efﬁcient spectral stochastic finite element …...east asian journal on applied mathematics vol....

East Asian Journal on Applied Mathematics Vol. 9, No. 3, pp. 601-621

doi: 10.4208/eajam.140119.160219 August 2019

Efficient Spectral Stochastic Finite Element

Methods for Helmholtz Equations with Random

Inputs

Guanjie Wang and Qifeng Liao∗

School of Information Science and Technology, ShanghaiTech University,

Shanghai, China.

Received 14 January 2019; Accepted (in revised version) 16 February 2019.

Abstract. The implementation of spectral stochastic Galerkin finite element approxi-

mation methods for Helmholtz equations with random inputs is considered. The cor-

responding linear systems have matrices represented as Kronecker products. The spar-

sity of such matrices is analysed and a mean-based preconditioner is constructed. Nu-

merical examples show the efficiency of the mean-based preconditioners for stochastic

Helmholtz problems, which are not too close to a resonant frequency.

AMS subject classifications: 65C30, 65F08, 65N30, 35J05

Key words: Helmholtz equations, PDEs with random data, generalised polynomial chaos, stochastic

finite elements, iterative solvers.

1. Introduction

During last decades there has been a rapid development in uncertainty quantification

for solving partial differential equations (PDEs) with random inputs. These random inputs

usually arise from lack of knowledge about the system or/and the measurements of realis-

tic model parameters such as the permeability coefficients in diffusion problems [29, 55],

the viscosity parameters of incompressible flows [12,42,47,49], and shape parameters in

acoustic scattering [58]. In particular, stochastic Helmholtz equations attracted a lot of

interest recently and this paper aims at the development of efficient strategies for their

solution.

The Helmholtz equation plays a fundamental role in ocean acoustics, optic and elec-

tromagnetic problems [28,33,37,51]. In acoustics wave problems, the uncertainties come

from the refractive indices (or wave number parameters), source functions, and the shapes

of scattering surfaces. Elman et al. [14] considered Helmholtz equations with random forc-

ing functions and boundary conditions and developed efficient multigrid solvers for the

∗Corresponding author. Email addresses: wanggj� shanghaite h.edu. n (G. Wang), liaoqf�

shanghaite h.edu. n (Q. Liao)

http://www.global-sci.org/eajam 601 c©2019 Global-Science Press

602 G. Wang and Q. Liao

corresponding stochastic finite element approximations. Xiu and Shen [58] developed gen-

eralised polynomial chaos (gPC) approximations [55, 56] (for polynomial chaos see [30])

based on stochastic collocation methods [2,54] to solve the problems with uncertain scat-

tering surface shapes. Tang and Zhou [60] investigated a stochastic collocation method

for scalar hyperbolic equations with a random wave speed and showed that the rate of

convergence depends on the regularity of solutions. Later on, multifidelity approaches to

stochastic optimisation problems, including stochastic wave numbers and impedance pa-

rameters have been studied [39, 40]. Papers [22, 23] deal with a Monte Carlo interior

penalty discontinuous method. Fang et al. [21] developed a stochastic Galerkin method for

Maxwell’s equations with random input.

Here we study spectral stochastic Galerkin finite element methods [3,30,52] for stochas-

tic Helmholtz equations with uncertainties in refractive indices. The stochastic parameter

space is discretised by gPC methods of [55, 56] and the physical space by finite element

methods of [6,15]. This leads to linear systems in Kronecker formulation [10,41,43]. We

note that for stochastic Galerkin linear systems, various efficient iterative solvers such as

mean-based preconditioning methods [41, 43], hierarchical preconditioners [47, 48], pre-

conditioned low-rank projection methods [36] are vigorously studied. Nevertheless, to the

best of our knowledge, in the case of stochastic Helmholtz problems these methods have

not been analysed. Here we investigate the sparsity of the stochastic Galerkin linear sys-

tems associated with stochastic Helmholtz problems and the corresponding mean-based

preconditioning scheme.

The outline of this work is as follows. In Section 2, we describe the problem, intro-

duce spectral stochastic Galerkin finite element approximations, discuss the sparsity of the

underling linear systems and present the linear systems associated with uniform random

inputs. In Section 3, iterative methods and mean-based preconditioning are discussed.

Section 4 contains numerical results and our conclusions are in Section 5.

2. A Stochastic Helmholtz Equation and its Discretisation

Let D ⊂ Rd , d = 2,3 denote a physical domain and x ∈ Rd the physical variable.

We assume that D is bounded, connected and has a polygonal boundary ∂ D. Moreover, let

ξ = [ξ1, · · · ,ξN ]T be the vector of real-valued random variables. The image of ξ is denoted

by Γ and the probability density function of ξ by π(ξ). Here, we consider the following

stochastic Helmholtz problem: Find an unknown function u(x ,ξ) such that

−∇2u(x ,ξ)− κ2(x ,ξ)u(x ,ξ) = f (x ) ∀(x ,ξ) ∈ D× Γ , (2.1)

u(x ,ξ) = 0 ∀(x ,ξ) ∈ ∂ DD × Γ , (2.2)

∂ u

∂ n− iκ(x ,ξ)u = 0 ∀(x ,ξ) ∈ ∂ DR × Γ , (2.3)

where κ ∈ R is the refractive index, ∂ u/∂ n the outward normal derivative of u on the

boundary and i =p−1. Moreover, we assume that the Dirichlet ∂ DD and the radiation

Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 603

(Sommerfeld) boundary ∂ DR satisfy the conditions

∂ DD ∪ ∂ DR = ∂ D, ∂ DD ∩ ∂ DR = ;.The refractive index in (2.1) has the form

κ(x ,ξ) =

N∑

m=0

κm(x )ξm

with real-valued deterministic functions {κm(x )}Nm=0and ξ0 = 1.

To ensure the well-posedness of this problem, we assume that there is a constant ε > 0

such that κ(x ,ξ) > ε for all (x ,ξ) ∈ D×Γ and the eigenvalues associated with deterministic

versions of (2.1) are greater than ε in magnitude. Thus for each realisation of ξwe consider

the following deterministic Helmholtz eigenvalue problem — cf. [27,34,35]:

−∇2u(x ,ξ)− κ2(x ,ξ)u(x ,ξ) = λ(ξ)u(x ,ξ) (2.4)

with the boundary conditions (2.2)–(2.3). Let us assign all its eigenvalues — i.e. all λ(ξ)

in (2.4), to a set Λξ and assume that |λ|> ε for all λ ∈ ∪ξ∈ΓΛξ.

2.1. Variational formulation

In this section we introduce the variational form of (2.1)–(2.3). Consider the space

L2(D) :=

�

v : D→ C�

�

�

�

∫

D

v v dx <∞�

of complex-valued square integrable functions ν and equip it with the norm

‖v‖2 :=

�∫

D

v vdx

�1/2

. (2.5)

Moreover, let

H10(D) :=

�

v ∈ H1(D) | v = 0 on ∂ DD

,

where H1(D) is the complex-valued Sobolev space

H1(D) :=�

v ∈ L2(D) , ∂ v/∂ x i ∈ L2(D), i = 1, · · · , d

.

Let us also recall that the expectation, or mean value, of a function g(ξ) : Γ → C is defined

by

E�

g(ξ)�

:=

∫

Γ

π(ξ)g(ξ)dξ,

where π(ξ) is the probability density function.

The solution and test spaces coincide and are

W := H10(D)⊗ L2

π(Γ ) =�

w(x ,ξ) : D× Γ → C | ‖w(x ,ξ)‖W <∞ and w|∂ DD×Γ = 0

,


where

L2π(Γ ) := {g : Γ → C | E[g g] <∞} ,

‖w(x ,ξ)‖2W :=

∫

Γ

π(ξ)

∫

D

|∇w|2 dx dξ.

According to Refs. [22, 43, 53], the problem (2.1)–(2.3) can be reformulated in the

following variational form: Find u ∈W , such that

E

�∫

D

∇u · ∇w−∫

D

κ2uw− i

∫

∂ DR

κuw

�

= E

�∫

D

f w

�

∀w ∈W. (2.6)

2.2. Discretisation

To obtain a discrete version of (2.6), we introduce a finite-dimensional subspace W h

of W . In particular, starting with the finite-dimensional subspaces of respective stochastic

and physical spaces — viz.

S = span�

Φ j(ξ)Nξ

j=1⊂ L2

π(Γ ), V h = span {vs(x )}Nx

s=1⊂ H1

0(D), (2.7)

where Φ j(ξ) and vs(x ) are basis functions, we then define the finite-dimensional subspace

of the solution and test function space W by

W h := V h ⊗ S := span�

v(x )Φ(ξ)|v ∈ V h,Φ ∈ S

.

According to discretisation methods used , one can choose various bases in (2.7) — e.g.

piecewise linear functions [3,6,9,15] or global orthogonal polynomials [5,30,45,53]. The

global orthogonal polynomial approximation for a stochastic space includes polynomial

chaos methods [29, 30], generalised polynomial chaos methods [56] and dynamically bi-

orthogonal methods [7,8,38,61]. Here, we use generalised polynomial chaos methods to

discretise the stochastic parameter space and finite element methods for the physical space.

For completeness let us provide a brief review of the generalised polynomial chaos methods

from [55,57].

According to [55], one can consider the following gPC approximation

u(x ,ξ) ≈ up(x ,ξ) :=

Nξ∑

j=1

u j(x )Φ j(ξ), (2.8)

of the solution u(x ,ξ), where S = {Φ j(ξ)}Nξj=1is an orthogonal basis with respect to the

inner product

E�

Φ j(ξ)Φk(ξ)�

=

∫

Γ

π(ξ)Φ j(ξ)Φk(ξ)dξ.


If π(ξ) is the probability density function for a one-dimensional random input, then the

basis functions in (2.8) have the form Φ j(ξ) = φ j−1(ξ), j = 1, · · · , Nξ, where {φ j}Nξ−1

j=0is

a sequence of polynomials orthogonal with respect to the inner product

E�

φ j(ξ)φk(ξ)�

=

∫

Γ

π(ξ)φ j(ξ)φk(ξ) dξ .

It is well known that the sequence of orthogonal polynomials {φ j(ξ)}Nξ−1

j=0can be generated

by a three-term recurrence relation — viz.

φ0(ξ) = 1,

φ1(ξ) = ξ−α1,

· · · · · · · · · · · · · · · · · · · · ·φ j+1(ξ) = (ξ−α j+1)φ j(ξ)− β j+1φ j−1(ξ), j ≥ 1,

(2.9)

where

α j+1 =

∫

Γ

ξπφ2j dξ

�∫

Γ

πφ2j dξ , β j+1 =

∫

Γ

ξπφ jφ j−1 dξ

�∫

Γ

πφ2j−1 dξ .

For more details about the orthogonal polynomials the reader can consult Refs. [1,45].

If ξ1, · · · ,ξN , N > 1 are independent random variables of a multi-dimensional random

input, then each stochastic basis function Φ j(ξ), j ∈ {1, · · · , Nξ} is a product of N univariate

orthogonal polynomials — i.e. Φ j(ξ) := φ(1)

j1(ξ1) · · ·φ(N)jN

(ξN ), where {φ(i)k(ξi)}∞k=0

, i =

1, · · · , N is the univariate orthogonal basis corresponding to ξi probability density function

πi(ξi). Every single-index j ∈ {1, · · · , Nξ} here can be represented by a multi-index j =

( j1, · · · , jN ) with total degree | j | = j1 + · · ·+ jN of Φ j(ξ). In order to define the functions

Φ j(ξ), bijections from single indices to the multi-indices are introduced so that

Mb : j←→ ( j1, · · · , jN ). (2.10)

Every specified bijection Mb determines the term Φ j(ξ). In particular, the multi-index j

can be arranged in a graded lexicographic order [53]— i.e. if |i| > | j | or |i| = | j | and the

first nonzero entry in the difference i − j is positive, we setM−1b(i) >M−1

b( j).

According to [59], for a given integer (the gPC order) p > 0, the gPC approximation

(2.8) can be rewritten in the multi-index form as

up(x ,ξ) =

Nξ∑

j=1

u j(x )Φ j(ξ) =

p∑

| j |=0

u j (x )Φ j (ξ), | j |= j1 + · · ·+ jN ,

where Nξ =�

N+pp

�

— cf. [55].

With respect to spatial variable, the functions u j(x ) are approximated as

u j(x ) ≈Nx∑

s=1

u jsvs(x ), vs(x ) ∈ V h, (2.11)

where {vs(x )}Nx

s=1is a finite element basis of V h.


Using (2.8) and (2.11), we can approximate the solution u(x ,ξ) by the aggregates

uph(x ,ξ) :=

Nξ∑

j=1

Nx∑

s=1

u jsvs(x )Φ j(ξ) . (2.12)

The unknown coefficients u js, j = 1, · · · , Nξ, s = 1, · · · , Nx in (2.12) can be determined

from the system of linear equations

Au = b, (2.13)

where

A = G00 ⊗ K −N∑

l=0

N∑

m=0

Glm ⊗Mlm−N∑

l=0

iGl0 ⊗ Ll , (2.14)

b = h⊗ f , (2.15)

the symbol ⊗ refers to the Kronecker tensor product and

h( j) = E�

Φ j(ξ)�

, f (s) =

∫

D

f vs dx ,

Mlm(s, t) =

∫

D

κlκmvsvt dx , Ll(s, t) =

∫

∂ DR

κl vsvt ds,

Glm( j, k) = E�

ξlξmΦ j(ξ)Φk(ξ)�

, K (s, t) =

∫

D

∇vs · ∇vt dx

(2.16)

with l, m = 0,1, · · · , N ; j, k = 1, · · · , Nξ and s, t = 1, · · · , Nx .

According to [43], the matrix A and the vectors u, b in (2.13) can be written in the

block form

A =

A11 A12 · · · A1Nξ

A21 A22 · · · A2Nξ...

.... . .

...

ANξ1 ANξ2 · · · ANξNξ

,

u =

u1

u2...

uNξ

, b =

b1

b2...

bNξ

,

(2.17)

where each A jk for j, k = 1, · · · , Nξ is a Nx × Nx matrix.

As soon as the approximation uph(x ,ξ) in (2.12) is determined, we can also approxi-

mate the mean and variance functions of the solution — viz.

E�

u(x ,ξ)� ≈ E �uph(x ,ξ)

�

,

Var(u(x ,ξ)) ≈ Var�

uph(x ,ξ)�

:= E��

�uph(x ,ξ)−E �uph(x ,ξ)��

�2�

.


2.3. Sparsity of the coefficient matrix

The Eqs. (2.14), (2.17) show that the coefficient matrix A has a block-form. It turns out

that every block has the same sparsity pattern as the corresponding deterministic problem

— cf. [43]. The general sparsity and structural properties of the coefficient matrix are

studied in [19] and one can use these results to find the number of nonzero entries of Glm,

l = 1, · · · , N , m = 0 for stochastic Helmholtz problems in the case even weight functions.

In what follows, we study a more general case — viz. the sparsity of Glm, l, m = 0,1, · · · , N

and the coefficient matrix A.

To evaluate the number of nonzero blocks in A of (2.17), we define the following matrix

G( j, k) =

¨

1, if there exist l, m ∈ {0,1, · · · , N} such that Glm( j, k) 6= 0,

0, otherwise,

where j, k = 1, · · · , Nξ. It is clear that the number of nonzero blocks in A does not exceed

the number of nonzero entries in G. Therefore, the nonzero entries of G can be counted.

Without loss of generality, we assume that the univariate basis functions are orthonormal

— i.e.∫

πi(ξi)φ j(ξi)φk(ξi) = δ jk, i = 1, · · · , N ,

where δ jk is the Kronecker delta function.

If l = m= 0, we have

G00( j, k) =

N∏

i=1

δ ji ki, j, k = 1, · · · , Nξ,

where ( j1, · · · , jN ) and (k1, · · · , kN ) are the multi-indices corresponding to j and k, so that

G00( j, k) 6= 0 if | ji − ki| = 0 for i ∈ {1, · · · , N}.If l = 0 or m = 0, we consider

G0l( j, k) = Gl0( j, k) = E�

ξlφ(l)

jl(ξl)φ

(l)

kl(ξl)

�N∏

i=1,i 6=l

δ ji ki

with l > 0. Since φ(l)

jl(ξl) and the polynomials of degree smaller than jl are orthogonal,

then G0l( j, k) 6= 0 — i.e. G( j, k) = 1 only if | jl − kl | ≤ 1 and ji = ki , i ∈ {1, · · · , N} \ {l},where ( j1, · · · , jN ) and (k1, · · · , kN ) are the multi-indices corresponding to j and k.

If l = m> 0, then

Gl l( j, k) = E�

ξ2l φ(l)

jl(ξl)φ

(l)

kl(ξl)

�N∏

i=1,k 6={l}δ ji ki

, Gl l( j, k) 6= 0

— i.e. G( j, k) = 1, only if | jl − kl | ≤ 2 and ji = ki for i ∈ {1, · · · , N} \ {l}.


If l 6= m and lm 6= 0, then

Glm( j, k) = E�

ξlφ(l)

jl(ξl)φ

(l)

kl(ξl)

�

E�

ξmφ(m)

jm(ξm)φ

(m)

km(ξm)

�N∏

i=1,i 6={l ,m}δ ji ki

,

so that Glm( j, k) 6= 0 — i.e. G( j, k) = 1, only if | jl − kl | ≤ 1 and | jm − km| ≤ 1 and ji = ki

for i ∈ {1, · · · , N} \ {l, m}.Summarising, we note that G( j, k) 6= 0 if and only if one of the following conditions

holds.

(a) ji = ki for i ∈ {1, · · · , N};

(b) for each l ∈ {1, · · · , N}, | jl − kl |= 1,2, and ji = ki for i ∈ {1, · · · , N} \ {l};

(c) for each pair l, m ∈ {1, · · · , N} such that l 6= m, the relations | jl−kl | = 1, | jm−km| = 1

and ji = ki , i ∈ {1, · · · , N} \ {l, m} hold.

The number of indices satisfying condition (a) is equal to the number of solutions of

the following problem: Find non-negative integers j1, · · · , jN , such that

j1 + · · ·+ jN ≤ p.

The number of solutions of this equation can be computed by the stars and bars method [20]

and is equal to�

N+pp

�

.

The situation | jl − kl | = 1 and ji = ki (i ∈ {1, · · · , N} \ {l}) in (b) has been studied

in [19]. For the sake of simplicity, here we use a different counting method. Let us consider

the situation jl = kl +1 and ji = ki (i ∈ {1, · · · , N}\{l}) to demonstrate the method. Since

the total degree of each gPC basis function does not exceed p, the multi-index of j satisfies

the inequality

j1 + · · ·+ jl + · · ·+ jN ≤ p,

or, equivalently,

j1 + · · ·+ (kl + 1) + · · ·+ jN ≤ p,

where j1, · · · , kl , · · · , jN are non-negative integers. Thus the number of the index pairs j, k

with the property jl = kl + 1 and ji = ki (i ∈ {1, · · · , N} \ {l}) is the same as the number of

the solutions of the following problem: Find non-negative integers j1, · · · , kl , · · · , jN , such

that

j1 + · · ·+ kl + · · ·+ jN ≤ p− 1.

The stars and bars method shows that this number is�

N+p−1p−1

�

. Analogously to the previous

considerations we show that the total number of indices satisfying condition (b) is equal to

2N

�

N + p− 1

p− 1

�

+ 2N

�

N + p− 2

p− 2

�

.


In case (c), the method of counting is similar and we consider the situation jl = kl +1,

jm = km + 1 and ji = ki (i ∈ {1, · · · , N} \ {l, m}) as an example. Since the total degree of

each gPC basis function does not exceed p, the multi-index of j satisfies the inequality

j1 + · · ·+ jl + · · ·+ jN ≤ p,

or, equivalently,

j1 + · · ·+ (kl + 1) + · · ·+ (km + 1) + jN ≤ p,

where j1, · · · , kl , · · · , km, · · · , jN are non-negative integers. Thus the number of index-pairs

j, k such that jl = kl+1, jm = km+1 and ji = ki (i ∈ {1, · · · , N}\{l, m}) is equal to the num-

ber of solutions of the following problem: Find non-negative integers j1, · · · , kl , · · · , km, · · · ,jN , such that

j1 + · · ·+ kl + · · ·+ km + · · ·+ jN ≤ p− 2.

Using again stars and bars method, we obtain that it is�

N+p−2p−2

�

. Therefore, the total number

of indices, which satisfy condition (c) is

(N2 − N )

�

N + p− 2

p− 2

�

+ (N2 − N )

�

N + p− 1

p− 1

�

,

and if p = 1, we set�

N+p−2p−2

�

:= 0.

Altogether, the total number of nonzero entries in the matrices G — i.e. the number of

nonzero blocks in the coefficient matrix (2.17) does not exceed the number

(N2 + N )

��

N − 1+ p

p− 1

�

+

�

N − 1+ p− 1

p− 2

��

+

�

N + p

p

�

=

�

(N2 + N )N + 2p− 2

N + p− 1

p

N + p+ 1

��

N + p

p

�

¬Cξ

�

N + p

p

�

= CξNξ,

where

Cξ =

�

(N2 + N )N + 2p− 2

N + p− 1

p

N + p+ 1

�

< 2(N2 + N ) + 1.

It is clear that for large N and p, the number Cξ is usually much smaller than Nξ =�

N+pp

�

.

Fig. 1 shows the ratio Cξ/Nξ. It is fast decreasing if N and p increase.

It was shown in [43] that every nonzero block of A in (2.17) has the same sparsity

pattern as the corresponding deterministic problem. Discretising the physical space D by

standard finite element method, one notes that the number of nonzero entries in each block

can be written as the product Cx Nx , where Cx ≪ Nx and Cx are the degrees of freedom

are independent of the finite element method. In particular, for bilinear rectangular finite

elements one has Cx = 9 — cf. [15]. It follows that the number of nonzero entries in the

matrix A does not exceed the number Cx CξNx Nξ and can be written as O (Nx Nξ), since

Cx ≪ Nx and Cξ≪ Nξ. Thus the matrix A of the size (Nx Nξ × Nx Nξ) is sparse and for the

system (2.13) it is of great interest to develop an iterative linear solver with cost O (Nx Nξ)

[15,44]. This problem will be again discussed in Section 3.


Figure 1: The sparsity of blo ks.

2.4. Detailed discrete formulation for uniform inputs

As soon as the coefficients α j ,β j of the three-term recurrence relation (2.9) are known,

the matrices Glm and vectors h can be calculated analytically. Here, we present a formula-

tion for independent identically distributed uniform random inputs.

We recall that for uniform distributions ξ in [−1,1], the probability density function

is π(ξ) = 1/2. It is well known that the Legendre polynomials form an orthogonal basis

in [−1,1] with respect to the probability density function π(ξ) = 1/2. Normalising the

Legendre polynomials, we obtain the three-term recurrence relation for the orthonormal

polynomial bases — i.e.

φi+1(ξ) =

p

(2i + 1)(2i + 3)

i + 1ξφi(ξ)−

ip

2i + 3

(i + 1)p

2i − 1φi−1(ξ),

where φ0(ξ) = 1 and φ1(ξ) =p

3ξ.

By the definition of h and Glm, we write

h( j) = E�

Φ j(ξ)�

=

�N∏

i=1

E�

φ ji(ξi)

�

�

=

�

1, if ji = 0,

0, otherwise,

and

1. If l = m= 0, then G00 = I .

2. If lm = 0 and l +m> 0, then

G0l( j, k) = Gl0( j, k) = E[ξlΦ j(ξ)Φk(ξ)]

=

N∏

i=1,i 6=l

E�

φ ji(ξi)φki

(ξi)�

!

E�

ξlφ jl(ξl)φkl

(ξl)�


=

jlq

4 j2l− 1

∏N

i=1,i 6=l δ ji ki, if kl = jl − 1,

klq

4k2l− 1

∏N

i=1,i 6=lδ ji ki

, if jl = kl − 1,

0, otherwise.

3. If l = m> 0,

Gl l( j, k) = E�

ξ2l Φ j(ξ)Φk(ξ)

�

=

N∏

i=1,i 6=l

E�

φ ji(ξi)φki

(ξi)�

!

E�

ξ2lφ jl(ξl)φkl

(ξl)�

=

�

( jl + 1)2

(2 jl + 1)(2 jl + 3)+

j2l

4 j2l− 1

�

∏N

i=1,i 6=l δ ji ki, if jl = kl ,

�

1p

(2 jl + 1)(2 jl − 3)

jl( jl − 1)

2 jl − 1

�

∏N

i=1,i 6=l δ ji ki, if kl = jl − 2,

�

1p

(2kl + 1)(2kl − 3)

kl(kl − 1)

2kl − 1

�

∏N

i=1,i 6=l δ ji ki, if jl = kl − 2,

0, otherwise.

4. If l 6= m and lm 6= 0,

Glm( j, k) = Gml( j, k) = E[ξlξmΦ j(ξ)Φk(ξ)]

=

N∏

i=1,i 6={l ,m}E�

φ ji(ξi)φki

(ξi)�

!

×E �ξlξmφ jl(ξl)φ jm

(ξm)φkl(ξl)φkm

(ξm)�

=

jlq

4 j2l− 1

jmÆ

4 j2m− 1

!

∏N

i=1,i 6={l ,m}δ ji ki, if

¨

kl = jl − 1,

km = jm − 1,

jlq

4 j2l− 1

kmÆ

4k2m − 1

!

∏N


¨

kl = jl − 1,

jm = km − 1,

klq

4k2l− 1

jmÆ

4 j2m− 1

!

∏N


¨

jl = kl − 1,

km = jm − 1,

klq

4k2l− 1

kmÆ

4k2m− 1

!

∏N


¨

jl = kl − 1,

jm = km − 1,

0, otherwise.

Combining the above representations with (2.10) and (2.14)–(2.15), we form the system

(2.13) for uniform random inputs.


3. Iterative Methods

As was shown in Section 2.2, the discretisation of the Helmholtz problem (2.1)–(2.3)

leads to the sparse linear system Au = b with A and b defined by (2.14) and (2.15),

respectively. For high accuracy solutions, one has to use large matrices A. In this section,

we will discuss efficient iterative methods for such large sparse linear systems. In particular,

we focus on Krylov subspace methods [15, 17, 31] based on the projection of the linear

system (2.13) into a consecutively constructed Krylov subspaces

Km

�

A, r (0)�

= span�

r (0), Ar (0), A2r (0), · · · , Am−1r (0)

,

where

r (0) = b− Au(0) (3.1)

and u(0) is an initial approximation.

For symmetric positive definite matrices, the conjugate gradient (CG) method [32] is

a popular choice. It generally uses only three vectors in memory and minimises the error

in the A-norm. But the linear system (2.13) is only complex-symmetric so that the di-

rect application of the CG method may not produce a convergent algorithm. On the other

hand, in case of non-symmetric nonsingular matrices one can use methods based on the

Lanczos bi-orthogonalization procedure — cf. [44], such as bi-conjugate gradient (Bi-CG)

method [24] and its modifications [46, 50]. Taking into account that for non-Hermitian

linear systems the Bi-CG methods can be unstable [16,25], Freund and Nachtigal [25] pro-

posed a more robust quasi-minimal residual (QMR) method. Here, we consider the QMR

method [25, 26] and compare it with a modification of the Bi-CG method [50] called the

bi-conjugate gradient stabilised (Bi-CGSTAB) method. Our implementation uses MATLAB

functions qmr and bi gstab for QMR and Bi-CGSTAB methods respectively.

To reduce the number of iterations, preconditioners are usually required. Therefore,

here we follow [15,44] and instead of the original problem (2.13), we will solve the prob-

lem

P−11 AP−1

2 u = P−11 b, u = P2u. (3.2)

The nonsingular matrices P1 and P2 are called preconditioners, and the systems Pix = b,

i = 1,2 are assumed to be solved at a low computational cost. Note that efficient precon-

ditioners correspond to well-clustered eigenvalues located not too close to the origin [18].

We next construct a mean-based preconditioner for the stochastic Helmholtz equation

following the ideas of [29, 41, 43]. Let ξ(0) denote the mean value of ξ and let P be the

matrix

P := G00 ⊗ (K +Mp + iLp), (3.3)

where G00, K are defined in (2.16) and

Mp(s, t) =

∫

D

κ2�

ξ(0)�

vsvt dx , Lp(s, t) =

∫

∂ DR

κ�

ξ(0)�

vsvt ds, s, t = 1, · · · , Nx .

Thus, the mean-based preconditioners are defined by choosing P1 = P and P2 = I in (3.2).

In addition, we let u(0) = P−1b for the initial approximation in (3.1).


At each iteration step, one has to solve the equation

P x = y , (3.4)

where

x =

x1

x2...

xNξ

, y =

y1

y2...

yNξ

, x i, yi ∈ CNx .

Since G00 = I is symmetric, (3.4) is equivalent to the problem

(K +Mp + iLp)XG00 = Y , (3.5)

where

X =�

x1, · · · , xNξ

�

, Y =�

y1, · · · , yNξ

�

.

Thus to find the solution of (3.5), one has to solve Nξ linear systems of the size Nx ×Nx . It

is much cheaper than direct solution of the system (2.13), whose size is Nx Nξ × Nx Nξ.

4. Numerical Results

We now consider two test problems: one where refractive index is a random field and

the other close to a resonant frequency. In both problems, the discretisation over physical

space uses a bilinear finite element approximation [6,15] and the implementation involves

IFISS [13] and S-IFISS [4] packages.

Example 4.1 (Refractive index modeled by a random field). We consider the physical do-

main D = [−1,1]× [−1,1], the boundary conditions (2.1)-(2.3) for ∂ DR := ∂ D, ∂ DD := ;,and the source term

f (x ) = 2�

0.5− x21 − x2

2

�

.

The refractive index is a truncated Karhunen–Loève (KL) expansion [11, 30] of a random

field with a mean function κ0(x ), standard deviation σ and covariance function Cov (x , y),

Cov (x , y) = σ2 exp

�

−|x1 − y1|c

− |x2 − y2|c

�

,

where x = [x1, x2]T , y = [y1, y2]

T and the correlation length is c = 4 . The KL expansion

has the form

κ(x ,ξ) = κ0(x ) +

N∑

i=1

κi(x )ξi = κ0(x ) +

N∑

i=1

Æ

λici(x )ξi ,

where {λi, ci(x )}Ni=1are the eigenpairs of Cov (x , y), {ξi}Ni=1

uncorrelated random variables

and N is the number of KL modes retained.


The error associated with the truncation of the KL expansion depends on the amount

of total variance captured δK L,

δK L :=

∑N

i=1λi

|D|σ2,

where |D| is the area of D [30, 43]. In what following, we choose κ0(x ) = 10, σ = 1,

assume that the random variables {ξi}Ni=1are independent uniform distributions with the

range [−1,1] and take N = 4, so that δK L > 89%.

Figs. 2(a)-2(c) show the gPC approximations of the mean and variance functions of

this problem. The order of gPC expansion is p = 10, the physical space is discretised

by the uniform 33× 33 grid, and the corresponding linear system (2.13) is solved by the

Bi-CGSTAB method with the mean-based preconditioner (3.3). We also use a stopping

criterion based on the relative residual ‖Au(k) − b‖/‖b‖, where ‖ · ‖ is the vector L2 norm

and the superscript k shows the iteration number. The iterations stop if the relative residual

is smaller than 10−8. For the sake of comparison, we also apply the Monte Carlo method

[54] and Figs. 2(d)-2(f) display the results obtained by the Monte Carlo method with 106

samples. Comparing both methods, we observe that they produce virtually the same mean

and variance functions.

To evaluate the accuracy of the gPC finite element approximation (2.12), we consider

the relative mean Em and variance Ev errors defined by

Em :=‖E(uph)−E(uref)‖2‖E(uref)‖2

, Ev :=‖Var(uph)− Var(uref)‖2

‖Var(uref)‖2,

where uref is a reference solution and ‖ · ‖2 is defined in (2.5).

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

(a) Mean function (real part)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

-0.01

0

0.01

0.02

(b) Mean function (imaginary part)-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

0.5

1

1.5

2

2.5

310 -4

(c) Variance function

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

(d) Mean function (real part)

-1 -0.5 0 0.5 1-1

-0.5

0

0.5

1

-0.01

0

0.01

0.02

(e) Mean function (imaginary part)-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

0.5

1

1.5

2

2.5

310 -4

(f) Variance function

Figure 2: Example 4.1. Top: gPC method, p = 10. Bottom: Monte Carlo method with 106samples.


For the Monte Carlo method, the relative mean and variance errors are defined analogously

but uph is replaced by the mean and variance function estimates generated by the Monte

Carlo method.

Figs. 3(a),3(b) show mean and variance errors for gPC and Monte Carlo methods with

the same uniform spatial grid. The reference solution we use, is obtained by gPC method for

p = 10 with the corresponding linear system solved by preconditioned Bi-CGSTAB. Fig. 3(a)

shows that the errors of the gPC approximation decrease quickly as the gPC order increases.

Note that the gPC method is more efficient than Monte Carlo method — e.g. the errors of

the former with p = 2 are smaller than the later with 106 samples. Figs. 3(c), 3(d) show

the the CPU time for gPC and Monte Carlo methods. In the gPC method the CPU time in-

cludes the time for constructing the linear system and solving a preconditioned Bi-CGSTAB

method. For the Monte Carlo method, the CPU time includes the time for constructing

and solving linear systems by the MATLAB backslash solver associated with deterministic

problems at all input sample points.

All computations are carried out in MATLAB environment on a desktop computer with

3.60GHz Intel Core i7 CPU. Figs. 3(c), 3(d) show that the gPC method requires significantly

less CPU time to achieve a given accuracy.

(a) Errors of the gPC method (b) Errors of the Monte Carlo method

(c) CPU time of the gPC method (d) CPU time of the Monte Carlo method

Figure 3: Example 4.1. Errors and CPU time.


Table 1: Example 4.1. Numbers of iterations and CPU time (in bra kets).

Iterative method h−1 p = 2 p = 4 p = 6 p = 8 p = 10

Bi-CGSTAB 16 8.5(0.36) 10(1.91) 10.5(6.35) 10.5(15.07) 11(31.88)

32 8.5(1.59) 10.5(9.83) 10.5(29.59) 10.5(68.69) 11(161.58)

64 8.5(10.78) 10.5(59.55) 10.5(181.17) 11(476.87) 11(975.86)

QMR 16 16(0.72) 19(3.74) 22(13.41) 21(30.13) 22(63.27)

32 16(3.27) 19(18.58) 21(61.76) 22(148.88) 22(346.83)

64 16(21.47) 19(110.59) 21(368.79) 22(982.89) 22(1975.09)

Table 1 shows the number of iterations and CPU time for the preconditioned iterative

method, including time for setting up the preconditioners and iteration time for meshes of

various size and different gPC orders. It is easily seen that the number of iterations does

not depend on the mesh size h and gPC order p and that these numbers are generally small.

Fig. 4 shows CPU time versus number of unknowns (Nx Nξ). The slope of the black line is 1,

so that CPU time grows almost linearly, in compliance with the properties of preconditioned

iterative methods.

10 5 10 6 10 7

10 0

10 2

10 4

10 6

slope = 1

Bi-CGSTAB QMR

Figure 4: Example 4.1. CPU time for pre onditioned iterative method.

Example 4.2 (A problem close to resonant frequency). We now consider the refractive

index of the form

κ(x ,ξ) = κ0 + κ1ξ, (4.1)

where ξ is uniformly distributed in [−1,1] and κi , i = 0,1 are constants specified in what

follows. We also consider the pure Dirichlet boundary condition ∂ D = ∂ DD. The eigenval-

ues of the corresponding problem

−∇2u(x ,ξ)− κ2(x ,ξ)u(x ,ξ) = λ(ξ)u(x ,ξ)

with the boundary condition (2.2) are

λi, j(ξ) =(i2 + j2)π2

4− κ2(ξ), i, j = 1,2, · · · . (4.2)


The right-hand side of (2.1) in this problem is the normalised eigenfunction correspond-

ing to λ1,1 — i.e.

f = cos

�πx1

2

�

cos

�πx2

2

�

.

The exact solution of this test problem is known — viz. u(x ,ξ) = f (x )/λ1,1(ξ) but we

solve the problem by using a stochastic Galerkin method to test the gPC approximations

and a mean-based preconditioning scheme. If κ(ξ) = (p

i2 + j2π)/2, i, j = 1,2, . . ., — i.e.

if λi, j = 0 in (4.2), the solution of the deterministic version of this problem is not unique

and we have a resonance state. Recalling (4.1), we focus on the case where κ is close to

the first resonant frequency π/p

2. In particular, for κ0 = π/p

2 + 0.41 we consider the

following values of κ1:

• κ1 = 0.1 with |λ1,1(ξ)| ∈ [1.47,2.53] for ξ ∈ [−1,1];

• κ1 = 0.4 with |λ1,1(ξ)| ∈ [0.04,4.26] for ξ ∈ [−1,1];

It is clear that for κ1 = 0.1 the stochastic Helmholtz problem (2.1)–(2.3) is not close to

resonance but for κ1 = 0.4 it is.

Fig. 5 shows mean and variance errors in Example 4.2. In both cases, the reference so-

lutions are obtained by the gPC method with p = 100 and the corresponding linear systems

are solved by a preconditioned Bi-CGSTAB. Fig. 5(a) shows that for the problem not close

to resonance — i.e. if κ1 = 0.1, the mean and variance errors of the gPC approximation

decrease quickly as p grows. On the other hand, for problems close to resonance, the errors

of the gPC approximation still decrease but much slower — cf. Fig. 5(b).

Fig. 6 shows the number of iterations for preconditioned solvers with the spatial 33×33

grid used. For κ1 = 0.1 the number of iterations in both Bi-CGSTAB and QMR is small —

viz. five for Bi-CGSTAB and nine for QMR, and they are independent of p. However, for the

close to resonance problem — i.e. for k1 = 0.4, both methods require a large number of

iterations. More precisely, one needs about sixty iterations to achieve the residual stopping

0 2 4 6 8p

10 -16

10 -14

10 -12

10 -10

10 -8

10 -6

10 -4

10 -2

10 0

Rel

ativ

e er

ror

meanvariance

(a) κ1 = 0.1

0 20 40 60 80p

10 -12

10 -10

10 -8

10 -6

10 -4

10 -2

10 0

Rel

ativ

e er

ror

meanvariance

(b) κ1 = 0.4

Figure 5: Example 4.2. Errors of gPC method.


0 20 40 60 80 100p

2

3

4

5

6

7

8

9

Num

ber

of it

erat

ions

Bi-CGSTAB QMR

(a) κ1 = 0.1

0 20 40 60 80 100p

0

20

40

60

80

100

Num

ber

of it

erat

ions

Bi-CGSTAB QMR

(b) κ1 = 0.4

Figure 6: Example 4.2. Number of iterations for pre onditioned iterative methods.

tolerance 10−8. We note that in this case the gPC order is 60. However, as the gPC order

p increases, the number of iterations of Bi-CGSTAB grows till p ≈ 50, whereas QMR keeps

going till p ≈ 90.

Note that the smallest magnitude of λ1,1(ξ) is much smaller for κ1 = 0.4 and since the

solution has the form u(x ,ξ) = f (x )/λ1,1(ξ), the magnitude of the variance function is

therefore much larger than for κ1 = 0.1. Nevertheless, the efficiency of the mean-based

preconditioner can deteriorate if the variance function grows.

5. Conclusions

We describe the mathematical framework and implementation of a spectral stochastic

finite element method for solving Helmholtz equations with random inputs. The sparsity of

the corresponding linear system is analysed and iterative methods combined with a mean-

based preconditioning scheme are investigated. The examples presented show that gPC

approximation and mean-based preconditioning scheme are more efficient if the stochastic

Helmholtz problem is not too close to a resonant frequency and less efficient for close to

resonance problems. The numerical studies here focus mainly on low frequency waves

— i.e. if the refractive index is small. More efficient gPC-based approximations and fast

iterative solvers for close to resonance and high-frequency wave problems will be reported

elsewhere.

Acknowledgments

The authors thank David Silvester and Catherine Powell for helpful suggestions and

discussions.

This work is supported by the Science Challenge Project (No. TZ2018001) and the

National Natural Science Foundation of China (No. 11601329).


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