Compressed modes for variational problems in mathematics and physics

By V. Ozoliņš, Rongjie Lai, Russel E. Caflisch, Stanley J. Osher. 2013

194 citations recorded by the source corpus.

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.

Subfields: Atomic and Molecular Physics, and Optics, Electrical and Electronic Engineering

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

In Electrical and Electronic Engineering

Citations in the corpus

Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations (2018, methodology): “More recently, sparsity [23] has been used to determine the governing dynamical system [24, 25, 26, 27, 28, 29, 30, 31, 32, 33].”
On the Compressive Spectral Method (2014, methodology): “In [21, 22, 23], eigenfunctions with compact support were constructed to efficiently solve problems in quantum mechanics.”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “There are many other interesting methods recently developed to incorporate sparsity and nonlinear dynamics (Schaeffer et al., 2013; Ozoliņš et al., 2013; Mackey et al., 2014).”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, unknown): “[39] V.”
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control (2015, background): “This follows a growing trend to exploit sparsity in dynamics [35, 36, 37] and dynamical systems [38, 39, 40].”
Chaos as an intermittently forced linear system (2016, background): “This work is part of a growing literature that is exploring the use of sparsity in dynamics [72, 84, 64] and dynamical systems [8, 78, 14].”