Regularization and numerical solution of the inverse scattering problem using shearlet frames

Kutyniok, Gitta ; Mehrmann, Volker und Petersen, Philipp C. (29. June 2016): Regularization and numerical solution of the inverse scattering problem using shearlet frames. In: Journal of Inverse and Ill-posed Problems, Vol. 25, No. 3: pp. 287-309

Abstract

Regularization techniques for the numerical solution of inverse scattering problems in two space dimensions are discussed. Assuming that the boundary of a scatterer is its most prominent feature, we exploit as model the class of cartoon-like functions. Since functions in this class are asymptotically optimally sparsely approximated by shearlet frames, we consider shearlets as a means for regularization. We analyze two approaches, namely solvers for the nonlinear problem and for the linearized problem obtained by the Born approximation. As example for the first class we study the acoustic inverse scattering problem, and for the second class, the inverse scattering problem of the Schrödinger equation. Whereas our emphasis for the linearized problem is more on the theoretical side due to the standardness of associated solvers, we provide numerical examples for the nonlinear problem that highlight the effectiveness of our algorithmic approach.

• Item Type: Journal article
• Keywords: Helmholtz equation; inverse medium scattering; regularization; Schrödinger equation, shearlets; sparse approximation
• Faculties: Mathematics, Computer Science and Statistics > Mathematics > Bavarian Chair for Mathematical Foundations of Artificial Intelligence
• Subjects: 500 Science > 510 Mathematics
• ISSN: 0928-0219
• Language: English