efficient spectral stochastic finite element …...east asian journal on applied mathematics vol....
TRANSCRIPT
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
,
604 G. Wang and Q. Liao
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.
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 605
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.
606 G. Wang and Q. Liao
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�
.
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 607
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}.
608 G. Wang and Q. Liao
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
�
.
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 609
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.
610 G. Wang and Q. Liao
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)�
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 611
=
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
i=1,i 6={l ,m}δ ji ki, if
¨
kl = jl − 1,
jm = km − 1,
klq
4k2l− 1
jmÆ
4 j2m− 1
!
∏N
i=1,i 6={l ,m}δ ji ki, if
¨
jl = kl − 1,
km = jm − 1,
klq
4k2l− 1
kmÆ
4k2m− 1
!
∏N
i=1,i 6={l ,m}δ ji ki, if
¨
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.
612 G. Wang and Q. Liao
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).
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 613
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.
614 G. Wang and Q. Liao
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.
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 615
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.
616 G. Wang and Q. Liao
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)
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 617
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.
618 G. Wang and Q. Liao
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).
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 619
References
[1] R. Askey and M. Ismail, Recurrence relations, continued fractions and orthogonal polynomials,
AMS (1984).
[2] I. Babuška, F. Nobile and R. Tempone, A stochastic collocation method for elliptic partial differ-
ential equations with random input data, SIAM J. Numer. Anal. 45, 1005–1034 (2007).
[3] I. Babuška, R. Tempone and G.E. Zouraris, Galerkin finite element approximations of stochastic
elliptic partial differential equations, SIAM J. Numer. Anal. 42, 800–825 (2004).
[4] A. Bespalov, C. Powell and D. Silvester, S-IFISS version 1.0, (2013). http://www.manchester.
ac.uk/ifiss/s-ifiss1.0.tar.gz.
[5] J.P. Boyd, Chebyshev and Fourier spectral methods, Courier Corporation (2001).
[6] D. Braess, Finite Elements, Cambridge University Press (1997).
[7] M. Cheng, T.Y. Hou and Z. Zhang, A dynamically bi-orthogonal method for time-dependent
stochastic partial differential equations I: Derivation and algorithms, J. Comput. Phys. 242,
843–868 (2013).
[8] M. Cheng, T.Y. Hou and Z. Zhang, A dynamically bi-orthogonal method for time-dependent
stochastic partial differential equations II: Adaptivity and generalizations, J. Comput. Phys. 242,
753–776 (2013).
[9] M.K. Deb, I.M. Babuška and J.T. Oden, Solution of stochastic partial differential equations us-
ing Galerkin finite element techniques, Comput. Methods Appl. Mech. Engrg. 190, 6359–6372
(2001).
[10] M. Eiermann, O.G. Ernst and E. Ullmann, Computational aspects of the stochastic finite element
method, Comput. Vis. Sci. 10, 3–15 (2007).
[11] H. Elman and D. Furnival, Solving the stochastic steady-state diffusion problem using multigrid,
IMA J. Numer. Anal. 27, 675–688 (2007).
[12] H. Elman and Q. Liao, Reduced basis collocation methods for partial differential equations with
random coefficients, SIAM/ASA J. Uncertain. Quantif. 1, 192–217 (2013).
[13] H. Elman, A. Ramage and D. Silvester, IFISS: A computational laboratory for investigating
incompressible flow problems, SIAM Rev. 56, 261–273 (2014).
[14] H.C. Elman, O.G. Ernst, D.P. O’Leary and M. Stewart, Efficient iterative algorithms for the
stochastic finite element method with application to acoustic scattering, Comput. Methods Appl.
Mech. Engrg. 194, 1037–1055 (2005).
[15] H.C. Elman, D.J. Silvester and A.J. Wathen, Finite elements and fast iterative solvers: with
applications in incompressible fluid dynamics, Oxford University Press (2014).
[16] Y.A. Erlangga, Advances in iterative methods and preconditioners for the Helmholtz equation,
Arch. Comput. Methods Eng. 15, 37–66 (2008).
[17] Y.A. Erlangga, C. Vuik and C.W. Oosterlee, On a class of preconditioners for solving the Helmholtz
equation, Appl. Numer. Math. 50, 409–425 (2004).
[18] O.G. Ernst and M.J. Gander, Why it is difficult to solve Helmholtz problems with classical iterative
methods, in: Numerical analysis of multiscale problems, I. Graham, T. Hou, L. O., R. Scheichl
(Eds), pp. 325–363, Springer (2012).
[19] O.G. Ernst and E. Ullmann, Stochastic Galerkin matrices, SIAM J. Matrix Anal. Appl. 31, 1848–
1872 (2010).
[20] W. Feller, An introduction to probability theory and its applications 1, John Wiley & Sons (1968).
[21] Z. Fang, J. Li, T. Tang and T. Zhou, Efficient stochastic Galerkin methods for Maxwell’s equations
with random input, J. Sci. Comput., 1–20, (2019).
[22] X. Feng, J. Lin and C. Lorton, An efficient numerical method for acoustic wave scattering in
random media, SIAM/ASA J. Uncertain. Quantif. 3, 790–822 (2015).
620 G. Wang and Q. Liao
[23] X. Feng, J. Lin and D. Nicholls, An efficient Monte Carlo-transformed field expansion method for
electromagnetic wave scattering by random rough surface, Commun. Comput. Phys. 23, 685-
705 (2018).
[24] R. Fletcher, Conjugate gradient methods for indefinite systems, in: Numerical analysis, pp. 73–
89, Springer (1976).
[25] R.W. Freund and N.M. Nachtigal, QMR: a quasi-minimal residual method for non-hermitian
linear systems, Numer. Math. 60, 315–339 (1991).
[26] R.W. Freund and N.M. Nachtigal, An implementation of the QMR method based on coupled
two-term recurrences, SIAM J. Sci. Comput. 15, 313–337 (1994).
[27] A. Frolov and E. Kartchevskiy, Integral equation methods in optical waveguide theory, in: Inverse
Problems and Large-Scale Computations, pp. 119–133, Springer (2013).
[28] M.J. Gander, I.G. Graham and E.A. Spence, Applying GMRES to the Helmholtz equation with
shifted laplacian preconditioning: what is the largest shift for which wavenumber-independent
convergence is guaranteed? Numer. Math. 131, 567–614 (2015).
[29] R.G. Ghanem and R.M. Kruger, Numerical solution of spectral stochastic finite element systems,
Comput. Methods Appl. Mech. Engrg. 129, 289–303 (1996).
[30] R.G. Ghanem and P.D. Spanos, Stochastic finite elements: a spectral approach, Courier Corpo-
ration (2003).
[31] G.H. Golub and C.F. Van Loan, Matrix computations, JHU Press (2013).
[32] M.R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, NBS
(1952).
[33] F.B. Jensen, W.A. Kuperman, M.B. Porter and H. Schmidt, Computational ocean acoustics,
Springer Science & Business Media (2011).
[34] E. Kartchevski, A. Nosich and G. Hanson, Mathematical analysis of the generalized natural
modes of an inhomogeneous optical fiber, SIAM J. Appl. Math. 65, 2033–2048 (2005).
[35] E. Karchevskii and S. Solov’ev, Investigation of a spectral problem for the Helmholtz operator
on the plane, Differ. Equ. 36, 631–634 (2000).
[36] K. Lee and H.C. Elman, A preconditioned low-rank projection method with a rank-reduction
scheme for stochastic partial differential equations, SIAM J. Sci. Comput. 39, S828–S850
(2017).
[37] R. März, Integrated optics: design and modeling, Artech House on Demand (1995).
[38] E. Musharbash, F. Nobile and T. Zhou, Error analysis of the dynamically orthogonal approxima-
tion of time dependent random PDEs, SIAM J. Sci. Comput. 37, A776–A810 (2015).
[39] L. Ng and K. Willcox, Multifidelity approaches for optimization under uncertainty, Internat.
J. Numer. Methods Engrg. 100, 746–772 (2014).
[40] L.W.T. Ng and M. Eldred, Multifidelity uncertainty quantification using non-intrusive polynomial
chaos and stochastic collocation, in: 53rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural
Dynamics and Materials Conference, pp. 1852-1868, American Institute of Aeronautics and
Astronautics Inc. (2012).
[41] M.F. Pellissetti and R.G. Ghanem, Iterative solution of systems of linear equations arising in the
context of stochastic finite elements, Adv. Eng. Softw. 31, 607–616 (2000).
[42] C. Powell and D. Silvester, Preconditioning steady-state Navier-Stokes equations with random
data, SIAM J. Sci. Comput. 34, A2482–A2506 (2012).
[43] C.E. Powell and H.C. Elman, Block-diagonal preconditioning for spectral stochastic finite element
systems, IMA J. Numer. Anal. 29, 350–375 (2009).
[44] Y. Saad, Iterative methods for sparse linear systems, SIAM (2003).
[45] J. Shen and T. Tang, Spectral and high-order methods with applications, Science Press (2006).
[46] P. Sonneveld, CGS, a fast Lanczos-type solver for nonsymmetric linear systems, SIAM J. Sci.
Efficient Spectral SFEM for Helmholtz Equations with Random Inputs 621
Statist. Comput. 10, 36–52 (1989).
[47] B. Sousedík and H. Elman, Stochastic Galerkin methods for the steady-state Navier-Stokes equa-
tions, J. Comput. Phys. 316, 435–452 (2016).
[48] B. Sousedík, R.G. Ghanem and E.T. Phipps, Hierarchical Schur complement preconditioner
for the stochastic Galerkin finite element methods, Numer. Linear Algebra Appl. 21, 136–151
(2014).
[49] L. Tamellini, O.L. Maître and A. Nouy, Model reduction based on proper generalized decomposi-
tion for the stochastic steady incompressible Navier-Stokes equations, SIAM J. Sci. Comput. 36,
A1089–A1117 (2014).
[50] H.A. Van der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution
of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 13, 631–644 (1992).
[51] C. Vassallo, Optical waveguide concepts, Elsevier (1991).
[52] D. Xiu, Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys.
5, 242–272 (2009).
[53] D. Xiu, Numerical methods for stochastic computations: a spectral method approach, Princeton
University Press (2010).
[54] D. Xiu and J. Hesthaven, High-order collocation methods for differential equations with random
inputs, SIAM J. Sci. Comput. 27, 1118–1139 (2005).
[55] D. Xiu and G.E. Karniadakis, Modeling uncertainty in steady state diffusion problems via gener-
alized polynomial chaos, Comput. Methods Appl. Mech. Engrg. 191, 4927–4948 (2002).
[56] D. Xiu and G.E. Karniadakis, The wiener-askey polynomial chaos for stochastic differential equa-
tions, SIAM J. Sci. Comput. 24, 619–644 (2002).
[57] D. Xiu and G.E. Karniadakis, Modeling uncertainty in flow simulations via generalized polyno-
mial chaos, J. Comput. Phys. 187, 137–167 (2003).
[58] D. Xiu and J. Shen, An efficient spectral method for acoustic scattering from rough surfaces,
Commun. Comput. Phys. 2, 54–72 (2007).
[59] D. Xiu and J. Shen, Efficient stochastic Galerkin methods for random diffusion equations, J. Com-
put. Phys. 228, 266–281 (2009).
[60] T. Tang and T. Zhou, Convergence analysis for stochastic collocation methods to scalar hyperbolic
equations, Commun. Comput. Phys. 8, 226–248 (2010).
[61] T. Zhou and T. Tang, Galerkin methods for stochastic hyperbolic problems using bi-orthogonal
polynomials, J. Sci. Comput. 51, 274-292 (2012).