Fast numerical solvers for parameter identification problems in mathematical biology

TL;DR

This paper introduces efficient discretization and iterative solution methods for nonlinear PDE-constrained optimization problems in biological pattern evolution, ensuring computational efficiency and accuracy.

Contribution

It develops discretization strategies that preserve the commutative property with optimization, enabling effective large-scale linear system solutions in biological modeling.

Findings

Numerical experiments confirm the efficiency of the proposed methods.

The approach effectively handles large-scale coupled linear systems.

Discretization and optimization operations are shown to be commutative.

Abstract

In this paper, we consider effective discretization strategies and iterative solvers for nonlinear PDE-constrained optimization models for pattern evolution within biological processes. Upon a Sequential Quadratic Programming linearization of the optimization problem, we devise appropriate time-stepping schemes and discrete approximations of the cost functionals such that the discretization and optimization operations are commutative, a highly desirable property of a discretization of such problems. We formulate the large-scale, coupled linear systems in such a way that efficient preconditioned iterative methods can be applied within a Krylov subspace solver. Numerical experiments demonstrate the viability and efficiency of our approach.