A globally convergent Carleman-Picard method for an inverse initial-value problem for a nonlinear diffusive coagulation-fragmentation equation coagulation-fragmentation equation
This work addresses an inverse initial-value problem in mathematical modeling of particle systems, providing a robust numerical solution for applications like materials science or environmental monitoring, though it appears incremental as it builds on existing Carleman and Picard techniques.
The authors tackled the inverse problem of recovering an unknown initial particle-size distribution from boundary observations of a nonlinear diffusive coagulation-fragmentation equation, developing a globally convergent Carleman-Picard method that achieved accurate and stable reconstructions in numerical experiments with noisy data.
We study an inverse initial-density problem for a nonlinear diffusive coagulation--fragmentation equation with known coagulation and fragmentation kernels. The objective is to recover the unknown initial particle-size distribution on a finite interval from time-dependent boundary observations of the solution and its size derivative. To solve this inverse problem, we develop a globally convergent numerical method based on a Legendre--exponential time reduction and a Carleman--Picard iteration. The time reduction transforms the original problem into a nonlinear coupled system for the spatial mode coefficients, while the Carleman weight and the corresponding Carleman estimate guaranty the global convergence of the Picard iteration without requiring a good initial guess. We prove the convergence of the proposed method and obtain a complete reconstruction procedure for the initial density. Numerical experiments with noisy boundary data demonstrate that the method yields accurate and stable reconstructions for several representative test profiles.