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Operator splitting algorithm for structured population models on metric spaces

Carolin Lindow, Christian Düll, Piotr Gwiazda, Błażej Miasojedow, Anna Marciniak-Czochra

TL;DR

The paper addresses numerical solvability and Bayesian parameter inference for structured population models formulated as measure-valued equations on Polish metric spaces.It introduces a particle-based operator splitting scheme that decouples transport from growth and non-local terms, approximating solutions by Dirac mixtures and proving an error bound that scales linearly with the spatial discretization $\varepsilon$ and as $\Delta t^{\alpha}$ in time.The authors extend the framework to Bayesian inverse problems, showing that the posterior derived from the discretized forward map remains close to the exact posterior in the flat metric under regularity assumptions.This approach enables linking data to finite-dimensional model parameters in non-Euclidean state spaces, offering a versatile tool for applications in biology and ecology.

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 $α$ in the time step size, assuming that the model functions are $α$ Hölder 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.

Operator splitting algorithm for structured population models on metric spaces

TL;DR

The paper addresses numerical solvability and Bayesian parameter inference for structured population models formulated as measure-valued equations on Polish metric spaces.It introduces a particle-based operator splitting scheme that decouples transport from growth and non-local terms, approximating solutions by Dirac mixtures and proving an error bound that scales linearly with the spatial discretization $\varepsilon$ and as $\Delta t^{\alpha}$ in time.The authors extend the framework to Bayesian inverse problems, showing that the posterior derived from the discretized forward map remains close to the exact posterior in the flat metric under regularity assumptions.This approach enables linking data to finite-dimensional model parameters in non-Euclidean state spaces, offering a versatile tool for applications in biology and ecology.

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 in the time step size, assuming that the model functions are Hölder 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.

Paper Structure

This paper contains 16 sections, 22 theorems, 130 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Problem Setting
  3. The numerical scheme
  4. Bayesian Inverse Method
  5. The data model and the posterior
  6. Numerical approximation of the posterior
  7. Approximation of model functions and of the initial measure
  8. Error due to approximation of the measure-valued model functions
  9. Error due to approximation of the initial measure
  10. Proving the convergence order of the numerical scheme
  11. Semigroup splitting
  12. Convergence Analysis
  13. Discussion
  14. Assumptions on model functions
  15. The space and solutions of structured population models on the space of Radon measures
  16. ...and 1 more sections

Key Result

Theorem 3.1

Assume that $N$ and $\eta$ are linear, i.e. do not depend on $\mu$. Denote by $(B_\varepsilon(z_l))_{l=1,...,L}$ an open covering of $K$ and let $(\varphi_l)_{l=1,...,L}$ be the corresponding partition of unity consisting of Lipschitz continuous functions, which exists due to Lemma Lemma:PartiotionO The functions $\hat{N}$ and $\hat{\eta}$ fulfill Assumptions assumptionsOnModelFunctionsassumptions

Figures (1)

  • Figure 1: Graphical overview of the convergence analysis conducted in this paper. $\mu$ always refers to a measure, $\Delta t$ is the size of the time steps, $\epsilon$ is the grid size and $r$ controls the cut off error.

Theorems & Definitions (51)

  • Definition 2.1
  • Theorem 3.1: Discrete approximation of $\eta$ and $N$
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4: Convergence of the numerical scheme
  • Remark 3.5
  • Remark 3.6
  • Remark 4.1
  • Lemma 4.2
  • proof
  • ...and 41 more