Weak error of finite element approximations of a
nonlinear stochastic heat equation
Adam Andersson
(joint work with Stig Larsson)
Department of Mathematical Sciences
Chalmers University of Technology and University of Gothenburg
April 19, 2012
1 / 11
A semi-linear stochastic heat equation
The stochastic evolution equation
(
dX (t) + [AX (t) − f (X (t))] dt = g (X (t)) dW (t),
t ∈ (0, T ],
X (0) = X0 .
D ⊂ Rd convex polygonal domain. H = L2 (D), Q ∈ L(H) selfadjoint
nonnegative operator, {W (t)}t≥0 an H-valued Q-Wiener process on
(Ω, F, {Ft }t≥0 , P), U0 = Q 1/2 (H), L02 = HS(U0 , H), A = −∆,
D(A) = H01 (D) ∩ H 2 (D).
2 / 11
A semi-linear stochastic heat equation
The stochastic evolution equation
(
dX (t) + [AX (t) − f (X (t))] dt = g (X (t)) dW (t),
t ∈ (0, T ],
X (0) = X0 .
D ⊂ Rd convex polygonal domain. H = L2 (D), Q ∈ L(H) selfadjoint
nonnegative operator, {W (t)}t≥0 an H-valued Q-Wiener process on
(Ω, F, {Ft }t≥0 , P), U0 = Q 1/2 (H), L02 = HS(U0 , H), A = −∆,
D(A) = H01 (D) ∩ H 2 (D).
For γ ≥ 0, Ḣ γ = D(Aγ/2 ) and Ḣ −γ = (Ḣ γ )∗ .
2 / 11
A semi-linear stochastic heat equation
The stochastic evolution equation
(
dX (t) + [AX (t) − f (X (t))] dt = g (X (t)) dW (t),
t ∈ (0, T ],
X (0) = X0 .
D ⊂ Rd convex polygonal domain. H = L2 (D), Q ∈ L(H) selfadjoint
nonnegative operator, {W (t)}t≥0 an H-valued Q-Wiener process on
(Ω, F, {Ft }t≥0 , P), U0 = Q 1/2 (H), L02 = HS(U0 , H), A = −∆,
D(A) = H01 (D) ∩ H 2 (D).
For γ ≥ 0, Ḣ γ = D(Aγ/2 ) and Ḣ −γ = (Ḣ γ )∗ .
Let β ∈ ( 12 , 1] and α ∈ [0, β)
2 / 11
A semi-linear stochastic heat equation
The stochastic evolution equation
(
dX (t) + [AX (t) − f (X (t))] dt = g (X (t)) dW (t),
t ∈ (0, T ],
X (0) = X0 .
D ⊂ Rd convex polygonal domain. H = L2 (D), Q ∈ L(H) selfadjoint
nonnegative operator, {W (t)}t≥0 an H-valued Q-Wiener process on
(Ω, F, {Ft }t≥0 , P), U0 = Q 1/2 (H), L02 = HS(U0 , H), A = −∆,
D(A) = H01 (D) ∩ H 2 (D).
For γ ≥ 0, Ḣ γ = D(Aγ/2 ) and Ḣ −γ = (Ḣ γ )∗ .
Let β ∈ ( 12 , 1] and α ∈ [0, β)
I
X0 ∈ Ḣ α ,
2 / 11
A semi-linear stochastic heat equation
The stochastic evolution equation
(
dX (t) + [AX (t) − f (X (t))] dt = g (X (t)) dW (t),
t ∈ (0, T ],
X (0) = X0 .
D ⊂ Rd convex polygonal domain. H = L2 (D), Q ∈ L(H) selfadjoint
nonnegative operator, {W (t)}t≥0 an H-valued Q-Wiener process on
(Ω, F, {Ft }t≥0 , P), U0 = Q 1/2 (H), L02 = HS(U0 , H), A = −∆,
D(A) = H01 (D) ∩ H 2 (D).
For γ ≥ 0, Ḣ γ = D(Aγ/2 ) and Ḣ −γ = (Ḣ γ )∗ .
Let β ∈ ( 12 , 1] and α ∈ [0, β)
I
X0 ∈ Ḣ α ,
I
f ∈ Cb2 (H, H),
2 / 11
A semi-linear stochastic heat equation
The stochastic evolution equation
(
dX (t) + [AX (t) − f (X (t))] dt = g (X (t)) dW (t),
t ∈ (0, T ],
X (0) = X0 .
D ⊂ Rd convex polygonal domain. H = L2 (D), Q ∈ L(H) selfadjoint
nonnegative operator, {W (t)}t≥0 an H-valued Q-Wiener process on
(Ω, F, {Ft }t≥0 , P), U0 = Q 1/2 (H), L02 = HS(U0 , H), A = −∆,
D(A) = H01 (D) ∩ H 2 (D).
For γ ≥ 0, Ḣ γ = D(Aγ/2 ) and Ḣ −γ = (Ḣ γ )∗ .
Let β ∈ ( 12 , 1] and α ∈ [0, β)
I
X0 ∈ Ḣ α ,
I
f ∈ Cb2 (H, H),
I
g ∈ Cb2 (Ḣ α , HS(U0 , Ḣ β−1 )), i.e., kA
β−1
2
1
g (x)Q 2 kHS < ∞, ∀x ∈ Ḣ α ,
2 / 11
A semi-linear stochastic heat equation
The stochastic evolution equation
(
dX (t) + [AX (t) − f (X (t))] dt = g (X (t)) dW (t),
t ∈ (0, T ],
X (0) = X0 .
D ⊂ Rd convex polygonal domain. H = L2 (D), Q ∈ L(H) selfadjoint
nonnegative operator, {W (t)}t≥0 an H-valued Q-Wiener process on
(Ω, F, {Ft }t≥0 , P), U0 = Q 1/2 (H), L02 = HS(U0 , H), A = −∆,
D(A) = H01 (D) ∩ H 2 (D).
For γ ≥ 0, Ḣ γ = D(Aγ/2 ) and Ḣ −γ = (Ḣ γ )∗ .
Let β ∈ ( 12 , 1] and α ∈ [0, β)
I
X0 ∈ Ḣ α ,
I
f ∈ Cb2 (H, H),
I
g ∈ Cb2 (Ḣ α , HS(U0 , Ḣ β−1 )), i.e., kA
I
g 00 = 0.
β−1
2
1
g (x)Q 2 kHS < ∞, ∀x ∈ Ḣ α ,
2 / 11
Mild solution
Let {E (t)}t≥0 be the analytic semigroup generated by −A.
∃! solution X ∈ C ([0, T ], L2 (Ω, Ḣ β )) to the mild equation
Z t
E (t − s)f (X (s)) ds
X (t) = E (t)X0 +
0
Z t
+
E (t − s)g (X (s)) dW (s), t ∈ (0, T ].
0
3 / 11
Approximation by the finite element method
A discretized equation:
(
dXh (t) + [Ah Xh (t) − Ph f (Xh (t))] dt = Ph g (Xh (t)) dW (t),
t ∈ (0, T ]
Xh (0) = Ph X0 .
Finite element spaces {Sh }h∈(0,1] of continuous piecewise linear functions
corresponding to a quasi-uniform family of triangulations of D.
Ah is the discrete Laplacian satisfying
hAh ψ, χiH = h∇ψ, ∇χiH ,
∀ψ, χ ∈ Sh .
Ph : H → Sh orthogonal projection w.r.t. h·, ·iH .
4 / 11
Mild solution of discretized equation
Let {Eh (t)}t≥0 be the analytic semigroup generated by −Ah .
For every h ∈ (0, 1] ∃! solution Xh ∈ C ([0, T ], L2 (Ω, Sh )) to the mild
equation
Z t
Xh (t) = Eh (t)Ph X0 +
Eh (t − s)Ph f (X (s)) ds
0
Z t
+
Eh (t − s)Ph g (X (s)) dW (s), t ∈ (0, T ].
0
5 / 11
Malliavin integration by parts formula
For all F ∈ D1,2 (H) and Φ ∈ L2 (Ω × [0, T ], L02 )
Z T
D Z T
E
E F,
Φ(t) dW (t) = E
Tr{QΦ(t)∗ Dt F } dt
0
0
= hDF , ΦiL2 (Ω×[0,T ],L02 ) .
In particular for u ∈ C 2 (H, R)
Z
D
E ux (Xh (t)),
0
T
Z
E
Φ(s) dW (s) = E
T
Tr{QΦ(s)∗ uxx (F )Ds Xh (t)} ds.
0
6 / 11
Malliavin integration by parts formula
For all F ∈ D1,2 (H) and Φ ∈ L2 (Ω × [0, T ], L02 )
Z T
D Z T
E
E F,
Φ(t) dW (t) = E
Tr{QΦ(t)∗ Dt F } dt
0
0
= hDF , ΦiL2 (Ω×[0,T ],L02 ) .
In particular for u ∈ C 2 (H, R)
Z
D
E ux (Xh (t)),
T
0
Dsu Xh (t)
T
Z
E
Φ(s) dW (s) = E
Tr{QΦ(s)∗ uxx (F )Ds Xh (t)} ds.
0
Z
t
= Eh (t − s)Ph σ(Xh (s))u +
Eh (t − r )Ph f 0 (Xh (r ))Dsu Xh (r ) dr
0
Z t
+
Eh (t − r )Ph σ 0 (Xh (r )) · Dsu Xh dW (r ).
0
6 / 11
Main result
Theorem
For every test function Φ ∈ Cb2 (H, R) and > 0 there exist a C > 0 such
that
E[Φ(X (T )) − Φ(Xh (T ))] ≤ Ch2β− .
7 / 11
Main result
Theorem
For every test function Φ ∈ Cb2 (H, R) and > 0 there exist a C > 0 such
that
E[Φ(X (T )) − Φ(Xh (T ))] ≤ Ch2β− .
Sketch of proof:
Method by A. Debussche (2011).
7 / 11
Main result
Theorem
For every test function Φ ∈ Cb2 (H, R) and > 0 there exist a C > 0 such
that
E[Φ(X (T )) − Φ(Xh (T ))] ≤ Ch2β− .
Sketch of proof:
Method by A. Debussche (2011).
Assume that f = 0 and X0 = 0.
7 / 11
Main result
Theorem
For every test function Φ ∈ Cb2 (H, R) and > 0 there exist a C > 0 such
that
E[Φ(X (T )) − Φ(Xh (T ))] ≤ Ch2β− .
Sketch of proof:
Method by A. Debussche (2011).
Assume that f = 0 and X0 = 0.
Let u(t, x) = EΦ(X (t, x)), t ∈ [0, T ] and x ∈ H where X (0, x) = x.
Satisfies the Kolmogorov equation
ut (t, x) + Lu(t, x) = 0,
u(0, x) = Φ(x),
t ∈ [0, T ),
x ∈ H,
x ∈ H.
Here
Lu(t, x) = hAx, ux (t, x)i −
1
Tr{g (x)Qg ∗ (x)uxx (t, x)}.
2
7 / 11
By Itô’s formula and the Kolmogorov equation
E[Φ(X (T )) − Φ(Xh (T ))]
= E[u(T , X0 ) − u(0, Xh (T ))]
hZ T
i
=E
ut (T − s, Xh (s)) + Lh u(T − s, Xh (s)) ds
0
T
hZ
=E
(Lh − L)u(T − s, Xh (s)) ds
i
0
T
hZ
=E
h(Ah − A)Xh (s), ux (T − s, Xh (s))i ds
0
1
−
2
Z
T
Tr{(Ph g (Xh (s))Qg ∗ (Xh (s))Ph
0
−g (Xh (s))Qg ∗ (Xh (s)))uxx (t, Xh (s))} ds
i
= I +J
8 / 11
Z
I ≤ E
T
Z
= E
T
Z
= E
T
0
0
D
E Ah Ph (I − Rh )ux (T − s, Xh (s)), Xh (s) ds (Rh = A−1
h Ph A)
D
ux (T − s, Xh (s)),
Z s
E (Ah Ph (I − Rh ))∗ Eh (s − r )Ph g (Xh (r )) dW (r ) ds 0
Z
s
n
Tr Qg ∗ (Xh (r ))Ph Eh (s − r )Ah Ph (I − Rh )
0
0
o
uxx (T − s, Xh (s))Dr Xh (s) dr ds Z TZ s
β−1
β−1
1−β
1
kQ 2 g ∗ (Xh (s))A 2 kHS kA 2 Ph Ah 2 kL(H)
≤E
0
×
0
kA1−
h Eh (s
× kA
× kA
1+β−
2
β−1
2
≤ Ch2β−3
Z
0
1−β+2
2
− r )Ph kL(H) kAh
uxx (T − s, Xh (s))A
1−β
2
Ph (I − Rh )A−
1+β−
2
kL(H)
kL(H)
1
2
Dr Xh (s)Q kHS dr ds
T Z s
2−
(T − s)− 2 (s − r )−1+ dr ds.
0
9 / 11
Estimates
Let 0 ≤ s ≤ r ≤ 1. Then
s
r
kAh2 Ph (I − Rh )A− 2 kL(H) ≤ Chr −s ,
h ∈ (0, 1].
10 / 11
Estimates
Let 0 ≤ s ≤ r ≤ 1. Then
s
r
kAh2 Ph (I − Rh )A− 2 kL(H) ≤ Chr −s ,
h ∈ (0, 1].
Let γ ≥ 0. Then
kAγ E (t)kL(H) + kAγh Eh (t)Ph kL(H) ≤ Ct −γ ,
∀t > 0.
10 / 11
Estimates
Let 0 ≤ s ≤ r ≤ 1. Then
s
r
kAh2 Ph (I − Rh )A− 2 kL(H) ≤ Chr −s ,
h ∈ (0, 1].
Let γ ≥ 0. Then
kAγ E (t)kL(H) + kAγh Eh (t)Ph kL(H) ≤ Ct −γ ,
∀t > 0.
Let λ ∈ [0, 1+β
2 ). Then
kAλ ux (t, x)kL(H) ≤ Ct −λ |Φ|Cb1 ,
∀t ∈ (0, T ],
∀x ∈ H.
10 / 11
Estimates
Let 0 ≤ s ≤ r ≤ 1. Then
s
r
kAh2 Ph (I − Rh )A− 2 kL(H) ≤ Chr −s ,
h ∈ (0, 1].
Let γ ≥ 0. Then
kAγ E (t)kL(H) + kAγh Eh (t)Ph kL(H) ≤ Ct −γ ,
∀t > 0.
Let λ ∈ [0, 1+β
2 ). Then
kAλ ux (t, x)kL(H) ≤ Ct −λ |Φ|Cb1 ,
∀t ∈ (0, T ],
∀x ∈ H.
Let λ, ρ ∈ [0, 1+β
2 ) be such that λ + ρ < 1. Then
kAρ uxx (t, x)Aλ kL(H) ≤ Ct −(ρ+λ) |G |Cb2 ,
∀t ∈ (0, T ],
∀x ∈ H.
10 / 11
Estimates
Let 0 ≤ s ≤ r ≤ 1. Then
s
r
kAh2 Ph (I − Rh )A− 2 kL(H) ≤ Chr −s ,
h ∈ (0, 1].
Let γ ≥ 0. Then
kAγ E (t)kL(H) + kAγh Eh (t)Ph kL(H) ≤ Ct −γ ,
∀t > 0.
Let λ ∈ [0, 1+β
2 ). Then
kAλ ux (t, x)kL(H) ≤ Ct −λ |Φ|Cb1 ,
∀t ∈ (0, T ],
∀x ∈ H.
Let λ, ρ ∈ [0, 1+β
2 ) be such that λ + ρ < 1. Then
kAρ uxx (t, x)Aλ kL(H) ≤ Ct −(ρ+λ) |G |Cb2 ,
∀t ∈ (0, T ],
∀x ∈ H.
∀t ∈ [0, T ],
∀s ∈ [0, t).
Let γ ∈ [0, β). Then
β−1+γ
2
E[kAh
1
Ds Xh (t)Q 2 k2HS ] ≤ C (t − s)−γ ,
10 / 11
Thank you for your attention!
11 / 11