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 � � d X ( t ) + AX ( t ) + F ( X ( t )) d t = d W ( t ); t ∈ ( 0 , T ] , (SPDE) X ( 0 ) = x 0 ∈ H , 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., ◮ x 0 ∈ 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 � t � t X ( t ) = S ( t ) x 0 − S ( t − σ ) F ( X ( σ )) d σ + S ( t − σ ) d W ( σ ) 0 0 P -a.s. for 0 ≤ t ≤ T . ◮ It holds sup t ∈ [ 0 , T ] � X ( t ) � L p (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 � t � t X ( t ) = S ( t ) x 0 − S ( t − σ ) F ( X ( σ )) d σ + S ( t − σ ) d W ( σ ) 0 0 P -a.s. for 0 ≤ t ≤ T . ◮ It holds sup t ∈ [ 0 , T ] � X ( t ) � L p (Ω; H ) < ∞ , for all p ≥ 2. 1 ◮ X takes values in dom ( A 2 ) provided that − A is self-adjoint, positive definite with compact inverse.
Computational goal For a given mapping ϕ ∈ C 2 p ( H ; R ) we want to estimate � � E ϕ ( X ( T )) . For this we need to discretize ◮ H (spatial discretization), ◮ [ 0 , T ] (temporal discretization), 1 ◮ H 0 = Q 2 ( H ) (discretization of the noise), ◮ Monte Carlo methods. . .
Computational goal For a given mapping ϕ ∈ C 2 p ( H ; R ) we want to estimate � � E ϕ ( X ( T )) . For this we need to discretize ◮ H (spatial discretization), ◮ [ 0 , T ] (temporal discretization), 1 ◮ H 0 = Q 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 , . . . , N k } , X j X j − 1 − kP h F ( X j − 1 h = ( I + kA h ) − 1 � ) + P h ∆ W j � , h h X 0 h = P h x 0 , where ◮ ( V h ) h ∈ ( 0 , 1 ] family of finite dimensional subspaces of H , ◮ A h : V h → V h discrete version of A , ◮ P h : H → V h orthogonal projector onto V h , ◮ k equidistant temporal step size, ◮ ∆ W j = W ( t j ) − W ( t j − 1 ) , t j = jk , j = 0 , 1 , . . . , N k .
The numerical scheme The numerical scheme is given by, j ∈ { 1 , . . . , N k } , X j X j − 1 − kP h F ( X j − 1 h = ( I + kA h ) − 1 � ) + P h ∆ W j � , h h X 0 h = P h x 0 , where ◮ ( V h ) h ∈ ( 0 , 1 ] family of finite dimensional subspaces of H , ◮ A h : V h → V h discrete version of A , ◮ P h : H → V h orthogonal projector onto V h , ◮ k equidistant temporal step size, ◮ ∆ W j = W ( t j ) − W ( t j − 1 ) , t j = jk , j = 0 , 1 , . . . , N k . Examples for ( V h ) 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 � 2 �� 1 2 , j = 1 , . . . , N k . � � X ( t j ) − X j �� � E h ⇒ Sample paths of X and X h are close to each other.
Strong vs. weak convergence Two different notions of convergence: Strong convergence: Estimates of the error � 2 �� 1 2 , j = 1 , . . . , N k . � � X ( t j ) − X j �� � E h ⇒ Sample paths of X and X h are close to each other. Weak convergence: Estimates of the error ϕ ( X ( t N k )) − ϕ ( X N k for ϕ ∈ C 2 � � �� � E h ) p ( H , R ) . � ⇒ Distribution of X N k converges to distribution of X ( t N k ) . h Both notions play an important role in setting up MLMC methods.
Main Idea: Gelfand triples Let us consider a Gelfand triple V ⊂ L 2 (Ω; H ) ⊂ V ∗ .
Main Idea: Gelfand triples Let us consider a Gelfand triple V ⊂ L 2 (Ω; H ) ⊂ V ∗ . By the mean value theorem, the weak error reads ϕ ( X ( t N k )) − ϕ ( X N k � = Φ N k h , X ( t N k ) − X N k � � �� � �� � � � E h ) � , h L 2 (Ω; H ) where � 1 Φ n ρ X ( t n ) + ( 1 − ρ ) X n ϕ ′ � � h = d ρ. h 0
Main Idea: Gelfand triples Let us consider a Gelfand triple V ⊂ L 2 (Ω; H ) ⊂ V ∗ . By the mean value theorem, the weak error reads ϕ ( X ( t N k )) − ϕ ( X N k � = Φ N k h , X ( t N k ) − X N k � � �� �� � � � � E h ) � , h L 2 (Ω; H ) where � 1 Φ n ρ X ( t n ) + ( 1 − ρ ) X n ϕ ′ � � h = d ρ. h 0 Then by duality ϕ ( X ( t N k )) − ϕ ( X N k � Φ N k � X ( t N k ) − X N k � ≤ � � �� � � � � h ) � E V ∗ . � � h h V
Road map to weak convergence In order to prove weak convergence, we 1. determine a “nice” subspace V , 2. prove � Φ N k h � V < ∞ , � X ( t N k ) − X N k � � 3. Then: Convergence of V ∗ → 0 implies weak � h 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 � Φ N k h � V < ∞ , � X ( t N k ) − X N k � � 3. Then: Convergence of V ∗ → 0 implies weak � h convergence with the same order. Simplest choice: V = L 2 (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: � 2 � � X ( t j ) − X j X ( t j ) − X j h , X ( t j ) − X j �� � � � E = h h L 2 (Ω; H ) � X ( t j ) − X j � X ( t j ) − X j � � � � ≤ V ∗ . � � h V h Thus: In order to prove strong convergence, we 1. determine a “nice” subspace V , 2. prove max j = 1 ,..., N k � X ( t j ) − X j h � V < ∞ , � X ( t j ) − X j � � 3. then: Convergence of max j = 1 ,..., N k V ∗ → 0 � h 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 ∈ C 2 b ( H ) , 1 2 ) and Q be of finite trace. Let ( V h ) h ∈ ( 0 , 1 ] be x 0 ∈ dom ( A suitable approximation spaces. Then for every ϕ ∈ C 2 p ( H , R ) and γ ∈ ( 0 , 1 ) there exists C such that � ≤ C ( k γ + h 2 γ ) ϕ ( X ( t N k )) − ϕ ( X N k � � �� � E h ) ∀ h , k ∈ ( 0 , 1 ] .
Weak convergence – Main result Theorem (Weak convergence) Let A be s.p.d. with compact inverse. Let F ∈ C 2 b ( H ) , 1 2 ) and Q be of finite trace. Let ( V h ) h ∈ ( 0 , 1 ] be x 0 ∈ dom ( A suitable approximation spaces. Then for every ϕ ∈ C 2 p ( H , R ) and γ ∈ ( 0 , 1 ) there exists C such that � ≤ C ( k γ + h 2 γ ) ϕ ( X ( t N k )) − ϕ ( X N k � � �� � E h ) ∀ 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, x 0 = 0, d X ( t ) + AX ( t ) d t = d W ( t ) , t ∈ ( 0 , T ] , (SPDE2) X ( 0 ) = 0 . Then, � t X ( t ) = W A ( t ) = S ( t − σ ) d W ( σ ) , 0 is the stochastic convolution.
Sketch of proof: Stochastic convolution For simplicity we consider the equation (SPDE) with F = 0, x 0 = 0, d X ( t ) + AX ( t ) d t = d W ( t ) , t ∈ ( 0 , T ] , (SPDE2) X ( 0 ) = 0 . Then, � t X ( t ) = W A ( t ) = S ( t − σ ) d W ( σ ) , 0 is the stochastic convolution. Numerical approximation � t j + 1 n − 1 X n � ( I + kA h ) n − j P h d W ( σ ) , n ∈ { 1 , . . . , N k } . h = t j j = 0
Sketch of proof: V = L 2 (Ω; H ) The It¯ o isometry gives � T � 2 1 � 2 � X ( T ) − X N k � � � 2 � L 2 (Ω , H ) = � E h , k ( T − σ ) Q L 2 ( H ) d σ h , k 0 � T ( T − σ ) − θ d σ ( h θ + k θ 2 ) 2 , ≤ C Tr ( Q ) 0 where E h , k ( t ) = S ( t ) − ( I + kA h ) j + 1 P h for t ∈ [ t j , t j + 1 ) ,
Sketch of proof: V = L 2 (Ω; H ) The It¯ o isometry gives � T � 2 1 � 2 � X ( T ) − X N k � � � 2 � L 2 (Ω , H ) = � E h , k ( T − σ ) Q L 2 ( H ) d σ h , k 0 � T ( T − σ ) − θ d σ ( h θ + k θ 2 ) 2 , ≤ C Tr ( Q ) 0 where E h , k ( t ) = S ( t ) − ( I + kA h ) j + 1 P h for t ∈ [ t j , t j + 1 ) , since � E h , k ( t ) � L ( H ) ≤ Ct − θ 2 ( h θ + k θ 2 ) . Therefore, θ ∈ [ 0 , 1 ) .
Sobolev-Malliavin spaces For p ∈ [ 2 , ∞ ) let D 1 , p ( H ) be the subspace of all H -valued random variables Z : Ω → H , such that � 1 p < ∞ , � � Z � p L p (Ω , H ) + � DZ � p � Z � D 1 , p ( H ) = L p (Ω , L 2 ([ 0 , T ] , L 2 ( H 0 , H )) where DZ denotes the Malliavin derivative of Z .
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