Approximation of Stochastic Partial Differential ... - MCQMC 2012
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
[email protected]
MCQMC 2012
February 2012
Introduction
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
MCQMC 2012
February 2012
Introduction
Meshfree Methods
Statistical Learning
[email protected]
Stochastic Analysis
MCQMC 2012
February 2012
Introduction
Books
Monographs Page 1 of 1 Page 1 of 1
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2010-12-11
MCQMC 2012
February 2012
Background
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
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 ( ∆u = f + ξ, in D ⊂ Rd , u = 0, on ∂D, where ∆=
∂2 j=1 ∂x 2 j
Pd
is the Laplacian operator,
suppose that u ∈ Sobolev space Hm (D) with m > 2 + d/2 a.s., f : D → R is a deterministic function, ξ : D × Ωξ → R is a Gaussian field with mean zero and covariance kernel W : D × D → R defined on a probability space (Ωξ , Fξ , Pξ ), i.e., E(ξx ) = 0, Cov (ξx , ξy ) = W (x, y). [email protected]
MCQMC 2012
February 2012
Background
The method in a nutshell
The proposed numerical method for solving a parabolic SPDE can be described as follows: 1
We choose a reproducing kernel K :D×D →R whose reproducing kernel Hilbert space HK (D) is embedded into Hm (D).
Noise Covariance Kernel W ↓ Convergent Rates
[email protected]
→ & ←
MCQMC 2012
Smoothness of Exact Solution u ↓ Reproducing Kernel K
February 2012
Background
2
The method in a nutshell
We simulate the Gaussian field with covariance structure W at a finite collection of predetermined collocation points XD := {x 1 , · · · , x N } ⊂ D,
X∂D := {x N+1 , · · · , x N+M } ⊂ ∂D,
i.e., yj := f (x j ) + ξx j ,
j = 1, · · · , N,
yN+j := 0,
j = 1, · · · , M,
and ξ := (ξx 1 , · · · , ξx N ) ∼ N (0, W) ,
N,N W := W (x j , x k ) j,k =1 .
We also let the random vector y ξ := (y1 , · · · , yN+M )T .
[email protected]
MCQMC 2012
February 2012
Background
3
The method in a nutshell
We also define its integral-type kernel Z ∗ K (x, y) := K (x, z)K (y, z)dz,
∗
K ∈ Hm,m (D × D).
D 4
The kernel-based collocation solution is written as ˆ (x) := u(x) ≈ u
N X
∗
ck ∆2 K (x, x k ) +
k =1
M X
∗
cN+k K (x, x N+k ),
k =1
where the unknown random coefficients c := (c1 , · · · , cN+M )T are obtained by solving a random system of linear equations, i.e., ∗
Kc = y ξ . [email protected]
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. [email protected]
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: Z
Z ξx φ(x)dx ≈
D
D
ξxn φ(x)dx
=
n X
Z p λk ek (x)φ(x)dx, ζk
k =1
D
where the truncated Gaussian noise ξx ≈ ξxn =
n X
ζk
p λk ek (x),
ζ1 , . . . , ζn ∼ i.i.d.N (0, 1),
k =1
and n
W (x, y) ≈ W (x, y) =
n X
λk ek (x)ek (y).
k =1
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MCQMC 2012
February 2012
Background
Difference for Finite Element Methods
Monte Carlo Methods: For each fixed sample path ω ∈ Ωξ , ξx (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., Z ξx φ(x)dx ≈ D
N X
ξx j φ(x j ).
j=1
Kernel-based Methods: ξx ≈ ξˆx := w (x)T W−1 ξ, where w (x) := (W (x, x 1 ), · · · , W (x, x N ))T ,
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MCQMC 2012
W := W (x j , x k )
N,N j,k =1
.
February 2012
Kernel-based Collocation Methods
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
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 ∈ HK (D), there exists a probability measure Pµ defined on (ΩK , FK ) = (HK (D), B(HK (D))) such that the stochastic fields ∆S, S given by ∆Sx (ω) = ∆S(x, ω) := (∆ω)(x), Sx (ω) = S(x, ω) := ω(x),
x ∈ D,
x ∈ D ∪ ∂D,
ω ∈ ΩK = HK (D), ω ∈ ΩK = HK (D), ∗
∗
are Gaussian with means ∆µ, µ and covariance kernels ∆1 ∆2 K , K defined on (ΩK , FK , Pµ ), respectively. For any fixed z ∈ R, we let Ex (z) := {ω ∈ ΩK : ω(x) = z} = {ω ∈ ΩK : Sx (ω) = z} .
[email protected]
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 := (∆Sx 1 , · · · , ∆Sx N , Sx N+1 , · · · , Sx N+M ) ∼ N (mµ , K), where mµ := (∆µ(x 1 ), · · · , ∆µ(x N ), µ(x N+1 ), · · · , µ(x N+M ))T ∗ ∗ N,N N,M ∗ (∆1 ∆2 K (x j , x k ))j,k =1 , (∆1 K (x j , x N+k ))j,k =1 . K := ∗ ∗ M,N M,M (∆2 K (x N+j , x k ))j,k =1 , (K (x N+j , x N+k ))j,k =1 For any given y = (y1 , · · · , yN+M )T ∈ RN+M , we let EX (y) := {ω ∈ ΩK : ∆ω(x 1 ) = y1 , . . . , ω(x N+M ) = yN+M } = {ω ∈ ΩK : S(ω) = y} .
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MCQMC 2012
February 2012
Kernel-based Collocation Methods
Approximation and Convergence
For each fixed x ∈ D and ω2 ∈ Ωξ , we obtain the "optimal" estimator ˆ (x, ω2 ) = argmax sup Pµξ Ex (z) × Ωξ EX y ξ (ω2 ) , u(x, ω2 ) ≈ u z∈R
µ∈HK (D)
= argmax sup Pµξ Sx = z S = y ξ (ω2 ) , z∈R
µ∈HK (D)
= argmax sup pxµ (z|y ξ (ω2 )), z∈R
µ∈HK (D) ∗
= k (x)T K−1 y ξ (ω2 ) ∗
∗
where k (x) := (∆2 K (x, x 1 ), · · · , K (x, x N+M ))T and ΩK ξ := ΩK × Ωξ ,
FK ξ := FK ⊗ Fξ ,
Pµξ := Pµ ⊗ Pξ ,
so that ∆S, S and ξ can be extended to the product space while preserving the original probability distributional properties. [email protected]
MCQMC 2012
February 2012
Kernel-based Collocation Methods
Approximation and Convergence
Error Bound Analysis For any > 0, we define n ˆ (x, ω2 )| ≥ , Ex := ω1 × ω2 ∈ ΩK × Ωξ : |ω1 (x) − u o s.t. ∆ω1 (x 1 ) = y1 (ω2 ), . . . , ω1 (x N+M ) = yN+M (ω2 ) .
Let the fill distance hX := sup
min
x∈D 1≤j≤N+M
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MCQMC 2012
kx − x j k2 .
February 2012
Kernel-based Collocation Methods
Approximation and Convergence
We can deduce that m−2−d/2
sup µ∈HK (D)
Pµξ (Ex )
=O
hX
!
,
where m is the order of the Sobolev space corresponded to the exact solution of the SPDE. ˆ (x, ω2 )| ≥ if and only if u ∈ Ex , we have Since |u(x, ω2 ) − u ˆ kL∞ (D) ≥ ≤ sup Pµξ ku − u sup Pµξ (Ex ) → 0, µ∈HK (D)
µ∈HK (D),x∈D
when hX → 0.
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MCQMC 2012
February 2012
Numerical Examples
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
MCQMC 2012
February 2012
Numerical Examples
Stochastic Laplace’s Equations
Let the domain D := (0, 1)2 ⊂ R2 . We choose the deterministic function f (x) := −2π 2 sin(πx1 ) sin(πx2 ) − 8π 2 sin(2πx1 ) sin(2πx2 ), and the covariance kernel of the Gaussian noise ξ to be W (x, y) :=4π 4 sin(πx1 ) sin(πx2 ) sin(πy1 ) sin(πy2 ) + 16π 4 sin(2πx1 ) sin(2πx2 ) sin(2πy1 ) sin(2πy2 ). Then the exact solution of the above elliptic SPDE has the form u(x) := sin(πx1 ) sin(πx2 ) + sin(2πx1 ) sin(2πx2 ) ζ2 + ζ1 sin(πx1 ) sin(πx2 ) + sin(2πx1 ) sin(2πx2 ), 2 where ζ1 , ζ2 ∼ i.i.d. N (0, 1). [email protected]
MCQMC 2012
February 2012
Numerical Examples
Stochastic Laplace’s Equations
For the collocation methods, we use the C4 -Matérn function with shape parameter θ > 0 gθ (r ) := (3 + 3θr + θ2 r 2 )e−θr ,
r > 0,
to construct the reproducing kernel (Sobolev-spline kernel) Kθ (x, y) := gθ (kx − yk2 ).
According to [Fasshauer and Ye 2011 Distributional Operators, Fasshauer and Ye 2011 Differential and Boundary Operators], we can deduce that HKθ (D) ∼ = H3+1/2 (D) ⊂ C2 (D).
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MCQMC 2012
February 2012
Numerical Examples
Stochastic Laplace’s Equations PDF, x1 = 0.52632, x2 = 0.52632
Collocation Points 1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
0 −4
−2
0
Empirical
Approximate Mean
2
4
Theoretical
Approximate Variance
2
1
1 0.5 0 −1 1 0.5
1
0 1
0.5
0 0 0.02 0.04 0.06 Relative Absolute Error
0.08
0.5
1 0.5
0 0 0.01 0.02 0.03 0.04 0.05 0.06 Relative Absolute Error
Figure: N = 65, M = 28 and θ = 0.9 [email protected]
MCQMC 2012
February 2012
Numerical Examples
Stochastic Laplace’s Equations
0.3
Relative Root−mean−square Error
0.25
0.2
Mean, θ = 0.9 Variance, θ = 0.9 Mean, θ = 1.9 Variance, θ = 1.9 Mean, θ = 2.9 Variance, θ = 2.9
0.15
0.1
0.05
0 0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
fill distance hX
Figure: Convergence of Mean and Variance [email protected]
MCQMC 2012
February 2012
Acknowledgments
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
MCQMC 2012
February 2012
Acknowledgments
T HANK YOU for the invitation and the NSF support from Prof. Owen.
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MCQMC 2012
February 2012
Appendix
References
References I R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd Ed.), Pure and Applied Mathematics, Vol. 140, Academic Press, 2003. A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic Publishers, 2004. M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press (Cambridge), 2003. G. E. Fasshauer, Meshfree Approximation Methods with M ATLAB, Interdisciplinary Mathematical Sciences, Vol. 6, World Scientific Publishers (Singapore), 2007. [email protected]
MCQMC 2012
February 2012
Appendix
References
References II L. Hörmander, The analysis of linear partial differential operators I, Classics in Mathematics, Springer, 2004. P. E. Kloeden and E. Platen Numerical Solution of Stochastic Differential Equations, Vol. 23, Springer, 2011. B. Øksendal Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, 2010. I. Steinwart and A. Christmann, Support Vector Machines, Springer Science Press, 2008. [email protected]
MCQMC 2012
February 2012
Appendix
References
References III G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics 59, SIAM (Philadelphia), 1990. H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005. G. E. Fasshauer and Q. Ye, Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operator, Numerische Mathematik, Volume 119, Number 3, Pages 585-611, 2011.
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February 2012
Appendix
References
References IV G. E. Fasshauer and Q. Ye, Reproducing Kernels of Sobolev Spaces via a Green Function Approach with Differential Operators and Boundary Operators, Advances in Computational Mathematics, 2011, to appear, DOI: 10.1007/s10444-011-9264-6. G. E. Fasshauer and Q. Ye, Kernel-based Collocation Methods versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations, in preparation. I. Cialenco, G. E. Fasshauer and Q. Ye, Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method, submitted. [email protected]
MCQMC 2012
February 2012
Appendix
References
References V
S. Koutsourelakis and J. Warner Learning Solutions to Multiscale Elliptic Problems with Gaussian Process Models, Research report at Cornell University, 2009. Q. Ye, Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operator, IIT technical report, 2010.
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MCQMC 2012
February 2012
February 2012
[email protected]
MCQMC 2012
February 2012
Introduction
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
MCQMC 2012
February 2012
Introduction
Meshfree Methods
Statistical Learning
[email protected]
Stochastic Analysis
MCQMC 2012
February 2012
Introduction
Books
Monographs Page 1 of 1 Page 1 of 1
Page 1 of 1
Page 1 of 1 Page 1 of 1
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Page 1 of 1
http://images.barnesandnoble.com/images/34380000/34380775.jpg 2010-12-11
2010-12-11
http://i.ebayimg.com/22/!B+SFnlw!Wk~$(KGrHqV,!jUEzKMBd-(3BM+JDHMq3w~~0... 2010-12-11
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[email protected]
http://images.barnesandnoble.com/images/8100000/8104072.jpg 2010-12-11
2010-12-11
MCQMC 2012
February 2012
Background
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
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 ( ∆u = f + ξ, in D ⊂ Rd , u = 0, on ∂D, where ∆=
∂2 j=1 ∂x 2 j
Pd
is the Laplacian operator,
suppose that u ∈ Sobolev space Hm (D) with m > 2 + d/2 a.s., f : D → R is a deterministic function, ξ : D × Ωξ → R is a Gaussian field with mean zero and covariance kernel W : D × D → R defined on a probability space (Ωξ , Fξ , Pξ ), i.e., E(ξx ) = 0, Cov (ξx , ξy ) = W (x, y). [email protected]
MCQMC 2012
February 2012
Background
The method in a nutshell
The proposed numerical method for solving a parabolic SPDE can be described as follows: 1
We choose a reproducing kernel K :D×D →R whose reproducing kernel Hilbert space HK (D) is embedded into Hm (D).
Noise Covariance Kernel W ↓ Convergent Rates
[email protected]
→ & ←
MCQMC 2012
Smoothness of Exact Solution u ↓ Reproducing Kernel K
February 2012
Background
2
The method in a nutshell
We simulate the Gaussian field with covariance structure W at a finite collection of predetermined collocation points XD := {x 1 , · · · , x N } ⊂ D,
X∂D := {x N+1 , · · · , x N+M } ⊂ ∂D,
i.e., yj := f (x j ) + ξx j ,
j = 1, · · · , N,
yN+j := 0,
j = 1, · · · , M,
and ξ := (ξx 1 , · · · , ξx N ) ∼ N (0, W) ,
N,N W := W (x j , x k ) j,k =1 .
We also let the random vector y ξ := (y1 , · · · , yN+M )T .
[email protected]
MCQMC 2012
February 2012
Background
3
The method in a nutshell
We also define its integral-type kernel Z ∗ K (x, y) := K (x, z)K (y, z)dz,
∗
K ∈ Hm,m (D × D).
D 4
The kernel-based collocation solution is written as ˆ (x) := u(x) ≈ u
N X
∗
ck ∆2 K (x, x k ) +
k =1
M X
∗
cN+k K (x, x N+k ),
k =1
where the unknown random coefficients c := (c1 , · · · , cN+M )T are obtained by solving a random system of linear equations, i.e., ∗
Kc = y ξ . [email protected]
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. [email protected]
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: Z
Z ξx φ(x)dx ≈
D
D
ξxn φ(x)dx
=
n X
Z p λk ek (x)φ(x)dx, ζk
k =1
D
where the truncated Gaussian noise ξx ≈ ξxn =
n X
ζk
p λk ek (x),
ζ1 , . . . , ζn ∼ i.i.d.N (0, 1),
k =1
and n
W (x, y) ≈ W (x, y) =
n X
λk ek (x)ek (y).
k =1
[email protected]
MCQMC 2012
February 2012
Background
Difference for Finite Element Methods
Monte Carlo Methods: For each fixed sample path ω ∈ Ωξ , ξx (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., Z ξx φ(x)dx ≈ D
N X
ξx j φ(x j ).
j=1
Kernel-based Methods: ξx ≈ ξˆx := w (x)T W−1 ξ, where w (x) := (W (x, x 1 ), · · · , W (x, x N ))T ,
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MCQMC 2012
W := W (x j , x k )
N,N j,k =1
.
February 2012
Kernel-based Collocation Methods
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
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 ∈ HK (D), there exists a probability measure Pµ defined on (ΩK , FK ) = (HK (D), B(HK (D))) such that the stochastic fields ∆S, S given by ∆Sx (ω) = ∆S(x, ω) := (∆ω)(x), Sx (ω) = S(x, ω) := ω(x),
x ∈ D,
x ∈ D ∪ ∂D,
ω ∈ ΩK = HK (D), ω ∈ ΩK = HK (D), ∗
∗
are Gaussian with means ∆µ, µ and covariance kernels ∆1 ∆2 K , K defined on (ΩK , FK , Pµ ), respectively. For any fixed z ∈ R, we let Ex (z) := {ω ∈ ΩK : ω(x) = z} = {ω ∈ ΩK : Sx (ω) = z} .
[email protected]
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 := (∆Sx 1 , · · · , ∆Sx N , Sx N+1 , · · · , Sx N+M ) ∼ N (mµ , K), where mµ := (∆µ(x 1 ), · · · , ∆µ(x N ), µ(x N+1 ), · · · , µ(x N+M ))T ∗ ∗ N,N N,M ∗ (∆1 ∆2 K (x j , x k ))j,k =1 , (∆1 K (x j , x N+k ))j,k =1 . K := ∗ ∗ M,N M,M (∆2 K (x N+j , x k ))j,k =1 , (K (x N+j , x N+k ))j,k =1 For any given y = (y1 , · · · , yN+M )T ∈ RN+M , we let EX (y) := {ω ∈ ΩK : ∆ω(x 1 ) = y1 , . . . , ω(x N+M ) = yN+M } = {ω ∈ ΩK : S(ω) = y} .
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MCQMC 2012
February 2012
Kernel-based Collocation Methods
Approximation and Convergence
For each fixed x ∈ D and ω2 ∈ Ωξ , we obtain the "optimal" estimator ˆ (x, ω2 ) = argmax sup Pµξ Ex (z) × Ωξ EX y ξ (ω2 ) , u(x, ω2 ) ≈ u z∈R
µ∈HK (D)
= argmax sup Pµξ Sx = z S = y ξ (ω2 ) , z∈R
µ∈HK (D)
= argmax sup pxµ (z|y ξ (ω2 )), z∈R
µ∈HK (D) ∗
= k (x)T K−1 y ξ (ω2 ) ∗
∗
where k (x) := (∆2 K (x, x 1 ), · · · , K (x, x N+M ))T and ΩK ξ := ΩK × Ωξ ,
FK ξ := FK ⊗ Fξ ,
Pµξ := Pµ ⊗ Pξ ,
so that ∆S, S and ξ can be extended to the product space while preserving the original probability distributional properties. [email protected]
MCQMC 2012
February 2012
Kernel-based Collocation Methods
Approximation and Convergence
Error Bound Analysis For any > 0, we define n ˆ (x, ω2 )| ≥ , Ex := ω1 × ω2 ∈ ΩK × Ωξ : |ω1 (x) − u o s.t. ∆ω1 (x 1 ) = y1 (ω2 ), . . . , ω1 (x N+M ) = yN+M (ω2 ) .
Let the fill distance hX := sup
min
x∈D 1≤j≤N+M
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MCQMC 2012
kx − x j k2 .
February 2012
Kernel-based Collocation Methods
Approximation and Convergence
We can deduce that m−2−d/2
sup µ∈HK (D)
Pµξ (Ex )
=O
hX
!
,
where m is the order of the Sobolev space corresponded to the exact solution of the SPDE. ˆ (x, ω2 )| ≥ if and only if u ∈ Ex , we have Since |u(x, ω2 ) − u ˆ kL∞ (D) ≥ ≤ sup Pµξ ku − u sup Pµξ (Ex ) → 0, µ∈HK (D)
µ∈HK (D),x∈D
when hX → 0.
[email protected]
MCQMC 2012
February 2012
Numerical Examples
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
[email protected]
MCQMC 2012
February 2012
Numerical Examples
Stochastic Laplace’s Equations
Let the domain D := (0, 1)2 ⊂ R2 . We choose the deterministic function f (x) := −2π 2 sin(πx1 ) sin(πx2 ) − 8π 2 sin(2πx1 ) sin(2πx2 ), and the covariance kernel of the Gaussian noise ξ to be W (x, y) :=4π 4 sin(πx1 ) sin(πx2 ) sin(πy1 ) sin(πy2 ) + 16π 4 sin(2πx1 ) sin(2πx2 ) sin(2πy1 ) sin(2πy2 ). Then the exact solution of the above elliptic SPDE has the form u(x) := sin(πx1 ) sin(πx2 ) + sin(2πx1 ) sin(2πx2 ) ζ2 + ζ1 sin(πx1 ) sin(πx2 ) + sin(2πx1 ) sin(2πx2 ), 2 where ζ1 , ζ2 ∼ i.i.d. N (0, 1). [email protected]
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Numerical Examples
Stochastic Laplace’s Equations
For the collocation methods, we use the C4 -Matérn function with shape parameter θ > 0 gθ (r ) := (3 + 3θr + θ2 r 2 )e−θr ,
r > 0,
to construct the reproducing kernel (Sobolev-spline kernel) Kθ (x, y) := gθ (kx − yk2 ).
According to [Fasshauer and Ye 2011 Distributional Operators, Fasshauer and Ye 2011 Differential and Boundary Operators], we can deduce that HKθ (D) ∼ = H3+1/2 (D) ⊂ C2 (D).
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Numerical Examples
Stochastic Laplace’s Equations PDF, x1 = 0.52632, x2 = 0.52632
Collocation Points 1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
0 −4
−2
0
Empirical
Approximate Mean
2
4
Theoretical
Approximate Variance
2
1
1 0.5 0 −1 1 0.5
1
0 1
0.5
0 0 0.02 0.04 0.06 Relative Absolute Error
0.08
0.5
1 0.5
0 0 0.01 0.02 0.03 0.04 0.05 0.06 Relative Absolute Error
Figure: N = 65, M = 28 and θ = 0.9 [email protected]
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Numerical Examples
Stochastic Laplace’s Equations
0.3
Relative Root−mean−square Error
0.25
0.2
Mean, θ = 0.9 Variance, θ = 0.9 Mean, θ = 1.9 Variance, θ = 1.9 Mean, θ = 2.9 Variance, θ = 2.9
0.15
0.1
0.05
0 0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
fill distance hX
Figure: Convergence of Mean and Variance [email protected]
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Acknowledgments
Outline
1
Introduction
2
Background
3
Kernel-based Collocation Methods
4
Numerical Examples
5
Acknowledgments
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Acknowledgments
T HANK YOU for the invitation and the NSF support from Prof. Owen.
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Appendix
References
References I R. A. Adams and J. J. F. Fournier, Sobolev Spaces (2nd Ed.), Pure and Applied Mathematics, Vol. 140, Academic Press, 2003. A. Berlinet and C. Thomas-Agnan, Reproducing Kernel Hilbert Spaces in Probability and Statistics, Kluwer Academic Publishers, 2004. M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press (Cambridge), 2003. G. E. Fasshauer, Meshfree Approximation Methods with M ATLAB, Interdisciplinary Mathematical Sciences, Vol. 6, World Scientific Publishers (Singapore), 2007. [email protected]
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Appendix
References
References II L. Hörmander, The analysis of linear partial differential operators I, Classics in Mathematics, Springer, 2004. P. E. Kloeden and E. Platen Numerical Solution of Stochastic Differential Equations, Vol. 23, Springer, 2011. B. Øksendal Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, 2010. I. Steinwart and A. Christmann, Support Vector Machines, Springer Science Press, 2008. [email protected]
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Appendix
References
References III G. Wahba, Spline Models for Observational Data, CBMS-NSF Regional Conference Series in Applied Mathematics 59, SIAM (Philadelphia), 1990. H. Wendland, Scattered Data Approximation, Cambridge University Press, 2005. G. E. Fasshauer and Q. Ye, Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Distributional Operator, Numerische Mathematik, Volume 119, Number 3, Pages 585-611, 2011.
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Appendix
References
References IV G. E. Fasshauer and Q. Ye, Reproducing Kernels of Sobolev Spaces via a Green Function Approach with Differential Operators and Boundary Operators, Advances in Computational Mathematics, 2011, to appear, DOI: 10.1007/s10444-011-9264-6. G. E. Fasshauer and Q. Ye, Kernel-based Collocation Methods versus Galerkin Finite Element Methods for Approximating Elliptic Stochastic Partial Differential Equations, in preparation. I. Cialenco, G. E. Fasshauer and Q. Ye, Approximation of Stochastic Partial Differential Equations by a Kernel-based Collocation Method, submitted. [email protected]
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Appendix
References
References V
S. Koutsourelakis and J. Warner Learning Solutions to Multiscale Elliptic Problems with Gaussian Process Models, Research report at Cornell University, 2009. Q. Ye, Reproducing Kernels of Generalized Sobolev Spaces via a Green Function Approach with Differential Operator, IIT technical report, 2010.
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