Related papers: Flexibly Enlarged Conjugate Gradient Methods
Conjugate Gradient (CG) methods are one of the most effective iterative methods to solve linear equations in Hilbert spaces. So far, they have been inherently bound to these spaces since they make use of the inner product structure. In more…
Subspace recycling iterative methods and other subspace augmentation schemes are a successful extension to Krylov subspace methods in which a Krylov subspace is augmented with a fixed subspace spanned by vectors deemed to be helpful in…
The efficient solution of large-scale multiterm linear matrix equations is a challenging task in numerical linear algebra, and it is a largely open problem. We propose a new iterative scheme for symmetric and positive definite operators,…
In recent years two Krylov subspace methods have been proposed for solving skew symmetric linear systems, one based on the minimum residual condition, the other on the Galerkin condition. We give new, algorithm-independent proofs that in…
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…
In this paper, based on the limited memory techniques and subspace minimization conjugate gradient (SMCG) methods, a regularized limited memory subspace minimization conjugate gradient method is proposed, which contains two types of…
It is shown in this paper that, almost all current prevalent iterative \mbox{methods} for solving linear system of equations can be classified as what we called extended Krylov subspace methods. In this paper a new type of iterative methods…
Block Krylov methods have recently gained a lot of attraction. Due to their increased arithmetic intensity they offer a promising way to improve performance on modern hardware. Recently Frommer et al. presented a block Krylov framework that…
Primal and dual block coordinate descent methods are iterative methods for solving regularized and unregularized optimization problems. Distributed-memory parallel implementations of these methods have become popular in analyzing large…
We present SKA-SGD (Streaming Krylov-Accelerated Stochastic Gradient Descent), a novel optimization approach that accelerates convergence for ill-conditioned problems by projecting stochastic gradients onto a low-dimensional Krylov…
Krylov subspace methods are extensively used in scientific computing to solve large-scale linear systems. However, the performance of these iterative Krylov solvers on modern supercomputers is limited by expensive communication costs. The…
In this article, we propose an accuracy-assuring technique for finding a solution for unsymmetric linear systems. Such problems are related to different areas such as image processing, computer vision, and computational fluid dynamics.…
This paper explores variants of the subspace iteration algorithm for computing approximate invariant subspaces. The standard subspace iteration approach is revisited and new variants that exploit gradient-type techniques combined with a…
Conjugated gradients on the normal equation (CGNE) is a popular method to regularise linear inverse problems. The idea of the method can be summarised as minimising the residuum over a suitable Krylov subspace. It is shown that using the…
This paper introduces a new class of algorithms for solving large-scale linear inverse problems based on new flexible and inexact Golub-Kahan factorizations. The proposed methods iteratively compute regularized solutions by approximating a…
Parallel implementations of Krylov subspace methods often help to accelerate the procedure of finding an approximate solution of a linear system. However, such parallelization coupled with asynchronous and out-of-order execution often…
Pipelined Krylov subspace methods typically offer improved strong scaling on parallel HPC hardware compared to standard Krylov subspace methods for large and sparse linear systems. In pipelined methods the traditional synchronization…
We investigate the application of Krylov space methods to the solution of shifted linear systems of the form (A+\sigma) x - b = 0 for several values of \sigma simultaneously, using only as many matrix-vector operations as the solution of a…
This paper presents two new augmented flexible (AF)-Krylov subspace methods, AF-GMRES and AF-LSQR, to compute solutions of large-scale linear discrete ill-posed problems that can be modeled as the sum of two independent random variables,…
Since numbers in the computer are represented with a fixed number of bits, loss of accuracy during calculation is unavoidable. At high precision where more bits (e.g. 64) are allocated to each number, round-off errors are typically small.…