A Spectral Method via Orthogonal Polynomial Expansions on Sparse Grids for Solving Stochastic Partial Differential Equations

Most mathematical models contain uncertainties that may be originated from various sources such as initial and boundary conditions, geometry representation of the domain and input parameters. When these sources are expressed as random processes or random fields, partial differential equations describing the underlying models become stochastic partial differential equations (SPDEs). Stochastic models are more complex than deterministic ones; as part of this complexity, the solution of an SPDE is not simply a function, but rather a random field which expresses the implicit variability of the system. This is the reason that SPDE are able to more fully capture the behavior of interesting phenomena. Numerical solutions for stochastic partial differential equations have received much attention in recent years. Several competitive methods have emerged, including generalized polynomial chaos method [21, 22, 23, 24], high order finite element method [6, 11, 12, 14], gPC based stochastic collocation method, spectral Galerkin method [3, 10, 16] , spectral collocation method [1, 2] and spectral collocation method on sparse grids [17, 18]. High dimensionality is the main bottleneck of any numerical methods for SPDEs. When the random inputs of a SPDE are represented by a finite number of random variables, the SPDE can be reformulated as a high dimensional deterministic problem where the dimension of the problem is the sum of the spatial dimension and the number of random variables. In practically applicable problems, the number of random variables is usually large. In such cases numerical methods based on the tensor products of one dimensional numerical algorithms suffer the so called “ curse of dimensionality” since the computing complexity increases exponentially as the dimension grows. Efforts have been made to overcome this difficulty, with the spectral collocation method on sparse grid as one of most prevailing approaches. Still obstacles remain. For instance, in the spectral collocation method on sparse grid, one must solve a very larger number of deterministic problems ([5]), which can be difficult when the complexity of solving the deterministic problem is large. In this paper we attempt to construct an efficient numerical algorithm for solving stochastic partial differential equations (SPDEs) through the construction of a fast algorithm for orthogonal polynomial expansions. Here the orthogonality is in terms of inner product of Lρ where ρ is a probability density weight. When solving SPDEs, the “curse of dimensionality ” is due to not only the exponential increase of the number of terms in the orthogonal expansion, but also the equal number of multiple dimensional integrals one must evaluate in order to compute the generalized Fourier coefficients. In order to overcome these obstacles, we first use the sparse grid idea to reduce the number of terms in the orthogonal polynomial expansion. Then we use the idea of sparse grid and fast Fourier transform to construct an efficient algorithm to evaluate the Fourier coefficients in the orthogonal polynomial expansion with Cheybshev weight. Through a basis transform on sparse grid we obtain the coefficients of orthogonal polynomial expansion with an arbitrary probability density weight. We shall demonstrate that the overall complexity of our algorithm is O(n ln n) where n is the number of terms of one dimensional expansion while the convergence rate of the algorithm is quasi-optimal. The total