A General Inverse Problem for the Growth-Fragmentation Equation

TL;DR

This paper develops a general theoretical and numerical framework for recovering the fragmentation rate in the growth-fragmentation equation from asymptotic solution observations, applicable across various contexts.

Contribution

It extends previous work to a fully general setting, providing new methods for inverse problems in growth-fragmentation models.

Findings

Theoretical conditions for unique recovery of fragmentation rates.

Numerical algorithms for estimating reaction rates from asymptotic data.

Discussion of open issues and future directions.

Abstract

The growth-fragmentation equation arises in many different contexts, ranging from cell division, protein polymerization, biopolymers, neurosciences etc. Direct observation of temporal dynamics being often difficult, it is of main interest to develop theoretical and numerical methods to recover reaction rates and parameters of the equation from indirect observation of the solution. Following the work done in (Perthame, Zubelli, 2006) and (Doumic, Perthame, Zubelli, 2009) for the specific case of the cell division equation, we address here the general question of recovering the fragmentation rate of the equation from the observation of the time-asymptotic solution, when the fragmentation kernel and the growth rates are fully general. We give both theoretical results and numerical methods, and discuss the remaining issues.