Recovery of rapidly decaying source terms from dynamical samples in evolution equations

TL;DR

This paper investigates methods to recover rapidly decaying source terms in evolution equations from noisy space-time samples of the solution, considering background sources and sampling strategies.

Contribution

It introduces a novel approach for recovering decaying source terms in evolution equations, accounting for noise and background sources in the sampling process.

Findings

Effective recovery of source terms demonstrated in theoretical analysis.

Robustness of the method against measurement noise shown.

Applicability to various decay functions and background sources.

Abstract

We analyze the problem of recovering a source term of the form $h(t)=\sum_{j}h_j\phi(t-t_j)\chi_{[t_j, \infty)}(t)$ from space-time samples of the solution $u$ of an initial value problem in a Hilbert space of functions. In the expression of $h$, the terms $h_j$ belong to the Hilbert space, while $\phi$ is a generic real-valued function with exponential decay at $\infty$. The design of the sampling strategy takes into account noise in measurements and the existence of a background source.