On the weak approximation of solutions to stochastic partial - - PowerPoint PPT Presentation

On the weak approximation of solutions to stochastic partial differential equations

Raphael Kruse (TU Berlin)

(joint work with Adam Andersson und Stig Larsson)

Berlin-Padova Young Researchers Meeting Stochastic Analysis and Applications in Biology, Finance and Physics Berlin, October 23, 2014

A semilinear SPDE with additive noise

We consider dX(t) +

• AX(t) + F(X(t))
• dt = dW(t);

t ∈ (0, T], X(0) = x0 ∈ H, (SPDE) where H is a separable Hilbert space,

◮ X : [0, T] × Ω → H, ◮ −A generator of an analytic semigroup S(t) on H, ◮ (W(t))t∈[0,T] is an H-valued Q-Wiener process, with

Q : H → H covariance operator, Tr(Q) < ∞,

◮ F : H → H nonlinear mapping, globally Lipschitz cont., ◮ x0 ∈ H deterministic initial value.

Existence and uniqueness

◮ Semigroup approach by [Da Prato, Zabczyk 1992]. ◮ There exists a unique mild solution X to (SPDE) given by

the variation of constants formula X(t) = S(t)x0 − t S(t − σ)F(X(σ)) dσ + t S(t − σ) dW(σ) P-a.s. for 0 ≤ t ≤ T.

◮ It holds supt∈[0,T] X(t)Lp(Ω;H) < ∞, for all p ≥ 2.

Existence and uniqueness

◮ Semigroup approach by [Da Prato, Zabczyk 1992]. ◮ There exists a unique mild solution X to (SPDE) given by

the variation of constants formula X(t) = S(t)x0 − t S(t − σ)F(X(σ)) dσ + t S(t − σ) dW(σ) P-a.s. for 0 ≤ t ≤ T.

◮ It holds supt∈[0,T] X(t)Lp(Ω;H) < ∞, for all p ≥ 2. ◮ X takes values in dom(A

1 2 ) provided that −A is self-adjoint,

positive definite with compact inverse.

Computational goal

For a given mapping ϕ ∈ C2

p(H; R) we want to estimate

E

• ϕ(X(T))
• .

For this we need to discretize

◮ H (spatial discretization), ◮ [0, T] (temporal discretization), ◮ H0 = Q

1 2 (H) (discretization of the noise),

◮ Monte Carlo methods. . .

Computational goal

For a given mapping ϕ ∈ C2

p(H; R) we want to estimate

E

• ϕ(X(T))
• .

For this we need to discretize

◮ H (spatial discretization), ◮ [0, T] (temporal discretization), ◮ H0 = Q

1 2 (H) (discretization of the noise),

◮ Monte Carlo methods. . .

In this talk:

◮ Discretization of H by Galerkin finite element methods, ◮ Temporal discretization of [0, T] by linearly implicit Euler

scheme.

The numerical scheme

The numerical scheme is given by, j ∈ {1, . . . , Nk}, X j

h = (I + kAh)−1

X j−1

h

− kPhF(X j−1

h

) + Ph∆W j , X 0

h = Phx0,

where

◮ (Vh)h∈(0,1] family of finite dimensional subspaces of H, ◮ Ah : Vh → Vh discrete version of A, ◮ Ph : H → Vh orthogonal projector onto Vh, ◮ k equidistant temporal step size, ◮ ∆W j = W(tj) − W(tj−1), tj = jk, j = 0, 1, . . . , Nk.

The numerical scheme

The numerical scheme is given by, j ∈ {1, . . . , Nk}, X j

h = (I + kAh)−1

X j−1

h

− kPhF(X j−1

h

) + Ph∆W j , X 0

h = Phx0,

where

◮ (Vh)h∈(0,1] family of finite dimensional subspaces of H, ◮ Ah : Vh → Vh discrete version of A, ◮ Ph : H → Vh orthogonal projector onto Vh, ◮ k equidistant temporal step size, ◮ ∆W j = W(tj) − W(tj−1), tj = jk, j = 0, 1, . . . , Nk.

Examples for (Vh)h∈(0,1]:

◮ standard finite element method, ◮ spectral Galerkin method, ◮ . . .

Strong vs. weak convergence

Two different notions of convergence: Strong convergence: Estimates of the error

• E
• X(tj) − X j

h

• 2 1

2 , j = 1, . . . , Nk.

⇒ Sample paths of X and Xh are close to each other.

Strong vs. weak convergence

Two different notions of convergence: Strong convergence: Estimates of the error

• E
• X(tj) − X j

h

• 2 1

2 , j = 1, . . . , Nk.

⇒ Sample paths of X and Xh are close to each other. Weak convergence: Estimates of the error

• E
• ϕ(X(tNk)) − ϕ(X Nk

h )

• for ϕ ∈ C2

p(H, R).

⇒ Distribution of X Nk

h

converges to distribution of X(tNk). Both notions play an important role in setting up MLMC methods.

Main Idea: Gelfand triples

Let us consider a Gelfand triple V ⊂ L2(Ω; H) ⊂ V ∗.

Main Idea: Gelfand triples

Let us consider a Gelfand triple V ⊂ L2(Ω; H) ⊂ V ∗. By the mean value theorem, the weak error reads

• E
• ϕ(X(tNk)) − ϕ(X Nk

h )

• =
• ΦNk

h , X(tNk) − X Nk h

• L2(Ω;H)
• ,

where Φn

h =

1 ϕ′ ρX(tn) + (1 − ρ)X n

h

• dρ.

Main Idea: Gelfand triples

Let us consider a Gelfand triple V ⊂ L2(Ω; H) ⊂ V ∗. By the mean value theorem, the weak error reads

• E
• ϕ(X(tNk)) − ϕ(X Nk

h )

• =
• ΦNk

h , X(tNk) − X Nk h

• L2(Ω;H)
• ,

where Φn

h =

1 ϕ′ ρX(tn) + (1 − ρ)X n

h

• dρ.

Then by duality

• E
• ϕ(X(tNk)) − ϕ(X Nk

h )

• ≤
• ΦNk

h

• V
• X(tNk) − X Nk

h

• V ∗.

Road map to weak convergence

In order to prove weak convergence, we

• 1. determine a “nice” subspace V,
• 2. prove ΦNk

h V < ∞,

• 3. Then: Convergence of
• X(tNk) − X Nk

h

• V ∗ → 0 implies weak

convergence with the same order.

Road map to weak convergence

In order to prove weak convergence, we

• 1. determine a “nice” subspace V,
• 2. prove ΦNk

h V < ∞,

• 3. Then: Convergence of
• X(tNk) − X Nk

h

• V ∗ → 0 implies weak

convergence with the same order. Simplest choice: V = L2(Ω; H) gives the well-known fact that strong convergence implies weak convergence.

Road map to strong convergence

The same steps also yield strong convergence: E

• X(tj) − X j

h

• 2

=

• X(tj) − X j

h, X(tj) − X j h

• L2(Ω;H)

≤

• X(tj) − X j

h

• V
• X(tj) − X j

h

• V ∗.

Thus: In order to prove strong convergence, we

• 1. determine a “nice” subspace V,
• 2. prove maxj=1,...,Nk X(tj) − X j

hV < ∞,

• 3. then: Convergence of maxj=1,...,Nk
• X(tj) − X j

h

• V ∗ → 0

implies strong convergence with half the order.

Weak convergence – Main result

Theorem (Weak convergence)

Let A be s.p.d. with compact inverse. Let F ∈ C2

b(H),

x0 ∈ dom(A

1 2 ) and Q be of finite trace. Let (Vh)h∈(0,1] be

suitable approximation spaces. Then for every ϕ ∈ C2

p(H, R)

and γ ∈ (0, 1) there exists C such that

• E
• ϕ(X(tNk)) − ϕ(X Nk

h )

• ≤ C(kγ + h2γ)

∀h, k ∈ (0, 1].

Weak convergence – Main result

Theorem (Weak convergence)

Let A be s.p.d. with compact inverse. Let F ∈ C2

b(H),

x0 ∈ dom(A

1 2 ) and Q be of finite trace. Let (Vh)h∈(0,1] be

suitable approximation spaces. Then for every ϕ ∈ C2

p(H, R)

and γ ∈ (0, 1) there exists C such that

• E
• ϕ(X(tNk)) − ϕ(X Nk

h )

• ≤ C(kγ + h2γ)

∀h, k ∈ (0, 1].

◮ Assumptions can be relaxed for white noise. ◮ Assumptions on F can be relaxed to allow for more

interesting Nemytskii operators.

Sketch of proof: Stochastic convolution

For simplicity we consider the equation (SPDE) with F = 0, x0 = 0, dX(t) + AX(t) dt = dW(t), t ∈ (0, T], X(0) = 0. (SPDE2) Then, X(t) = W A(t) = t S(t − σ) dW(σ), is the stochastic convolution.

Sketch of proof: Stochastic convolution

For simplicity we consider the equation (SPDE) with F = 0, x0 = 0, dX(t) + AX(t) dt = dW(t), t ∈ (0, T], X(0) = 0. (SPDE2) Then, X(t) = W A(t) = t S(t − σ) dW(σ), is the stochastic convolution. Numerical approximation X n

h = n−1

• j=0

tj+1

tj

(I + kAh)n−jPh dW(σ), n ∈ {1, . . . , Nk}.

Sketch of proof: V = L2(Ω; H)

The It¯

• isometry gives
• X(T) − X Nk

h,k

• 2

L2(Ω,H) =

T

• Eh,k(T − σ)Q

1 2

2

L2(H) dσ

≤ CTr(Q) T (T − σ)−θ dσ(hθ + k

θ 2 )2,

where Eh,k(t) = S(t) − (I + kAh)j+1Ph for t ∈ [tj, tj+1),

Sketch of proof: V = L2(Ω; H)

The It¯

• isometry gives
• X(T) − X Nk

h,k

• 2

L2(Ω,H) =

T

• Eh,k(T − σ)Q

1 2

2

L2(H) dσ

≤ CTr(Q) T (T − σ)−θ dσ(hθ + k

θ 2 )2,

where Eh,k(t) = S(t) − (I + kAh)j+1Ph for t ∈ [tj, tj+1), since Eh,k(t)L(H) ≤ Ct− θ

2 (hθ + k θ 2 ).

Therefore, θ ∈ [0, 1).

Sobolev-Malliavin spaces

For p ∈ [2, ∞) let D1,p(H) be the subspace of all H-valued random variables Z : Ω → H, such that ZD1,p(H) =

• Zp

Lp(Ω,H) + DZp Lp(Ω,L2([0,T],L2(H0,H))

1

p < ∞,

where DZ denotes the Malliavin derivative of Z.

Sobolev-Malliavin spaces

For p ∈ [2, ∞) let D1,p(H) be the subspace of all H-valued random variables Z : Ω → H, such that ZD1,p(H) =

• Zp

Lp(Ω,H) + DZp Lp(Ω,L2([0,T],L2(H0,H))

1

p < ∞,

where DZ denotes the Malliavin derivative of Z. We have DW A(t) =

and hence DW A(t)Lp(Ω,L2([0,T],L2(H0,H)) < ∞.

Refined Sobolev-Malliavin spaces

For p, q ∈ [2, ∞) let M1,p,q(H) be the subspace of all H-valued random variables Z : Ω → H, such that ZM1,p,q(H) =

• Zp

Lp(Ω,H) + DZp Lp(Ω,Lq([0,T],L2(H0,H))

1

p < ∞,

where DZ denotes the Malliavin derivative of Z.

Refined Sobolev-Malliavin spaces

For p, q ∈ [2, ∞) let M1,p,q(H) be the subspace of all H-valued random variables Z : Ω → H, such that ZM1,p,q(H) =

• Zp

Lp(Ω,H) + DZp Lp(Ω,Lq([0,T],L2(H0,H))

1

p < ∞,

where DZ denotes the Malliavin derivative of Z. We have DW A(t) =

and hence DW A(t)Lp(Ω,Lq([0,T],L2(H0,H)) < ∞.

Burkholder-Davis-Gundy-type inequality

Theorem (Andersson, K., Larsson, 2013)

If Φ ∈ L2([0, T] × Ω, L0

2) is predictable, then

• T

Φ(t) dW(t)

• M1,p,q(H)∗ ≤ ΦLp′(Ω,Lq′([0,T],L0

2)),

where 1

p + 1 p′ = 1 and 1 q + 1 q′ = 1.

Sketch of proof: V = M1,p,p(H)

The theorem then gives

• X(T) − X Nk

h,k

• M1,p,p(H)∗ ≤

T

• Eh,k(T − σ)Q

1 2

p′

L2(H) dσ

1

p′

≤ C T (T − σ)− θp′

2 dσ

1

p′ (hθ + k θ 2 ),

where Eh,k(t) = S(t) − (I + kAh)j+1Ph for t ∈ [tj, tj+1), since Eh,k(t)L(H) ≤ Ct− θ

2 (hθ + k θ 2 )

Sketch of proof: V = M1,p,p(H)

The theorem then gives

• X(T) − X Nk

h,k

• M1,p,p(H)∗ ≤

T

• Eh,k(T − σ)Q

1 2

p′

L2(H) dσ

1

p′

≤ C T (T − σ)− θp′

2 dσ

1

p′ (hθ + k θ 2 ),

where Eh,k(t) = S(t) − (I + kAh)j+1Ph for t ∈ [tj, tj+1), since Eh,k(t)L(H) ≤ Ct− θ

2 (hθ + k θ 2 )

Since p′ can be chosen arbitrary close to 1 we obtain convergence of order (hθ + k

θ 2 ) for any

θ ∈ [0, 2) .

Summary

Conclusions:

◮ Used a Gelfand triple for the proof of weak convergence. ◮ A proof of strong convergence is automatically included

with half the order of convergence.

◮ Derived a Burkholder-Davis-Gundy-type inequality for the

weak norm. (Malliavin calculus)

◮ Convergence in the weak norm is then proved by following

the same path as for strong convergence –> standard Gronwall arguments.

◮ Allows treatment of equations with stochastic coefficients.

Summary

Conclusions:

◮ Used a Gelfand triple for the proof of weak convergence. ◮ A proof of strong convergence is automatically included

with half the order of convergence.

◮ Derived a Burkholder-Davis-Gundy-type inequality for the

weak norm. (Malliavin calculus)

◮ Convergence in the weak norm is then proved by following

the same path as for strong convergence –> standard Gronwall arguments.

◮ Allows treatment of equations with stochastic coefficients.

Open problems:

◮ multiplicative noise, ◮ noise approximation, ◮ more general noise, ◮ more general generators (e.g. A(t, ω)).

References

◮ Andersson, K., Larsson, arXiv 1312.5893, 2013. ◮ K., LNM, Springer, 2014.

Thank you for your attention!