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Compressed Modes for Variational Problems in Mathematics and Physics
Vidvuds Ozoliņš, Rongjie Lai, Russel Caflisch, Stanley Osher
TL;DR
The paper tackles the problem of constructing localized functions for variational problems whose conventional eigenfunctions are spatially extended. It uses L1-regularized variational optimization to produce compressed modes and compressed plane waves, with numerical results showing compact support and approximate eigenspace representation, while CPW completeness remains a conjecture.
Problem
Conventional eigenfunctions are often spatially extended, while existing localized Wannier-function methods require precomputed eigenfunctions, non-convex optimization, and manual cutoffs.
Method
The paper adds an L1 penalty to a variational formulation to construct compactly supported compressed modes, then develops compressed plane waves and associated numerical transforms.
Results
Numerical experiments find that compressed modes are compactly supported, orthonormal, and approximately span the eigenspace of the Schrödinger operator.
Takeaways & Limitations
Compressed plane waves provide localized position- and scale-dependent modes derived from a differential equation, offering a potential natural basis for solving PDEs.
Takeaways & Limitations
Completeness and orthonormal-basis status of the compressed plane waves are formulated as a conjecture for future study.
Abstract
from arXiv · showhide
This paper describes a general formalism for obtaining localized solutions to a class of problems in mathematical physics, which can be recast as variational optimization problems. This class includes the important cases of Schrödinger's equation in quantum mechanics and electromagnetic equations for light propagation in photonic crystals. These ideas can also be applied to develop a spatially localized basis that spans the eigenspace of a differential operator, for instance, the Laplace operator, generalizing the concept of plane waves to an orthogonal real-space basis with multi-resolution capabilities.
1 Introduction
The paper addresses the difficulty of obtaining spatially localized functions from variational problems when conventional eigenfunctions are extended and dense. It proposes L1-regularized compressed modes and develops compressed plane waves as localized orthonormal functions with multiresolution capabilities.
- Motivation: Sparsity methods traditionally seek sparse coefficients in a chosen basis, whereas this paper seeks modes that are themselves sparse and localized in space.The proposal extends sparsity techniques from information-science representations to variational problems in mathematics and physics.
- Motivation: Conventional Schrödinger eigenfunctions often have infinite support, creating computational challenges because orthogonalization requires O(N^3) operations for large systems.The paper connects this computational issue with the physical intuition that screened correlations in condensed matter are short-ranged.
- Existing approaches: Wannier functions localize the eigenspace through unitary combinations of eigenfunctions, but maximally localized Wannier functions require precomputed eigenfunctions and a difficult non-convex optimization.Their spatial range must also be cut off manually, potentially causing numerical errors when the range is unknown in advance.
- Proposed method: The proposed compressed modes modify the variational objective with L1 regularization, producing compactly supported functions while approximating total energy without calculating eigenfunctions.The parameter µ controls the trade-off between localization and energy accuracy: larger µ favors accuracy and more extended functions, while smaller µ favors localization and larger errors.
- Proposed method: The paper also introduces compressed plane waves, a spatially localized orthonormal function set with multiresolution capabilities, together with fast forward and inverse transforms.These are presented as a second major contribution alongside the compressed-mode method and its numerical algorithm.
2 Variational Model for Compressed Modes
The compressed-mode variational model adds L1 regularization to obtain localized, orthonormal functions while controlling the trade-off between localization and energy accuracy. Numerical results show compact support, approximate eigenspace spanning, and convergence of eigenvalue errors under increasing µ or N.
- Variational model: L1 regularization localizes solutions of the Schrödinger variational problem, interpolating between a Dirac delta as µ → 0 and the first eigenfunction as µ → ∞.The parameter µ controls the balance between sparsity and accuracy.
- Variational model: Theoretical free-electron solutions have compact support, with smaller µ producing a smaller support region.The support is [L/2 − l, L/2 + l] when the compact-support condition holds.
- Variational model: Compressed modes are defined as the first N functions minimizing the L1-regularized variational model and are expected to form localized orthonormal combinations of eigenmodes.The paper formulates a conjecture that these modes approximately capture the low-energy eigenspace.
- Numerical algorithms: The SOC algorithm solves the constrained optimization problem by splitting variables and applying split Bregman iteration.The formulation introduces auxiliary variables for the L1 norm and orthogonality constraints.
- Numerical algorithms: Compact support limits each mode’s overlap to finitely many neighbors, replacing full orthogonality with banded constraints and reducing factorization cost from O(N^3) to 8p^3O(N).The reduced computation uses N factorizations of 2p × 2p matrices.
- Numerical results: Numerical experiments show compactly supported orthonormal CMs that approximately span the Schrödinger operator’s low-energy eigenspace.For fixed M = N = 50, relative error converges to zero as µ → ∞; for fixed µ = 10 and M = 50, it converges to zero as N → ∞.
3 Compressed Plane Waves (CPWs)
Compressed plane waves (CPWs) form a spatially localized, orthonormal basis with scale and shift parameters analogous to wavelets. Numerical experiments show sparse representations of localized states, accurate eigenvalue approximations, and fast transforms, while completeness remains conjectural.
- Construction and properties: The CPW construction is designed for localized basis functions and can be extended from the one-dimensional periodic setting to higher dimensions.The multidimensional construction differs from the usual tensor-product generalization, while the fast transform algorithms can be extended to higher dimensions.
- Open question: Completeness and orthonormal-basis status for CPWs are conjectured rather than established, with the proposed claim requiring µ ≥ µ0 for some constant µ0.The conjecture is motivated by numerical experiments and is identified as future work.
- Construction and properties: CPWs are constructed from localized orthonormal modes, with index n controlling support size and scale and index j controlling spatial shift.These parameters parallel the scale and shift parameters of wavelet theory.
- Spectral properties: Each of the first six basic compressed plane waves occupies a distinct Fourier-space region, while their combined spectral weight forms a step-like distribution.This distribution approximately covers the same local Fourier space as plane waves below a given kinetic-energy cutoff.
- Localized-function representations: CPWs represent localized impurity Kronig–Penney energy states more sparsely than Fourier basis functions, requiring significantly fewer basis functions for comparable accuracy.The comparison uses the first 120 CPWs generated from the first six basic compressed plane waves and examines the largest-magnitude coefficients and ℓ2 representation errors.
- Localized-function representations: CPW representations accurately approximate the first few impurity Kronig–Penney eigenvalues by reducing the original eigenvalue problem to a smaller matrix problem.The reported comparison uses 120 CPWs generated from the first six basic compressed plane waves.
- Fast transforms: The CPW transform accurately reproduces the first four lowest-energy impurity Kronig–Penney states, and windowed transforms permit different mesh resolutions across spatial regions.The transform comparison uses direct diagonalization as the reference; windowed inverse transforms are evaluated over regions where the functions are nonzero.
4 Discussions and Conclusions
The paper presents compressed modes for the Laplace operator plus a potential and compressed plane waves when the potential vanishes. It outlines extensions to simulations, PDE discretizations, broader geometries, and rigorous analysis.
- Contributions: Compressed modes are sparse and localized modes constructed variationally for the Laplace operator plus a potential using L1 penalization and the SOC algorithm.With V = 0, the method also produces compressed plane waves parameterized by position and scale.
- Future applications: The authors plan to construct compressed modes for a variety of potentials.
- Future applications: Compressed modes may support an accelerated O(N) simulation method for density functional theory.
- Future applications: The proposed modes may serve as Galerkin bases for PDEs such as Maxwell’s equations and extend to heat-type equations derived from variational gradient descent.
- Future applications: The framework is intended for higher dimensions and geometries, including Laplace-Beltrami equations on manifolds and discrete Laplacians on networks.
- Open questions: A planned analysis will rigorously examine the existence and qualitative properties of compressed modes and compressed plane waves, including hypothesized properties.