Fast Iterative Solution of Reaction-Diffusion Control Problems Arising from Chemical Processes

Abstract

PDE-constrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix systems, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matrix systems that arise from the optimal control of reaction-diffusion equations, which themselves result from chemical processes. Important aspects of our solvers are saddle point theory, mass matrix representation, and effective Schur complement approximation, as well as the incorporation of control constraints and application of the outer (Newton) iteration to take into account the nonlinearity of the underlying PDEs. Introduction.A class of problems which has numerous applications within mathematical and physical problems is that of PDE-constrained optimization problems. One field in which these problems can be posed is that of chemical processes [4,19,20,21,22]. In this case the underlying PDEs are reaction-diffusion equations, and therefore the PDE constraints in our formulation are nonlinear PDEs.When solving such reaction-diffusion control problems using a finite element method, and employing a Lagrange-Newton iteration to take account of the nonlinearity involved in the PDEs, the resulting matrix system for each Newton iteration will be large, sparse, and of saddle point structure. It is therefore desirable to devise preconditioned iterative methods to solve these systems efficiently and in such a way that the structure of the matrix is exploited. Work in constructing preconditioners for PDE-constrained optimization problems has been considered for simpler problems previously, for instance, Poisson control [46,47,53], convection-diffusion control [45], Stokes control [36,50,56], and heat equation control [44,54].In this paper, we will consider an optimal control formulation of a reactiondiffusion problem, which generates a symmetric matrix system upon each Newton iteration. (Such an iteration is required to take into account the nonlinear terms within the underlying PDEs.) We will generally search for block triangular preconditioners for the matrix systems we examine, to be used in conjunction with a suitable iterative solver. In order to do this, we will need to approximate the (1, 1)-block by accurately representing the inverse of mass matrices amongst other things, as well as

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