Fast numerical solvers for parameter identification problems in mathematical biology

TL;DR

This paper formulation the large-scale, coupled linear systems in such a way that efficient preconditioned iterative methods can be applied within a Krylov subspace solver and demonstrates the viability and efficiency of this approach.

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.

Figures (3)

• Figure 1: Snapshot of the solution at $t=0.7$ for the desired states $\hat{u}$, $\hat{v}$ (left) and computed state variables $u$, $v$ (middle), and at $t=0.695$ for the computed control variables $a$, $b$ (right), with $\beta=10^{-2}$. The colorbar takes account of values from the whole time interval.
• Figure 2: Snapshot of the solution at $t=2$ for the desired states $\hat{u}$, $\hat{v}$ (left) and computed state variables $u$, $v$ (middle), and at $t=1.995$ for the computed control variables $a$, $b$ (right), with $\beta=10^{-2}$. The colorbar takes account of values from the whole time interval.
• Figure 3: Evolution of the mean of the numerical solution for the control variables $a$, $b$ across the time interval compared to those of the control variables $a_G$, $b_G$ used to generate the target state at $t=5$, with $\beta=10^{-2}$.

Theorems & Definitions (1)

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