An inverse problem: recovering the fragmentation kernel from the short-time behaviour of the fragmentation equation. (Un problème inverse: estimer le noyau de fragmentation à partir du comportement en temps court de la solution de l’équation de fragmentation.) (English. French summary) Zbl 07914802
Summary: 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.
MSC:
| 35Q92 | PDEs in connection with biology, chemistry and other natural sciences |
| 92C40 | Biochemistry, molecular biology |
| 35R30 | Inverse problems for PDEs |
| 35R06 | PDEs with measure |
| 35R09 | Integro-partial differential equations |
| 45Q05 | Inverse problems for integral equations |
| 46F12 | Integral transforms in distribution spaces |
| 30D05 | Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable |
| 35C05 | Solutions to PDEs in closed form |
| 35C20 | Asymptotic expansions of solutions to PDEs |
| 28C05 | Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
measure-valued solutions; size-structured partial differential equation; fragmentation equation; inverse problemReferences:
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