L2
ν(Γ )⊗RN
M≤C||RA(Kv)||L2
ν(Γ )⊗RN
M+||RA(sP
A(Kv))||L2
ν(Γ )⊗RN
M
+RAs2−||v||2
RN
ML2
ν(Γ ),(71)
where μh:Γ→Ris the smallest eigenvalue of M−1K(y)and uh:Γ→RNis the
corresponding eigenvector normalized in || · ||RN
M(and with appropriate sign). Here
C depends only on s∗,ˆ
λ∗,μ∗
h:= supy∈Γμh(y),K
∗=supy∈Γ||K−1(y)||RN
M, and
M∗=||M−1||RN
M.
Proof With rdefined as in Proposition 4we have
||r||L2
ν(Γ )⊗RN
M≤K∗||RA(Kv)||L2
ν(Γ )⊗RN
M+s−1
∗||RA(sP
A(Kv))||L2
ν(Γ )⊗RN
M,
(72)
and
||M−1RA(sP
A(Kv))(y)||L2
ν(Γ )⊗RN
M≤M∗||RA(sP
A(Kv))(y)||L2
ν(Γ )⊗RN
M.(73)
The bounds (70) and (71) now follow from Proposition 4.
By Proposition 1the exact eigenvalue and eigenvector of problem (9) are analytic
functions of the parameter vector y∈Γ. This suggests that the residuals on the right
hand side of Eqs. (70) and (71) can be asymptotically estimated from Proposition 2.
5.2.3 Convergence of the spectral inverse iteration to the dominant fixed point
The classical inverse iteration converges to the dominant eigenpair of the inverse
matrix at a speed characterized by the gap between the two largest eigenvalues. Here
we will establish a similar asymptotic result for the convergence of the spectral inverse
iteration towards the dominant fixed point.
Fixed points of the spectral inverse iteration may be characterized using the tensor
notation of Algorithm 2.Let ˆ
uA∈RPN be a fixed point of the algorithm, i.e., ˆ
uA=Sˆ
v
and (ˆs,ˆ
v)∈RP×RPN are such that
ˆ
v=T(ˆs)Sˆ
v
F(ˆs,ˆ
v)=0.(74)
Define a linear operator R(ˆs,ˆ
v):RPN →RPN by
R(ˆs,ˆ
v)ˆ
w:= ˆ
w−TΔ−1(ˆs)Fv(ˆ
v,ˆ
w)T−1(ˆs)ˆ
v.
The convergence of the spectral inverse iteration to the fixed point ˆ
uAcan now be
related to the ratio of the norms of Δ−1(ˆs)and R(ˆs,ˆ
v)S−1.
123
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Asymptotic convergence of spectral inverse iterations for… 597
Theorem 3 Let ˆ
uA∈RPN be a fixed point of the Algorithm 2and (ˆs,ˆ
v)∈RP×
RPN a corresponding solution to (74). Assume that Δ(ˆs)is invertible. Let ˆμA=
Δ−1(ˆs)ˆe1, where ˆe1={δα0}α∈A∈RP. Set φmin := ||Δ−1(ˆs)||−1
RPand ψmax :=
||R(ˆs,ˆ
v)S−1||RP⊗RN
M. Then for any ε>0the iterates of Algorithm 2satisfy
||ˆ
u(k)−ˆ
uA||RP⊗RN
M≤ψmax
φmin +ε||ˆ
u(k−1)−ˆ
uA||RP⊗RN
M,k∈N(75)
whenever ˆ
u(k)is sufficiently close to ˆ
uA. Furthermore, there exists C >0such that
||ˆμ(k)−ˆμA||RP≤C||ˆ
u(k)−ˆ
uA||RP⊗RN
M,k∈N.(76)
Proof The partial derivative (Jacobian) of the function F(ˆs+ˆ
t,ˆ
v+ˆ
w)with respect
to ˆ
tat ˆ
t=0 is given by 2Δ(ˆs). The implicit function theorem now guarantees that
there is a unique differentiable function ˆ
t(ˆ
w)defined in a neighbourhood of ˆ
w=0
such that F(ˆs+ˆ
t(ˆ
w), ˆ
v+ˆ
w)=0. Computing the first order approximation of this
function we see that for ˆ
wsmall enough
ˆ
t(ˆ
w)=Δ−1(ˆs)Fv(ˆ
v,ˆ
w)+h.o.t. in ˆ
w,(77)
where h.o.t. stands for higher order terms. From (77) we obtain
T−1ˆs+ˆ
t(w)=T(ˆs)+TΔ−1(ˆs)Fv(ˆ
v,ˆ
w)+h.o.t. in ˆ
w−1
=T−1(ˆs)−T−1(ˆs)TΔ−1(ˆs)Fv(ˆ
v,ˆ
w)T−1(ˆs)+h.o.t. in ˆ
w.
(78)
Set ˆ
v(k)=S−1ˆ
u(k)and ˆ
w(k)=ˆ
v(k)−ˆ
v.Now
Sˆ
w(k)=T−1ˆs+ˆ
t(w(k−1))ˆ
v+ˆ
w(k−1)−Sˆ
v
=T−1(ˆs)ˆ
v+ˆ
w(k−1)−T−1(ˆs)TΔ−1(ˆs)Fv(ˆ
v,ˆ
w(k−1))T−1(ˆs)ˆ
v−Sˆ
v
+h.o.t. in ˆ
w(k−1)
=T−1(ˆs)ˆ
w(k−1)−TΔ−1(ˆs)Fv(ˆ
v,ˆ
w(k−1))T−1(ˆs)ˆ
v+h.o.t. in ˆ
w(k−1)
=T−1(ˆs)R(ˆs,ˆ
v)ˆ
w(k−1)+h.o.t. in ˆ
w(k−1).(79)
Since Sˆ
w(k)=ˆ
u(k)−ˆ
uAwe have that
ˆ
u(k)−ˆ
uA=T−1(ˆs)R(ˆs,ˆ
v)S−1ˆ
u(k−1)−ˆ
uA+h.o.t. in ˆ
u(k−1)−ˆ
uA.(80)
Equations (75) and (76) now follow from (80) and the fact that ˆμ(k)is asymptotically
given as a linear function of ˆ
u(k).
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598 H. Hakula, M. Laaksonen
Adapting Theorem 3to the context of Algorithm 1we obtain the following Corol-
lary.
Corollary 1 Let uA∈WN
Abe a fixed point of the Algorithm 1and (s,v)∈WA×WN
A
a corresponding solution to (45). Let μA∈WAbe such that PA(sμA)=1. Assume
that s∗:= inf y∈Γs(y)>0and let ψmax be as in Theorem 3. Then for any ε>0the
iterates of Algorithm 1satisfy
||u(k)−uA||L2
ν(Γ )⊗RN
M≤ψmax
s∗+ε||u(k−1)−uA||L2
ν(Γ )⊗RN
M,k∈N
whenever u(k)is sufficiently close to uA. Furthermore, there exists C >0such that
||μ(k)−μA||L2
ν(Γ ) ≤C||u(k)−uA||L2
ν(Γ )⊗RN
M,k∈N.(81)
Proof Interpret Theorem 3in the context of Algorithm 1. The bound φmin ≥s∗is a
consequence of Lemma 1.
Obviously the previous Corollary has practical value only if ψmax <s∗.Herewe
will briefly discuss the value of ψmax in the case that (ˆs,ˆ
v)∈RP×RPN is associated
to the dominant fixed point of Algorithm 2. Observe that the equation ˆ
z=R(ˆs,ˆ
v)ˆ
w
is equivalent to the system
z(y)=w(y)−PA(tuA)(y)
PA(st)(y)=PAv,wRN
M(y)(82)
for all y∈Γ. We see that, if w=vthen z=0, whereas, if w(y), v(y)RN
M=0for
all y∈Γthen z=w. Thus, the matrix R(ˆs,ˆ
v)acts as a deflation that shrinks vectors
that are close to v(y)and preserves vectors that are almost orthogonal to v(y).From
Proposition 4we know that s−1(y)is an approximation of the smallest eigenvalue
of M−1K(y)and v(y)is an approximation of the corresponding eigenvector. By
Lemma 1the operator norm of S−1is bounded by supy∈Γλ−1
1(y), where λ1(y)is the
smallest eigenvalue of M−1K(y). Analogously, since the eigenvector corresponding
to this smallest eigenvalue is deflated by R(ˆs,ˆ
v), we expect the norm of R(ˆs,ˆ
v)S−1
to be bounded by a value close to supy∈Γλ−1
2(y), where λ2(y)is the second smallest
eigenvalue of M−1K(y). With this reasoning, if the deflation is sufficient, there is
ψ∗
max ∈Rsuch that
ψmax
s∗≤ψ∗
max
s∗≈λ1/2:= supy∈Γλ−1
2(y)
inf y∈Γλ−1
1(y)=supy∈Γλ1(y)
inf y∈Γλ2(y).(83)
One might suspect that the speed of convergence of the spectral inverse iteration is
characterized by the largest value of the ratio λ1(y)/λ2(y). The bound obtained from
(83) is slightly more pessimistic, though not necessarily optimal.
123
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Asymptotic convergence of spectral inverse iterations for… 599
5.2.4 Combined error analysis
Let (μ, u)∈L2
ν(Γ ) ×L2
ν(Γ ) ⊗H1
0(D)be the smallest eigenvalue and the associated
eigenfunction of the continuous problem (9). Let (μh,uh)∈L2
ν(Γ ) ×L2
ν(Γ ) ⊗Vh
be the corresponding eigenpair of the semi-discrete problem (13). Assume that there
exists a dominant fixed point uA∈WN
Aof Algorithm 1and an associated eigenvalue
approximation μh,A:= μA∈WAas in Proposition 4. Denote by u(k)∈WN
Athe k:th
iterate of Algorithm 1and by μh,A,k:= μ(k)∈WAthe associated solution to (37).
Let uh,Aand uh,A,kdenote the functions in WA⊗Vh, whose coordinates are defined
by the vectors uAand u(k)respectively. The spatial, stochastic, and iteration errors
may now be separated in the following sense:
||μ−μh,A,k||L2
ν(Γ ) ≤||μ−μh||L2
ν(Γ )
+||μh−μh,A||L2
ν(Γ ) +||μh,A−μh,A,k||L2
ν(Γ ) (84)
and
||u−uh,A,k||L2
ν(Γ )⊗L2(D)≤||u−uh||L2
ν(Γ )⊗L2(D)+||uh−uh,A||L2
ν(Γ )⊗L2(D)
+||uh,A−uh,A,k||L2
ν(Γ )⊗L2(D).(85)
Under sufficient conditions we may now bound each term in the Eqs. (84) and (85)
separately using the theory developed earlier in this section. The first term may be
approximated using Theorem 1, the second term may be approximated using Theo-
rem 2and Proposition 2, and the third term may be approximated using Corollary 1of
Theorem 3and the hypothesis (83). We therefore expect that, with an optimal choice
the multi-index sets Afor >0, the output of the spectral inverse iteration converges
to the exact solution according to
||u−uh,A,k||L2
ν(Γ )⊗L2(D)h1+l+(#A)−r+λk
1/2(86)
and similarly
||μ−μh,A,k||L2
ν(Γ ) h2l+(#A)−r+λk
1/2(87)
for certain rates r>0 and l>0.
5.3 Numerical examples
We present numerical evidence to verify the Eqs. (86) and (87). In each of the following
examples we compute the smallest eigenvalue and the corresponding eigenfunction
of the model problem (4) in the unit square D=[0,1]2using the Algorithm 1.We
use the smallest eigenvector at y=0 as an initial guess. For the diffusion coefficient
we assume the form (5) with a0:= 1 and
am(x):= (m+1)−ςsin(mπx1), m=1,3,...
(m+1)−ςsin(mπx2), m=2,4,... x=(x1,x2)∈D,
123
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600 H. Hakula, M. Laaksonen
(a)(b)
Fig. 1 The mean and variance of the eigenfunction as computed by Algorithm 1
where we set ς=3.2. Now ||am||L∞(D)≤Cm−ςand ||am||W2,∞(D)≤Cm−ς+2so
that the assumptions (6)–(8)fors=2 are satisfied with p0>ς
−1and p2>(ς−2)−1.
We therefore expect the regions of analyticity in Proposition 1to increase according
to τm≥Cmς−1.
The deterministic mesh is a uniform grid of second order quadrilateral elements
in all computations. The discretization in the parameter space is obtained by setting
τm:= (m+1)ς−1for m=1,2,... and using the multi-index sets Aas defined in
Proposition 2. Multi-index sets of this form have been introduced in [7] and in [5]an
algorithm for generating them has been suggested.
We use a matrix free formulation of the conjugate gradient method for solving the
linear systems (41) and (43). The preconditioner is constructed using the mean of the
parametric matrix in question [17] and as an initial guess we set the solution of the
system from the previous iteration. We wish to note that in this setting only a very
few iterations of the conjugate gradient method are needed at each step of the spectral
inverse iteration.
In the lack of an exact solution we compute an overkill solution (μ∗,u∗)for which
the number of deterministic degrees of freedom is N=36741, the parameter is
chosen such that #A=264, and the number of iterations is k=16. This results
in roughly 107total degrees of freedom. The number of active dimensions in the
overkill solution is M(A)=113. All the numerical examples in this section have
been computed using this overkill solution as a reference. The expected value and
variance of the eigenfunction are presented in Fig. 1.
5.3.1 Convergence in space
Keeping the number of stochastic degrees of freedom #A=264 and the number of
iterations k=16 fixed, we may investigate the convergence of the solution (μ∗,h,u∗,h)
as a function of the spatial discretization parameter h. This convergence for piecewise
quadratic basis functions is illustrated in Fig. 2. We observe algebraic convergence
rates of order 3 and 4 for the eigenfunction and eigenvalue respectively, exactly as
123
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Asymptotic convergence of spectral inverse iterations for… 601
(a)(b)
Fig. 2 Convergence of the spatial errors for the eigenfunction and eigenvalue as computed by Algorithm 1.
The points represent a log–log plot of the errors as a function of h. The dashes lines represent the rates h3
and h4respectively
predicted by Theorem 1. Thus, the error behaves like N−3/2and N−2with respect to
the number of deterministic degrees of freedom.
5.3.2 Convergence in the parameter domain
Keeping the number of spatial degrees of freedom N=36,741 and the number of iter-
ations k=16 fixed, we may investigate the convergence of the solution (μ∗,A,u∗,A)
as a function of #Aas →0. This convergence is illustrated in Fig. 3. We observe
approximate algebraic convergence rates of order −r=−1.9 with respect to the
number of stochastic degrees of freedom #A.
In Fig. 4we have presented the norms of the Legendre coefficients of the overkill
solution. The ordering of the coefficients is the same as the order in which they would
appear in the multi-index set #Aas →0. We see that the norms converge at the
rate −r−1/2=−2.4 exactly as we would expect from the proof of Proposition 2.
In Fig. 5we have presented the norms of the same Legendre coefficients sorted by
decreasing magnitude. From this Figure we estimate that, with an optimal selection
of the multi-index sets we could in fact observe a rate of convergence −r=−2.3for
(a)(b)
Fig. 3 Convergence of the stochastic errors for the eigenfunction and eigenvalue as computed by Algo-
rithm 1. The points represent a log–log plot of the errors as a function of #A. The dashed lines represent
the rate (#A)−1.9
123
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602 H. Hakula, M. Laaksonen
(a)(b)
Fig. 4 A log–log plot of the norms of the Legendre coefficients of the overkill solution. The dashed lines
represent the algebraic rate −2.4
(a)(b)
Fig. 5 A log–log plot of the norms of the Legendre coefficients of the overkill solution sorted by decrasing
magnitude. The dashed lines represent the algebraic rate −2.8
the error of the solution. This ideal rate of convergence is somewhat faster than the
asymptotic theoretical bound of −r=−ς+3/2=−1.7 predicted by Proposition 2.
Interestingly we observe two well separated clusters of values in Fig. 4b. It seems
that many of the multi-indices that correspond to relatively large Legendre coefficients
of the eigenfunction, account only for a marginal contribution to the eigenvalue.
5.3.3 Convergence of the iteration error
Keeping the number of spatial basis functions N=36,741 and the parameter fixed
so that #A=264, we may investigate the convergence of the solution (μ∗,k,u∗,k)
as a function of the number of iterations k. This convergence is illustrated in Fig. 6.
Assuming that the variation in the eigenvalues within the parameter space is small,