Solving elliptic problems with non-Gaussian spatially-dependent random coefficients

We propose a simple and effective numerical procedure for solving elliptic problems with non-Gaussian random coefficients, assuming that samples of the non-Gaussian random inputs are available from a statistical model. Given a correlation function, the Karhunen–Loève (K–L) expansion is employed to reduce the dimensionality of random inputs. Using the kernel density estimation technique, we obtain the marginal probability density functions (PDFs) of the random variables in the K–L expansion, based on which we define an auxiliary joint PDF. We then implement the generalized polynomial chaos (gPC) method via a collocation projection according to the auxiliary joint PDF. Based on the observation that the solution has an analytic extension in the parametric space, we ensure that the polynomial interpolation achieves point-wise convergence in the parametric space regardless of the PDF, where the energy norm is employed in the physical space. Hence, we can sample the gPC solution using the joint PDF instead of the auxiliary one to obtain the correct statistics. We also implement Monte Carlo methods to further refine the statistics using the gPC solution for variance reduction. Numerical results are presented to demonstrate the efficiency of the proposed approach.