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Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method Qi Ye Department of Applied Mathematics Illinois Institute of Technology Joint work with Prof. I. Cialenco and Prof. G. E. Fasshauer February 2012 qye3@iit.edu MCQMC 2012 February 2012

Introduction Outline Introduction 1 Background 2 Kernel-based Collocation Methods 3 Numerical Examples 4 5 Acknowledgments qye3@iit.edu MCQMC 2012 February 2012

Introduction Meshfree Methods Statistical Learning Stochastic Analysis qye3@iit.edu MCQMC 2012 February 2012

Introduction Books Monographs j jpg jpg qye3@iit.edu MCQMC 2012 February 2012 g

Background Outline Introduction 1 Background 2 Kernel-based Collocation Methods 3 Numerical Examples 4 5 Acknowledgments qye3@iit.edu MCQMC 2012 February 2012

Background The method in a nutshell Parabolic Stochastic Equations = ⇒ Elliptic Stochastic Equations Here, we only consider the simple high-dimensional elliptic SPDE � in D ⊂ R d , ∆ u = f + ξ, u = 0 , on ∂ D , where ∂ 2 ∆ = � d j is the Laplacian operator, j = 1 ∂ x 2 suppose that u ∈ Sobolev space H m ( D ) with m > 2 + d / 2 a.s., f : D → R is a deterministic function, ξ : D × Ω ξ → R is a Gaussian field with mean zero and covariance kernel W : D × D → R defined on a probability space (Ω ξ , F ξ , P ξ ) , i.e., E ( ξ x ) = 0 , Cov ( ξ x , ξ y ) = W ( x , y ) . qye3@iit.edu MCQMC 2012 February 2012

Background The method in a nutshell The proposed numerical method for solving a parabolic SPDE can be described as follows: We choose a reproducing kernel 1 K : D × D → R whose reproducing kernel Hilbert space H K ( D ) is embedded into H m ( D ) . → Noise Covariance Kernel W Smoothness of Exact Solution u ↓ ց ↓ ← Convergent Rates Reproducing Kernel K qye3@iit.edu MCQMC 2012 February 2012

Background The method in a nutshell We simulate the Gaussian field with covariance structure W at a 2 finite collection of predetermined collocation points X D := { x 1 , · · · , x N } ⊂ D , X ∂ D := { x N + 1 , · · · , x N + M } ⊂ ∂ D , i.e., j = 1 , · · · , N , j = 1 , · · · , M , y j := f ( x j ) + ξ x j , y N + j := 0 , and � N , N � ξ := ( ξ x 1 , · · · , ξ x N ) ∼ N ( 0 , W ) , W := W ( x j , x k ) j , k = 1 . We also let the random vector y ξ := ( y 1 , · · · , y N + M ) T . qye3@iit.edu MCQMC 2012 February 2012

Background The method in a nutshell We also define its integral-type kernel 3 ∗ � ∗ K ∈ H m , m ( D × D ) . K ( x , y ) := K ( x , z ) K ( y , z ) d z , D The kernel-based collocation solution is written as 4 N M ∗ ∗ � � u ( x ) ≈ ˆ u ( x ) := c k ∆ 2 K ( x , x k ) + K ( x , x N + k ) , c N + k k = 1 k = 1 where the unknown random coefficients c := ( c 1 , · · · , c N + M ) T are obtained by solving a random system of linear equations, i.e., ∗ K c = y ξ . qye3@iit.edu MCQMC 2012 February 2012

Background Advantages Advantages The kernel-based collocation method is a meshfree approximation method. It does not require an underlying triangular mesh as the Galerkin finite element method does. The kernel-based collocation method can be applied to a high-dimensional domain D with complex boundary ∂ D . To obtain the truncated Gaussian noise ξ n for the finite element method, it is difficult for us to compute the eigenvalues and eigenfunctions of the noise covariance kernel W . For the kernel-based collocation method we need not worry about this issue. Once the reproducing kernel is fixed, the error of the collocation solution only depends on the collocation points. qye3@iit.edu MCQMC 2012 February 2012

Background Difference for Finite Element Methods Given a finite element basis φ , we shall compute the right-hand side for the Galerkin finite element methods. Popular Methods: n � � � � ξ n � ξ x φ ( x ) d x ≈ x φ ( x ) d x = ζ k λ k e k ( x ) φ ( x ) d x , D D D k = 1 where the truncated Gaussian noise n � � ξ x ≈ ξ n ζ 1 , . . . , ζ n ∼ i.i.d. N ( 0 , 1 ) , x = ζ k λ k e k ( x ) , k = 1 and n � W ( x , y ) ≈ W n ( x , y ) = λ k e k ( x ) e k ( y ) . k = 1 qye3@iit.edu MCQMC 2012 February 2012

Background Difference for Finite Element Methods Monte Carlo Methods: For each fixed sample path ω ∈ Ω ξ , ξ x ( ω ) is a function defined on D . However, we do not know its exact form. We can only use Monte Carlo methods to approximate the right-hand side, i.e., N � � ξ x φ ( x ) d x ≈ ξ x j φ ( x j ) . D j = 1 Kernel-based Methods: ξ x ≈ ˆ ξ x := w ( x ) T W − 1 ξ , where � N , N w ( x ) := ( W ( x , x 1 ) , · · · , W ( x , x N )) T , � W := W ( x j , x k ) j , k = 1 . qye3@iit.edu MCQMC 2012 February 2012

Kernel-based Collocation Methods Outline Introduction 1 Background 2 Kernel-based Collocation Methods 3 Numerical Examples 4 5 Acknowledgments qye3@iit.edu MCQMC 2012 February 2012

Kernel-based Collocation Methods Gaussian Fields According to [Cialenco, Fasshauer and Ye 2011 SPDE, Theorem 3.1], for a given µ ∈ H K ( D ) , there exists a probability measure P µ defined on (Ω K , F K ) = ( H K ( D ) , B ( H K ( D ))) such that the stochastic fields ∆ S , S given by x ∈ D , ω ∈ Ω K = H K ( D ) , ∆ S x ( ω ) = ∆ S ( x , ω ) := (∆ ω )( x ) , S x ( ω ) = S ( x , ω ) := ω ( x ) , x ∈ D ∪ ∂ D , ω ∈ Ω K = H K ( D ) , ∗ ∗ are Gaussian with means ∆ µ , µ and covariance kernels ∆ 1 ∆ 2 K , K defined on (Ω K , F K , P µ ) , respectively. For any fixed z ∈ R , we let E x ( z ) := { ω ∈ Ω K : ω ( x ) = z } = { ω ∈ Ω K : S x ( ω ) = z } . qye3@iit.edu MCQMC 2012 February 2012

Kernel-based Collocation Methods Gaussian Fields [Cialenco, Fasshauer and Ye 2011 SPDE, Corollary 3.2], shows that the random vector ∗ S := (∆ S x 1 , · · · , ∆ S x N , S x N + 1 , · · · , S x N + M ) ∼ N ( m µ , K ) , where m µ := (∆ µ ( x 1 ) , · · · , ∆ µ ( x N ) , µ ( x N + 1 ) , · · · , µ ( x N + M )) T ∗ ∗   K ( x j , x k )) N , N K ( x j , x N + k )) N , M  (∆ 1 ∆ 2 j , k = 1 , (∆ 1 ∗ j , k = 1  . K := ∗ ∗ K ( x N + j , x k )) M , N K ( x N + j , x N + k )) M , M (∆ 2 j , k = 1 , ( j , k = 1 For any given y = ( y 1 , · · · , y N + M ) T ∈ R N + M , we let E X ( y ) := { ω ∈ Ω K : ∆ ω ( x 1 ) = y 1 , . . . , ω ( x N + M ) = y N + M } = { ω ∈ Ω K : S ( ω ) = y } . qye3@iit.edu MCQMC 2012 February 2012

Kernel-based Collocation Methods Approximation and Convergence For each fixed x ∈ D and ω 2 ∈ Ω ξ , we obtain the "optimal" estimator P µ � u ( x , ω 2 ) ≈ ˆ � � �� u ( x , ω 2 ) = argmax sup E x ( z ) × Ω ξ � E X y ξ ( ω 2 ) , ξ z ∈ R µ ∈ H K ( D ) P µ � � � = argmax sup S x = z � S = y ξ ( ω 2 ) , ξ z ∈ R µ ∈ H K ( D ) p µ = argmax sup x ( z | y ξ ( ω 2 )) , z ∈ R µ ∈ H K ( D ) = k ( x ) T ∗ K − 1 y ξ ( ω 2 ) ∗ ∗ K ( x , x N + M )) T and where k ( x ) := (∆ 2 K ( x , x 1 ) , · · · , ξ := P µ ⊗ P ξ , P µ Ω K ξ := Ω K × Ω ξ , F K ξ := F K ⊗ F ξ , so that ∆ S , S and ξ can be extended to the product space while preserving the original probability distributional properties. qye3@iit.edu MCQMC 2012 February 2012

Kernel-based Collocation Methods Approximation and Convergence Error Bound Analysis For any ǫ > 0, we define � E ǫ ω 1 × ω 2 ∈ Ω K × Ω ξ : | ω 1 ( x ) − ˆ x := u ( x , ω 2 ) | ≥ ǫ, � s.t. ∆ ω 1 ( x 1 ) = y 1 ( ω 2 ) , . . . , ω 1 ( x N + M ) = y N + M ( ω 2 ) . Let the fill distance h X := sup 1 ≤ j ≤ N + M � x − x j � 2 . min x ∈D qye3@iit.edu MCQMC 2012 February 2012

Kernel-based Collocation Methods Approximation and Convergence We can deduce that � h m − 2 − d / 2 � P µ ξ ( E ǫ X sup x ) = O , ǫ µ ∈ H K ( D ) where m is the order of the Sobolev space corresponded to the exact solution of the SPDE. u ( x , ω 2 ) | ≥ ǫ if and only if u ∈ E ǫ Since | u ( x , ω 2 ) − ˆ x , we have P µ P µ ξ ( E ǫ � � u − ˆ � sup u � L ∞ ( D ) ≥ ǫ ≤ sup x ) → 0 , ξ µ ∈ H K ( D ) µ ∈ H K ( D ) , x ∈D when h X → 0. qye3@iit.edu MCQMC 2012 February 2012

Numerical Examples Outline Introduction 1 Background 2 Kernel-based Collocation Methods 3 Numerical Examples 4 5 Acknowledgments qye3@iit.edu MCQMC 2012 February 2012

Numerical Examples Stochastic Laplace’s Equations Let the domain D := ( 0 , 1 ) 2 ⊂ R 2 . We choose the deterministic function f ( x ) := − 2 π 2 sin ( π x 1 ) sin ( π x 2 ) − 8 π 2 sin ( 2 π x 1 ) sin ( 2 π x 2 ) , and the covariance kernel of the Gaussian noise ξ to be W ( x , y ) := 4 π 4 sin ( π x 1 ) sin ( π x 2 ) sin ( π y 1 ) sin ( π y 2 ) + 16 π 4 sin ( 2 π x 1 ) sin ( 2 π x 2 ) sin ( 2 π y 1 ) sin ( 2 π y 2 ) . Then the exact solution of the above elliptic SPDE has the form u ( x ) := sin ( π x 1 ) sin ( π x 2 ) + sin ( 2 π x 1 ) sin ( 2 π x 2 ) + ζ 1 sin ( π x 1 ) sin ( π x 2 ) + ζ 2 2 sin ( 2 π x 1 ) sin ( 2 π x 2 ) , where ζ 1 , ζ 2 ∼ i.i.d. N ( 0 , 1 ) . qye3@iit.edu MCQMC 2012 February 2012