Globally Constructed Adaptive Local Basis Set for Spectral Projectors of Second Order Differential Operators

TL;DR

This paper introduces an adaptive local basis set for spectral projectors of second order differential operators, significantly reducing computational costs while accurately capturing local features, applicable to various dimensions and potential types.

Contribution

The authors develop a systematically improvable, randomized basis construction method that efficiently approximates spectral projectors with fewer degrees of freedom, even with nonlocal potentials.

Findings

Effective in 1D, 2D, and 3D problems with local potentials.

Accurately approximates spectral projectors with reduced degrees of freedom.

Applicable to nonlinear problems with nonlocal potentials.

Abstract

Spectral projectors of second order differential operators play an important role in quantum physics and other scientific and engineering applications. In order to resolve local features and to obtain converged results, typically the number of degrees of freedom needed is much larger than the rank of the spectral projector. This leads to significant cost in terms of both computation and storage. In this paper, we develop a method to construct a basis set that is adaptive to the given differential operator. The basis set is systematically improvable, and the local features of the projector is built into the basis set. As a result the required number of degrees of freedom is only a small constant times the rank of the projector. The construction of the basis set uses a randomized procedure, and only requires applying the differential operator to a small number of vectors on the global domain, while each basis function itself is supported on strictly local domains and is discontinuous across the global domain. The spectral projector on the global domain is systematically approximated from such a basis set using the discontinuous Galerkin (DG) method. The global construction procedure is very flexible, and allows a local basis set to be consistently constructed even if the operator contains a nonlocal potential term. We verify the effectiveness of the globally constructed adaptive local basis set using one-, two- and three-dimensional linear problems with local potentials, as well as a one dimensional nonlinear problem with nonlocal potentials resembling the Hartree-Fock problem in quantum physics.