Constructing a variational quasi‐reversibility method for a Cauchy problem for elliptic equations

Abstract

In the recent developments of regularization theory for inverse and ill‐posed problems, a variational quasi‐reversibility (QR) method has been designed to solve a class of time‐reversed quasi‐linear parabolic problems. Known as a PDE‐based approach, this method relies on adding a suitable perturbing operator to the original problem and consequently, on gaining the corresponding fine stabilized operator, which leads us to a forward‐like problem. In this work, we establish new conditional estimates for such operators to solve a prototypical Cauchy problem for elliptic equations. This problem is based on the stationary case of the inverse heat conduction problem, where one wants to identify the heat distribution in a certain medium, given the partial boundary data. Using the new QR method, we obtain a second‐order initial value problem for a wave‐type equation, whose weak solvability can be deduced using a priori estimates and compactness arguments. Weighted by a Carleman‐like function, a new type of energy estimates is explored in a variational setting when we investigate the Hölder convergence rate of the proposed scheme. Besides, a linearized version of this scheme is analyzed. Numerical examples are provided to corroborate our theoretical analysis.