Direct algorithms for solving some inverse source problems in 2D elliptic equations

Abstract

This paper deals with the resolution of some inverse source problems in the 2D elliptic equation ∆u + µu = F from Cauchy data. Two types of sources are considered, pointwise sources and sources having compact support within a finite number of small subdomains. An identification direct algorithm, based on an algebraic approach, is proposed. This is a new result, as far as we know, except in the case µ = 0 which is already considered in [14].

Dispersion–current adjoint functions for monitoring accidental sources in 3D transport equations

“…In the literature, many authors have addressed similar inverse source problems in different PDEs, for instance [2,3,4,18,19,20,31]. In these works, the developed identification approaches are mainly either iterative based on the minimisation of cost functions such as least squares and Kohn-Vogelius or quasi-direct such as the algebraic method especially used for elliptic equations [1,32]. Besides, the underlined inverse source problem becomes more challenging in n = 2, 3 dimensions where the involved PDE admits an advection term.…”

“…), where λ (k) ∈ L 2 (0, T ) fulfilling ( 6) and S (k) is an interior point in Ω i . We denote by w = u (2) − u (1) . Then, from assuming…”

“…Using v 0i = 1 in (47) givesλ (2) =λ (1) . Afterwards, from (47) and ( 35), we get Ψ m,n i (S (2) ) = Ψ m,n i (S (1) ) =⇒ S (2) = S (1) .…”

A regularization operator for source identification for elliptic PDEs

“…Introduction. We will study the problem of identifying the source in a prototypical elliptic PDE from Dirichlet boundary data: (1) min…”

“…Even though most source identification tasks for elliptic PDEs are ill-posed, several methods for computing reliable results have been developed. Typically, one assumes a priori that f is composed of a finite number of pointwise sources or sources having compact support within a small number of finite subdomains, see, e.g., [1,4,7,10,22] and references therein. Such approaches lead to involved mathematical issues, but in many cases optimization procedures and/or explicit regularization can be avoided.…”