A two-stage numerical approach for the sparse initial source identification of a diffusion-advection equation

TL;DR

This paper introduces a two-stage numerical method to identify sparse initial sources in diffusion-advection equations, effectively addressing the ill-posedness and achieving accurate source reconstruction in short time horizons.

Contribution

A novel two-stage approach combining sparsity-promoting optimal control and refinement for initial source identification in diffusion-advection PDEs.

Findings

Effective in short time horizons

Handles ill-posedness via sparsity-promoting terms

Numerical experiments validate efficiency

Abstract

We consider the problem of identifying a sparse initial source condition to achieve a given state distribution of a diffusion-advection partial differential equation after a given final time. The initial condition is assumed to be a finite combination of Dirac measures. The locations and intensities of this initial condition are required to be identified. This problem is known to be exponentially ill-posed because of the strong diffusive and smoothing effects. We propose a two-stage numerical approach to treat this problem. At the first stage, to obtain a sparse initial condition with the desire of achieving the given state subject to a certain tolerance, we propose an optimal control problem involving sparsity-promoting and ill-posedness-avoiding terms in the cost functional, and introduce a generalized primal-dual algorithm for this optimal control problem. At the second stage, the initial condition obtained from the optimal control problem is further enhanced by identifying its locations and intensities in its representation of the combination of Dirac measures. This two-stage numerical approach is shown to be easily implementable and its efficiency in short time horizons is promisingly validated by the results of numerical experiments. Some discussions on long time horizons are also included.