School on
Stochastic Partial Differential Equations
Part 2
Stig Larsson
Department of Mathematical Sciences
Chalmers University of Technology and University of Gothenburg
Organized by EU Horizon 2020 Research and Innovation Staff Exchange program CHiPS
March 7–10, 2017
1 / 59
Outline
I
I
Quick review of strong convergence analysis.
Weak convergence analysis.
2 / 59
Semigroup approach
3 / 59
Semigroup approach
Linear SPDE with additive noise:
(
dX (t) + AX (t) dt = B dW (t), t > 0
X (0) = X0
I
(Ω, F, P, {Ft }t≥0 ), filtered probability space
I
H, U Hilbert spaces
I
{W (t)}t≥0 , Q-Wiener process in U with respect to {Ft }t≥0
I
{X (t)}t≥0 , stochastic process in H
I
B : U → H, bounded linear operator
I
E (t) = e−tA , t ≥ 0, C0 -semigroup of bounded linear operators on H
I
X0 is an F0 -measurable H-valued random variable
3 / 59
Mild solution
(
dX (t) + AX (t) dt = B dW (t), t > 0
X (0) = X0
The unique solution is given by (mild solution)
Z
t
E (t − s)B dW (s)
X (t) = E (t)X0 +
0
4 / 59
Stochastic heat equation

∂u


 ∂t (ξ, t) − ∆u(ξ, t) = Ẇ (ξ, t),
u(ξ, t) = 0,



u(ξ, 0) = u0 ,
(
dX + AX dt = dW ,
ξ ∈ D ⊂ Rd , t > 0
ξ ∈ ∂D, t > 0
ξ∈D
t>0
X (0) = X0
I
I
I
I
I
I
H = U = L2 (D), k·k, (·, ·), D ⊂ Rd , bounded domain
A = Λ = −∆, D(Λ) = H 2 (D) ∩ H01 (D), B = I
probability space (Ω, F, P)
W (t), Q-Wiener process on H
X (t), H-valued stochastic process
E (t) = e −tA analytic semigroup generated by −A
Mild solution (stochastic convolution):
Z t
X (t) = E (t)X0 +
E (t − s) dW (s),
t≥0
0
5 / 59
The finite element method
(
I
triangulations {Th }0<h<1 , mesh size h
I
finite element spaces {Sh }0<h<1 ,
I
Sh continuous piecewise poly degree ≤ r − 1, r ≥ 2
I
Xh (t) ∈ Sh ; (dXh , χ) + (∇Xh , ∇χ) dt = (dW , χ) ∀χ ∈ Sh , t > 0
I
Λh : Sh → Sh , discrete Laplacian, (Λh ψ, χ) = (∇ψ, ∇χ) ∀ψ, χ ∈ Sh
Sh ⊂ H01 (D) = Ḣ 1
I
Ah = Λh
I
Ph : L2 → Sh , orthogonal projection,
Xh (t) ∈ Sh ,
(Ph f , χ) = (f , χ) ∀χ ∈ Sh
Xh (0) = Ph X0
dXh + Ah Xh dt = Ph dW ,
t>0
Ph W (t) is Qh -Wiener process with Qh = Ph QPh .
Mild solution, with Eh (t) = e −tAh ,
Z t
Xh (t) = Eh (t)Ph X0 +
Eh (t − s)Ph dW (s)
0
6 / 59
Regularity and strong convergence
|v |β = kΛβ/2 v k =
∞
X
λβj (v , φj )2
1/2
,
Ḣ β = D(Λβ/2 ),
β∈R
j=1
Theorem. If kΛ(β−1)/2 Q 1/2 kHS < ∞ for some β ∈ [0, r ], then
kX (t)kL2 (Ω,Ḣ β ) ≤ C kX0 kL2 (Ω,Ḣ β ) + kΛ(β−1)/2 Q 1/2 kHS
kXh (t) − X (t)kL2 (Ω,H) ≤ Chβ kX0 kL2 (Ω,Ḣ β ) + kΛ(β−1)/2 Q 1/2 kHS
7 / 59
Regularity and strong convergence
|v |β = kΛβ/2 v k =
∞
X
λβj (v , φj )2
1/2
,
Ḣ β = D(Λβ/2 ),
β∈R
j=1
Theorem. If kΛ(β−1)/2 Q 1/2 kHS < ∞ for some β ∈ [0, r ], then
kX (t)kL2 (Ω,Ḣ β ) ≤ C kX0 kL2 (Ω,Ḣ β ) + kΛ(β−1)/2 Q 1/2 kHS
kXh (t) − X (t)kL2 (Ω,H) ≤ Chβ kX0 kL2 (Ω,Ḣ β ) + kΛ(β−1)/2 Q 1/2 kHS
Two cases:
I
If kQ 1/2 k2HS = Tr(Q) < ∞, then β = 1.
I
∂
(β−1)/2
If Q = I , d = 1, Λ = − ∂ξ
kHS < ∞ for β < 1/2.
2 , then kΛ
2
7 / 59
Proofs
The proofs are based on
I
Itô isometry
Z t
Z t
2
E
F (s) dW (s) = E
kF (s)Q 1/2 k2HS ds
0
I
0
Smoothing property (heat equation)
Z
t
kΛ1/2 E (s)v k2 ds ≤ C kv k2
0
I
Error estimates for the approximation of the semigroup
8 / 59
Proofs
The proofs are based on
I
Itô isometry
Z t
Z t
2
E
F (s) dW (s) = E
kF (s)Q 1/2 k2HS ds
0
I
0
Smoothing property (heat equation)
Z
t
kΛ1/2 E (s)v k2 ds ≤ C kv k2
0
I
Error estimates for the approximation of the semigroup
Similar for the stochastic wave equation.
8 / 59
Weak convergence
The law of Xh (T ):
µXh (T ) = P ◦ Xh (T )−1
converges weakly to the law of X (T ), if
hµXh (T ) , ϕi → hµX (T ) , ϕi
as h → 0
∀ϕ ∈ Cb (H, R)
9 / 59
Weak convergence
The law of Xh (T ):
µXh (T ) = P ◦ Xh (T )−1
converges weakly to the law of X (T ), if
hµXh (T ) , ϕi → hµX (T ) , ϕi
as h → 0
∀ϕ ∈ Cb (H, R)
Since
Z
hµXh (T ) , ϕi =
Z
ϕ(x) dµXh (T ) (x) =
H
ϕ(Xh (T , ω)) dP(ω) = E ϕ(Xh (T )) ,
Ω
this means
E ϕ(Xh (T )) → E ϕ(X (T )) as h → 0
∀ϕ ∈ Cb (H, R).
9 / 59
Weak convergence
Test functions:
ϕ ∈ Cb (H, R) = continuous and bounded functions (functionals)
But we will use
ϕ ∈ Cb2 (H, R) = not necessarily bounded but with
continuous and bounded Fréchet derivatives Dϕ and D 2 ϕ
10 / 59
Weak convergence
Test functions:
ϕ ∈ Cb (H, R) = continuous and bounded functions (functionals)
But we will use
ϕ ∈ Cb2 (H, R) = not necessarily bounded but with
continuous and bounded Fréchet derivatives Dϕ and D 2 ϕ
Our goal is now to show (essentially)
E ϕ(Xh (T )) − E ϕ(X (T )) = O(h2β )
as h → 0
∀ϕ ∈ Cb2 (H, R).
The weak rate is (essentially) twice the strong rate of convergence.
10 / 59
Weak convergence
We will prove this for linear problems (heat and wave equations).
But first we will perform formal calculations for the nonlinear problem
dX (t) + [AX (t) − F (X (t))] dt = G (X (t)) dW (t), t ∈ (0, T ];
X (0) = X0 ,
or in mild form
Z t
X (t) = E (t)X0 +
E (t − s)F (X (s)) ds
0
Z t
+
E (t − s)G (X (s)) dW (s), t ∈ [0, T ].
0
The semidiscrete approximation is
Z t
Xh (t) = Eh (t)Ph X0 +
Eh (t − s)Ph F (Xh (s)) ds
0
Z t
+
Eh (t − s)Ph G (Xh (s)) dW (s), t ∈ [0, T ].
0
11 / 59
Weak error representation: preliminaries
dX (t) + [AX (t) − F (X (t))] dt = G (X (t)) dW (t), t ∈ (0, T ];
X (0) = X0 ,
Auxiliary process Z (s) = Z (s; t, ξ): if ξ is Ft -measurable and
0≤t≤s≤T
Z s
Z s
Z (s) = E (s − t)ξ +
E (s − r )F (Z (r )) dr +
E (s − r )G (Z (r )) dW (r )
t
t
Define u : H × [0, T ] → R by
h
i
u(x, t) = E ϕ Z (T ; t, x) .
If ϕ ∈ Cb2 (H, R), then u is a solution to Kolmogorov’s equation
 0
00
∗
0
1

 ut (x, t) − hux (x, t), Ax − F (x)i + 2 Tr (uxx (x, t)G (x)QG (x) ) = 0,
t ∈ [0, T ), x ∈ D(A),


u(x, T ) = ϕ(x)
12 / 59
Weak convergence
If ξ is Ft -measurable and 0 ≤ t ≤ s ≤ T :
Z s
Z s
Z (s; t, ξ) = E (s − τ )ξ +
E (s − r )F (X (r )) dr +
E (t − r )G (X (r )) dW (r )
t
t
Define u : H × [0, T ] → R by
h
i
u(x, t) = E ϕ Z (T ; t, x) .
With random Ft -measurable input ξ:
h
i
u(ξ, t) = E ϕ Z (T ; t, ξ) |Ft
Hence
h
h h
i
ii
E[u(ξ, t)] = E E ϕ Z (T ; t, ξ) |Ft = E ϕ Z (T ; t, ξ) .
13 / 59
Weak convergence
So we have
h
i
E[u(ξ, t)] = E ϕ Z (T ; t, ξ) .
Note also
Z (T ; t, ξ) = Z (T ; s, Z (s; t, ξ))
Then
h
i
E[u(ξ, t)] = E ϕ Z (T ; t, ξ)
h
h
i
i
= E ϕ Z (T ; s, Z (s; t, ξ)) = E u(s, Z (s; t, ξ)) ,
that is, the expected value of u is constant along trajectories
y = Z (s; t, ξ),
s ∈ [t, T ].
14 / 59
Weak convergence
Assume Xh (0) = X (0) for simplicity.
E ϕ(Xh (T )) − ϕ(X (T )) = E u(Xh (T ), T ) − u(X (T ), T )
= E u(Xh (T ), T ) − u(X (0), 0) = E u(Xh (T ), T ) − u(Xh (0), 0)
Z T
00
d[Xh , Xh ])
Itô’s formula: = E
(ut0 dt + hux0 , dXh i + 12 uxx
0
Z
T
n
=E
ut0 (Xh (t), t) − hux0 (Xh (t), t), Ah Xh (t) − Ph F (Xh (t))i
0
o
00
1
+ 2 Tr[uxx
(Xh (t), t)Ph G (Xh (t))QG (Xh (t))∗ Ph ] dt
Kolm. eq: ut0 (Xh (t), t) = hux0 (Xh (t), t), AXh (t) − F (Xh (t))i
00
− 12 Tr[uxx
(Xh (t), t)G (Xh (t))QG (Xh (t))∗ ]
Z Tn
=E
− hux0 (·, t), (Ah − A)Xh (t) − (Ph − I )F (Xh (t))i
0
00
o
+ 21 Tr uxx
(·, t)[Ph G (·)QG (·)∗ Ph − G (·)QG (·)∗ ] dt,
where · = Xh (t).
15 / 59
Weak convergence
h
i
u(x, t) = E ϕ Z (T ; t, x) .
The derivative ux0 (x, t) ∈ H is given by
h
i
hux0 (x, t), φi = E hϕ0 Z (T ; t, x) , Zx0 (T ; t, x)φi
h
i
= E hZx0 (T ; t, x)∗ ϕ0 Z (T ; t, x) , φi
So, in order to bound norms of
ux0 (x, t) = E Zx0 (T ; t, x)∗ ϕ0 Z (T ; t, x) = E[η(t; t, x)],
we must study the linearized adjoint equation:
∗
0
Z
η(s) = E (T − s) ϕ (Z (T ; t, x)) +
T
E (T − r )∗ F 0 (Z (r ; t, x))η(r ) dr
s
Z
+
T
E (T − r )∗ G 0 (Z (r ; t, x))η(r ) dW (r )
s
The second derivative is related to the second adjoint variation.
16 / 59
Weak convergence
Let us compute ux0 (x, t) in the simplest case, the linear case:
Z
h
h i
u(x, t) = E ϕ Z (T ; t, x) = E ϕ E (T − t)x +
T
i
E (T − s)B dW (s)
t
Then
Z T
hD Ei
hux0 (x, t), φi = E ϕ0 E (T − t)x +
E (T − s)B dW (s) , E (T − t)φ
t
h
i
∗ 0
= E hE (T − t) ϕ Z (T ; t, x) , φi
h
i
so that ux0 (x, t) = E E (T − t)∗ ϕ0 Z (T ; t, x) = E η(t; t, x) . Here
h
i
η(s) = η(s; t, x) = E E (T − s)∗ ϕ0 Z (T ; t, x) is the solution of the
∗
adjoint equation, recall E (T − s)∗ = e−(T −s)A ,
η̇(s) − A∗ η(s) = 0, s ≤ T ;
η(T ) = ϕ0 Z (T ; t, x) .
h
i
00
Similarly, we have uxx
(x, t) = E E (T − t)∗ ϕ00 Z (T ; t, x) E (T − t) .
17 / 59
Weak convergence
Another difficulty: the Kolmogorov equation is proved only for x ∈ D(A).
 0
0
00
∗
1

 ut (x, t) − hux (x, t), Ax − f (x)i + 2 Tr (uxx (x, t)g (x)Qg (x) ) = 0,
t ∈ [0, T ), x ∈ D(A),


u(x, T ) = ϕ(x)
Project onto the eigenspaces of A. Auxiliary process Zm (s) = Zm (s; t, x):
Z s
Zm (s) = Em (s − t)Pm ξ +
Em (s − r )Pm f (Zm (r )) dr
t
Z s
+
Em (s − r )Pm g (Zm (r )) dW (r ).
t
Define um : H × [0, T ] → R by
h
i
um (x, t) = E ϕ Zm (T ; t, x) .
Then um (x, t) = um (Pm x, t), to be used with x = Xh (t).
The Kolmogorov equation is now well-defined.
Must verify that additional terms vanish as m → ∞.
18 / 59
Weak convergence
The first term in the weak error:
Z T
E
−hux0 (Xh (t), t), (Ah − A)Xh (t) − (Ph − I )F (Xh (t))i dt,
0
For the heat equation, we have here A = Λ, Ah = Λh , so that
hux0 , (Ah − A)Xh i = h(Ah Ph − A)ux0 , Xh i
0
= hAh Ph (A−1 − A−1
h Ph )Aux , Xh i
0
= h(A−1 − A−1
h Ph )Aux , Ah Xh i
This is expressed with the “elliptic” error (A−1 − A−1
h Ph ), which is small.
But the operators are badly distributed between the factors. For the heat
equation this can be handled (to some extent) by rewriting by means of
Malliavin calculus.
19 / 59
Weak convergence
Here we try to explain why the operators on the previous slide are “badly
distributed”. We compute for the linear heat equation:
0
hux0 , (Ah − A)Xh i = h(A−1 − A−1
h Ph )Aux , Ah Xh i,
h
i
ux0 (x, t) = E E (T − t)ϕ0 Z (T ; t, x) ,
Aux0 (Xh (t), t) = E AE (T − t)ϕ0 Z (T ; t, Xh (t)) |Ft ,
2
kA−1 − A−1
h Ph kL(H) ≤ Ch ,
kAE (T − t)kL(H) ≤ C (T − t)−1 .
20 / 59
Weak convergence
Hence, the bad term becomes
Z T
hux0 (Xh (t), t), (Ah − A)Xh (t)i dt E
0
Z
= E
T
Z
= E
T
D
E 0
(A−1 − A−1
P
)Au
(X
(t),
t),
A
X
(t)
dt h
h
h h
x
h
D
E 0
(A−1 − A−1
h Ph )E AE (T − t)ϕ Z (T ; t, Xh (t)) |Ft , Ah Xh (t) dt 0
Z
0
T
0
kA−1 − A−1
h Ph kL(H) kAE (T − t)kL(H) sup kϕ (x)kH
≤C
x∈H
0
× kAh Xh (t)kL2 (Ω,H) dt
≤ Ch2
Z
0
T
(T − t)−1 dt |ϕ|Cb1 sup kAh Xh (t)kL2 (Ω,H) .
t∈[0,T ]
Here: sup kAh Xh (t)kL2 (Ω,H) < ∞ if kA
β−1
2
1
Q 2 kHS with β = 2
t∈[0,T ]
(regularity of order β = 2). But the rate is only h2 = hβ , not h2β .
21 / 59
Weak convergence
Here we have not been able to exploit the possibility for the integral to
absorb a singularity also at t = 0, i.e.,
Z
T
(T − t)−1 t −1 dt
(almost convergent).
0
This can be achieved by an integration by parts from the Malliavin
calculus.
22 / 59
Weak convergence: the linear case
This explains some difficulties encountered in connection with the
nonlinear problem.
The story is more complete for the linear problem:
(
dX (t) + AX (t) dt = B dW (t), t > 0
X (0) = X0
I will present this now for the heat and wave equations.
23 / 59
Weak convergence: the linear case
This explains some difficulties encountered in connection with the
nonlinear problem.
The story is more complete for the linear problem:
(
dX (t) + AX (t) dt = B dW (t), t > 0
X (0) = X0
I will present this now for the heat and wave equations.
We use a trick introduced by
De Bouard and Debussche (nonlinear Schrödinger equation)
[de Bouard and Debussche(2006)].
Debussche and Printems (linear heat equation)
[Debussche and Printems(2009)].
23 / 59
Weak convergence: the linear case
This explains some difficulties encountered in connection with the
nonlinear problem.
The story is more complete for the linear problem:
(
dX (t) + AX (t) dt = B dW (t), t > 0
X (0) = X0
I will present this now for the heat and wave equations.
We use a trick introduced by
De Bouard and Debussche (nonlinear Schrödinger equation)
[de Bouard and Debussche(2006)].
Debussche and Printems (linear heat equation)
[Debussche and Printems(2009)].
The trick is: Remove the troublesome term (Ah − A)Xh by means of an
integrating factor. We present this before turning to the Malliavin
calculus.
23 / 59
Weak error representation: preliminaries
Apply the integrating factor E (T − t) to get Y (t) = E (T − t)X (t):
dY (t) = E (T − t)B dW (t), t ∈ (0, T ]; Y (0) = E (T )X0 ,
with mild solution
Z
t
E (T − s)B dW (s).
Y (t) = E (T )X0 +
0
Similarly, consider
dYh (t) = Eh (T − t)B dW (t), t ∈ (0, T ]; Yh (0) = Eh (T )Ph X0 ,
with mild solution
Z
t
Eh (T − s)Bh dW (s).
Yh (t) = Eh (T )Ph X0 +
0
Note: X (T ) = Y (T ), Xh (T ) = Yh (T ).
No drift term in eq. for Y and Yh .
24 / 59
Weak error representation: preliminaries
Auxiliary problem: Z (s) = Z (s; t, ξ), ξ is a Ft -measurable,
dZ (s) = E (T − s)B dW (s), s ∈ (t, T ]; Z (t) = ξ.
Z
s
E (T − r )B dW (r ).
Unique mild solution: Z (s; t, ξ) = ξ +
t
Define u : H × [0, T ] → R by u(x, t) = E [ϕ(Z (T ; t, x))] .
The partial derivatives are:
ux0 (x, t) = E ϕ0 (Z (T ; t, x)) ,
00
uxx
(x, t) = E ϕ00 (Z (T ; t, x)) .
If ϕ ∈ Cb2 (H, R), then u is a solution to Kolmogorov’s equation
(
00
ut0 (x, t) + 12 Tr (uxx
(x, t)E (T − t)BQ[E (T − t)B]∗ ) = 0, t ∈ [0, T ), x ∈ H
u(x, T ) = ϕ(x)
25 / 59
Weak error representation
THEOREM. If
Tr
Z
T
E (t)BQ[E (t)B]∗ dt < ∞
0
and ϕ ∈
Cb2 (H, R),
then the weak error
eh (T ) = E[ϕ(Xh (T ))] − E[ϕ(X (T ))]
has the representation
eh (T ) = E [u(Yh (0), 0) − u(Y (0), 0)]
Z T 1
00
+ E
Tr uxx
(Yh (t), t)
2 0
× [Eh (T − t)Bh + E (T − t)B]Q[Eh (T − t)Bh − E (T − t)B]∗ dt
= E [u(Yh (0), 0) − u(Y (0), 0)]
Z T 1
00
Tr uxx
(Yh (t), t)
+ E
2 0
× [Eh (T − t)Bh − E (T − t)B]Q[Eh (T − t)Bh + E (T − t)B]∗ dt.
26 / 59
Weak convergence: proof
Use Itô formula and Kolmogorov equation as before:
E[ϕ(Xh (T ))] − E[ϕ(X (T ))]
= E[ϕ(Yh (T ))] − E[ϕ(Y (T ))]
h
i
= E u(Yh (T ), T ) − u(Y (T ), T )
h
i
h
i
= E u(Yh (T ), T ) − u(Yh (0), 0) + E u(Yh (0), 0) − u(Y (0), 0)
Z Tn
ut0 (Yh (t), t)
= E [u(Yh (0), 0) − u(Y (0), 0)] + E
0
o
1 00
+ Tr uxx
(Yh (t), t)[Eh (T − t)Bh ]Q[Eh (T − t)Bh ]∗ dt
2
Z T 1
00
= E [u(Yh (0), 0) − u(Y (0), 0)] + E
Tr uxx
(Yh (t), t)
2 0
× [Eh (T − t)Bh ]Q[Eh (T − t)Bh ]∗ − E (T − t)BQB ∗ E (T − t)∗ dt.
27 / 59
Weak convergence: proof
Here the expression
00
[S, T ] = Tr(uxx
SQT ∗ )
is symmetric:
00
00
[S, T ] = Tr(uxx
SQT ∗ ) = Tr(SQT ∗ uxx
)
00 ∗
00
= Tr([SQT ∗ uxx
] ) = Tr(uxx
TQS ∗ ) = [T , S],
00
because Q, uxx
are selfadjoint and Tr(S ∗ ) = Tr(S), Tr(ST ) = Tr(TS).
Hence, we have a conjugate rule
[S + T , S − T ] = [S, S] − [T , T ].
Therefore,
00
Tr uxx
(ξ, r ) [Eh (s)Bh ]Q[Eh (s)Bh ]∗ − [E (s)B]Q[E (s)B]∗
00
= Tr uxx
(ξ, r )[Eh (s)Bh + E (s)B]Q[Eh (s)Bh − E (s)B]∗
00
= Tr uxx
(ξ, r )[Eh (s)Bh − E (s)B]Q[Eh (s)Bh + E (s)B]∗ .
Note, by the way, that Bh ∈ L(U, H) with Bh : U → Sh , Eh (s) : Sh → Sh ,
and we consider Eh (s)Bh ∈ L(U, H). Hence, [Eh (s)Bh ]∗ 6= Bh∗ Eh (s)∗ .
28 / 59
Weak convergence: heat equation
Here A = Λ, B = I , Ah = Λh , Bh = Ph .
dX + ΛX dt = dW , t > 0;
X (0) = X0 ,
(1)
dXh + Λh Xh dt = Ph dW , t > 0;
Xh (0) = Ph X0 .
(2)
Theorem
Let X and Xh be the solutions of (1) and (2), respectively. Let
β−1
β−1
1
1
ϕ ∈ Cb2 (H, R) and assume that kA 2 Q 2 kHS = kΛ 2 Q 2 kHS < ∞ for
some β ∈ (0, 1]. Then there are C > 0, h0 > 0, depending on ϕ, X0 , Q,
β, and T but not on h, such that for h ≤ h0 ,
E ϕ(Xh (T )) − ϕ(X (T )) ≤ Ch2β | log(h)|.
If, in addition X0 ∈ L1 (Ω, Ḣ 2β ), then C is independent of T as well.
29 / 59
Proof
eh (T ) = E [u(Yh (0), 0) − u(Y (0), 0)]
Z T 1
00
Tr uxx
(Yh (t), t)
+ E
2 0
× [Eh (T − t)Ph + E (T − t)]Q[Eh (T − t)Ph − E (T − t)]∗ dt
Approximation of the semigroup:
k(Eh (t)Ph − E (t))v k = kFh (t)v k ≤ Chs t −
s−γ
2
|v |γ ,
0 ≤ γ ≤ s ≤ r.
30 / 59
Proof
In the initial error we have
Yh (0) − Y (0) = Eh (T )Ph X0 − E (T )X0 = Fh (T )X0 ,
so that
E (u(Yh (0), 0) − u(Y (0), 0))
Z 1
=E
hux0 (Y (0) + s(Yh (0) − Y (0)), 0), Yh (0) − Y (0)i ds
0
Z
=E
1
hux0 (E (T )X0 + sFh (T )X0 , 0), Fh (T )X0 i ds.
0
Thus, recalling ux0 (x, t) = E ϕ0 (Z (T ; t, x)) ,
|E (u(Yh (0), 0) − u(Y (0), 0))| ≤ sup kux0 (x, 0)k E kFh (T )X0 k
x∈H
2β
≤ Ch T
− 2β−γ
2
E |X0 |γ sup kϕ0 (x)k,
0 ≤ γ ≤ 2β.
x∈H
If γ = 2β there is no dependence on T .
31 / 59
Proof
The main term: use | Tr(ST )| ≤ kSkHS kT kHS
Z T 00
Tr uxx
(Yh (t), t)
E
0
× [Eh (T − t)Ph + E (T − t)]Q[Eh (T − t)Ph − E (T − t)]∗ dt Z T 00
= E
Tr uxx
(Yh (t), t)[Eh (T − t)Ph + E (T − t)]∗
0
β−1
β−1
1−β
1−β
1
1
× A 2 A 2 Q 2 Q 2 A 2 A 2 Fh (T − t) dt Z T 1−β
00
= E
Tr uxx
(Yh (t), t)(A 2 [Eh (T − t)Ph + E (T − t)])∗
0
β−1
β−1
1−β
1
1
× A 2 Q 2 Q 2 A 2 A 2 Fh (T − t) dt Z T
1−β
β−1
1
00
kuxx
(Yh (t), t)(A 2 [Eh (T − t)Ph + E (T − t)])∗ A 2 Q 2 kHS
≤E
0
1
× kQ 2 A
β−1
2
A
1−β
2
Fh (T − t)kHS dt
32 / 59
Proof
Use kST kHS ≤ kSkL kT kHS :
Z T
β−1
1−β
1
00
··· ≤ E
kuxx
(Yh (t), t)(A 2 [Eh (T − t)Ph + E (T − t)])∗ A 2 Q 2 kHS
0
1
× kQ 2 A
≤
β−1
2
A
1−β
2
Fh (T − t)kHS dt
00
kuxx
(x, t)kL(H) kA
sup
β−1
2
1
Q 2 k2HS
(x,t)∈H×[0,T ]
T
Z
×
kA
1−β
2
(Eh (t)Ph + E (t))kL(H) kA
1−β
2
Fh (t)kL(H) dt.
0
Here
sup
00
kuxx
(x, t)kL(H) ≤ sup kϕ00 (x)kL(H) .
x∈H
(x,t)∈H×[0,T ]
Recall
1
1
kA 2 vh k = k∇vh k = kAh2 vh k,
δ
kA vh k ≤
kAδh vh k,
vh ∈ Sh ,
vh ∈ Sh , δ ∈ [0, 12 ],
kAδ (Eh (t)Ph + E (t))kL(H) ≤ Ce −ωt t −δ ,
δ=
1−β
2
∈ [0, 12 ].
33 / 59
Proof
Now consider kA
1−β
2
Fh (t)kL(H) . Analyticity:
kAδ Fh (t)kL(H) ≤ Ct −δ ,
δ ∈ [0, 12 ].
Approximation:
kFh (t)v k ≤ Chs t −
s−γ
2
|v |γ ,
0 ≤ γ ≤ s ≤ r.
Hence
kA
1−β
2
1
2β −
Fh (t)kL(H) ≤ kFh (t)kβL(H) kA 2 Fh (t)k1−β
L(H) ≤ Ch t
1+β
2
,
β ∈ [0, 1].
Therefore, for β ∈ (0, 1] one may estimate the above integral:
Z T
1−β
1−β
kA 2 (Eh (t)Ph + E (t))kL(H) kA 2 Fh (t)kL(H) dt
0
Z 2 Z !
h
T
kA
+
=
≤C
(Eh (t)Ph + E (t))kL(H) kA
1−β
2
Fh (t)kL(H) dt
h2
0
Z
1−β
2
h2
t
− 1−β
− 1−β
2
2
t
Z
T
dt + C
0
e −ωt t −
1−β
2
h2β t −
1+β
2
dt ≤ Ch2β | log(h)|
h2
and the proof is complete.
34 / 59
Weak convergence: heat equation
By inspection of the above proof we see that the error estimate is
E ϕ(Xh (T )) − ϕ(X (T )) 2β−γ
≤ Ch2β T − 2 E |X0 |γ sup kϕ0 (x)kH
x∈H
2β
+ Ch | log(h)|β
−1
sup kϕ00 (x)kL(H) kA
x∈H
β−1
2
1
Q 2 k2HS .
35 / 59
Weak convergence: heat equation
By inspection of the above proof we see that the error estimate is
E ϕ(Xh (T )) − ϕ(X (T )) 2β−γ
≤ Ch2β T − 2 E |X0 |γ sup kϕ0 (x)kH
x∈H
2β
+ Ch | log(h)|β
−1
sup kϕ00 (x)kL(H) kA
x∈H
β−1
2
1
Q 2 k2HS .
The previous theorem does not allow β > 1.
This is satisfactory if the order of the FEM is r = 2.
Under a slightly stronger condition on A and Q we now extend the result
to the case β > 1.
35 / 59
Weak convergence: heat equation
Theorem
Let X and Xh be the solutions of (1) and (2), respectively. Let
ϕ ∈ Cb2 (H, R) and assume that kAβ−1 QkTr = kΛβ−1 QkTr < ∞ for some
β ∈ [1, 2r ]. Then there are C > 0, h0 > 0, depending on ϕ, X0 , Q, β, and
T but not on h, such that for h ≤ h0 ,
E ϕ(Xh (T )) − ϕ(X (T )) ≤ Ch2β | log(h)|.
If, in addition X0 ∈ L1 (Ω, Ḣ 2β ), then C is independent of T as well.
This theorem differs in the assumption about Q. According to the
theorem on “alternative conditions” in the first part of my lectures we
have
kΛ
β−1
2
1
Q 2 k2HS ≤ kΛβ−1 QkTr .
Thus, the new condition implies the previous one. If Λ and Q commute,
then they coincide.
36 / 59
Proof
The initial error term is treated as before. For the main term we
distribute factors differently:
Z T 00
Tr uxx
(Yh (t), t)
E
0
× [Eh (T − t)Ph − E (T − t)]Q[Eh (T − t)Bh + E (T − t)B]∗ dt Z T 00
= E
Tr uxx
(Yh (t), t)
0
× Fh (t)A1−β Aβ−1 Q[Eh (T − t)Ph + E (T − t)]∗ dt Z T
00
β−1
≤C
sup
kuxx (x, t)kL(H) kA
QkTr
kFh (t)Aβ−1 kL(H) e −ωt dt.
(x,t)∈H×[0,T ]
0
Hence,
Z T
Z
1−β
−ωt
kFh (t)A
kL(H) e
dt =
0
0
Z
≤C
h2β
dt + Ch
0
2β
Z
h2β
Z
T
+
kFh (t)A1−β kL(H) e −ωt dt
h2β
T
t −1 e −ωt dt ≤ Ch2β | log(h)|.
h2β
37 / 59
Weak convergence: the wave equation
Recall the notation:
0 −I
A :=
,
Λ 0
0
B :=
,
I
E (t) = e−tA =
X
X := 1 ,
X2
C (t)
−Λ1/2 S(t)
X0,1
X0 :=
,
X0,2
Λ−1/2 S(t)
,
C (t)
where C (t) = cos(tΛ1/2 ) and S(t) = sin(tΛ1/2 ).
Spatially discrete:
0 −I
0
Ah :=
, Bh :=
, Xh0 = Ph X0 .
Λh 0
Ph
"
Eh (t) = e
1/2
−tAh
Ch (t)
=
1/2
−Λh Sh (t)
#
Sh (t)
,
Ch (t)
−1/2
Λh
1/2
with Ch (t) = cos(tΛh ), Sh (t) = sin(tΛh ).
38 / 59
Weak convergence: the wave equation
Theorem
1
1
Let g ∈ Cb2 (Ḣ 0 , R) and assume that kΛβ− 2 QΛ− 2 kTr < ∞ and that
X0 ∈ L1 (Ω, H 2β ) for some β ∈ [0, r +1
2 ]. Then, there are C > 0, h0 > 0,
depending on g , X0 , Q, and T but not on h, such that for h ≤ h0 ,
r
2β
E g (Xh,1 (T )) − g (X1 (T )) ≤ Ch r +1
.
Note: the test function g depends on the first component X1 only.
Again the new condition on Q implies the previous one:
kΛ
β−1
2
1
1
1
Q 2 k2HS ≤ kΛβ− 2 QΛ− 2 kTr .
Therefore, the rate of weak convergence is twice the rate of strong
convergence.
39 / 59
Proof
We only make a brief discussion of the main term.
The error operator for the first component is
−1
1
Kh (t) := Λh 2 Sh (t)Ph − Λ− 2 S(t).
We have
r
kKh (t)w k ≤ C (T )h r +1 s |w |s−1 ,
or
kKh (t)Λ
1−s
2
w ∈ Ḣ s−1 , s ∈ [0, r + 1],
r
v k ≤ C (T )h r +1 s kv k,
v ∈ Ḣ 1−s .
We use s = 2β:
1
r
kKh (t)Λ 2 −β kL(Ḣ 0 ) ≤ C (T )h r +1 2β ,
t ∈ [0, T ], 2β ∈ [0, r + 1].
We use a test function of the form
ϕ(x) := g (P1 x) = g (x1 ),
for x = [x1 , x2 ]> ∈ H = Ḣ 0 × Ḣ −1 .
40 / 59
Proof
The main term is
Z T
00
Tr uxx
(Yh (t), t)
E
0
× [Eh (T − t)Bh + E (T − t)B]Q[Eh (T − t)Bh − E (T − t)B]∗ dt The integrand simplifies to (with s = T − t)
00
(Yh (t), t)[Eh (s)Bh + E (s)B]Q[Eh (s)Bh − E (s)B]∗ E Tr uxx
00
(Yh (t), t)∗ = E Tr [Eh (s)Bh − E (s)B]Q[Eh (s)Bh + E (s)B]∗ uxx
1
−1
= E Tr Kh (s)Q[Λh 2 Sh (s)Ph + Λ− 2 S(s)]g 00 (P1 Z (T ; t, Yh (t))) 1
1
1
≤ kKh (s)Λ 2 −β kL(Ḣ 0 ) kΛβ− 2 QΛ− 2 kTr
1
−1
1
× kΛ 2 [Λh 2 Sh (s)Ph + Λ− 2 S(s)]kL(Ḣ 0 ) sup kg 00 (x)kL(Ḣ 0 ) .
x∈Ḣ 0
≤ C (T )h
r
r +1 2β
β− 12
kΛ
QΛ
− 12
kTr sup kg 00 (x)kL(Ḣ 0 ) .
x∈Ḣ 0
41 / 59
Weak convergence: completely discrete
This weak error representation formula has been generalized so that it
applies to completely discrete approximations. Recall
Z t
X (t) = E (t)X0 +
E (t − s)B dW (s),
0
Z t
Y (t) = E (T − t)X (t) = E (T )X0 +
E (T − s)B dW (s),
0
X (T ) = Y (T ).
Assume that X̃ (T ) is the result of some temporal and spatial
approximation. Construct a process {Ỹ (t)}t∈[0,T ] of the form
Z
t
Ẽ (T − s)B̃ dW (s)
Ỹ (t) = Ẽ (T )X̃0 +
with X̃ (T ) = Ỹ (T ).
0
Here {Ẽ (t)}t∈[0,T ] ⊂ B(S, S) and B̃ ∈ B(U, S), where S is a Hilbert
subspace of H with the same norm (typically S = H or S is a
finite-dimensional subspace of H). Ẽ (t) can be obtained by time
interpolation of the time stepping operator.
42 / 59
Weak convergence
Theorem
If ϕ ∈ Cb2 (H, R), then the weak error e(T ) has the representation
h
i
e(T ) = E u(Ỹ (0), 0) − u(Y (0), 0)
Z T 00
Tr uxx
(Ỹ (t), t)O(t) dt,
+ 12 E
0
where
∗
O(t) = Ẽ (T − t)B̃ + E (T − t)B Q Ẽ (T − t)B̃ − E (T − t)B ,
or
∗
O(t) = Ẽ (T − t)B̃ − E (T − t)B Q Ẽ (T − t)B̃ + E (T − t)B .
This has been applied to fully discrete schemes for the linear heat, wave
and Cahn-Hilliard-Cook equations, [Debussche and Printems(2009)],
[Kovács et al.(2012a)], [Kovács et al.(2012b)],
[Lindner and Schilling(2012)].
43 / 59
Weak convergence: Malliavin calculus
I will now explain how the integration by parts from the Malliavin
calculus can be used. As we have seen this is not needed for linear
problems, but the main difficulty occurs already there, so I will present
the argument for the linear heat equation.
Assume for simplicity that X0 = 0, F = 0, G = I , so that
Z t
X (t) =
E (t − s) dW (s),
0
Z t
Eh (t − s)Ph dW (s)
Xh (t) =
0
and the weak error
h
i
E ϕ(Xh (T )) − ϕ(X (T ))
Z Tn
=E
− hux0 (Xh (t), t), (Ah − A)Xh (t)i
0
00
o
(Xh (t), t)[Ph QPh − Q] dt.
+ 12 Tr uxx
44 / 59
Weak convergence: Malliavin calculus
We assume as usual, for some β ∈ [0, 1],
kA
β−1
2
1
Q 2 kHS = kA
β−1
2
kL02 < ∞.
To be specific, let β = 1:
kA
β−1
2
1
1
1
Q 2 kHS = kQ 2 kHS = kI kL02 = Tr(Q) 2 < ∞.
We have strong order hβ = h.
We want to obtain weak order h2β− = h2− .
45 / 59
Malliavin integration by parts
Theorem
For any random variable F ∈ D1,2 (H) and any predictable process
Φ ∈ L2 ([0, T ] × Ω, L02 (H)) the following integration by parts formula is
valid.
hD Z T
E i
hZ T i
E F,
Φ(s) dW (s)
=E
Ds F , Φ(s) L0 (H) ds
H
0
T
D Z
F,
Φ dW
0
E
2
0
D
E
= DF , Φ
L2 (Ω,H)
L2 ([0,T ]×Ω,L02 (H))
We will use this (essentially) with Φ(s) = Eh (t − s)Ph and
F = ux0 (Xh (t), t),
00
Ds F = (DF )(s) = Ds ux0 (Xh (t), t) = uxx
(Xh (t), t)Ds Xh (t),
where
Z
t
Eh (t − s)Ph dW (s),
Xh (t) =
Ds Xh (t) = Eh (t − s)Ph ,
0
and
00
uxx
(Xh (t), t) = E E (T − t)ϕ00 Z (T ; t, Xh (t))E (T − t) |Ft .
46 / 59
Malliavin
The difficult term:
Z T
hux0 (Xh (t), t), (Ah − A)Xh (t)i dt E
0
Z
= E
T
Z
=
Z t
hD
Ei E Kh Aux0 (Xh (t), t), Ah
Eh (t − s)Ph dW (s) dt 0
T
0
h
i
0
h(A−1 − A−1
Kh = A−1 − A−1
h Ph )Aux , Ah Xh i dt h Ph
0
Z
=
T
hZ t D
E i
Kh ADs ux0 (Xh (t), t), Ah Eh (t − s)Ph 0 ds dt E
Z
=
T
hZ t
i
00
E
hKh Auxx
(Xh (t), t)Ds Xh (t), Ah Eh (t − s)Ph iL02 ds dt 0
Z
≤
0
0
0
T
L2
0
hZ t
i
00
E
kKh Auxx
(Xh (t), t)Ds Xh (t)kL02 kAh Eh (t − s)Ph kL02 ds dt
0
47 / 59
Malliavin
Z
··· ≤ 0
T
hZ t
i
00
E
kKh Auxx
(Xh (t), t)Ds Xh (t)kL02 kAh Eh (t − s)Ph kL02 ds dt 0
hZ t
00
≤
E
kKh kL kAuxx
(Xh (t), t)kL kDs Xh (t)kL kI kL02
0
0
i
× kAh Eh (t − s)Ph kL kI kL02 ds dt
00
uxx
(Xh (t), t) = E E (T − t)ϕ00 Z (T ; t, Xh (t))E (T − t) |Ft
Z
T
Ds Xh (t) = Eh (t − s)Ph
Z TZ t
kKh kL kAE (T − t)kL |ϕ|Cb2 kE (T − t)kL kEh (t − s)Ph kL
≤
0
0
× kAh Eh (t − s)Ph kL ds dt kI k2L0
2
Z T
Z t
≤ Ch2
(T − t)−1
(t − s)−1 ds dt kI k2L0 |ϕ|Cb2
0
0
2
Almost convergent: correct it and lose .
48 / 59
Malliavin
Try again, with loss:
Z
··· = T
0
Z
=
Z
T
0
T
0
i
hZ t 00
(Xh (t), t)Ds Xh (t), A− 2 Ah Eh (t − s)Ph iL02 ds dt
E
hA 2 Kh A 2 A1− 2 uxx
0
Z
t
kA 2 Kh A 2 kL kA1− 2 E (T − t)kL |ϕ|Cb2 kE (T − t)kL kEh (t − s)Ph kL
≤
0
hZ t
i
00
E
hKh Auxx
(Xh (t), t)Ds Xh (t), Ah Eh (t − s)Ph iL02 ds dt 0
1− × kA− 2 Ah2 kL kAh 2 Eh (t − s)Ph kL ds dt kI k2L0
2
Z T
Z t
≤ Ch2−2
(T − t)−1+ 2
(t − s)−1+ 2 ds dt kI k2L0 |ϕ|Cb2 ≤ Ch2−2 .
0
0
2
Here kA− 2 Ah2 kL ≤ C , if we have a quasi-uniform mesh family, for
example.
49 / 59
Malliavin
00
In the nonlinear case we do not have formulas for uxx
(Xh (t), t) and
Ds Xh (t) and so we must write down the equations that they satisfy and
prove bounds for
00
(Xh (t), t)kL ,
kA1− 2 uxx
kDs Xh (t)kL02 .
The remaining term is easier:
Z T 00
Tr uxx
(Xh (t), t)[Ph QPh − Q] dt E
0
Z
= E
T
Z
= E
T
0
Z
0
T
≤E
00
Tr uxx
(Xh (t), t)[(Ph + I )Q(Ph − I )] dt 00
Tr A1− 2 uxx
(Xh (t), t)[(Ph + I )Q(Ph − I )A−1+ 2 ] dt 00
kA1− 2 uxx
(Xh (t), t)kL kPh + I kL Tr(Q)k(Ph − I )A−1+ 2 kL dt
0
≤ Ch2−
Z
T
(T − t)−1+ 2 dt Tr(Q).
0
50 / 59
Weak convergence: Malliavin
The above argument is not rigorous because the Kolmogorov equation is
not valid for x ∈ H. To handle this we project onto the eigenspaces of A
in order to get a finite dimensional Kolmogorov equation. Auxiliary
process Zm (s) = Zm (s; t, x):
Z s
Zm (s) = Em (s − t)Pm ξ +
Em (s − r )Pm dW (r ).
t
Define um : H × [0, T ] → R by
h
i
um (x, t) = E ϕ Zm (T ; t, x) .
Then um (x, t) = um (Pm x, t), to be used with x = Xh (t). The partial
derivatives are
h
i
0
um,x
(x, t) = E Em (T − t)Pm ϕ0 Zm (T ; t, x) ,
h
i
00
um,xx
(x, t) = E Em (T − t)Pm ϕ00 Zm (T ; t, x) Em (T − t)Pm .
51 / 59
Malliavin
Auxiliary process:
Z
Zm (s) = Em (s − t)Pm ξ +
s
Em (s − r )Pm dW (r ).
t
Define um : H × [0, T ] → R by
h
i
um (x, t) = E ϕ Zm (T ; t, x) .
Kolmogorov’s equation:
 0
0

 um,t (x, t) − hum,x (x, t), Am xi +


1
2
00
Tr um,xx
(x, t)Pm QPm = 0,
t ∈ [0, T ), x ∈ H,
u(x, T ) = ϕ(Pm x)
52 / 59
Malliavin
This leads to the weak error formula:
h
i
E ϕ(Xh (T )) − ϕ(X (T ))
Z Tn
0
=E
− hum,x
(Xh (t), t), (Ah − Am )Xh (t)i
0
00
o
(Xh (t), t)[Ph QPh − Pm QPm ] dt.
+ 12 Tr um,xx
In the first term we write
0
0
hum,x
, (Ah − Am )Xh i = hum,x
, (Ph Ah − Am Ph )Xh i
0
= h(Ah Ph − Ph Am )um,x
, Xh i
0
= hAh Ph (I − A−1
h Ph Am )um,x , Xh i
0
= hAh Ph (I −Pm + A−1 APm − A−1
h Ph APm )um,x , Xh i
0
= h(A−1 − A−1
h Ph )APm um,x , Ah Xh i
0
+ h(I − Pm )um,x
, Ah Xh i.
53 / 59
Malliavin
Similar treatment of the other term:
00
00
Tr um,xx
[Ph QPh − Pm QPm ] = Tr um,xx
[Ph + Pm ]Q[Ph − Pm ]
00
= Tr um,xx
[Ph + Pm ]Q[Ph − I + I − Pm ]
00
00
= Tr um,xx
[Ph + Pm ]Q[Ph − I ] + Tr um,xx
[Ph + Pm ]Q[I − Pm ] .
In both cases we get an extra term containing I − Pm .
For fixed h, let m → ∞, show that extra terms → 0. Then let h → 0.
54 / 59
Malliavin
The main term is
Z T
0
h(A−1 − A−1
E
h Ph )APm um,x , Ah Xh i dt Malliavin integration by parts...
0
T
Z
Z
≤
0
0
t
kA 2 Kh A 2 kL kA− 2 Am Em (T − t)Pm kL |ϕ|Cb2 kEm (T − t)Pm kL
1− × kEh (t − s)Ph kL kA− 2 Ah2 kL kAh 2 Eh (t − s)Ph kL ds dt kI k2L0
2
Z t
Z T
(t − s)−1+ 2 ds dt kI k2L0 |ϕ|Cb2 ≤ Ch2−2 ,
≤ Ch2−2
(T − t)−1+ 2
2
0
0
which is independent of m. The extra term becomes
Z T
0
h(I − Pm )um,x
, Ah Xh i dt E
0
Z
≤E
0
T
k(I − Pm )A−1+ kL kA1− Em (T − t)Pm kL |ϕ|Cb1 kAh Xh (t)kH dt
≤ C λ−1+
m
Z
0
T
(T − t)−1+ dt |ϕ|Cb1 kAh Ph kL sup kXh (t)kL2 (Ω,H) → 0,
t∈[0,T ]
as m → ∞ for fixed h.
55 / 59
Malliavin
More precisely,
h
i
E ϕ(Xh (T )) − ϕ(X (T )) ≤ Ch2−2 + Ch−2 λ−1+
+ other terms of the same form.
m
Therefore
h
i
E ϕ(Xh (T )) − ϕ(X (T )) ≤ Ch2−2 .
56 / 59
Malliavin
More precisely,
h
i
E ϕ(Xh (T )) − ϕ(X (T )) ≤ Ch2−2 + Ch−2 λ−1+
+ other terms of the same form.
m
Therefore
h
i
E ϕ(Xh (T )) − ϕ(X (T )) ≤ Ch2−2 .
This type of analysis has been carried out for the nonlinear heat equation:
I
I
I
Debussche [Debussche(2011)], multiplicative noise in 1–D,
time-stepping,
Wang and Gan [Wang and Gan(2012)], additive noise in multi–D,
time-stepping,
Andersson and L [Andersson and Larsson(2012)], additive noise in
multi–D, multiplicative noise in 1–D, spatial discretization.
56 / 59
References
A. Andersson and S. Larsson.
Weak convergence for a spatial approximation of the nonlinear
stochastic heat equation.
preprint 2012.
A. de Bouard and A. Debussche.
Weak and strong order of convergence of a semidiscrete scheme for
the stochastic nonlinear Schrödinger equation.
Appl. Math. Optim., 54:369–399, 2006.
A. Debussche.
Weak approximation of stochastic partial differential equations: the
nonlinear case.
Math. Comp., 80:89–117, 2011.
A. Debussche and J. Printems.
Weak order for the discretization of the stochastic heat equation.
Math. Comp., 78:845–863, 2009.
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References
A. Grorud and É. Pardoux.
Intégrales hilbertiennes anticipantes par rapport à un processus de
Wiener cylindrique et calcul stochastique associé.
Appl. Math. Optim., 25:31–49, 1992.
M. Kovács, S. Larsson, and F. Lindgren.
Weak convergence of finite element approximations of linear
stochastic evolution equations with additive noise.
BIT Numer. Math., 52:85–108, 2012a.
M. Kovács, S. Larsson, and F. Lindgren.
Weak convergence of finite element approximations of linear
stochastic evolution equations with additive noise ii. fully discrete
schemes.
BIT Numer. Math., 2012b.
doi:10.1007/s10543-012-0405-1.
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References
F. Lindner and R. L. Schilling.
Weak order for the discretization of the stochastic heat equation
driven by impulsive noise.
Potential Anal., 2012.
doi:10.1007/s11118-012-9276-y.
X. Wang and S. Gan.
Weak convergence analysis of the linear implicit euler method for
semilinear stochastic partial differential equations with additive
noise.
Journal of Mathematical Analysis and Applications, 2012.
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