An adaptive method for solving stochastic equations based on interpolants over Voronoi cells

An adaptive collocation-based surrogate model is developed for the solution of equations with random coefficients, referred to as stochastic equations. The surrogate model is defined on a Voronoi tessellation of the samples of the random parameters with centers chosen to be statistically representative of these samples. We investigate various interpolants over Voronoi cells in order to formulate surrogates and analyze their convergence properties. Unlike Monte Carlo solutions, relatively small numbers of deterministic calculations are needed to implement surrogate models. These models can be used to generate large sets of solution samples with a minimum computational effort. In this work, we propose a framework for an adaptive construction of the surrogate such that by refining the Voronoi cells, the mapping between the random parameters and the solution is incorporated. A rigorous refinement measure which is quantitatively indicative of the performance of the surrogate is used to drive adaptivity. We present numerical examples that compare this surrogate with other collocation-based surrogates and demonstrate the theoretical aspects of the adaptive method.