Iterative Solvers for a Spectral Galerkin Approach to Elliptic Partial Differential Equations with Fuzzy Coefficients

Mathematical models of physical systems often contain parameters with an imprecisely known and uncertain character. It is quite common to represent these parameters by means of random variables. Numerous methods have been developed to compute accurate approximations to solutions of equations with such parameters. This approach, however, may not be entirely justified when the uncertainty is due to vagueness or incomplete knowledge. For such cases, alternative uncertainty representations using tools from imprecise probability theory have been suggested. Among those, the fuzzy representation is probably most popular. In this paper, we consider numerical methods for solving partial differential equations with fuzzy coefficients. We demonstrate that spectral expansion methods, quite common in the random variable approach, can also be used effectively for solving fuzzy equations. We motivate the use of Chebyshev polynomials in the spectral representation and apply a Galerkin projection to convert the fuzzy problem into a high-dimensional deterministic one. Two preconditioners are proposed in order to efficiently solve the resulting high-dimensional algebraic system. A Fourier analysis demonstrates that both preconditioners yield a convergence rate that is independent of the spatial resolution and independent of the number of fuzzy variables and the polynomial order. The practical applicability of the algorithm is illustrated by means of two numerical experiments: a fuzzy heat transfer problem on an L-shaped domain and a fuzzy elasticity problem.