Communications in Mathematical Sciences

Volume 13 (2015)

Number 8

PDEs with compressed solutions

Pages: 2155 – 2176

DOI: http://dx.doi.org/10.4310/CMS.2015.v13.n8.a8

Authors

Russel E. Caflisch (Department of Mathematics, University of California at Los Angeles, Los Angeles)

Stanley J. Osher (Department of Mathematics, University of California at Los Angeles, Los Angeles)

Hayden Schaeffer (Department of Computing Mathematical Sciences, California Institute of Technology, Pasadena, Calif., U.S.A.)

Giang Tran (Department of Mathematics, University of California at Los Angeles, Los Angeles)

Abstract

Sparsity plays a central role in recent developments in signal processing, linear algebra, statistics, optimization, and other fields. In these developments, sparsity is promoted through the addition of an $L^1$ norm (or related quantity) as a constraint or penalty in a variational principle. We apply this approach to partial differential equations that come from a variational quantity, either by minimization (to obtain an elliptic PDE) or by gradient flow (to obtain a parabolic PDE). Also, we show that some PDEs can be rewritten in an $L^1$ form, such as the divisible sandpile problem and signum-Gordon. Addition of an $L^1$ term in the variational principle leads to a modified PDE where a subgradient term appears. It is known that modified PDEs of this form will often have solutions with compact support, which corresponds to the discrete solution being sparse. We show that this is advantageous numerically through the use of efficient algorithms for solving $L^1$ based problems.

Keywords

sparsity, compressive sensing, PDE, free boundary

2010 Mathematics Subject Classification

35A99, 65K10, 65M22

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Published 3 September 2015