Heat Source Identification Based on L1 Constrained Minimization

Heat Source Identification Based on L1 Constrained Minimization

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.

Simulation

Recovery of the heat source $u_0$ from 60 randomly selected measurements with 1% noise on a $32 \times 32$ grid.
Source recovery with a smooth spatially varying thermal conductivity. Left: distribution of thermal conductivity (shades of orange); sampling locations are red stars, heat source locations are blue dots. Middle: heat distribution at time $T$ (shades of blue). Right: recovered source.
Source recovery with successive sampling. The $(k+1)^{th}$ measurement is added in step $k$. The estimate of the source term is in blue, the exclusion region is in gray, sample locations are shown as red stars.

Publications

  • Y. Li, S. Osher, and R. Tsai. "Heat Source Identification Based Onconstrained Minimization." Inverse Problems and Imaging 8.1 (2014): 199-221.
  • M. Burger, Y. Landa, N. Tanushev, and R. Tsai. "Discovering a point source in unknown environments." Algorithmic Foundation of Robotics VIII. Springer, Berlin, Heidelberg, 2009. 663-678.

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