Related papers: How to Make the Preconditioned Conjugate Gradient …

We analyze the conjugate gradient (CG) method with variable preconditioning for solving a linear system with a real symmetric positive definite (SPD) matrix of coefficients $A$. We assume that the preconditioner is SPD on each step, and…

We present and analyze several strategies for improving the performance of stochastic variance-reduced gradient (SVRG) methods. We first show that the convergence rate of these methods can be preserved under a decreasing sequence of errors…

Many problems in geophysical and atmospheric modelling require the fast solution of elliptic partial differential equations (PDEs) in "flat" three dimensional geometries. In particular, an anisotropic elliptic PDE for the pressure…

This study aims to reconstruct the complete flow field from spatially restricted domain data by utilizing an Enhanced Super-Resolution Generative Adversarial Network (ESRGAN) model. The difficulty in flow field reconstruction lies in…

We consider adaptive finite element methods (AFEMs) with inexact algebraic solvers for second-order symmetric linear elliptic diffusion problems. Optimal complexity of AFEM, i.e., optimal convergence rates with respect to the overall…

Recently, most of state-of-the-art single image super-resolution (SISR) methods have attained impressive performance by using deep convolutional neural networks (DCNNs). The existing SR methods have limited performance due to a fixed…

The conjugate gradient method is a widely used algorithm for the numerical solution of a system of linear equations. It is particularly attractive because it allows one to take advantage of sparse matrices and produces (in case of infinite…

We introduce two new stochastic conjugate frameworks for a class of nonconvex and possibly also nonsmooth optimization problems. These frameworks are built upon Stochastic Recursive Gradient Algorithm (SARAH) and we thus refer to them as…

Current Retrieval-Augmented Generation (RAG) systems typically employ a traditional two-stage pipeline: an embedding model for initial retrieval followed by a reranker for refinement. However, this paradigm suffers from significant…

In this contribution, we present a numerical analysis of the continuous stochastic gradient (CSG) method, including applications from topology optimization and convergence rates. In contrast to standard stochastic gradient optimization…

Multiphysics problems such as multicomponent diffusion, phase transformations in multiphase systems and alloy solidification involve numerical solution of a coupled system of nonlinear partial differential equations (PDEs). Numerical…

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,…

By reducing the number of global synchronization bottlenecks per iteration and hiding communication behind useful computational work, pipelined Krylov subspace methods achieve significantly improved parallel scalability on present-day HPC…

This work is based on the seminar titled ``Resiliency in Numerical Algorithm Design for Extreme Scale Simulations'' held March 1-6, 2020 at Schloss Dagstuhl, that was attended by all the authors. Naive versions of conventional resilience…

We study sample efficient reinforcement learning (RL) under the general framework of interactive decision making, which includes Markov decision process (MDP), partially observable Markov decision process (POMDP), and predictive state…

While RAG has greatly enhanced LLMs, extending this paradigm to Time-Series Foundation Models (TSFMs) remains a challenge. This is exemplified in the Predictive Maintenance of the Pressure Regulating and Shut-Off Valve (PRSOV), a…

Recurrent neural networks (RNNs) are valued for their computational efficiency and reduced memory requirements on tasks involving long sequence lengths but require high memory-processor bandwidth to train. Checkpointing techniques can…

Generalized Chinese Remainder Theorem (CRT) is a well-known approach to solve ambiguity resolution related problems. In this paper, we study the robust CRT reconstruction for multiple numbers from a view of statistics. To the best of our…

Background: Recursive reasoning models achieve strong performance through iterative refinement, allowing small networks to match large language models. However, training is computationally expensive, often requiring 36 GPU-hours for Sudoku…

We consider the preconditioned conjugate gradient method (PCG) with optimal preconditioner in the frame of the boundary element method (BEM) for elliptic first-kind integral equations. Our adaptive algorithm steers the termination of PCG as…