Numerical Methods for Partial Differential Equations with Random Data Howard Elman University of Maryland
Outline I. Problem statement and discretization • Example: diffusion equation with random diffusion coefficient • Discretization by stochastic Galerkin method • Discretization by stochastic collocation method II. Solution algorithms • Multigrid-style methods for various discretizations • Comparison of solution costs for different discretizations 1
I. Stochastic Differential Equations with Random Data Example: diffusion equation D d - ( ) in a u f R D D D D on , ( ) 0 on \ u g a u n D D N D Uncertainty / randomness: a = a (x, ω ) a random field For each fixed x , a ( x, ω ) a random variable Other possibly uncertain quantities : Forcing function f g Boundary data D Viscosity ν in Navier-Stokes equations 2 ( grad ) grad u u u p f div 0 u 2
Depictions: Random Data on Unit Square 3
Diffusion Equation with Random Diffusion Coefficient D - ( ) in a u f Assumptions: D , : 1. Spatial correlation of random field: For x y Random field a ( x, ω ) Mean μ ( x ) = E ( a ( x,· )) 2 ) 2 ( ) ( ( , ) Variance x E a x Covariance function c ( x,y ) = E ( ( a ( x,· )- μ ( x )) ( a ( y,· )- μ ( y )) ) is finite vs. white noise , where c is a δ -function 0 2. Coercivity a 1 2 Problem is well-posed 4
Monte-Carlo Simulation D Sample a ( x, ω ) at all x , solve in usual way 1 D ( ) Standard weak formulation: find such that u H E ( , ) ( ) a u v v 0 D 1 ( ), v H for all E ( , ) , ( ) a u v a u v dx v f v dx D D Multiple realizations (samples) of a ( x, · ) Multiple realizations of u Statistical properties of u Problem: convergence is slow, requires many solves 5
Another Point of View D - ( ) in a u f Covariance function is finite random field (diffusion coefficient) has Karhunen-Loève expansion: ( , ) ( ) ( ) ( ) a x a x a x 0 r r r 1 r ( ) ( ) ( ( , )) mean a x x E a x 0 ( ), a r x = eigenfunctions/eigenvalues of covariance operator r C C ( )( ) ( ), ( ) ( ) ( , ) ( ) a x a x a x c x y a y dy D ( ) = identically distributed uncorrelated random r variables with mean 0 and variance 1 6
Finite Noise Assumption D - ( ) in a u f Truncated Karhunen-Loève expansion: m ( , ) ( ) ( ) ( ) a x a x a x 0 r r r 1 r ~ Principal components analysis Requires: m large enough so that the fluctuation of a 1 / is well-represented, i.e. is small 1 m D m 2 | | j More precisely: error from truncation is 1 j D 2 | | Choose m to make this small 7
Various Ways to Use This m ( , ) ( ) ( ) ( ) a x a x a x 0 r r r 1 r 1. Stochastic Finite Element (Galerkin) Method: Introduce a weak formulation analogous to finite elements in space that handles the “stochastic” component of the problem 2. Stochastic Collocation Method: Devise a special strategy for sampling ξ that converges more quickly than Monte Carlo simulation; derived from interpolation Ghanem, Spanos, Babuška , Deb, Oden, Matthies, Keese, Karniadakis, Xiu, Hesthaven, Tempone, Nobile, Webster, Schwab, Todor, Ernst, Powell, Furnival, E., Ullmann, Rosseel, Vandewalle 8
Stochastic Finite Element (Stochastic Galerkin) Method Probability space ( Ω , F , P ) 2 {square integrable functions wrt dP ( ω )} ( ) L P 2 , ( ) ( ) ( ) ( ) Inner product on : ( ) v w E vw v w dP L P D 1 2 Use to concoct weak formulation on product space ( ) ( ) H L E P D 1 2 ( ) ( ) Find such that u H L E P ( , ) ( ) a u v v ( ) a u v dx dP D 1 2 ( ) ( ) v H L for all D E P 0 Solution u=u ( x, ω ) is itself a random field 9
For Computation: Return to Finite Noise Assumption Truncated Karhunen-Loève expansion m ( , ( )) ( ) ( ) ( ) a x a x a x 0 r r r 1 r Stochastic weak formulation uses ( , ) ( ) ( , ) ( ) a u v a u v dx dP a x u v dx d ( ) D D Bilinear form entails ξ plays the role of a integral over image of random variables ξ Require joint Cartesian coordinate density function associated with ξ 10
Statement of Problem Becomes D 1 2 ( ) ( ) Find such that u H L E P ( , ) ( ) ( ) a x u v dx d fv dx d D D D 1 2 ( ) ( ) ( ( )) v H L for all E P 0 Like an ordinary Galerkin (or Petrov-Galerkin) problem on a ( d+m )- dimensional “continuous” space d = dimension of spatial domain m = dimension of stochastic space 11
Discretization ( , ) ( ) ( ) a x u v dx d f v dx d D D Finite dimensional spaces: ( D 1 N • spatial discretization: ), spanned by { } S H x 0 1 h j j for example: piecewise linear on triangles N • stochastic discretization: 2 ( ), spanned by { } T L 1 p l l for example: polynomial chaos = m -variate Hermite polynomials (orthogonal wrt Gaussian measure) Discrete weak formulation: ( , ) ( ) for all a u v v v S T hp hp hp hp h p N N x ( ) ( ) u u x hp jl j l 1 1 12 j l
Basis Functions for Stochastic Space { | } 2 2 ( ) v( ) v( ) ( ) Underlying space: L d ( ) ( ) ( ) ( ) 1 1 2 2 M M ( ) k ( ) q Let polynomial of degree j orthogonal wrt k j k Examples: if ρ k ~ Gaussian measure Hermite polynomials ρ k ~ uniform distribution Legendre polynomials Any ρ k can be handled computationally (Gautschi) Rys polynomials ( 1 ) ( 2 ) ( ) m 2 { ( ) ( ) ( )} q q q ( ) spanned by T p L 1 2 j j j m 1 2 m Orthogonality of basis functions sparsity of coefficient matrix 13
m Matrix Equation Au=f ( , ( )) ( ) ( ) ( ) a x a x a x 0 r r r 1 r m A G A G A 0 0 r r r 1 [ ] ( ) ( ) ( ) A a x x x dx 0 0 jk j k D [ ] ( ) ( ) ( ) ( ) A x a x x x dx r r jk r j k D [ ] , , [ ] , G G 0 lq l q r lq r l q [ ] ( , ) ( ) ( ) ( ) f f x x dx d kq k q D Γ Properties of A: • order = N x x N ξ = (size of spatial basis) x (size of stochastic basis) • sparsity: inherited from that of { } and { } G A r r 14
Dimensions of Discrete Stochastic Space ( 1 ) ( 2 ) ( ) m 2 { ( ) ( ) ( )} q q q ( ) spanned by T p L 1 2 j j j m 1 2 m Full tensor product basis: 0 , 1 j i p i , ,m m Dimension: ( 1 ) p Too large “Complete” polynomial basis : j j j p 1 2 m m+p (m+p)! More ( ) Dimension: = p m! p! manageable Order these in a systematic way ( ), ( ), , ( ) 1 2 N 15
Example ( 1 ) ( 2 ) ( ) m 2 { ( ) ( ) ( )} q q q ( ) spanned by T p L 1 2 j j j m 1 2 m “Complete” polynomial basis : j j j p 1 2 m m+p 5 ( ) ( ) m =2, p =3 =10 = p 2 Orthogonal (Hermite) polynomials in 1D: 2 3 ( ) 1 , ( ) , ( ) 1 , ( ) 3 H H H H 0 1 2 3 ( ) ( ) 1 Gives basis set: 6 1 2 1 2 ( ) ( ) ( 1 ) 2 1 7 1 2 2 2 ( ) 1 ( ) ( 1 ) 3 1 8 2 3 2 ( ) 3 ( ) ( 1 ) 4 1 1 9 2 1 ( ) 3 ( ) 3 5 2 10 2 2 16
Example of Sparsity Pattern For m -variate polynomials of total degree p : ( m+p) ! N ξ = m ! p ! 10! = 6 ! 4 ! = 210 17
Uses of the Computed Solution: N N N x ( ) ( ) ( ) ( ) u u x u x hp jl j l l l 1 1 1 l j l ( ) u x l 1. Moments: First moment of u (expected value): M ( ) ( ) ( ) ( ) E u u x d hp l l 1 l Free! N ( ) ( ) u x u x 1 1 j j 1 j using orthogonality of stochastic basis functions Similarly for second moment / covariance 18
Uses of the Computed Solution: N N N x ( ) ( ) ( ) ( ) u u x u x hp jl j l l l 1 1 1 l j l ( ) u x l 2. Cumulative distribution functions E.g.: ( ( , ) ) P u x at some point x hp Sample ξ N ( , ) ( ) ( ) u x u x Evaluate hp l l 1 l Precomputed Repeat Not free, but no solves required 19
Stochastic Collocation Method m ( , ) ( ) ( ) a x a x a x Given as above 0 r r r 1 r Let ξ be a specified realization (~ Monte Carlo) Weak formulation: m ( ( ) ( ) ) a x a x u v dx f v dx 0 r r r 1 r D D Discretize in space in usual way. ( ) N ( 1 ) ( 2 ) , , , Stochastic collocation: choose special set from considerations of interpolation Advantage: Spatial systems are decoupled 20
Multi-Dimensional Interpolation ( 1 ) ( 2 ) ( ) N Given and v ( ξ ), consider an interpolant , , , , N ( ) k ( )( ) ( ) ( ) ( ), Iv v L v k 1 k ( ) j where Lagrange interpolating polynomial ( ) , L k jk ( k ) If solves the discrete (in space) version of u h m ( ( ) ( ) ) a x a x u v dx f v dx 0 r r r 1 r D D ( k ) , with then the collocated solution is N ( ) k ( , ) ( ) ( ) u x u x L hp h k 1 k 21
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