Asymptotic approaches in inverse problems for depolymerization estimation

Marie Doumic; Philippe Moireau

doi:10.3934/ipi.2025020

Article Contents

2026, Volume 20: 105-155. Doi: 10.3934/ipi.2025020

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Asymptotic approaches in inverse problems for depolymerization estimation

Marie Doumic^1, and
Philippe Moireau^1, ,

1.
CMAP, Inria, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, France

^*Corresponding author: Philippe Moireau
^*Corresponding author: Philippe Moireau

Received: October 2024

Revised: April 2025

Early access: May 16, 2025

Published online: May 16, 2025

This research was partially supported by the French Grant ANR ODISSE (ANR-19-CE48-0004-01).

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Abstract

Depolymerisation 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, 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 the origin. 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 studied for the first order asymptotic model, and we analyse here the second order asymptotic. Thanks to Carleman inequalities and 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.

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Mathematics Subject Classification: Primary: 93C20, 93B07, 92-10, 35B40; Secondary: 35Q92, 37N25.

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Figure 1. Moments computed from the initial Becker-Döring model (BD, green), the Transport model (Tr, dashed-blue), or the Advection-Diffusion model (AD, dashed-red)

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Figure 2. Snapshots of the resulting Depolymerisation obtained using the initial Becker-Döring model (BD, green), the Transport model (Tr, dashed-blue), or the Advection-Diffusion model (AD, dashed-red)

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Figure 3. Snapshots of the identity approximation (88) for the Transport model (Tr, dashed-blue), or the Advection-Diffusion model (AD, dashed-red)

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Figure 4. (Left) Boundary observation generated by the Becker-Döring model (BD, green), the Transport model (Tr, dashed-blue), or the Advection-Diffusion model (AD, dashed-red). (Right) Second moment observation from Becker-Döring model with additive noise

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Figure 5. Snapshots of the reconstruction from a boundary observation generated from the Becker-Döring model using the Becker-Döring model (BD, green), the Transport model (Tr, dashed-blue), or the Advection-Diffusion model (AD, dashed-red)

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Figure 6. Snapshots of the reconstruction from a noisy boundary observation generated from the Becker-Döring model using the Becker-Döring model (BD, green), the Transport model (Tr, dashed-blue), or the Advection-Diffusion model (AD, dashed-red)

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Figure 7. Snapshots of the reconstruction from a noisy second moment observation generated from the Becker-Döring model using the Becker-Döring model (BD, green), the Transport model (Tr, dashed-blue), or the Advection-Diffusion model (AD, dashed-red)

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Table 1. Direct models and associated parameters

Physics
Parameters	Becker-Döring (BD)	Transport (Tr)	Advection-diffusion (AD)
$ N_{\text{poly}} $	100	-	-
$ i_0 $	1	-	-
$ L $	-	1	1
$ T $	100	1	1
$ b $	1	1	1
$ \varepsilon $	-	0	$ 10^{-2} $
Discretisation
$ \Delta t $	$ 10^{-3} $	$ 10^{-2} $	$ 5.10^{-3} $
$ h $	-	$ 10^{-2} $	$ 10^{-2} $

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