# Fast Numerical Solvers for Parameter Identification Problems in Mathematical Biology

## TL;DR

This paper introduces efficient numerical methods for solving complex optimization problems in mathematical biology.

## Contribution

The paper proposes commutative discretization and optimization techniques for PDE-constrained optimization in biology.

## Key findings

- The proposed discretization ensures commutativity with optimization operations.
- Efficient preconditioned iterative methods are applied to solve large-scale systems.
- Numerical experiments confirm the approach's viability and efficiency.

## Abstract

In this paper, we consider effective discretization strategies and iterative solvers for nonlinear PDE-constrained optimization models of 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.