Carleman--Picard and time-dimensional reduction for inverse initial-data problems in nonlinear transport with memory
We study an inverse initial-data problem for a quasilinear transport equation with nonlinear and memory effects. The unknown initial state is reconstructed from time-dependent measurements on the outflow boundary, with prescribed inflow data. We first apply a Legendre--exponential time-dimensional reduction to transform the governing equation into a finite nonlinear system in space. We then develop a Carleman-weighted and Tikhonov-regularized Picard method. At each iteration, the nonlinear terms are evaluated using the previous iterate, leading to a linear minimization problem with a unique solution. A Carleman estimate for the principal transport operator is used to prove that, for a sufficiently large Carleman parameter, the resulting Picard map is contractive on a prescribed admissible set. Consequently, the method converges from an arbitrary initial guess in that set. We also establish stability with respect to noisy outflow data. These analytical results concern the truncated and regularized reduced problem. Two-dimensional numerical experiments demonstrate accurate reconstruction of single and multiple inclusions, robustness with respect to noise, and rapid convergence of the Picard iteration.