Living Reference Work Entry

Handbook of Geomathematics

pp 1-25

Date: Latest Version

Sparsity in Inverse Geophysical Problems

  • Markus GrasmairAffiliated withDepartment of Mathematics, Norwegian University of Science and Technology Email author 
  • , Markus HaltmeierAffiliated withInstitute of Mathematics, University of Innsbruck
  • , Otmar ScherzerAffiliated withComputational Science Center, University of Vienna

Abstract

Many geophysical imaging problems are ill-posed in the sense that the solution does not depend continuously on the measured data. Therefore, their solutions cannot be computed directly but instead require the application of regularization. Standard regularization methods find approximate solutions with small L 2 norm. In contrast, sparsity regularization yields approximate solutions that have only a small number of nonvanishing coefficients with respect to a prescribed set of basis elements. Recent results demonstrate that these sparse solutions often much better represent real objects than solutions with small L 2 norm. In this survey, recent mathematical results for sparsity regularization are reviewed. As an application of the theoretical results, synthetic focusing in Ground Penetrating Radar is considered, which is a paradigm of inverse geophysical problem.