Compressed modes for variational problems in mathematics and physics

Vidvuds Ozoliņš, Rongjie Lai, Russel Caflisch, Stanley Osher

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
Pages (from-to)18368-18373
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America
Volume110
Issue number46
DOIs
StatePublished - Nov 12 2013

Fingerprint

Mathematics
Physics
Mechanics

ASJC Scopus subject areas

  • General

Cite this

Compressed modes for variational problems in mathematics and physics. / Ozoliņš, Vidvuds; Lai, Rongjie; Caflisch, Russel; Osher, Stanley.

In: Proceedings of the National Academy of Sciences of the United States of America, Vol. 110, No. 46, 12.11.2013, p. 18368-18373.

Research output: Contribution to journalArticle

Ozoliņš, Vidvuds ; Lai, Rongjie ; Caflisch, Russel ; Osher, Stanley. / Compressed modes for variational problems in mathematics and physics. In: Proceedings of the National Academy of Sciences of the United States of America. 2013 ; Vol. 110, No. 46. pp. 18368-18373.
@article{edcaa2e1b2ff485ba11a25c712512391,
title = "Compressed modes for variational problems in mathematics and physics",
abstract = "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 Schr{\"o}dinger'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.",
author = "Vidvuds Ozoliņš and Rongjie Lai and Russel Caflisch and Stanley Osher",
year = "2013",
month = "11",
day = "12",
doi = "10.1073/pnas.1318679110",
language = "English (US)",
volume = "110",
pages = "18368--18373",
journal = "Proceedings of the National Academy of Sciences of the United States of America",
issn = "0027-8424",
number = "46",

}

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.

UR - http://www.scopus.com/inward/record.url?scp=84887425211&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887425211&partnerID=8YFLogxK

U2 - 10.1073/pnas.1318679110

DO - 10.1073/pnas.1318679110

M3 - Article

C2 - 24170861

AN - SCOPUS:84887425211

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

SN - 0027-8424

IS - 46

ER -