Related papers: Techniques for Accelerating the Convergence of Res…
This paper deals with the definition and optimization of augmentation spaces for faster convergence of the conjugate gradient method in the resolution of sequences of linear systems. Using advanced convergence results from the literature,…
In this paper, we develop a new Randomized Global Generalized Minimum Residual (RGlGMRES) algorithm for efficiently computing solutions to large scale linear systems with multiple right hand sides.The proposed method builds on a recently…
Enlarged Krylov subspace methods and their s-step versions were introduced [7] in the aim of reducing communication when solving systems of linear equations Ax = b. These enlarged CG methods consist of enlarging the Krylov subspace by a…
Process Reward Models (PRMs), which assign fine-grained scores to intermediate reasoning steps within a solution trajectory, have emerged as a promising approach to enhance the reasoning quality of Large Language Models (LLMs). However,…
Hardware trends have motivated the development of mixed precision algo-rithms in numerical linear algebra, which aim to decrease runtime while maintaining acceptable accuracy. One recent development is the development of an adaptive…
This work proposes a new class of preconditioners for the low rank Generalized Minimal Residual Method (GMRES) for multiterm matrix equations arising from implicit timestepping of linear matrix differential equations. We are interested in…
We propose new restarting strategies for accelerated gradient and accelerated coordinate descent methods. Our main contribution is to show that the restarted method has a geometric rate of convergence for any restarting frequency, and so it…
In the numerical treatment of large-scale Sylvester and Lyapunov equations, projection methods require solving a reduced problem to check convergence. As the approximation space expands, this solution takes an increasing portion of the…
Multimodal Large Language Models (MLLMs) have shown remarkable success in comprehension tasks such as visual description and visual question answering. However, their direct application to embedding-based tasks like retrieval remains…
Block Krylov subspace methods (KSMs) comprise building blocks in many state-of-the-art solvers for large-scale matrix equations as they arise, e.g., from the discretization of partial differential equations. While extended and rational…
Many practical imaging systems suffer from uncertainty in acquisition geometry -- such as projection angles in computed tomography or sensor positions in photoacoustic tomography -- leading to nonlinear inverse problems that require joint…
In this paper, two efficient iterative algorithms based on the simpler GMRES method are proposed for solving shifted linear systems. To make full use of the shifted structure, the proposed algorithms utilizing the deflated restarting…
We consider algebraic iterative reconstruction methods with applications in image reconstruction. In particular, we are concerned with methods based on an unmatched projector/backprojector pair; i.e., the backprojector is not the exact…
In this paper, we consider an efficient iterative approach to the solution of the discrete Helmholtz equation with Dirichlet, Neumann and Sommerfeld-like boundary conditions based on a compact sixth order approximation scheme and…
Low-precision computing is essential for efficiently utilizing memory bandwidth and computing cores. While many mixed-precision algorithms have been developed for iterative sparse linear solvers, effectively leveraging half-precision (fp16)…
Over the past two decades, descent methods have received substantial attention within the multiobjective optimization field. Nonetheless, both theoretical analyses and empirical evidence reveal that existing first-order methods for…
Preconditioned Krylov subspace (KSP) methods are widely used for solving large-scale sparse linear systems arising from numerical solutions of partial differential equations (PDEs). These linear systems are often nonsymmetric due to the…
Many Krylov subspace methods for shifted linear systems take advantage of the invariance of the Krylov subspace under a shift of the matrix. However, exploiting this fact in the non-Hermitian case introduces restrictions; e.g., initial…
Designing modern photonic devices often involves traversing a large parameter space via an optimization procedure, gradient based or otherwise, and typically results in the designer performing electromagnetic simulations of correlated…
The residual cutting (RC) method has been proposed for efficiently solving linear equations obtained from elliptic partial differential equations. Based on the RC, we have introduced the generalized residual cutting (GRC) method, which can…