TY - JOUR
T1 - Compressed modes for variational problems in mathematics and physics
AU - Ozoliņš, Vidvuds
AU - Lai, Rongjie
AU - Caflisch, Russel
AU - Osher, Stanley
PY - 2013/11/12
Y1 - 2013/11/12
N2 - This article describes a general formalism for obtaining spatially localized ("sparse") solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger's equation in quantum mechanics. Sparsity is achieved by adding an L1 regularization term to the variational principle, which is shown to yield solutions with compact support ("compressed modes"). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size.
AB - This article describes a general formalism for obtaining spatially localized ("sparse") solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems, such as the important case of Schrödinger's equation in quantum mechanics. Sparsity is achieved by adding an L1 regularization term to the variational principle, which is shown to yield solutions with compact support ("compressed modes"). Linear combinations of these modes approximate the eigenvalue spectrum and eigenfunctions in a systematically improvable manner, and the localization properties of compressed modes make them an attractive choice for use with efficient numerical algorithms that scale linearly with the problem size.
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U2 - 10.1073/pnas.1318679110
DO - 10.1073/pnas.1318679110
M3 - Article
C2 - 24170861
AN - SCOPUS:84887425211
SN - 0027-8424
VL - 110
SP - 18368
EP - 18373
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America
IS - 46
ER -