On convex modified output least-squares for elliptic inverse problems: stability, regularization, applications, and numerics. (3rd June 2017) - Jadamba, Baasansuren; Khan, Akhtar A.; Sama, Miguel; Tammer, Christiane
Abstract: Inverse problems of identifying parameters in partial differential equations constitute an important class of problems with diverse real-world applications. These identification problems are commonly explored in an optimization framework and there are many optimization formulations having their own advantages and disadvantages. Although a non-convex output least-squares (OLS) objective is commonly used, a convex-modified output least-squares (MOLS) has shown encouraging results in recent years. In this work, we focus on various aspects of the MOLS approach. We devise a rigorous (quadratic and non-quadratic) regularization framework for the identification of smooth as well as discontinuous coefficients. This framework subsumes the total variation regularization that has attracted a great deal of attention in identifying sharply varying coefficients and also in image processing. We give new existence results for the regularized optimization problems for OLS and MOLS. Restricting to the Tikhonov (quadratic) regularization, we carry out a detailed study of various stability aspects of the inverse problem under data perturbation and give new stability estimates for general inverse problems using OLS and MOLS formulations. We give a discretization scheme for the continuous inverse problem and prove the convergence of the discrete inverse problem to the continuous one. We collect discrete formulas for OLS and MOLS and compute their gradients and Hessians. We present applications of our theoretical results. To show the feasibility of the MOLS framework, we also provide computational results for the inverse problem of identifying parameters in three different classes of partial differential equations .