Operator splitting algorithm for structured population models on metric spaces

TL;DR

This paper introduces a numerical operator splitting scheme for structured population models on metric spaces, enabling accurate approximation of measure-valued solutions and linking models to Bayesian inverse problems.

Contribution

The paper presents a novel operator splitting algorithm for measure-valued population models on metric spaces, with proven convergence and application to Bayesian inverse problems.

Findings

Linear convergence in spatial discretization

Polynomial order convergence in time with Hölder regularity

Effective approximation of posterior measures in Bayesian inference

Abstract

In this paper, we propose a numerical scheme for structured population models defined on a separable and complete metric space. In particular, we consider a generalized version of a transport equation with additional growth and non-local interaction terms in the space of nonnegative Radon measures equipped with the flat metric. The solutions, given by families of Radon measures, are approximated by linear combinations of Dirac measures. For this purpose, we introduce a finite-range approximation of the measure-valued model functions, provided that they are linear. By applying an operator splitting technique, we are able to separate the effects of the transport from those of growth and the non-local interaction. We derive the order of convergence of the numerical scheme, which is linear in the spatial discretization parameters and polynomial of order $\alpha$ in the time step size, assuming that the model functions are $\alpha$ H\"older regular in time. In a second step, we show that our proposed algorithm can approximate the posterior measure of Bayesian inverse models, which will allow us to link model parameters to measured data in the future.