Newton-Multigrid for Biological Reaction-Diffusion Problems with Random Coefficients

Abstract

An algebraic Newton-multigrid method is proposed in order to efficiently solve systems of nonlinear reaction-diffusion problems with stochastic coefficients. These problems model the conversion of starch into sugars in growing apples. The stochastic system is first converted into a large coupled system of deterministic equations by applying a stochastic Galerkin finite element discretization. This method leads to high-order accurate stochastic solutions. A stable and high-order time discretization is obtained by applying a fully implicit Runge-Kutta method. After Newton linearization, a point-based algebraic multigrid solution method is applied. In order to decrease the computational cost, alternative multigrid preconditioners are presented. Numerical results demonstrate the convergence properties, robustness and efficiency of the proposed multigrid methods.

A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

“…The model problem (2.1) is a prototype stationary reaction-diffusion problem that can be found in many chemical and biological applications. For example, it appears in the semi-discretization in time of the nonlinear stochastic reaction-diffusion problem modeling the conversion of starch into sugars in growing apples [34]. We introduce some notations which will be used later.…”

Convergence Analysis for Spectral Approximation to a Scalar Transport Equation with a Random Wave Speed

“…In numerical simulation, accounting for uncertainties in input quantities (such as model parameters, initial and boundary conditions, and geometry) becomes an important issue in recent years, especially in risk analysis, safety, and optimal design, see, e.g., [1,7,9,20,23,27]. Many works have been recently devoted to the analysis and the implementation of the Stochastic Galerkin (SG) methods and Stochastic Collocation (SC) techniques for such problems.…”