### 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 |

### ASJC Scopus subject areas

- General

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## 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