A time reversal based algorithm for solving initial data inverse problems

Abstract

We propose an iterative algorithm to solve initial data inverse problems for a class of linear evolution equations, including the wave, the plate, the Schrödinger and the Maxwell equations in a bounded domain Ω. We assume that the only available information is a distributed observation (i.e. partial observation of the solution on a sub-domain ω during a finite time interval (0,τ)). Under some quite natural assumptions (essentially: the exact observability of the system for some time τ obs > 0, τ ≥ τ obs and the existence of a time-reversal operator for the problem), an iterative algorithm based on a Neumann series expansion is proposed. Numerical examples are presented to show the efficiency of the method.