Stochastic Galerkin approximations of elliptic PDEs driven by spatial white noise Xiaoliang Wan Division of Applied Mathematics, Brown University Sea Grant College Program, MIT (with G. E. Karniadakis, B. Rozovsky) Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 1
Two stochastic elliptic models Model I: −∇ · (( E [ a ]( x ) + ǫ ( x, ω )) ∇ u ) = f ǫ ( x, ω ) is a colored noise with a known correlation function. Model II: −∇ · (( E [ a ]( x ) + ˙ W ( x, ω )) ⋄ ∇ u ) = f ˙ W ( x, ω ) is spatial white noise on L 2 ( D ) . ‘ ⋄ ’ indicates the Wick product corresponding to Itô-Skorokhod integral. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 2
Outline A brief overview of numerical methods for Model I Karhunen-Loève expansion of the noise. Polynomial chaos methods and variants. A stochastic finite element method for Model II Spectral expansion of the white noise. Weighted Wiener chaos space. A stochastic FEM method. A simple comparison between Model I and II. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 3
Karhunen-Loève expansion of the colored noise Let ǫ ( x , ω ) be a second-order random process with zero mean and unit variance, i.e., E [ ǫ ]( x ) = 0 , E [ ǫ 2 ]( x ) = 1 . If the correlation function R ( x , y ) = E [ ǫ ( x , ω ) ǫ ( y , ω )] is known, the noise can be expressed as ∞ � � ǫ ( x, ω ) = λ i h i ( x ) ξ i i =1 Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4
Karhunen-Loève expansion of the colored noise Let ǫ ( x , ω ) be a second-order random process with zero mean and unit variance, i.e., E [ ǫ ]( x ) = 0 , E [ ǫ 2 ]( x ) = 1 . If the correlation function R ( x , y ) = E [ ǫ ( x , ω ) ǫ ( y , ω )] is known, the noise can be expressed as ∞ � � ǫ ( x, ω ) = λ i h i ( x ) ξ i i =1 { ( √ λ i , h i ( x )) } ∞ i =1 are eigen-pairs of R ( x , y ) , where � � R ( x , y ) h i ( y )d y = λ i h i ( x ) , h i ( x ) h j ( x )d x = δ ij . D D Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4
Karhunen-Loève expansion of the colored noise Let ǫ ( x , ω ) be a second-order random process with zero mean and unit variance, i.e., E [ ǫ ]( x ) = 0 , E [ ǫ 2 ]( x ) = 1 . If the correlation function R ( x , y ) = E [ ǫ ( x , ω ) ǫ ( y , ω )] is known, the noise can be expressed as ∞ � � ǫ ( x, ω ) = λ i h i ( x ) ξ i i =1 { ( √ λ i , h i ( x )) } ∞ i =1 are eigen-pairs of R ( x , y ) , where � � R ( x , y ) h i ( y )d y = λ i h i ( x ) , h i ( x ) h j ( x )d x = δ ij . D D { ξ i } is a set of mutually uncorrelated random variables with zero mean and unit variance. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4
Karhunen-Loève expansion of the colored noise Let ǫ ( x , ω ) be a second-order random process with zero mean and unit variance, i.e., E [ ǫ ]( x ) = 0 , E [ ǫ 2 ]( x ) = 1 . If the correlation function R ( x , y ) = E [ ǫ ( x , ω ) ǫ ( y , ω )] is known, the noise can be expressed as ∞ � � ǫ ( x, ω ) = λ i h i ( x ) ξ i i =1 { ( √ λ i , h i ( x )) } ∞ i =1 are eigen-pairs of R ( x , y ) , where � � R ( x , y ) h i ( y )d y = λ i h i ( x ) , h i ( x ) h j ( x )d x = δ ij . D D { ξ i } is a set of mutually uncorrelated random variables with zero mean and unit variance. The convergence of K-L expansion is optimal in the L 2 sense. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 4
Approximate the problem using a series of random variables −∇ · ( a ( x , ω ) ∇ u ( x , ω )) = f ( x ) on D, u ( x , ω ) = 0 on ∂D. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 5
Approximate the problem using a series of random variables −∇ · ( a ( x , ω ) ∇ u ( x , ω )) = f ( x ) on D, u ( x , ω ) = 0 on ∂D. −∇ · ( a M ( x , ξ ) ∇ u ( x , ξ )) = f ( x ) on D, u ( x , ξ ) = 0 on ∂D. √ λ i h i ( x ) ξ i . a M ( x , ξ ) = E [ a ]( x ) + σ � M i =1 ξ = ( ξ 1 , . . . , ξ M ) , and σ indicates the degree of perturbation. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 5
Approximate the problem using a series of random variables −∇ · ( a ( x , ω ) ∇ u ( x , ω )) = f ( x ) on D, u ( x , ω ) = 0 on ∂D. −∇ · ( a M ( x , ξ ) ∇ u ( x , ξ )) = f ( x ) on D, u ( x , ξ ) = 0 on ∂D. √ λ i h i ( x ) ξ i . a M ( x , ξ ) = E [ a ]( x ) + σ � M i =1 ξ = ( ξ 1 , . . . , ξ M ) , and σ indicates the degree of perturbation. Need to approximate the random function u ( x , ξ ) efficiently! Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 5
(Generalized) polynomial chaos (gPC) Polynomial chaos: [Wiener, 38; Cameron and Martin, 47] Let C be the space of continuous functions induced by the Wiener process { W t , 0 < t < 1 } . If F is a functional of L 2 ( C ) , i.e., E [ F 2 ] < ∞ , then the Fourier-Hermite expansion � 1 ∞ ∞ ∞ � � � F = f α φ α ( ξ ) = H j,α j ( ξ j ) , ξ j = b j ( t )d W t f α 0 j =1 | α | =0 | α | =0 converges in the L 2 ( C ) sense, where α i ∈ N 0 , | α | = � ∞ i =1 α i , { b i ( t ) } is a complete orthonormal set of real functions in L 2 (0 , 1) . Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 6
(Generalized) polynomial chaos (gPC) Polynomial chaos: [Wiener, 38; Cameron and Martin, 47] Let C be the space of continuous functions induced by the Wiener process { W t , 0 < t < 1 } . If F is a functional of L 2 ( C ) , i.e., E [ F 2 ] < ∞ , then the Fourier-Hermite expansion � 1 ∞ ∞ ∞ � � � F = f α φ α ( ξ ) = H j,α j ( ξ j ) , ξ j = b j ( t )d W t f α 0 j =1 | α | =0 | α | =0 converges in the L 2 ( C ) sense, where α i ∈ N 0 , | α | = � ∞ i =1 α i , { b i ( t ) } is a complete orthonormal set of real functions in L 2 (0 , 1) . Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 6
(Generalized) polynomial chaos (gPC) - cont’d. Generalized polynomial chaos: [Xiu and Karniadakis, 02] ∞ ∞ � � u α ( x ) � M u ( x ; ξ ) = u α ( x ) φ α ( ξ ) = j =1 φ j,α j ( ξ j ) , | α | =0 | α | =0 0 , ξ ∈ R M , | α | = � M α ∈ N M i =1 α i . ξ i are independent ; E [ φ α φ β ] = δ αβ w.r.t. the PDF f ( ξ ) Correspondence between PDF and classical orthogonal polynomials: uniform - Legendre; Gaussian -Hermite, etc. L 2 completeness of orthogonal polynomials. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 7
Apply gPC to the stochastic elliptic problem L ( x , u ; ξ ) = f ( x ) Galerkin projection: PC expansions: u = P p | α | =0 u α φ α . Residual: R ( ξ ) = L ( x , P p | α | =0 u α φ α ) − f ( x ) . E ˆ R ( ξ ) φ β ( ξ ) ˜ = 0 , | β | = 0 , . . . , p. Deterministic system of u α : # of φ α : ( M + p )! M ! p ! Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 8
Apply gPC to the stochastic elliptic problem L ( x , u ; ξ ) = f ( x ) Galerkin projection: PC expansions: u = P p | α | =0 u α φ α . Residual: R ( ξ ) = L ( x , P p | α | =0 u α φ α ) − f ( x ) . E ˆ R ( ξ ) φ β ( ξ ) ˜ = 0 , | β | = 0 , . . . , p. Deterministic system of u α : # of φ α : ( M + p )! M ! p ! Collocation projection: Interpolation operator: { ξ ( j ) } N g j =1 : a set of grid points in the parametric space. L ( x , u ; ξ ( j ) ) = f ( x ) . Deterministic system on grid points: Choices of { ξ ( j ) } : full tensor-products of Gauss quadrature points - O ( N M ) , sparse grids - O ( N log( N ) M − 1 ) Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 8
Comments on (generalized) polynomial chaos Advantages of gPC: Fast convergence due to spectral expansion. Efficiency due to orthogonality. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 9
Comments on (generalized) polynomial chaos Advantages of gPC: Fast convergence due to spectral expansion. Efficiency due to orthogonality. Disadvantages of gPC: Efficency decreases as the number of random dimensions increases. Inefficient for problems with low regularity in the parametric space. May diverge for long-time integrations. Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 9
Variants of the polynomial chaos method Choices of global approximation bases. Sparse polynomial chaos bases. [Schwab et al., Webster et al.] = φ α α ≤ V : { p } ( ) ( ) < dim dim ����� V s V = φ α α ≤ β : { } V s i i Industrial Mathematics Institute, Department of Mathematics, University of South Carolina, 04/02/2008 – p. 10
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