Compressed modes for variational problems in mathematics and physics
Significance Intuition suggests that many interesting phenomena in physics, chemistry, and materials science are “short-sighted”—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’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 (“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 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.