### 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 language | English (US) |
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Pages (from-to) | 18368-18373 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 110 |

Issue number | 46 |

DOIs | |

State | Published - Nov 12 2013 |

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### ASJC Scopus subject areas

- General

### Cite this

*Proceedings of the National Academy of Sciences of the United States of America*,

*110*(46), 18368-18373. https://doi.org/10.1073/pnas.1318679110

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

Research output: Contribution to journal › Article

*Proceedings of the National Academy of Sciences of the United States of America*, vol. 110, no. 46, pp. 18368-18373. https://doi.org/10.1073/pnas.1318679110

}

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

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 -