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This paper presents a parallel preconditioning approach based on incomplete LU (ILU) factorizations in the framework of Domain Decomposition (DD) for general sparse linear systems. We focus on distributed memory parallel architectures,…

Numerical Analysis · Mathematics 2023-03-17 Tianshi Xu , Ruipeng Li , Daniel Osei-Kuffuor

We present a deep learning-based iterative approach to solve the discrete heterogeneous Helmholtz equation for high wavenumbers. Combining classical iterative multigrid solvers and convolutional neural networks (CNNs) via preconditioning,…

Machine Learning · Computer Science 2024-06-07 Bar Lerer , Ido Ben-Yair , Eran Treister

Efficient numerical solvers for partial differential equations empower science and engineering. One of the commonly employed numerical solvers is the preconditioned conjugate gradient (PCG) algorithm which can solve large systems to a given…

Numerical Analysis · Mathematics 2023-09-07 Yichen Li , Peter Yichen Chen , Tao Du , Wojciech Matusik

The generalized Dryja--Smith--Widlund (GDSW) preconditioner is a two-level overlapping Schwarz domain decomposition (DD) preconditioner that couples a classical one-level overlapping Schwarz preconditioner with an energy-minimizing coarse…

Numerical Analysis · Mathematics 2023-04-12 Ichitaro Yamazaki , Alexander Heinlein , Sivasankaran Rajamanickam

It is now a noticeable trend in High Performance Computing that the systems are becoming more and more heterogeneous. Compute nodes with a host CPU are being equipped with accelerators, the latter being a GPU or FPGA cards or both. In many…

Distributed, Parallel, and Cluster Computing · Computer Science 2020-01-16 G. Korcyl , P. Korcyl

With the hardware support for half-precision arithmetic on NVIDIA V100 GPUs, high-performance computing applications can benefit from lower precision at appropriate spots to speed up the overall execution time. In this paper, we investigate…

Mathematical Software · Computer Science 2020-07-16 Kyaw L. Oo , Andreas Vogel

Sparse linear systems are typically solved using preconditioned iterative methods, but applying preconditioners via sparse triangular solves introduces bottlenecks due to irregular memory accesses and data dependencies. This work leverages…

Performance · Computer Science 2025-08-08 Atharva Gondhalekar , Kjetil Haugen , Thomas Gibson , Wu-chun Feng

We present and release in open source format a sparse linear solver which efficiently exploits heterogeneous parallel computers. The solver can be easily integrated into scientific applications that need to solve large and sparse linear…

Distributed, Parallel, and Cluster Computing · Computer Science 2023-07-26 Massimo Bernaschi , Alessandro Celestini , Pasqua D'Ambra , Flavio Vella

Modern graphics hardware is designed for highly parallel numerical tasks and promises significant cost and performance benefits for many scientific applications. One such application is lattice quantum chromodyamics (lattice QCD), where the…

High Energy Physics - Lattice · Physics 2010-12-06 M. A. Clark , R. Babich , K. Barros , R. C. Brower , C. Rebbi

Adaptive multi-grid methods have proven very successful in dealing with critical slow down for the Wilson-Dirac solver in lattice gauge theory. Multi-grid algorithms developed for Staggered fermions using the K\"ahler-Dirac…

High Energy Physics - Lattice · Physics 2023-04-28 Venkitesh Ayyar , Richard Brower , M. A. Clark , Mathias Wagner , Evan Weinberg

Over the past five years, graphics processing units (GPUs) have had a transformational effect on numerical lattice quantum chromodynamics (LQCD) calculations in nuclear and particle physics. While GPUs have been applied with great success…

High Energy Physics - Lattice · Physics 2016-11-15 R. Babich , M. A. Clark , B. Joó , G. Shi , R. C. Brower , S. Gottlieb

Differentiable model predictive control (MPC) offers a powerful framework for combining learning and control. However, its adoption has been limited by the inherently sequential nature of traditional optimization algorithms, which are…

Optimization and Control · Mathematics 2025-10-08 Emre Adabag , Marcus Greiff , John Subosits , Thomas Lew

Operators with fractional perturbations are crucial components for robust preconditioning of interface-coupled multiphysics systems. However, in case the perturbation is strong, standard approaches can fail to provide scalable approximation…

Numerical Analysis · Mathematics 2022-12-01 Miroslav Kuchta

A structured preconditioned conjugate gradient (PCG) solver is developed for the Newton steps in second-order methods for a class of constrained network optimal control problems. Of specific interest are problems with discrete-time dynamics…

Systems and Control · Electrical Eng. & Systems 2020-10-13 Armaghan Zafar , Michael Cantoni , Farhad Farokhi

In this paper, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers,…

Numerical Analysis · Mathematics 2021-01-18 Luca Bergamaschi , Jacek Gondzio , Ángeles Martínez , John W. Pearson , Spyridon Pougkakiotis

Deep neural networks (DNNs) are currently predominantly trained using first-order methods. Some of these methods (e.g., Adam, AdaGrad, and RMSprop, and their variants) incorporate a small amount of curvature information by using a diagonal…

Machine Learning · Computer Science 2022-10-28 Achraf Bahamou , Donald Goldfarb , Yi Ren

Linear solvers are key components in any software platform for scientific and engineering computing. The solution of large and sparse linear systems lies at the core of physics-driven numerical simulations relying on partial differential…

Numerical Analysis · Mathematics 2026-04-16 Massimo Bernaschi , Mauro G. Carrozzo , Alessandro Celestini , Giacomo Piperno , Pasqua D'Ambra

We present an efficient, robust and fully GPU-accelerated aggregation-based algebraic multigrid preconditioning technique for the solution of large sparse linear systems. These linear systems arise from the discretization of elliptic PDEs.…

Numerical Analysis · Mathematics 2014-03-10 Rajesh Gandham , Ken Esler , Yongpeng Zhang

Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…

Numerical Analysis · Mathematics 2025-05-14 Nunzio Dimola , Nicola Rares Franco , Paolo Zunino

We investigate the performance of multigrid preconditioners for solving linear systems arising from finite element discretizations of elliptic interface problems using the Fictitious Domain with Distributed Lagrange Multipliers (FD-DLM)…

Numerical Analysis · Mathematics 2025-03-04 Najwa Alshehri , Daniele Boffi , Chayapol Chaoveeraprasit