An $L^\infty$ Regularization Strategy to the Inverse Source Identification Problem for Elliptic Equations

Abstract

In this paper we utilise new methods of Calculus of Variations in L ∞ to provide a regularisation strategy to the ill-posed inverse problem of identifying the source of a non-homogeneous linear elliptic equation, satisfying Dirichlet data on a domain. One of the advantages over the classical Tykhonov regularisation in L 2 is that the approximated solution of the PDE is uniformly close to the noisy measurements taken on a compact subset of the domain.

On the absolute stable difference scheme for the space‐wise dependent source identification problem for elliptic‐telegraph equation

“…Source identification problems (SIPs) for partial differential equations (PDEs) are used to model physical and biological system engineering and sociological processes and have been studied by many authors (see [1–12] and the references given therein).…”

“…The first and second order of accuracy difference schemes (DSs) were investigated and the numerical results were given. In [10], new methods of calculus of variations in L ∞ to study the ill‐posed inverse problem of identifying the source of a nonhomogeneous linear elliptic equation with Dirichlet conditions were investigated. The inverse problem for elliptic–hyperbolic equation with a nonlocal boundary condition was discussed.…”

Stability of the space identification problem for the elliptic‐telegraph differential equation

“…Source identification problems for partial differential equations are used to model biological, physical, system engineering, and sociological processes and have been studied by many authors (see other studies [1][2][3][4][5][6][7][8][9][10][11][12] and the references given therein).…”

“…The stability, almost coercive stability, and coercive stability inequalities for its solution have been obtained. Katzourakis 10 investigated new methods of calculus of variations in L ∞ to study the ill‐posed inverse problem of identifying the source of a nonhomogeneous linear elliptic equation for Dirichlet conditions. Sabitov and Martem'yanova 11 studied the inverse problem for an equation of elliptic‐hyperbolic type with a nonlocal boundary condition.…”

Inverse Optical Tomography through PDE Constrained Optimization $L^\infty$

“…In this paper we utilise novel methods of Calculus of Variations in L ∞ in order to lay the rigorous mathematical foundations of the FOT problem. Motivated by developments underpinning the papers [38,40,41,42], we pose FOT as a minimisation problem in L ∞ with PDE constraints as well as unilateral constraints, studying the direct as well as the inverse FOT problem, both in L p for finite p and in L ∞ . Further, we derive the relevant variational inequalities in L p for finite p and in L ∞ that the constrained minimisers satisfy, which involve (generalised) Lagrange multipliers.…”