Probabilistic models for stochastic elliptic partial differential equations

Abstract

Mathematical requirements that the random coefficients of stochastic elliptical partial differential equations must satisfy such that they have unique solutions have been studied extensively. Yet, additional constraints that these coefficients must satisfy to provide realistic representations for physical quantities, referred to as physical requirements, have not been examined systematically. It is shown that current models for random coefficients constructed solely by mathematical considerations can violate physical constraints and, consequently, be of limited practical use. We develop alternative models for the random coefficients of stochastic differential equations that satisfy both mathematical and physical constraints. Theoretical arguments are presented to show potential limitations of current models and establish properties of the models developed in this study. Numerical examples are used to illustrate the construction of the proposed models, assess the performance of these models, and demonstrate the sensitivity of the solutions of stochastic differential equations to probabilistic characteristics of their random coefficients.