Inverse Problems and Imaging (IPI)

Heat source identification based on $l_1$ constrained minimization

Pages: 199 - 221, Volume 8, Issue 1, February 2014      doi:10.3934/ipi.2014.8.199

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Yingying Li - University of California, Los Angeles, Los Angeles, CA 90095, United States (email)
Stanley Osher - Department of Mathematics, University of California, Los Angeles, CA 90095-1555, United States (email)
Richard Tsai - The University of Texas at Austin, Austin, TX 78712, United States (email)

Abstract: We consider the inverse problem of finding sparse initial data from the sparsely sampled solutions of the heat equation. The initial data are assumed to be a sum of an unknown but finite number of Dirac delta functions at unknown locations. Point-wise values of the heat solution at only a few locations are used in an $l_1$ constrained optimization to find the initial data. A concept of domain of effective sensing is introduced to speed up the already fast Bregman iterative algorithm for $l_1$ optimization. Furthermore, an algorithm which successively adds new measurements at specially chosen locations is introduced. By comparing the solutions of the inverse problem obtained from different number of measurements, the algorithm decides where to add new measurements in order to improve the reconstruction of the sparse initial data.

Keywords:  $L1$, source identification, heat equation.
Mathematics Subject Classification:  Primary: 49N45, 65M32.

Received: January 2011;      Revised: November 2012;      Available Online: March 2014.