Content uploaded by Andreas Petersson
Author content
All content in this area was uploaded by Andreas Petersson on Jul 23, 2021
Content may be subject to copyright.
APPROXIMATION OF SPDE COVARIANCE OPERATORS BY FINITE
ELEMENTS: A SEMIGROUP APPROACH
MIH ´
ALY KOV´
ACS, ANNIKA LANG, AND ANDREAS PETERSSON
Abstract. The problem of approximating the covariance operator of the mild solution to
a linear stochastic partial differential equation is considered. An integral equation involving
the semigroup of the mild solution is derived and a general error decomposition formula is
proven. This formula is applied to approximations of the covariance operator of a stochastic
advection-diffusion equation and a stochastic wave equation, both on bounded domains. The
approximations are based on finite element discretizations in space and rational approxima-
tions of the exponential function in time. Convergence rates are derived in the trace class
and Hilbert–Schmidt norms with numerical simulations illustrating the results.
1. Introduction
This paper considers stochastic partial differential equations (SPDEs) formulated as linear
stochastic evolution equations on a Hilbert space H. That is to say, equations of the form
dX(t) + AX(t) dt=F X (t) dt+BdW(s) for t∈(0, T ], T < ∞,
X(0) = ξ.
(1)
Here Xis an H-valued stochastic process, Fand Bare linear operators, Wis a Wiener process
in Hwith covariance operator Qand −Ais the generator of a C0-semigroup S= (S(t))t∈[0,T ]
of linear operators on H. Since (1) is linear and the noise term is additive and Gaussian, the
solution X(t) to (1) at time t∈(0, T ] is an H-valued Gaussian random variable when the
initial value ξis Gaussian. The distribution of X(t) is therefore completely determined by
its mean value E[X(t)] and covariance (operator) Cov(X(t)) = E[(X(t)−E[X(t)]) ⊗(X(t)−
E[X(t)])]. Computing these quantities is therefore vital for understanding of X(t). In general,
there are no analytic solutions, so numerical approximations are needed. In this paper, we
focus on the approximation of Cov(X(t)).
The literature on the numerical analysis of approximations of SPDE covariance operators
is sparse. We are only aware of [14, 15, 20]. Therein, the authors consider SPDEs of parabolic
2010 Mathematics Subject Classification. 60H15, 65M12, 65M60, 65R20, 45N05, 35C15.
Key words and phrases. stochastic partial differential equations, integral equations, covariance operators,
finite element method, stochastic advection-diffusion equations, stochastic wave equations.
M. Kov´acs acknowledges the support of the Marsden Fund of the Royal Society of New Zealand through
grant. no. 18-UOO-143, the Swedish Research Council (VR) through project no. 2017-04274 and the NKFIH
through grant no. 131545. The work of A. Lang was partially supported by the Swedish Research Council
(VR) (project no. 2020-04170), by the Wallenberg AI, Autonomous Systems and Software Program (WASP)
funded by the Knut and Alice Wallenberg Foundation, and by the Chalmers AI Research Centre (CHAIR).
The work of A. Petersson was supported in part by the Research Council of Norway (RCN) through project no.
274410, the Swedish Research Council (VR) through reg. no. 621-2014-3995 and the Knut and Alice Wallenberg
foundation.
1
2 M. KOV´
ACS, A. LANG, AND A. PETERSSON
type and solve a tensorized equation related to the concept of a weak solution to (1). They
assume the operator Ato be self-adjoint. In this paper, we take a different approach. We
work with the mild solution of an SPDE to derive an operator-valued integral equation for
the covariance, expressed in terms of the semigroup S.
This approach allows us to treat parabolic SPDEs where Ais not self-adjoint. One example
of such an equation comes from the modeling of sea surface temperature dynamics [12]. The
equation, posed in the space H=L2(D) of square integrable functions on some domain
D ⊂ R2, is given by
(2) dX(t, x) + AX(t, x) dt= dW(t, x) for t∈(0, T ], x ∈ D.
Here AX(t) corresponds to AX(t, ·) = δX(t, ·)−D∆X(t, ·)−a·∇X(t, ·), where X(t, x) is the
sea surface temperature at time tand point x∈ D,a∈R2is the velocity vector field of the
upper ocean layer, D > 0 is a diffusion coefficient and δ∈Ris a feedback parameter. When
a6= 0, the operator Aceases to be self-adjoint. The Wiener process Wmodels small time
scale fluctuations in the heat flux across the ocean-atmosphere interface and Qis an integral
operator with a kernel qhaving small correlation length. Similar models have recently been
considered for reconstructing the evolution of cloud systems from discrete measurements [24].
Moreover, our approach extends the parabolic setting to any SPDE which has a mild
solution in terms of a semigroup. This include s hyperbolic SPDEs, such as a stochastic
equation for the vertical displacement U(t, x) of a strand of DNA suspended in a liquid at
time tand space x∈ D ⊂ Rd,d= 1,2,3 [8]. It is given by
(3) d ˙
U(t, x)−∆U(t, x) dt=−(QU)(t, x) dt+ dW(t, x) for t∈(0, T ], x ∈ D.
The first term on the right hand side models friction due to viscosity of the fluid, while the
Wiener process term Wcorresponds to random bombardment of the DNA string by the fluid’s
molecules. Writing X= [U, ˙
U]>, the equation can be put in the form of (1) by considering it
on a product space, see Section 3.2. Figure 1 shows realizations of the solutions to (2) and (3)
for the domain D= (0,1), see Examples 3.5 and 3.15.
-0.25
0.1
0
0.25
X(t,x)
0.5
t
0.05 1
0.75
x
0.5
0.25
00
(a) A stochastic advection-diffusion equation. (b) A stochastic wave equation.
Figure 1. Realizations of the solutions Xand Uto (2) and (3) when D= (0,1).
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 3
Let us now outline the content of the paper. In Section 2, we formulate an operator-valued
integral equation for the covariance of the mild solution Xto (1) in an abstract Hilbert space
framework, and give assumptions that ensure that it has a unique solution. We confirm that X
is Gaussian and that the process [0, T ]3t7→ Cov(X(t)) is a solution to the integral equation.
The mild Itˆo formula [6] is key for this. We finish the section by giving an abstract error
decomposition formula for approximations of this process. The error is, for t∈[0, T ], analyzed
with respect to the norms k·kL1(H)and k·kL2(H). Here L1(H) and L2(H) denote the spaces
of trace class and Hilbert–Schmidt operators, respectively. The first norm is a natural choice
since if (Qj)∞
j=1 is a sequence of covariances of some Gaussian H-valued random variables
(Xj)∞
j=1 with zero mean, then Qj→Cov(X(t)) in L1(H) if and only if Xj→X(t) weakly,
i.e., E[f(Xj)] →E[f(X(t))] for all continuous and bounded functionals fon H[5]. The
norm of L2(H) is weaker. It has a natural meaning when H=L2(D): if X(t) = X(t, ·) is
F × B(D)-measurable,
hCov(X(t))u, viH=ZD×D
Cov(X(t, x), X(t, y))u(x)v(y) dxdy
and kCov(X(t))k2
L2(H)=kCov(X(t, ·), X(t, ·))k2
L2(D×D). Therefore we may, formally at least,
view convergence in L2(H) as convergence in L2(D × D) of underlying covariance functions
on D. Figure 2 shows the covariance functions for the solutions to (2) and (3) at T= 0.1.
0.555
1
0.56
0.565
c(x,y)
0.75
0.57
0.575
y
0.5 1
0.75
x
0.25 0.5
0.25
00
(a) Covariance function for a stochastic advection-
diffusion equation.
(b) Covariance function for a stochastic wave equa-
tion.
Figure 2. Plot of the covariance functions c(x, y) = Cov(X(0.1, x), X(0.1, y))
and c(x, y) = Cov(U(0.1, x), U (0.1, y)), x, y ∈ D = (0,1), for the solutions X
and Uto (2) and (3).
In Section 3 we apply our abstract framework to the two concrete equations (2) and (3). In
both cases, the covariance integral equations are discretized by finite elements in space and
rational approximations of the driving semigroup in time. The resulting approximations are
expressed as integral equations based on a fully discrete approximation ˜
Sof S.
Consider a discrete approximation ˜
X(t) of X(t), t∈[0, T ]. Suppose, without loss of
generality, that E[X(t)] = E[˜
X(t)] = 0. Then, by properties of the norms k·kLi(H),i∈ {1,2},
4 M. KOV´
ACS, A. LANG, AND A. PETERSSON
and the H¨older inequality,
kCov(X(t)) −Cov(˜
X(t))kLi(H)
=kE[X(t)⊗X(t)] −E[˜
X(t)⊗˜
X(t)]kLi(H)
=1
2
Eh(X(t) + ˜
X(t)) ⊗(X(t)−˜
X(t))i+Eh(X(t)−˜
X(t)) ⊗(X(t) + ˜
X(t))i
Li(H)
≤max E[kX(t)k2
H]1
2,E[k˜
X(t)k2
H]1
2EhkX(t)−˜
X(t)k2
Hi1
2.
(4)
Therefore, if Cov(˜
X(t)) can be calculated, we immediately obtain an approximation scheme
for Cov(X(t)) for which the error can be bounded by the strong error EhkX(t)−˜
X(t)k2
Hi1/2.
In Section 3.1, where a stochastic advection-diffusion equation is considered, we demonstrate
that this bound is suboptimal in the sense that the convergence rate we derive for an ap-
proximation based on replacing Sby a fully discrete approximation ˜
Sin the operator-valued
integral equation is higher than the strong rate EhkX(t)−˜
X(t)k2
Hi1/2when ˜
Xis based on
the same discretization ˜
S, see Remark 3.6. We also demonstrate, in theory and by a numerical
simulation, that there are cases when the L2error decays faster than the stronger L1error.
In Section 3.2 we work with the stochastic wave equation and provide, to the best of our
knowledge, the first results on convergence rates for the approximation of covariance operators
of hyperbolic SPDEs. We first consider a temporally semidiscrete approximation, which is
then used for analyzing a fully discrete approximation. Numerical simulations finish the
section.
Throughout the paper, we adopt the notion of generic constants, which may vary from
occurrence to occurrence and are independent of any parameter of interest, such as spatial or
temporal step sizes. By a.bwe denote the existence of a generic constant such that a≤Cb.
2. Covariance operators of stochastic evolution equations
In this section we prove existence and uniqueness of the solution to an operator-valued
integral equation. We then show that the covariance of the mild solution of an SPDE is a
solution of this equation. First, however, we introduce our setting and reiterate some facts
from operator theory and probability theory in Hilbert spaces.
2.1. Operator theory. Let (H, h·,·iH) and (U, h·,·iU) be real and separable Hilbert spaces.
We denote by L(H, U) the space of bounded linear operators from Hto Uequipped with the
usual operator norm and by L1(H, U ) and L2(H, U ) the spaces of trace class and Hilbert–
Schmidt operators, respectively. We use the shorthand notations L(H), L1(H) and L2(H)
when U=H. Additionally, we denote by Σ(H)⊂ L(H) the set of symmetric bounded
operators on H. An operator Γ ∈ L1(H, U ) if and only if there are two orthonormal sequences
(ej)∞
j=1 ⊂H, (fj)∞
j=1 ⊂Uand a sequence (µj)∞
j=1 ∈1such that
(5) Γx=
∞
X
j=1
µjhx, ejifjfor x∈H.
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 5
The space L1(H, U ) is a separable Banach space with norm
(6) kΓkL1(H,U)= inf
(ej)⊂H
(fj)⊂U
∞
X
j=1 kejkHkfjkU: Γ =
∞
X
j=1h·, ejiHfj
,
see [26, Sections 47-48]. Moreover, L2(H, U ) is a separable Hilbert space with an inner
product, for an arbitrary orthonormal basis (ej)∞
j=1 of H, given by
hΓ1,Γ2iL2(H,U)=
∞
X
j=1hΓ1ej,Γ2ejifor Γ1,Γ2∈ L2(H, U ),
We have L1(H, U)⊂ L2(H, U )⊂ L∞(H, U ), the space of compact operators Γ ∈ L(H, U),
and Γ ∈ Li(H, U ) if and only if Γ∗∈ Li(U, H) with
(7) kΓkLi(H,U)=kΓ∗kLi(U,H), i ∈ {1,2}.
We identify the spaces L2(H, U) and U⊗H, the Hilbert tensor product, with equivalent norms.
The tensor u⊗vis regarded as an element of L(H, U ) by the relation (u⊗v)w=hv, wiHu
for v, w ∈Hand u∈U. It can be seen that u⊗v∈ L1(H, U ) with ku⊗vkL1(H,U)=
ku⊗vkL2(H,U)=kukUkvkH. Moreover,
(8) hΓ, u ⊗viL2(H,U )=hΓv, uiUfor Γ ∈ L2(H, U ).
If Vand Gare two other real and separable Hilbert spaces, then
(9) Γ1u⊗Γ2v= Γ1(u⊗v)Γ∗
2
for u∈U,v∈H, Γ1∈ L(U, V ) and Γ2∈ L(H, G), with Γ∗
2denoting the adjoint of Γ2.
The space Li(H, U ), i∈ {1,2}, is an operator ideal: if Γ1∈ L(G, V ), Γ2∈ Li(U, G) and
Γ3∈ L(H, U ) then Γ3Γ2Γ1∈ Li(H, V ) with
(10) kΓ1Γ2Γ3kLi(H,V )≤ kΓ1kL(G,V )kΓ2kLi(U,G)kΓ3kL(H,U ).
If Γ1∈ L2(U, V ) and Γ2∈ L2(H, U ), then Γ1Γ2∈ L1(H, V ) and
(11) kΓ1Γ2kL1(H,V )≤ kΓ1kL2(U,V )kΓ2kL2(H,U ).
The trace of Γ ∈ L1(H) is, for an arbitrary orthonormal basis (ej)∞
j=1 of H, defined by
Tr(Γ) =
∞
X
j=1hΓej, ejiH.
If Γ ∈Σ+(H)⊂Σ(H), the space of all positive semidefinite operators, then Tr(Γ) = kΓkL1(H).
2.2. Probability theory in Hilbert spaces. Below we work on the bounded interval [0, T ],
T < ∞. Let (Ω,A,(Ft)t∈[0,T ], P ) be a complete filtered probability space satisfying the usual
conditions, which is to say that F0contains all P-null sets and Ft=∩s>tFsfor all t∈[0, T ].
By Lp(Ω, H), p∈[1,∞) we denote the space of all H-valued random variables Ywith norm
kYkLp(Ω,H)= (E[kYkp
H])1/p. For Y, Y 0∈L2(Ω, H), the cross-covariance (operator) of Y, Y 0
is defined by Cov(Y, Y 0) = E[(Y−E[Y]) ⊗(Y0−E[Y0])] ∈ L1(H) and the covariance of Y
by Cov(Y) = Cov(Y, Y )∈ L1(H)∩Σ+(H). Note that Cov(Y, Y 0) is uniquely determined by
hCov(Y, Y 0)u, viH=Cov(hY, viH,hY0, uiH).
6 M. KOV´
ACS, A. LANG, AND A. PETERSSON
An H-valued random variable Yis said to be Gaussian if hY, viHis a Gaussian real-valued
random variable for all v∈H. Then Y∈Lp(Ω, H) for all p≥1. A pair Y , Y 0of H-valued
random variables is said to be jointly Gaussian if Y⊕Y0is an H⊕H-valued Gaussian random
variable. Then Yand Y0are independent if and only if Cov(Y, Y 0) = 0, cf. [23].
Let Wbe a generalized Wiener process in U(see [7, Chapter 4]) with covariance Q∈Σ+(U),
not necessarily of trace class. The Hilbert space (Q1/2(U),hQ−1/2·, Q−1/2·iU) is denoted by U0,
where Q1/2is the unique positive semidefinite square root of Qand Q−1/2its pseudoinverse.
2.3. Covariance integral equations and mild solutions to SPDEs. Let ˆ
Hbe another
Hilbert space such that H →ˆ
Hcontinuously and densely. The main topic of study in this
paper are integral equations of the form
K(t) = S(t)QξS(t)∗+Zt
0
S(t−s)F K (s)S(t−s)∗+S(t−s)K(s)(S(t−s)F)∗ds
+Zt
0
S(t−s)B(S(t−s)B)∗ds,
(12)
taking values in the space L1(H), with the adjoint being taken with respect to the inner
product of H. Here S= (S(t))t∈[0,T ]is a family of L(ˆ
H, H)-valued operators, not yet assumed
to be a semigroup, while F∈ L(H, ˆ
H), B∈ L2(U0,ˆ
H) and Qξ∈ L1(H)∩Σ+(H). We assume
that for any v∈H, the mappings t7→ S(t)v∈H,t7→ S(t)F v ∈Hand t7→ S(t)B∈
L2(U0, H) are continuous on [0, T ]. Below, we make an assumption on the boundedness of
these mappings, which is used to deduce existence and uniqueness of solutions to (12) and
the stochastic evolution equation
(13) X(t) = S(t)X(0) + Zt
0
S(t−s)F X (s) ds+Zt
0
S(t−s)BdW(s),for t∈(0, T ].
When Sis a semigroup, this is the mild solution of (1). Here X(0) = ξis a Gaussian (possibly
deterministic) F0-measurable H-valued random variable. The stochastic integral is of the Itˆo
kind [7, Chapter 4].
Assumption 2.1. There is a constant C < ∞and functions a∈L1([0, T ],R), b ∈L2([0, T ],R)
such that kS(t)kL(H)≤C,kS(t)FkL(H)≤a(t) and kS(t)BkL2(U0,H)≤b(t) for all t∈[0, T ].
We look for a solution Kto (12) in the space C([0, T ],L1(H)) of continuous mappings with
values in L1(H). This is a Banach space with norm kfk∞,L1(H)= supt∈[0,T ]kf(t)kL1(H).
Proposition 2.2. Under Assumption 2.1, there is a unique solution K∈ C([0, T ],L1(H))
to (12) such that K(t)∈Σ(H)for all t∈[0, T ].
Proof. First we note that for v∈Hfixed, we may write
S(t)QξS(t)∗v−S(s)QξS(s)∗v=
∞
X
j=1
(hv, (S(t)−S(s))ejiH)S(t)µjej
+
∞
X
j=1hv, S(s)ejiH(S(s)−S(t))µjej,
(14)
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 7
where (ej)∞
j=1 is an orthonormal eigenbasis of Qξwith corresponding eigenvalues (µj)∞
j=1.
Since (µj)∞
j=1 ∈1, both sums are well-defined operators in L(H) applied to v. By (6) we get
kS(t)QξS(t)∗−S(s)QξS(s)∗kL1(H)≤
∞
X
j=1 k(S(t)−S(s))ejkHkµjS(t)ejkH
+
∞
X
j=1 k(S(t)−S(s))µjejkHkS(s)ejkH.
Hence S(·)QξS(·)∗∈ C([0, T ],L1(H)) as a consequence of Assumption 2.1 and the dominated
convergence theorem. By (11), the mapping s7→ S(t−s)B(S(t−s)B)∗takes values in the
separable Banach space L1(H) and it is continuous on [0, t] so that the Bochner integral of
it is well-defined. Similarly, for K ∈ C([0, T ],L1(H)), the mapping s7→ S(t−s)FK(s)S(t−
s)∗+S(t−s)K(s)(S(t−s)F)∗takes values in L1(H) and it can be seen to be continuous by
applying (5) along with a calculation similar to (14). The mapping
K 7→S(·)QξS(·)∗+Z·
0
S(· − s)FK(s)S(· − s)∗+S(· − s)K(s)(S(· − s)F)∗ds
+Z·
0
S(· − s)B(S(· − s)B)∗ds
(15)
from C([0, T ],L1(H)) into itself is therefore well-defined. By Assumption 2.1,
Zt
0
S(t−s)FK1(s)S(t−s)∗+S(t−s)K1(s)(S(t−s)F)∗ds
−Zt
0
S(t−s)FK2(s)S(t−s)∗+S(t−s)K2(s)(S(t−s)F)∗ds
L1(H)
.eσt Zt
0
a(t−s)e−σ(t−s)e−σskK1(s)− K2(s)kL1(H)ds
for arbitrary σ∈Rand K1,K2∈ C([0, T ],L1(H)). Therefore, since limσ→∞ RT
0a(s)e−σs ds=
0 by the dominated convergence theorem, the mapping (15) is a contraction with respect
to the norm defined by supt∈[0,T ]e−σtkK(t)kL1(H)for sufficiently large σ≥0. This norm is
equivalent to k·k∞,L1(H), so the Banach fixed point theorem yields existence and uniqueness
of Kas the limit of the sequence (Kn)∞
n=0 ⊂ C([0, T ],L1(H)). Here K0= 0 and Kn,n≥1,
is given by
Kn(t) = S(t)QξS(t)∗+Zt
0
S(t−s)F Kn−1(s)S(t−s)∗+S(t−s)Kn−1(s)(S(t−s)F)∗ds
+Zt
0
S(t−s)B(S(t−s)B)∗ds, t ∈[0, T ].
Clearly K0(t)∈Σ(H) for all t∈[0, T ]. By induction, Kn(t)∈Σ(H) for all n∈N,t∈[0, T ].
Since convergence of (Kn)∞
n=0 in C([0, T ],L1(H)) yields convergence of (Kn(t))∞
n=0 in L(H) we
have hK(t)u, viH= limnhKn(t)u, viHfor all t∈[0, T ] and u, v ∈H. Therefore, K(t)∈Σ(H)
for all t∈[0, T ].
Remark 2.3.It is possible to relax the assumption on Sto include the case that the mappings
t7→ S(t)v∈H,t7→ S(t)F v ∈Hand t7→ S(t)B∈ L2(U0, H ) are discontinuous on a
subset of [0, T ] with Lebesgue measure zero. One reason for doing this could be to treat the
8 M. KOV´
ACS, A. LANG, AND A. PETERSSON
advection term in (2) as a linear perturbation F(cf. [19, Example 2.22]), which would lead
to t7→ S(t)F v ∈Hbeing discontinuous at t= 0, Sbeing the semigroup generated by the
elliptic operator of (2). For notational convenience, we instead choose to assume continuity
of the mappings in this section and to, in Section 3.1, treat the advection term in (2) as part
of an elliptic operator.
The next proposition confirms that the solution Xto the stochastic evolution equation (13)
is Gaussian at all times t∈[0, T ]. As a consequence, K(t) therefore determines the distri-
bution of X(t) when Sis a semigroup, since by Theorem 2.5 below, K(t) = Cov(X(t)) for
t∈[0, T ].
Proposition 2.4. Under Assumption 2.1, there is a unique solution X∈ C([0, T ], L2(Ω, H))
to (13) and X(t)is Gaussian for all t∈[0, T ].
Proof. In the case that Sis a semigroup, ξis deterministic and F= 0, the result is well-
known, see, e.g., [7, Theorem 5.2]. We only sketch the proof in our general case. Existence and
uniqueness of a solution to (13) follow from a Banach fixed point theorem as in Proposition 2.2,
using the Itˆo isometry for the stochastic integral [19, Theorem 2.25]. In particular, we have
existence and uniqueness of the process
X1=S(·)ξ+Z·
0
S(· − s)F X1(s) ds
in C([0, T ], L2(Ω, H )) and of the process
X2=Z·
0
S(· − s)F X2(s) ds+Z·
0
S(· − s)BdW(s)
in C([0, T ], L2(Ω, H )) as limits of iterative sequences (Xn
1)∞
n=0 and (Xn
2)∞
n=0 as in the proof
of Proposition 2.2, with X0
1=X0
2= 0. Since Xn
1(t) is obtained from a linear and bounded
transformation of ξ, it is Gaussian for each t∈[0, T ]. For Xn
2, one can use an induc-
tive argument along with the stochastic Fubini theorem [19, Theorem 4.18] to see that
there is a function ψn: [0, T ]×[0, T ]→ L2(U0, H ), continuous in each argument, with
supt∈[0,T ]kψn(t, ·)kL2([0,t],L2(U0,H)) <∞such that Xn
2(t) = Rt
0ψn(t, s) dW(s) for all t∈[0, T ].
Therefore Xn
2(t) is also Gaussian for each t∈[0, T ]. Since limnXn
1(t)⊕Xn
2(t) = X1(t)⊕X2(t)
in L2(Ω, H ⊕H), (X1, X2) is a jointly Gaussian pair, from which the result follows.
We now prove our main result, connecting the equations (12) and (13).
Theorem 2.5. Let Cov(ξ) = Qξand let Sbe a C0-semigroup satisfying Assumption 2.1.
Then, the process K= (Cov(X(t)))t∈[0,T ], where Xis given by (13), is the solution of (12).
Proof. We first suppose that X(0) = ξ= 0 so that Cov(X(t)) = E[X(t)⊗X(t)]. By an argu-
ment analogous to (4), the continuity of Ximplies that Cov(X(·)) ∈ C([0, T ],L1(H)). Below,
we will make several interchanges of integration, summation and expectation. These are al-
lowed by Fubini’s theorem, using Assumption 2.1 and the fact that supt∈[0,T ]kX(t)kL2(Ω,H)<
∞. Let (ej)∞
j=1 and (fj)∞
j=1 be orthonormal bases of Hand U0, respectively. By Assump-
tion 2.1, the mild stochastic integration by parts formula [6, Example 1] is applicable with
a continuous bilinear functional ϕgiven by ϕ(x, y) = hx, eiiHhy, ejiHfor x, y ∈H,i, j ∈N.
Using also the zero expectation property of the Itˆo integral and (8), we find that
hCov(X(t)), ei⊗ejiL2=E[hX(t), eiiHhX(t), ejiH]
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 9
=Zt
0
E[hS(t−s)F X (s), eiiHhS(t−s)X(s), ejiH] ds
+Zt
0
E[hS(t−s)X(s), eiiHhS(t−s)F X (s), ejiH] ds
+
∞
X
n=1 Zt
0hS(t−s)Bfn, eiiHhS(t−s)Bfn, ejiHds,
for t∈[0, T ], i, j ∈N. By (8), (9) and the definition of h·,·iU0, this is equal to
Zt
0hE[S(t−s)F X (s)⊗S(t−s)X(s)], ei⊗ejiL2(H)ds
+Zt
0hE[S(t−s)X(s)⊗S(t−s)F X (s)], ei⊗ejiL2(H)ds
+Zt
0h(S(t−s)B)∗ei,(S(t−s)B)∗ejiHds
=Zt
0hS(t−s)FCov(X(s))S(t−s)∗, ei⊗ejiL2(H)ds
+Zt
0hS(t−s)Cov(X(s))(S(t−s)F)∗, ei⊗ejiL2(H)ds
+Zt
0hS(t−s)B(S(t−s)B)∗, ei⊗ejiL2(H)ds.
By applying h·, ei⊗ejiHto (12) and using (8), we obtain similarly that
hK(t), ei⊗ejiL2(H)=Zt
0hS(t−s)F K (s)S(t−s)∗, ei⊗ejiL2(H)ds
+Zt
0hS(t−s)K(s)(S(t−s)F)∗, ei⊗ejiL2(H)ds
+Zt
0hS(t−s)B(S(t−s)B)∗, ei⊗ejiL2(H)ds.
Combining this with the previous result yields
hK(t)−Cov(X(t)), ei⊗ejiL2(H)
=Zt
0hS(t−s)F(K(s)−Cov(X(s))) S(t−s)∗, ei⊗ejiL2(H)ds
+Zt
0hS(t−s) (K(s)−Cov(X(s))) (S(t−s)F)∗, ei⊗ejiL2(H)ds.
This shows that
K(t)−Cov(X(t)) =
∞
X
i,j=1hK(t)−Cov(X(t)), ei⊗ejiL2(H)ei⊗ej
=Zt
0
S(t−s)F(K(s)−Cov(X(s))) S(t−s)∗
+S(t−s) (K(s)−Cov(X(s))) (S(t−s)F)∗ds
10 M. KOV´
ACS, A. LANG, AND A. PETERSSON
so that K(t) = Cov(X(t)) for all t∈[0, T ] by uniqueness Kin (12).
For the general case ξ6= 0, we write X=X1+X2as in the proof of Proposition 2.4.
From the fact that X1(t)⊕X2(t) = limnXn
1(t)⊕Xn
2(t) in L2(Ω, H ⊕H), we find that
Cov(hX1(t), uiH,hX2(t), viH) = limnCov(hXn
1(t), uiH,hXn
2(t), viH) = 0. This implies that
Cov(X1(t), X2(t)) = Cov(X2(t), X1(t)) = 0 for arbitrary t∈[0, T ]. Since X2(0) = 0,
Cov(X2(t)) = Zt
0
S(t−s)FCov(X2(s))S(t−s)∗+S(t−s)Cov(X2(s))(S(t−s)F)∗ds
+Zt
0
S(t−s)B(S(t−s)B)∗ds
as a consequence of what we have already shown. A similar argument using the mild Itˆo
formula of [6] yields
Cov(X1(t)) = S(t)QξS(t)∗
+Zt
0
S(t−s)FCov(X1(s))S(t−s)∗+S(t−s)Cov(X1(s))(S(t−s)F)∗ds
for all t∈[0, T ]. The proof is completed by noting that
Cov(X1(t) + X2(t)) = Cov(X1(t)) + Cov(X1(t), X2(t)) + Cov(X2(t), X1(t)) + Cov(X2(t)).
Remark 2.6.As a consequence of the theorem, K(t)∈Σ+(H) for all t∈[0, T ].
We finish this section with a general error decomposition formula with respect to the
integral equation (12) and an approximation of the semigroup S. For this we consider a
family ˜
S= ( ˜
S(t))t∈[0,T ]of operators in L(ˆ
H, H) such that the mappings t7→ ˜
S(t)v∈H,
t7→ ˜
S(t)F v ∈Hand t7→ ˜
S(t)B∈ L2(U0, H) are continuous almost everywhere on [0, T ]
for all v∈H. If ˜
Salso satisfies Assumption 2.1 and we consider a function ˆ
K: [0, T ]→
L1(H)∩Σ(H)) that is continuous almost everywhere on [0, T ], then ˜
K(t), given by
˜
K(t) = ˜
S(t)Qξ˜
S(t)∗+Zt
0
˜
S(t−s)Fˆ
K(s)˜
S(t−s)∗+˜
S(t−s)ˆ
K(s)( ˜
S(t−s)F)∗ds
+Zt
0
˜
S(t−s)B(˜
S(t−s)B)∗ds,
(16)
is well-defined since the integrands are measurable mappings with values in the separable
Banach space L1(H). In the error decomposition formula we consider ˜
K(t), defined by (16),
as an approximation of K(t), t∈[0, T ].
Proposition 2.7. Let, for t∈[0, T ],K(t)be given by (12) and ˜
K(t)by (16). Then, with
O+(s) = S(s) + ˜
S(s)and O−(s) = S(s)−˜
S(s)for s∈[0, t],
kK(t)−˜
K(t)kLi(H)≤ kO−(t)QξO+(t)kLi(H)
+ 2 Zt
0kS(t−s)F(K(s)−ˆ
K(s))S(t−s)∗kLi(H)ds
+Zt
0kO−(t−s)Fˆ
K(s)(O+(t−s))∗kLi(H)ds
+Zt
0kO+(t−s)Fˆ
K(s)(O−(t−s))∗kLi(H)ds
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 11
+Zt
0kO−(t−s)B(O+(t−s)B)∗kLi(H)dsfor i∈ {1,2}.
Proof. The proposition is a straightforward consequence of the triangle inequality, (7), the
fact that K(t),˜
K(t)∈Σ(H) for t∈[0, T ], and the identity
(17) Γ1˜
ΓΓ∗
1−Γ2˜
ΓΓ∗
2=1
2(Γ1+ Γ2)˜
Γ(Γ1−Γ2)∗+ (Γ1−Γ2)˜
Γ(Γ1+ Γ2)∗
for Γ1,Γ2,˜
Γ∈ L(H).
3. Applications
We apply the theory of the previous section to two concrete stochastic equations, a sto-
chastic advection-diffusion equation and the stochastic wave equation. Fully discrete approx-
imation schemes are analysed and numerical simulations are provided for illustration.
3.1. A stochastic advection–diffusion equation. Let D ⊂ Rd,d= 1,2,3 be a bounded
domain. The SPDE we consider in this section is formally given by
(18) dX(t, x) + AX(t, x) dt= dW(t, x) for t∈(0, T ], x ∈ D,
X(0, x) = ξ(x),for x∈ D,
for a random initial condition X(0) = ξand an operator
A=−
d
X
i,j=1
∂
∂xi
ai,j
∂
∂xj
+
d
X
j=1
aj
∂
∂xj
+a0.
We consider either Dirichlet, Neumann or Robin boundary conditions and we let Wbe a
generalized Wiener process in H=L2(D) with covariance operator Q∈Σ+(H). We assume
that ξis an H-valued F0-measurable Gaussian random variable with covariance Qξ. The
coefficients ai,j, ai,i, j = 1, . . . , d, and a0are functions on ¯
Dfulfilling ai,j =aj,i. We assume
that for there is some λ0>0 such that Pd
i,j=1 ai,j (x)yiyj≥λ0|y|2for all y∈Rdand x∈¯
D,
so that Ais elliptic.
To put (18) into our framework, we follow [9] and introduce the spaces Vand Has subspaces
of the Sobolev spaces H1=H1(D) and H2=H2(D), respectively. In the Dirichlet case we set
V=H=H1
0={u∈H1:u= 0 on ∂D}. In the Robin case (Neumann boundary conditions
being a special case thereof ), we set V=H1(D) and H={u∈H2:∂u/∂νΛ+σu = 0 on ∂D},
where σ:∂D → Ris a sufficiently smooth function and
∂u
∂νΛ
=
d
X
i,j=1
niai,j
∂u
∂xj
,
with n= (n1, . . . , nd) being the outward unit normal to ∂D. We define a bilinear form
λ:V×V→Rassociated with Aby
λ(u, v) = ZD
d
X
i,j=1
ai,j
∂u
∂xi
∂v
∂xj
+
d
X
j=1
aj
∂u
∂xj
v+a0uv dx+Z∂D
σuv dx,
where the last term is dropped in the Dirichlet case. If the coefficients are bounded, then
|λ(u, v)|≤kukVkvkVso that we may associate an operator Λ: V→V∗to λby λ(u, v) =
12 M. KOV´
ACS, A. LANG, AND A. PETERSSON
V∗hΛu, viV. By Riesz’s representation theorem, we obtain a Gelfand triple V⊂H⊂V∗. We
restrict Λ to dom(Λ) = {u∈V: Λu∈H}without changing notation. When Dhas Lipschitz
boundary, one may show (using the trace inequality [11, Theorem 1.5.1.3] if necessary) that
there are two constants ˜
λ0>0, c0≥0 such that λ(u, u)≥˜
λ0kuk2
H1−c0kuk2
Hfor u∈V.
We add the term c0X(t, x) to both sides of (18). Then the associated bilinear form a(·,·) =
λ(·,·) + c0h·,·iHis coercive. In the case that σ≥0, the constant c0may, for example, be
chosen as any number fulfilling
(19) c0>supx∈D Pd
j=1 |aj(x)|
4λ0−inf
x∈D c(x),
for an arbitrary ∈(0,1). In the Dirichlet case, we may pick c0= 0 if aj= 0 for all
j= 1, . . . , d and c(x)≥0. With this, the SPDE (18) is put into the form of (1) by letting
A= Λ + c0Iand F=c0I. The adjoint operator A∗is defined by associating it with the
bilinear form a∗, given by a∗(u, v) = a(v, u) for u, v ∈V. For smooth coefficients, we use
Green’s formula to see that A∗can be identified with the formal adjoint of Aperturbed by
c0(cf. [27, Section 2.1.3]). It is given by
A∗=−
d
X
i,j=1
∂
∂xi
ai,j
∂
∂xj−
d
X
j=1
aj
∂
∂xj−
d
X
j=1
∂aj
∂xj
+a0+c0.
In the Robin case, the boundary conditions of A∗change to the ones of the space
H∗=
u∈H2:∂u
∂νΛ
+
d
X
j=1
ajnj+σ
u= 0 on ∂D
,
while in the Dirichlet case, H∗=H. We also need the symmetrized operator A0associated
with the bilinear form a0= (a+a∗)/2. Like A∗,A0is identified with a differential operator
A0=−
d
X
i,j=1
∂
∂xi
ai,j
∂
∂xj−1
2
d
X
j=1
∂aj
∂xj
+a0+c0.
We set
H0=
u∈H2:∂u
∂νΛ
+
1
2
d
X
j=1
ajnj+σ
u= 0 on ∂D
,
in the Robin case and H0=Hin the Dirichlet case. With these notions in place, we introduce
an assumption of elliptic regularity.
Assumption 3.1. The coefficients ai,j, aj, a0, σ,i, j = 1, . . . , d, are sufficiently smooth and
Dis sufficiently regular, with ∂Dat least Lipschitz, to guarantee that dom(A) = H2∩H,
dom(A∗) = H2∩H∗and dom(A0) = H2∩H0. The equalities hold with equivalence of k·kH2
and the graph norms kA· kL(H),kA∗· kL(H)and kA0· kL(H), respectively.
We refer to [11] for details on when this assumption holds. It is satisfied when Dis a convex
polygon and ai,j , aj, a0, σ,i, j = 1, . . . , d are infinitely differentiable [9].
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 13
Since ais coercive, Ais a sectorial operator. Negative fractional powers of Aare therefore
well-defined as elements of L(H) given by
A−s
2=1
2πi Zγ
λ−s
2(λ−A)−1dλ,
where γis a counterclockwise oriented contour surrounding the spectrum of A. Positive
fractional powers are densely defined closed operators on Hdefined by As/2= (A−s/2)−1[27,
Section 2.1.7]. We note that (A∗)s/2= (As/2)∗for all s∈R. Moreover, by [13, Theorem 3.1]
and [22, Th´eor`eme 6.1] (applicable since Dhas Lipschitz boundary) we have dom(As/2) =
dom((A∗)s/2) = dom(As/2
0) for all s∈[0,1] with norm equivalence. For s= 1, these spaces
can also be identified with V.
It is convenient to express our regularity assumptions on Qnot in terms of fractional powers
of A, which is usually the case when Ais self-adjoint, but of A0. As A0is positive definite with
a compact inverse (a consequence of [27, Theorem 1.38] since dom(A0)⊂H2), its fractional
powers can be characterized in a simple way by the spectral theorem, cf. [19, Appendix B.2].
For s≥0, we write ˙
Hsfor the Hilbert space dom(As/2
0). Moreover, Hilbert spaces ˙
H−sare
well-defined as completions of sequences in Hwith respect to k·k˙
H−s=kA−s/2
0· kH. In
this way, we obtain a set ( ˙
Hs)s∈Rof Hilbert spaces, with ˙
Hα→˙
Hsfor α > s, continuously
and densely. Lemma 2.1 in [4] allows us to, for all α, s ∈R, extend As/2
0to an operator in
L(˙
Hα,˙
Hα−s), and we do so without changing notation. Note, that for v∈Hand s∈[0,1],
kA−s
2
0vkH= sup
w∈H
kwkH=1 hA−s
2
0v, wiH
= sup
w∈H
kwkH=1 hA−s
2v, (As
2)∗A−s
2
0wiH
≤ kA−s
2vkHsup
w∈H
kwkH=1
k(As
2)∗A−s
2
0wkH.kA−s
2vkH,
by the equivalence dom(As/2
0) = dom((A∗)s/2). Similarly,
(20) kA−s
2vkH.kA−s
2
0vkH.
This is true also for A∗. Therefore, As/2and (A∗)s/2can be considered as operators in
L(˙
Hs, H) for s∈[−1,1].
The operator −Ais the infinitesimal generator of a uniformly bounded analytic semigroup
Son H[9], which yields a mild solution (13) to the SPDE (18). The semigroup maps into
dom(As) for all s≥0, with Sand Ascommutative on dom(As). The stability estimate
(21) kAsS(t)kL(H).t−s
is satisfied for t > 0. Moreover, as seen in [27, Section 2.7.7],
kA−s(S(t)−I)kL(H).ts, s ∈[0,1], t ≥0.(22)
We now make the following assumption on Q.
14 M. KOV´
ACS, A. LANG, AND A. PETERSSON
Assumption 3.2. There is a constant r∈(0,1] such that
kA
r−1
2
0Q1
2kL2(H)=kQ1
2kL2(H, ˙
Hr−1)<∞.
With this assumption in place, we have for t > 0 that
kS(t)kL2(U0,H)=kA1−r
2S(t)Ar−1
2Q1
2kL2(H)≤ kA1−r
2S(t)kL(H)kAr−1
2Q1
2kL2(H).tr−1
2.
The last inequality is a consequence of (20) and (21). To have Assumption 2.1 fulfilled, we
set B=Iand F=c0I. Here, the inclusion Bis regarded as an operator in U0=Q1/2(H)
to ˆ
H=˙
Hr−1,Fas an operator from Hto ˙
Hr−1and S(t) as an operator from ˙
Hr−1to H.
Continuity of (0, T ]3t7→ S(t)B∈ L2(U0, H) is a consequence of the fact kS(t)kL2(U0,H)<∞
for all t∈(0, T ] along with continuity of the mapping [0, T ]3t7→ S(t)vfor all v∈U0=
Q1/2(H)⊂H. With this Assumption 2.1 is fulfilled, so that by Proposition 2.4 and the
fact that Sis a semigroup, a predictable mild solution to (1) exists. Proposition 2.2 and
Theorem 2.5 yield a unique solution Kto (12) such that K(t) = Cov(X(t)), t∈[0, T ].
We now move on to approximation of (12). For this, we consider the same approximation of
Sas in [9] and assume from here on that Dis a convex polygon. For the spatial discretization,
we let (Vh)h∈(0,1] ⊂H1be a standard family of finite element spaces consisting of piecewise
linear polynomials with respect to a regular family of triangulations of Dwith maximal mesh
size h, vanishing on ∂Din the Dirichlet case. We assume the mesh to be quasi-uniform.
The spaces are equipped with h·,·iVh=h·,·iH. On this space, let Ah:Vh→Vhbe given
by hAhvh, uhi˙
H0=a(vh, uh) for all vh, uh∈Vh. Since Ahis sectorial, fractional powers
of it are defined in the same way as for A. We write A0,h for the operator defined in the
same way using the bilinear form a0. Note that A∗
hcoincides with the operator defined
using a∗. By Ph:˙
H−1→Vhwe denote the generalized orthogonal projector defined by
hPhx, yhiH=hA−1/2
0x, A1/2
0yhiHfor x∈˙
H−1, yh∈Vh. Since a0is a symmetric form, we have
kA1/2
0,h PhvkH=kA1/2
0PhvkH.kPhvkV. The mesh of Vhis assumed to be quasi-uniform, so
kPhkL(V)<∞[9, Proposition 3.2], [19, Example 3.6]. The interpolation arguments of [2]
then imply that for s∈[−1,1] there is a constant C < ∞such that for all h∈(0,1],
(23) kA
s
2
0,hPhA−s
2
0kL(H)≤C.
Moreover, by [13, Theorem 3.1] (see also the proof of [9, Theorem 5.3]), for all s∈[0,1) there
is a constant C < ∞such that for all h∈(0,1], kA
s
2
hA−s
2
0,h PhkL(H)≤Cand
(24) kA−s
2
hA
s
2
0,hPhkL(H)≤C.
We use the backward Euler method for the temporal discretization of S. For a time step
∆t∈(0,1] let (tj)j∈N0⊂Rbe given by tj= ∆tj and N∆t+ 1 = inf{j∈N:tj/∈[0, T ]}.
We write Sh,∆t= (I+ ∆tAh)−1. The discrete family (Sj
h,∆t)j∈{0,...,N∆t}of powers of Sh,∆t
acts as a fully discrete approximation of S. For brevity, we write Sj
h,∆tfor Sj
h,∆tPh. For all
s∈[0,1), there is a constant C < ∞such that for all h, ∆t∈(0,1] and j= 1, . . . , N∆t,
kAs
hSj
h,∆tkL(H)≤Ct−s
j, see [9, (8.7)]. We define an interpolation ˜
Sh,∆t: [0, T ]→ L(H) of
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 15
(Sj
h,∆t)j∈{0,...,N∆t}by
(25) ˜
Sh,∆t(t) = χ{0}(t) +
N∆t
X
j=1
χ(tj−1,tj](t)Sj
h,∆t, t ∈[0, T ],
where χdenotes the indicator function. We immediately obtain that for all s∈[0,1) there is
a constant C < ∞such that for all h, ∆t∈(0,1] and t∈(0, T ]
(26) kAs
h˜
Sh,∆t(t)kL(H)≤Ct−s.
The next lemma gives an error bound for the semigroup approximation. In the Dirichlet
case, a proof is given in [1, Lemma 5.1]. In our general case, it is a consequence of the split
k(S(t)−˜
Sh,∆t(t))vkH≤ k(S(t)−S(tj))vkH+k(S(tj)−Sh(tj)Ph)vkH
+k(Sh(tj)Ph−˜
Sh,∆t(tj))vkH,
for t∈(tj−1, tj], j= 1, . . . , N∆t, and v∈H. Here Shis the semigroup on Vhgenerated
by Ah[9, Section 7]. The first term can be bounded by (21), (22) and (20) and the third
by using an integral representation of the error as in the proof of [9, Theorem 8.2]. For the
second term, one employs a similar interpolation argument as in [1, Lemma 5.1], making use
of [9, Theorem 7.1] and [19, Lemma 3.8(iii)]. The latter result is, again, proven for Dirichlet
boundary conditions, but the arguments are the same in our general case, cf. [9, Remark 7.2].
Lemma 3.3. For all θ∈[0,2] and s∈[0,2−θ]∩[0,1), there is a constant C < ∞such that
for all h, ∆t∈(0,1] and t > 0
k(S(t)−˜
Sh,∆t(t))kL(˙
H−s,H)=k(S(t)−˜
Sh,∆t(t))A
s
2
0kL(H)≤C(hθ+ ∆tθ
2)t−θ+s
2.
We now define an approximation ( ˜
Kh,∆t(tj))N∆t
j=0 of (12) by ˜
Kh,∆t(0) = PhQξPhand
(I+ ∆tAh)˜
Kh,∆t(tj)(I+ ∆tAh)∗
=˜
Kh,∆t(tj−1)+∆tF ˜
Kh,∆t(tj−1)+∆t˜
Kh,∆t(tj−1)F∗+ ∆tPhB(PhB)∗, j ≥1.
With our choice of F, this can equivalently be written as
(27) (I+ ∆tAh)˜
Kh,∆t(tj)(I+ ∆tAh)∗= (1 + 2c0∆t)˜
Kh,∆t(tj−1)+∆tPhQPh.
To implement this scheme, we have to know the value of c0explicitly. Several choices are
possible, see (19) for an example. A closed form of ˜
Kh,∆t(tj), j = 0, . . . , N∆t,is given by
˜
Kh,∆t(tj) = ˜
Sh,∆t(tj)Qξ˜
Sh,∆t(tj)∗
+Ztj
0
˜
Sh,∆t(tj−s)F˜
Kh,∆t(bsc∆t)˜
S∗
h,∆t(tj−s) ds
+Ztj
0
˜
Sh,∆t(tj−s)˜
Kh,∆t(bsc∆t)( ˜
Sh,∆t(tj−s)F)∗ds
+Ztj
0
˜
Sh,∆t(tj−s)B(˜
Sh,∆t(tj−s)B)∗ds.
(28)
16 M. KOV´
ACS, A. LANG, AND A. PETERSSON
Here b·c∆t=b·/∆tc∆t, with b·c denoting the floor function. The L1(H)-norm of the last
term can be bounded by (11), Assumption 3.2 and (26) via
Ztj
0
˜
Sh,∆t(tj−s)B(˜
Sh,∆t(tj−s)B)∗ds
L1(H)
≤Ztj
0k˜
Sh,∆t(tj−s)Bk2
L2(U0,H)ds
=Ztj
0k˜
Sh,∆t(tj−s)A
1−r
2
hA
r−1
2
hA
1−r
2
0,h A
r−1
2
0,h PhA
1−r
2
0A
r−1
2
0Q1
2k2
L2(H)ds
.kA
r−1
2
hA
1−r
2
0,h k2
L(H)kA
r−1
2
0,h PhA
1−r
2
0k2
L(H)kA
r−1
2
0Q1
2k2
L2(H)Ztj
0
(t−s)r−1ds.tr.Tr.
(29)
Here we also made use of (23) and (24). Using this result along with (26) applied to the other
terms of (28) we find that
k˜
Kh,∆t(tj)kL1(H).1 +
j−1
X
j=0 k˜
Kh,∆t(tj)kL1(H).
Along with the fact that Qξ∈ L1(H), the discrete Gronwall lemma [10, 2.2 (9)] implies that
sup
h,∆t∈(0,1]
sup
j=0,...,N∆tk˜
Kh,∆t(tj)kL1(H)<∞.
With this estimate in place, we move on to the main result of this section.
Theorem 3.4. Let Assumption 3.2 be satisfied. For all θ < 2r, there is a constant C < ∞
such that for all h, ∆t∈(0,1]
sup
j=0,...,N∆tkK(tj)−˜
Kh,∆t(tj)kL1(H)≤C(hθ+ ∆tθ
2).
Moreover, for all θ < 1 + r, there is a constant C < ∞such that for all h, ∆t∈(0,1]
sup
j=0,...,N∆tkK(tj)−˜
Kh,∆t(tj)kL2(H)≤C(hθ+ ∆tθ
2).
Proof. The result is a consequence of Proposition 2.7 applied to kK(tj)−˜
Kh,∆t(tj)kLi(H).
We start with i= 1. For the last term of the error decomposition in this proposition, the
properties (10), (7) and (11), Lemma 3.3 and an argument similar to that of (29) imply that
Ztj
0kO−(tj−s)B(O+(tj−s)B)∗kL1(H)ds
≤Ztj
0kO−(tj−s)BkL2(H)k(O+(tj−s)B)∗kL2(H)ds
.Ztj
0k(S(tj−s)−˜
Sh,∆t(tj−s))A
1−r
2
0kL(H)kA
r−1
2
0Q1
2k2
L2(H)
×k˜
Sh,∆t(tj−s)A
1−r
2
hkL(H)+kS(tj−s)A1−r
2kL(H)ds
.(hθ+ ∆tθ
2)Ztj
0
(tj−s)r−θ−1ds.Tr−θ(hθ+ ∆tθ
2).
(30)
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 17
Similarly, the first, third and fourth term of the split in Proposition 2.7 can be bounded by
a constant times hθ+ ∆tθ/2, noting that F=c0I. By (21), the second term can be bounded
by
Ztj
0kS(tj−s)F(K(bsc∆t)−˜
Kh,∆t(bsc∆t))S(tj−s)∗kL1(H)ds
.∆t
j−1
X
j=0 k(K(tj)−˜
Kh,∆t(tj))kL1(H).
The proof for i= 1 is now completed by an application of the discrete Gronwall inequality.
In the case that i= 2, the only major difference is the calculation in (30). For this, let
us note that since hv, O+(tj−s)BuiH=hQO+(tj−s)∗v, uiU0for v∈H, u ∈U0, we have
O−(tj−s)B(O+(tj−s)B)∗=O−(tj−s)QO+(tj−s)∗. The adjoint on the right hand side
is taken with respect to L(H). Using this observation and the fact that Q1/2∈ L(H) yields
Ztj
0kO−(tj−s)B(O+(tj−s)B)∗kL2(H)ds
≤Ztj
0kO−(tj−s)Q1
2kL(H)kO+(tj−s)Q1
2kL2(H)ds
.Ztj
0k(S(tj−s)−˜
Sh,∆t(tj−s))kL(H)kQ1
2kL(H)
×k˜
Sh,∆t(tj−s)A
1−r
2
hkL(H)+kS(tj−s)A1−r
2kL(H)kA
r−1
2
0Q1
2kL2(H)ds
.(hθ+ ∆tθ
2)Ztj
0
(tj−s)r−1−θ
2ds.Tr+1−θ
2(hθ+ ∆tθ
2).
Example 3.5. We conclude this section by demonstrating the results of Theorem 3.4 in
the case that D= (0,1) and T= 1. We choose deterministic initial conditions so that
Qξ= 0. Homogeneous Neumann boundary conditions are considered and we set a1,1(x) = 4,
a1(x) = sin(2πx) and a0(x) = 0 for all x∈ D. With this, we may take c0= 1/8. From (27),
we obtain a matrix recursion
(Mh+ ∆tAh)Kj,h,∆t(Mh+ ∆tAh)∗= (1 + 2c0∆t)MhKj−1,h,∆tMh+ ∆tQh.
Here Kj,h,∆tis the matrix of coefficients (kj,m,n)Nh
m,n=1 in the expansion
˜
Kh,∆t(tj) =
Nh
X
m,n=1
kj,m,nφh
m⊗φh
n,
where (φh
m)Nh
m=1 is the usual ”hat function” basis of Vhwith Nh= dim(Vh). Moreover,
(Mh)i,j =hφh
i, φh
jiH, (Ah)i,j =a(φh
i, φh
j) and (Qh)i,j =hQφh
i, φh
jiHfor i, j = 1, . . . , Nh. We
solve this system of matrix equations for two choices of Qand decreasing values of h, ∆t.
First, we consider the white noise case, i.e., Q=I. By our choice of b, the operator A0
retains the Neumann boundary conditions of A. Lemma 2.3 in [17] then implies that As-
sumption 3.2 holds for all r < 1/2. By Theorem 3.4, we therefore expect to see a convergence
rate essentially of order 1 and 3/2, respectively, if we plot the errors kK(T)−˜
Kh,∆t(T)kLi(H),
i∈ {1,2}, for h=√∆t= 2−1,...,2−7. This agrees with the results of Figure 3(a).
18 M. KOV´
ACS, A. LANG, AND A. PETERSSON
10-2 10 -1 10 0
h
10-5
10-4
10-3
10-2
10-1
error
(a) Errors with white noise.
10-2 10 -1 10 0
h
10-6
10-5
10-4
10-3
10-2
10-1
error
(b) Errors with exponential kernel noise.
Figure 3. Approximate errors kK(T)−˜
Kh,∆t(T)kLi(H),i∈ {1,2}, for the
equations of Example 3.5.
In place of K(T) we used ˜
Kh0,∆t0(T) at h0=√∆t0= 2−8. The errors were computed by
k˜
Kh,∆t(T)−˜
Kh0,∆t0(T)kL1(H)= Tr pNh,h0KN∆t,h,∆t0
0−KN∆t0,h0,∆t0pNh,h0
and
k˜
Kh,∆t(T)−˜
Kh0,∆t0(T)k2
L2(H)= Tr (KN∆t,h,∆tMh)2
−2 Tr KN∆t,h,∆tMh,h0KN∆t0,h0,∆t0Mh0,h
+ Tr KN∆t0,h0,∆t0Mh02.
Here (Mh,h0)i,j =hφh
i, φh0
jiHfor i= 1, . . . , Nh,j= 1, . . . , Nh0while
Nh,h0=MhMh0,h
Mh,h0Mh0.
Next, we consider the case that Qis an integral operator defined by
hQu, viH=ZD×D
q(x, y)u(x)v(y) dxdy, u, v ∈H.
We set q(x, y) = exp(−2|x−y|) for x, y ∈ D. Figure 1(a) shows an approximate realization
of the solution to (18) for this q. Figure 2(a) shows its covariance function, corresponding
to ˜
Kh,∆t(t), at t= 0.1. Since Tr(Q)<∞, Assumption 3.2 holds for r≤1, so we expect a rate
of order 2 for both norms, in the same setup as before. Again, this agrees with Figure 3(b).
APPROXIMATION OF SPDE COVARIANCES BY FINITE ELEMENTS 19
Remark 3.6.If one would instead directly compute an approximation ˜
Xh,∆tbased on the same
semigroup approximation ˜
Sh,∆tand compute its covariance by the method of [25], we would
directly get a bound on the covariance error kK(T)−Cov(˜
Xh,∆t(T))kLi(H)by the strong error
kX(t)−˜
Xh,∆t(T)kL2(Ω,H), see (4). In the first case above, when Q=I, this means that the
error would be bounded by hr+kr/2for r < 1/2, which is a lower rate than the estimates
obtained