@article {Ozolin{\v s}18368,
author = {Ozolin{\v s}, Vidvuds and Lai, Rongjie and Caflisch, Russel and Osher, Stanley},
title = {Compressed modes for variational problems in mathematics and physics},
volume = {110},
number = {46},
pages = {18368--18373},
year = {2013},
doi = {10.1073/pnas.1318679110},
publisher = {National Academy of Sciences},
abstract = {Intuition suggests that many interesting phenomena in physics, chemistry, and materials science are {\textquotedblleft}short-sighted{\textquotedblright}{\textemdash}that is, perturbation in a small spatial region only affects its immediate surroundings. In mathematical terms, near-sightedness is described by functions of finite range. As an example, the so-called Wannier functions in quantum mechanics are localized functions, which contain all the information about the properties of the system, including its spectral properties. This work{\textquoteright}s main research objective is to develop theory and numerical methods that can systematically derive functions that span the energy spectrum of a given quantum-mechanical system and are nonzero only in a finite spatial region. These ideas hold the key for developing efficient methods for solving partial differential equations of mathematical physics.This article describes a general formalism for obtaining spatially localized ({\textquotedblleft}sparse{\textquotedblright}) 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{\textquoteright}s equation in quantum mechanics. Sparsity is achieved by adding an regularization term to the variational principle, which is shown to yield solutions with compact support ({\textquotedblleft}compressed modes{\textquotedblright}). 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.},
issn = {0027-8424},
URL = {https://www.pnas.org/content/110/46/18368},
eprint = {https://www.pnas.org/content/110/46/18368.full.pdf},
journal = {Proceedings of the National Academy of Sciences}
}