High-Order Collocation Methods for Differential Equations with Random Inputs

TLDR

Interest has surged in efficient solvers for differential equations with random inputs, where stochastic Galerkin methods offer superior convergence but can be complex to implement, whereas sampling approaches are simple but slower. This work introduces a high‑order stochastic collocation method. The method exploits solution smoothness in random space for rapid convergence, requires only repeated runs of a deterministic solver, and employs sparse‑grid collocation points that scale weakly with input dimensionality for high‑accuracy applications. Numerical experiments confirm the method’s accuracy and efficiency.

Abstract

Recently there has been a growing interest in designing efficient methods for the solution of ordinary/partial differential equations with random inputs. To this end, stochastic Galerkin methods appear to be superior to other nonsampling methods and, in many cases, to several sampling methods. However, when the governing equations take complicated forms, numerical implementations of stochastic Galerkin methods can become nontrivial and care is needed to design robust and efficient solvers for the resulting equations. On the other hand, the traditional sampling methods, e.g., Monte Carlo methods, are straightforward to implement, but they do not offer convergence as fast as stochastic Galerkin methods. In this paper, a high-order stochastic collocation approach is proposed. Similar to stochastic Galerkin methods, the collocation methods take advantage of an assumption of smoothness of the solution in random space to achieve fast convergence. However, the numerical implementation of stochastic collocation is trivial, as it requires only repetitive runs of an existing deterministic solver, similar to Monte Carlo methods. The computational cost of the collocation methods depends on the choice of the collocation points, and we present several feasible constructions. One particular choice, basedon sparse grids, depends weakly on the dimensionality of the random space and is more suitable for highly accurate computations of practical applications with large dimensional random inputs. Numerical examples are presented to demonstrate the accuracy and efficiency of the stochastic collocation methods.