相关论文: Convergence rates of individual Ritz values in blo…

Preconditioned gradient iterations for very large eigenvalue problems are efficient solvers with growing popularity. However, only for the simplest preconditioned eigensolver, namely the preconditioned gradient iteration (or preconditioned…

The performance of eigenvalue problem solvers (eigensolvers) depends on various factors such as preconditioning and eigenvalue distribution. Developing stable and rapidly converging vectorwise eigensolvers is a crucial step in improving the…

Convergence analysis of block iterative solvers for Hermitian eigenvalue problems and the closely related research on properties of matrix-based signal filters are challenging, and attract increasing attention due to their recent…

This paper provides a comprehensive and detailed analysis of the local convergence behavior of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for computing the extreme eigenvalue of a…

Preconditioned eigenvalue solvers (eigensolvers) are gaining popularity, but their convergence theory remains sparse and complex. We consider the simplest preconditioned eigensolver--the gradient iterative method with a fixed step size--for…

Convergence rates of block iterations for solving eigenvalue problems typically measure errors of Ritz values approximating eigenvalues. The errors of the Ritz values are commonly bounded in terms of principal angles between the initial or…

Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) is widely used to compute eigenvalues of large sparse symmetric matrices. The algorithm can suffer from numerical instability if it is not implemented with care. This is…

A new iteration bound for the preconditioned conjugate gradient (PCG) method is presented that more accurately captures convergence for systems with clustered eigenspectra, where the classical condition number-based bound is too…

We present an iterative algorithm for computing an invariant subspace associated with the algebraically smallest eigenvalues of a large sparse or structured Hermitian matrix A. We are interested in the case in which the dimension of the…

This paper is concerned with the convergence analysis of an extended variation of the locally optimal preconditioned conjugate gradient method (LOBPCG) for the extreme eigenvalue of a Hermitian matrix polynomial which admits some extended…

Projected policy gradient (PPG) is a basic policy optimization method in reinforcement learning. Given access to exact policy evaluations, previous studies have established the sublinear convergence of PPG for sufficiently small step sizes…

We present a novel class of projected gradient (PG) methods for minimizing a smooth but not necessarily convex function over a convex compact set. We first provide a novel analysis of the constant-stepsize PG method, achieving the…

In symmetric block eigenvalue algorithms, such as the subspace iteration algorithm and the locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm, a large block size is often employed to achieve robustness and rapid…

The adaptive $s$-step CG algorithm is a solver for sparse, symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we…

The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positive definite matrix A. In this work,…

We study the solution of large symmetric positive-definite linear systems in a matrix-free setting with a limited iteration budget. We focus on the preconditioned conjugate gradient (PCG) method with spectral preconditioning. Spectral…

In this paper, we compute the stationary states of the multicomponent phase-field crystal model by formulating it as a block constrained minimization problem. The original infinite-dimensional non-convex minimization problem is approximated…

Subspace methods are commonly used for finding approximate eigenvalues and singular values of large-scale matrices. Once a subspace is found, the Rayleigh-Ritz method (for symmetric eigenvalue problems) and Petrov-Galerkin projection (for…

The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind of all-at-once linear systems, which are solved via the parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and…

Block coordinate descent is a powerful algorithmic template suitable for big data optimization. This template admits a lot of variants including block gradient descent (BGD), which performs gradient descent on a selected block of variables,…