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A hierarchical solver is proposed for solving sparse ill-conditioned linear systems in parallel. The solver is based on a modification of the LoRaSp method, but employs a deferred-compression technique, which provably reduces the…
Large language models (LLMs) have achieved impressive results on multi-step mathematical reasoning, yet at the cost of high computational overhead. This challenge is particularly acute for test-time scaling methods such as parallel…
Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelising the temporal domain. Given approximate initial values at each temporal sub-interval, the algorithm locates a…
Many challenging tasks in sensor networks, including sensor calibration, ranking of nodes, monitoring, event region detection, collaborative filtering, collaborative signal processing, {\em etc.}, can be formulated as a problem of solving a…
Reasoning-augmented search agents such as Search-R1, trained via reinforcement learning with verifiable rewards (RLVR), demonstrate remarkable capabilities in multi-step information retrieval from external knowledge sources. These agents…
The linear FEAST algorithm is a method for solving linear eigenvalue problems. It uses complex contour integration to calculate the eigenvectors whose eigenvalues that are located inside some user-defined region in the complex plane. This…
This paper presents a highly-parallelizable parallel-in-time algorithm for efficient solution of nonlinear time-periodic problems. It is based on the time-periodic extension of the Parareal method, known to accelerate sequential…
Efficient solutions of large-scale, ill-conditioned and indefinite algebraic equations are ubiquitously needed in numerous computational fields, including multiphysics simulations, machine learning, and data science. Because of their…
In Density Functional Theory simulations based on the LAPW method, each self-consistent field cycle comprises dozens of large dense generalized eigenproblems. In contrast to real-space methods, eigenpairs solving for problems at distinct…
To analyze large sets of grid states, e.g. when evaluating the impact from the uncertainties of the renewable generation with probabilistic Monte Carlo simulation or in stationary time series simulation, large number of power flow…
We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters;…
The study of fractional order differential operators is receiving renewed attention in many scientific fields. In order to accommodate researchers doing work in these areas, there is a need for highly scalable numerical methods for solving…
Multiresolution Matrix Factorization (MMF) was recently introduced as a method for finding multiscale structure and defining wavelets on graphs/matrices. In this paper we derive pMMF, a parallel algorithm for computing the MMF…
The eigenvalue problem for a linear function L centers on solving the eigen-equation Lx = rx. This paper generalizes the eigenvalue problem from a single linear function to an iterated function system F consisting of possibly an infinite…
Many large-scale scientific computations require eigenvalue solvers in a scaling regime where efficiency is limited by data movement. We introduce a parallel algorithm for computing the eigenvalues of a dense symmetric matrix, which…
Eigensolvers involving complex moments can determine all the eigenvalues in a given region in the complex plane and the corresponding eigenvectors of a regular linear matrix pencil. The complex moment acts as a filter for extracting…
Discrete variational methods show excellent performance in numerical simulations of different mechanical systems. In this paper, we introduce an iterative procedure for the solution of discrete variational equations for boundary value…
A parallel algorithm for computing the finite difference solution to the elliptic equations with non-separable variables is presented. The resultant matrix is symmetric positive definite, thus the preconditioning conjugate gradient or the…
A class of abstract nonlinear time-periodic evolution problems is considered which arise in electrical engineering and other scientific disciplines. An efficient solver is proposed for the systems arising after discretization in time based…
High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em…