Implicit algorithms for eigenvector nonlinearities

Jarlebring, Elias; Upadhyaya, Parikshit

doi:10.1007/s11075-021-01189-4

Cited by 6 publications

(2 citation statements)

References 29 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The selection strategy ususally depends on the context provided by the application. For example, for the solution of the Gross-Pitaevskii equation in quantum physics [17], we would select the eigenpair corresponding to the smallest eigenvalue because we are interested in computing the ground state of bosons. For this paper, we will focus on applications that require computing the eigenpair corresponding to the second smallest eigenvalue, that is, a partition into two clusters.…”

Section: Nepv1 Formulationmentioning

confidence: 99%

The self-consistent field iteration for p-spectral clustering

Upadhyaya¹,

Jarlebring²,

Tudisco³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

The self-consistent field (SCF) iteration, combined with its variants, is one of the most widely used algorithms in quantum chemistry. We propose a procedure to adapt the SCF iteration for the p-Laplacian eigenproblem, which is an important problem in the field of unsupervised learning. We formulate the p-Laplacian eigenproblem as a type of nonlinear eigenproblem with one eigenvector nonlinearity , which then allows us to adapt the SCF iteration for its solution after the application of suitable regularization techniques. The results of our numerical experiments confirm the viablity of our approach.

show abstract

Section: Nepv1 Formulationmentioning

confidence: 99%

The self-consistent field iteration for p-spectral clustering

Upadhyaya¹,

Jarlebring²,

Tudisco³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…A well-known iteration scheme for the nonlinear eigenvector problem is the self-consistent field iteration (SCF). Each SFC iteration step involves the solution of a linear eigenvalue problem, see, e.g., [9,10,13] and [21] for its connection to Newton's method. On the Riemmanian side, the direct constrained minimization algorithm (DCM) is very popular.…”

Section: Introductionmentioning

confidence: 99%

Energy-adaptive Riemannian optimization on the Stiefel manifold

2022

View full text Add to dashboard Cite

This paper addresses the numerical solution of nonlinear eigenvector problems such as the Gross-Pitaevskii and Kohn-Sham equation arising in computational physics and chemistry. These problems characterize critical points of energy minimization problems on the infinite-dimensional Stiefel manifold. To efficiently compute minimizers, we propose a novel Riemannian gradient descent method induced by an energyadaptive metric. Quantified convergence of the methods is established under suitable assumptions on the underlying problem. A non-monotone line search and the inexact evaluation of Riemannian gradients substantially improve the overall efficiency of the method. Numerical experiments illustrate the performance of the method and demonstrates its competitiveness with well-established schemes.

show abstract

Riemannian Newton Methods for Energy Minimization Problems of Kohn–Sham Type

Altmann,

Peterseim,

Stykel

2024

J Sci Comput

View full text Add to dashboard Cite

This paper is devoted to the numerical solution of constrained energy minimization problems arising in computational physics and chemistry such as the Gross–Pitaevskii and Kohn–Sham models. In particular, we introduce Riemannian Newton methods on the infinite-dimensional Stiefel and Grassmann manifolds. We study the geometry of these two manifolds, its impact on the Newton algorithms, and present expressions of the Riemannian Hessians in the infinite-dimensional setting, which are suitable for variational spatial discretizations. A series of numerical experiments illustrates the performance of the methods and demonstrates their supremacy compared to other well-established schemes such as the self-consistent field iteration and gradient descent schemes.

show abstract

Implicit algorithms for eigenvector nonlinearities

Cited by 6 publications

References 29 publications

The self-consistent field iteration for p-spectral clustering

The self-consistent field iteration for p-spectral clustering

Energy-adaptive Riemannian optimization on the Stiefel manifold

Riemannian Newton Methods for Energy Minimization Problems of Kohn–Sham Type

Contact Info

Product

Resources

About