Elliptic stochastic partial differential equations: An orthonormal - PowerPoint PPT Presentation

Elliptic stochastic partial differential equations: An orthonormal vector basis approach S Adhikari 1 1 Swansea University, UK Uncertainty Quantification Workshop, Edinburgh, 26 May, 2010 Adhikari (SU) Reduced methods for SPDE 26 May 2010 1 / 33

Outline of the talk Introduction 1 Stochastic elliptic PDEs Spectral decomposition in a vector space 2 Projection in a finite dimensional vector-space Properties of the spectral functions Error minimization in the Hilbert space 3 The Galerkin approach Computational method Numerical illustration 4 ZnO nanowires Conclusions 5 Adhikari (SU) Reduced methods for SPDE 26 May 2010 2 / 33

Introduction Stochastic elliptic PDEs Stochastic elliptic PDE We consider the stochastic elliptic partial differential equation (PDE) − ∇ [ a ( r , ω ) ∇ u ( r , ω )] = p ( r ); r in D (1) with the associated boundary condition u ( r , ω ) = 0 ; r on ∂ D (2) Here a : R d × Ω → R is a random field, which can be viewed as a set of random variables indexed by r ∈ R d . We assume the random field a ( r , ω ) to be stationary and square integrable. Based on the physical problem the random field a ( r , ω ) can be used to model different physical quantities. Adhikari (SU) Reduced methods for SPDE 26 May 2010 3 / 33

Introduction Stochastic elliptic PDEs Discretized Stochastic PDE The random process a ( r , ω ) can be expressed in a generalized fourier type of series known as the Karhunen-Lo` eve expansion ∞ � √ ν i ξ i ( ω ) ϕ i ( r ) a ( r , ω ) = a 0 ( r ) + (3) i = 1 Here a 0 ( r ) is the mean function, ξ i ( ω ) are uncorrelated standard Gaussian random variables, ν i and ϕ i ( r ) are eigenvalues and eigenfunctions satisfying the integral equation � D C a ( r 1 , r 2 ) ϕ j ( r 1 ) d r 1 = ν j ϕ j ( r 2 ) , ∀ j = 1 , 2 , · · · . Truncating the series (3) upto the M -th term, substituting a ( r , ω ) in the governing PDE (1) and applying the boundary conditions, the discretized equation can be written as � � M � A 0 + ξ i ( ω ) A i u ( ω ) = f (4) i = 1 Adhikari (SU) Reduced methods for SPDE 26 May 2010 4 / 33

Introduction Stochastic elliptic PDEs Polynomial Chaos expansion After the finite truncation, concisely, the polynomial chaos expansion can be written as P � ˆ u ( ω ) = H k ( ξ ( ω )) u k (5) k = 1 where H k ( ξ ( ω )) are the polynomial chaoses. The value of the number of terms P depends on the number of basic random variables M and the order of the PC expansion r as r � ( M + j − 1 )! P = (6) j !( M − 1 )! j = 0 Adhikari (SU) Reduced methods for SPDE 26 May 2010 5 / 33

Introduction Stochastic elliptic PDEs Some basics of linear algebra Definition (Linearly independent vectors) A set of vectors { φ 1 , φ 2 , . . . , φ n } is linearly independent if the expression � n k = 1 α k φ k = 0 if and only if α k = 0 for all k = 1 , 2 , . . . , n. Remark (The spanning property) Suppose { φ 1 , φ 2 , . . . , φ n } is a complete basis in the Hilbert space H. Then for every nonzero u ∈ H, it is possible to choose α 1 , α 2 , . . . , α n � = 0 uniquely such that u = α 1 φ 1 + α 2 φ 2 + . . . α n φ n . Adhikari (SU) Reduced methods for SPDE 26 May 2010 6 / 33

Introduction Stochastic elliptic PDEs Polynomial Chaos expansion We can ‘split’ the Polynomial Chaos type of expansions as n P � � ˆ u ( ω ) = H k ( ξ ( ω )) u k + H k ( ξ ( ω )) u k (7) k = 1 k = n + 1 According to the spanning property of a complete basis in R n it is always possible to project ˆ u ( ω ) in a finite dimensional vector basis for any ω ∈ Ω . Therefore, in a vector polynomial chaos expansion (7), all u k for k > n must be linearly dependent. This is the motivation behind seeking a finite dimensional expansion. Adhikari (SU) Reduced methods for SPDE 26 May 2010 7 / 33

Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space Theorem There exist a finite set of functions Γ k : ( R m × Ω) → ( R × Ω) and an orthonormal basis φ k ∈ R n for k = 1 , 2 , . . . , n such that the series n � ˆ u ( ω ) = Γ k ( ξ ( ω )) φ k (8) k = 1 converges to the exact solution of the discretized stochastic finite element equation (4) with probability 1. Outline of the proof: The first step is to generate a complete orthonormal basis. We use the eigenvectors φ k ∈ R n of the matrix A 0 such that A 0 φ k = λ 0 k φ k ; k = 1 , 2 , . . . n (9) Adhikari (SU) Reduced methods for SPDE 26 May 2010 8 / 33

Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space Suppose the solution of Eq. (4) is given by � � − 1 M � ˆ u ( ω ) = A 0 + ξ i ( ω ) A i f (10) i = 1 Using the eigenvector matrix and the orthonormality of Φ one has � � − 1 M � Φ − T Λ 0 Φ − 1 + ξ i ( ω ) Φ − T � A i Φ − 1 f = ΦΨ ( ξ ( ω )) Φ T f ˆ u ( ω ) = i = 1 (11) where � � − 1 M � ξ i ( ω ) � Ψ ( ξ ( ω )) = Λ 0 + A i (12) i = 1 and the M -dimensional random vector ξ ( ω ) = { ξ 1 ( ω ) , ξ 2 ( ω ) , . . . , ξ M ( ω ) } T (13) Adhikari (SU) Reduced methods for SPDE 26 May 2010 9 / 33

Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space Now we separate the diagonal and off-diagonal terms of the � A i matrices as � A i = Λ i + ∆ i , i = 1 , 2 , . . . , M (14) Here the diagonal matrix � � � � � ∈ R n × n Λ i = diag A = diag λ i 1 , λ i 2 , . . . , λ i n (15) and ∆ i = � A i − Λ i is an off-diagonal only matrix.   − 1     M M � �     Ψ ( ξ ( ω )) = Λ 0 + ξ i ( ω ) Λ i + ξ i ( ω ) ∆ i (16)     i = 1 i = 1   � �� � � �� � Λ ( ξ ( ω ) ) ∆ ( ξ ( ω ) ) where Λ ( ξ ( ω )) ∈ R n × n is a diagonal matrix and ∆ ( ξ ( ω )) is an off-diagonal only matrix. Adhikari (SU) Reduced methods for SPDE 26 May 2010 10 / 33

Spectral decomposition in a vector space Projection in a finite dimensional vector-space Projection in a finite dimensional vector-space We rewrite Eq. (16) as � � �� − 1 I n + Λ − 1 ( ξ ( ω )) ∆ ( ξ ( ω )) Ψ ( ξ ( ω )) = Λ ( ξ ( ω )) (17) The above expression can be represented using a Neumann type of matrix series as ( − 1 ) s � � s ∞ � Λ − 1 ( ξ ( ω )) ∆ ( ξ ( ω )) Λ − 1 ( ξ ( ω )) Ψ ( ξ ( ω )) = (18) s = 0 Adhikari (SU) Reduced methods for SPDE 26 May 2010 11 / 33

Spectral decomposition in a vector space Projection in a finite dimensional vector-space Polynomial Chaos expansion Taking an arbitrary r -th element of ˆ u ( ω ) , Eq. (11) can be rearranged to have   � � n n � �  φ T  ˆ u r ( ω ) = Φ rk Ψ kj ( ξ ( ω )) j f (19) k = 1 j = 1 Defining � � n � φ T Γ k ( ξ ( ω )) = Ψ kj ( ξ ( ω )) j f (20) j = 1 and collecting all the elements in Eq. (19) for r = 1 , 2 , . . . , n one has n � ˆ u ( ω ) = Γ k ( ξ ( ω )) φ k (21) k = 1 Adhikari (SU) Reduced methods for SPDE 26 May 2010 12 / 33

Spectral decomposition in a vector space Properties of the spectral functions Spectral functions Definition The functions Γ k ( ξ ( ω )) , k = 1 , 2 , . . . n are called the spectral functions as they are expressed in terms of the spectral properties of the coefficient matrices of the governing discretized equation. The main difficulty in applying this result is that each of the spectral functions Γ k ( ξ ( ω )) contain infinite number of terms and they are highly nonlinear functions of the random variables ξ i ( ω ) . For computational purposes, it is necessary to truncate the series after certain number of terms. Different order of spectral functions can be obtained by using truncation in the expression of Γ k ( ξ ( ω )) Adhikari (SU) Reduced methods for SPDE 26 May 2010 13 / 33

Spectral decomposition in a vector space Properties of the spectral functions First-order spectral functions Definition The first-order spectral functions Γ ( 1 ) k ( ξ ( ω )) , k = 1 , 2 , . . . , n are obtained by retaining one term in the series (18). Retaining one term in (18) we have δ kj Ψ ( 1 ) ( ξ ( ω )) = Λ − 1 ( ξ ( ω )) Ψ ( 1 ) or kj ( ξ ( ω )) = λ 0 k + � M i = 1 ξ i ( ω ) λ i k (22) Using the definition of the spectral function in Eq. (20), the first-order spectral functions can be explicitly obtained as � � n � φ T k f Γ ( 1 ) Ψ ( 1 ) φ T ( ξ ( ω )) = kj ( ξ ( ω )) j f = (23) λ 0 k + � M k i = 1 ξ i ( ω ) λ i k j = 1 From this expression it is clear that Γ ( 1 ) ( ξ ( ω )) are non-Gaussian k random variables even if ξ i ( ω ) are Gaussian random variables. Adhikari (SU) Reduced methods for SPDE 26 May 2010 14 / 33