Asymptotic approaches in inverse problems for depolymerization estimation

TL;DR

This paper develops asymptotic models for depolymerization reactions to link observed average quantities to initial size distributions, analyzing inverse problems with regularization and observer methods, and demonstrating improved accuracy of second-order models.

Contribution

It introduces second-order asymptotic models for inverse depolymerization problems and compares their effectiveness to first-order models using regularization and observer techniques.

Findings

Second-order models provide more accurate initial distribution estimates.

Carleman inequalities enable observability and error bounds.

Numerical simulations confirm the superiority of second-order approaches.

Abstract

Depolymerization reactions constitute frequent experiments, for instance in biochemistry for the study of amyloid fibrils. The quantities experimentally observed are related to the time dynamics of a quantity averaged over all polymer sizes, such as the total polymerised mass or the mean size of particles. The question analysed here is to link this measurement to the initial size distribution. To do so, we first derive, from the initial reaction system $$\mathcal{C}_i \stackrel{b}{\longrightarrow} \mathcal{C}_{i-1} + \mathcal{C}_1,\qquad i\geq i_0\geq 2,$$ two asymptotic models: at first order, a backward transport equation, and at second order, an advection-diffusion/Fokker-Planck equation complemented with a mixed boundary condition at $x = 0$. We estimate their distance to the original system solution. We then turn to the inverse problem, i.e., how to estimate the initial size distribution from the time measurement of an average quantity, given by a moment of the solution. This question has been already studied for the first order asymptotic model, and we analyse here the second order asymptotic. Thanks to Carleman inequalities and to log-convexity estimates, we prove observability results and error estimates for a Tikhonov regularization. We then develop a Kalman-based observer approach, and implement it on simulated observations. Despite its severely ill-posed character, the second-order approach appears numerically more accurate than the first-order one.