An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation

Doumic, Marie; Escobedo, Miguel; Tournus, Magali

Annales Henri Lebesgue, Tome 7 (2024), pp. 621-671

Given a phenomenon described by a self-similar fragmentation equation, how to infer the fragmentation kernel from experimental measurements of the solution? To answer this question at the basis of our work, a formal asymptotic expansion suggested us that using short-time observations and initial data close to a Dirac measure should be a well-adapted strategy. As a necessary preliminary step, we study the direct problem, i.e. we prove existence, uniqueness and stability with respect to the initial data of non negative measure-valued solutions when the initial data is a compactly supported, bounded, non negative measure. A representation of the solution as a power series in the space of Radon measures is also shown. This representation is used to propose a reconstruction formula for the fragmentation kernel, using short-time experimental measurements when the initial data is close to a Dirac measure. We prove error estimates in Total Variation and Bounded Lipshitz norms; this gives a quantitative meaning to what a “short” time observation is. For general initial data in the space of compactly supported measures, we provide estimates on how the short-time measurements approximate the convolution of the fragmentation kernel with a suitably-scaled version of the initial data. The series representation also yields a reconstruction formula for the Mellin transform of the fragmentation kernel $\kappa$ and an error estimate for such an approximation. Our analysis is complemented by a numerical investigation.

Comment, à partir de données expérimentales, retrouver le noyau de fragmentation $\kappa$ associé à une population dont l’évolution est décrite par une équation de fragmentation  ? Un développement asymptotique formel autour de la donnée initiale nous laisse penser que la réponse réside dans le comportement en temps court de la solution, avec comme donnée initiale une masse de Dirac. Pour exploiter cette piste, notre première étape consiste à étudier le problème direct, et plus particulièrement à établir des résultats d’existence, unicité et stabilité par rapport à la donnée initiale de solutions mesures, et ce, dans le cas où la donnée initiale est mesure positive bornée à support compact. Au cours de cette première étape, nous exhibons la solution sous la forme d’une série entière dans l’espace des mesures de Radon. Nous nous servons cette représentation de la solution pour fournir une formule de reconstruction pour le noyau de fragmentation $\kappa$, en utilisant le profil de la solution à un temps $t$ suffisamment court, dans le cas où la donnée initiale est une masse de Dirac. Dans ce cadre, nous contrôlons rigoureusement l’erreur sur l’estimation du noyau $\kappa$ en variation totale (TV) et en norme “Bounded–Lipschitz”. Cette erreur dépendant en partie du temps $t$ auquel la mesure est faite, ceci clarifie ce qu’une observation “en temps court” signifie. Dans le cas où la donnée initiale est une mesure de Radon générique (mais positive et à support compact), nous montrons qu’une observation de la solution en un temps suffisamment court permet d’approcher le produit de convolution entre le noyau $\kappa$ et une homothétie de la donnée initiale. La représentation de la solution en série entière nous fournit également une formule de reconstruction pour la transformée de Mellin $K$ du noyau de fragmentation $\kappa$, ainsi qu’une estimation de l’erreur pour cette formule approchée. Nous complétons notre analyse par des explorations numériques.

Classification : 35Q92, 35R06, 35R09, 45Q05, 46F12, 30D05
Keywords: Measure-valued solutions, Size-structured partial differential equation, Fragmentation equation, Inverse problem

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