Numerical solution of fuzzy elliptic partial differential equations by a polynomial Galerkin approximation

Mathematical models in science and engineering often contain parameters that are uncertain. These parameters are usually represented by random numbers, fields or processes. However, when the stochastic characteristics of these parameters are not precisely known, an interval representation, or, more generally, a fuzzy representation may be more appropriate. This leads to so-called fuzzy differential equations. Unfortunately, there is no real consensus in the literature on how to define and interpret the solution to such equations. In this paper, we introduce a precise definition of fuzzy fields that allows for a straightforward application of Zadeh’s extension principle to define a solution to a fuzzy differential equation. Next, we describe and analyze a Galerkin method to construct a response surface for the solution of elliptic partial differential equations with a fuzzy diffusion coefficient. To assess the accuracy of this Galerkin approximation, we derive a-priori error bounds using the fuzzy supremum distance measure. By imposing some assumptions on the fuzzy diffusion coefficient, we prove (sub-)exponential convergence for approximations constructed from multivariate Chebyshev polynomials. We end with a numerical experiment that confirms our theoretical findings. Timings of the computations also show a substantial speed-up in comparison to direct fuzzification of the parameterized PDE, justifying the use of response surface methods to solve fuzzy differential equations.