Explanipedia

Neural Approximate Inverse Preconditioners Open

Tianshi Xu, Ruipeng Li, Yuanzhe Xi · 2025

In this paper, we propose a data-driven framework for constructing efficient approximate inverse preconditioners for elliptic partial differential equations (PDEs) by learning the Green's function of the underlying operator with neural net…

Rethinking Neural-based Matrix Inversion: Why can't, and Where can Open

Yuliang Ji, Jian Wu, Yuanzhe Xi · 2025

Deep neural networks have achieved substantial success across various scientific computing tasks. A pivotal challenge within this domain is the rapid and parallel approximation of matrix inverses, critical for numerous applications. Despit…

Mixed Precision Orthogonalization-Free Projection Methods for Eigenvalue and Singular Value Problems Open

Tianshi Xu, Ziqi Zhang, Jie Chen, Yousef Saad, Yuanzhe Xi · 2025

Mixed-precision arithmetic offers significant computational advantages for large-scale matrix computation tasks, yet preserving accuracy and stability in eigenvalue problems and the singular value decomposition (SVD) remains challenging. T…

HiGP: A high-performance Python package for Gaussian Process Open

Huang Hua, Tianshi Xu, Yuanzhe Xi, Edmond Chow · 2025

Gaussian Processes (GPs) are flexible, nonparametric Bayesian models widely used for regression and classification tasks due to their ability to capture complex data patterns and provide uncertainty quantification (UQ). Traditional GP impl…

Multiscale Neural Networks for Approximating Green's Functions Open

Wenrui Hao, Ruipeng Li, Yuanzhe Xi, Tianshi Xu, Yahong Yang · 2024

Neural networks (NNs) have been widely used to solve partial differential equations (PDEs) in the applications of physics, biology, and engineering. One effective approach for solving PDEs with a fixed differential operator is learning Gre…

Reducing Operator Complexity of Galerkin Coarse-grid Operators with Machine Learning Open

Ru Huang, Kai Chang, Huan He, Ruipeng Li, Yuanzhe Xi · 2024

Here, we propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in multigrid (MG) methods, addressing the well-known issue of increasing operator complexity. Guided by the MG theory on spect…

Straggler-tolerant stationary methods for linear systems Open

Vassilis Kalantzis, Yuanzhe Xi, Lior Horesh, Yousef Saad · 2024

In this paper, we consider the iterative solution of linear algebraic equations under the condition that matrix-vector products with the coefficient matrix are computed only partially. At the same time, non-computed entries are set to zero…

Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows Open

Shuhao Cao, Francesco Brarda, Ruipeng Li, Yuanzhe Xi · 2024

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training exp…

Anderson Acceleration with Truncated Gram-Schmidt Open

Ziyuan Tang, Tianshi Xu, Huan He, Yousef Saad, Yuanzhe Xi · 2024

Anderson Acceleration (AA) is a popular algorithm designed to enhance the convergence of fixed-point iterations. In this paper, we introduce a variant of AA based on a Truncated Gram-Schmidt process (AATGS) which has a few advantages over …

NLTGCR: A class of Nonlinear Acceleration Procedures based on Conjugate Residuals Open

Huan He, Ziyuan Tang, Shifan Zhao, Yousef Saad, Yuanzhe Xi · 2023

This paper develops a new class of nonlinear acceleration algorithms based on extending conjugate residual-type procedures from linear to nonlinear equations. The main algorithm has strong similarities with Anderson acceleration as well as…

An Adaptive Factorized Nyström Preconditioner for Regularized Kernel Matrices Open

Shifan Zhao, Tianshi Xu, Edmond Chow, Yuanzhe Xi · 2023

The spectrum of a kernel matrix significantly depends on the parameter values of the kernel function used to define the kernel matrix. This makes it challenging to design a preconditioner for a regularized kernel matrix that is robust acro…

MedDiff: Generating Electronic Health Records using Accelerated Denoising Diffusion Model Open

Huan He, Shifan Zhao, Yuanzhe Xi, Joyce C. Ho · 2023

Due to patient privacy protection concerns, machine learning research in healthcare has been undeniably slower and limited than in other application domains. High-quality, realistic, synthetic electronic health records (EHRs) can be levera…

MuG: A Multimodal Classification Benchmark on Game Data with Tabular, Textual, and Visual Fields Open

Jiaying Lu, Yongchen Qian, Shifan Zhao, Yuanzhe Xi, Carl Yang · 2023

Previous research has demonstrated the advantages of integrating data from multiple sources over traditional unimodal data, leading to the emergence of numerous novel multimodal applications. We propose a multimodal classification benchmar…

Data-Driven Linear Complexity Low-Rank Approximation of General Kernel Matrices: A Geometric Approach Open

Difeng Cai, Edmond Chow, Yuanzhe Xi · 2022

A general, {\em rectangular} kernel matrix may be defined as $K_{ij} = κ(x_i,y_j)$ where $κ(x,y)$ is a kernel function and where $X=\{x_i\}_{i=1}^m$ and $Y=\{y_i\}_{i=1}^n$ are two sets of points. In this paper, we seek a low-rank approxim…

An Efficient Nonlinear Acceleration method that Exploits Symmetry of the Hessian Open

Huan He, Shifan Zhao, Ziyuan Tang, Joyce C. Ho, Yousef Saad , et al. · 2022

Nonlinear acceleration methods are powerful techniques to speed up fixed-point iterations. However, many acceleration methods require storing a large number of previous iterates and this can become impractical if computational resources ar…

Learning Optimal Multigrid Smoothers via Neural Networks Open

Ru Huang, Ruipeng Li, Yuanzhe Xi · 2022

Multigrid methods are one of the most efficient techniques for solving large sparse linear systems arising from partial differential equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of mul…

parGeMSLR: A parallel multilevel Schur complement low-rank preconditioning and solution package for general sparse matrices Open

Tianshi Xu, Vassilis Kalantzis, Ruipeng Li, Yuanzhe Xi, Geoffrey Dillon , et al. · 2022

AUTM Flow: Atomic Unrestricted Time Machine for Monotonic Normalizing Flows Open

Difeng Cai, Yuliang Ji, Huan He, Qiang Ye, Yuanzhe Xi · 2022

Nonlinear monotone transformations are used extensively in normalizing flows to construct invertible triangular mappings from simple distributions to complex ones. In existing literature, monotonicity is usually enforced by restricting fun…

Data-driven Construction of Hierarchical Matrices with Nested Bases Open

Difeng Cai, Hua Huang, Edmond Chow, Yuanzhe Xi · 2022

Hierarchical matrices provide a powerful representation for significantly reducing the computational complexity associated with dense kernel matrices. For general kernel functions, interpolation-based methods are widely used for the effici…

parGeMSLR: A Parallel Multilevel Schur Complement Low-Rank Preconditioning and Solution Package for General Sparse Matrices Open

Tianshi Xu, Vassilis Kalantzis, Ruipeng Li, Yuanzhe Xi, Geoffrey Dillon , et al. · 2022

This paper discusses parGeMSLR, a C++/MPI software library for the solution of sparse systems of linear algebraic equations via preconditioned Krylov subspace methods in distributed-memory computing environments. The preconditioner impleme…

GDA-AM: On the effectiveness of solving minimax optimization via Anderson Acceleration Open

Huan He, Shifan Zhao, Yuanzhe Xi, Joyce C. Ho, Yousef Saad · 2021

Many modern machine learning algorithms such as generative adversarial networks (GANs) and adversarial training can be formulated as minimax optimization. Gradient descent ascent (GDA) is the most commonly used algorithm due to its simplic…

Solve Minimax Optimization by Anderson Acceleration Open

Huan He, Shifan Zhao, Yuanzhe Xi, Joyce C. Ho, Yousef Saad · 2021

Many modern machine learning algorithms such as generative adversarial networks (GANs) and adversarial training can be formulated as minimax optimization. Gradient descent ascent (GDA) is the most commonly used algorithm due to its simplic…

Fast randomized non-Hermitian eigensolver based on rational filtering and matrix partitioning Open

Vassilis Kalantzis, Yuanzhe Xi, Lior Horesh · 2021

This paper describes a set of rational filtering algorithms to compute a few eigenvalues (and associated eigenvectors) of non-Hermitian matrix pencils. Our interest lies in computing eigenvalues located inside a given disk, and the propose…

Learning optimal multigrid smoothers via neural networks Open

Ru Huang, Ruipeng Li, Yuanzhe Xi · 2021

Multigrid methods are one of the most efficient techniques for solving linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. One of the key components of multigrid is smo…

Fast deterministic approximation of symmetric indefinite kernel matrices with high dimensional datasets Open

Difeng Cai, J.I. Nagy, Yuanzhe Xi · 2021

Kernel methods are used frequently in various applications of machine learning. For large-scale high dimensional applications, the success of kernel methods hinges on the ability to operate certain large dense kernel matrix K. An enormous …

Fast and stable deterministic approximation of general symmetric kernel matrices in high dimensions. Open

Difeng Cai, James G. Nagy, Yuanzhe Xi · 2021

Kernel methods are used frequently in various applications of machine learning. For large-scale applications, the success of kernel methods hinges on the ability to operate certain large dense kernel matrix K. To reduce the computational c…

Generating a Doppelganger Graph: Resembling but Distinct Open

Yuliang Ji, Ru Huang, Jie Chen, Yuanzhe Xi · 2021

Deep generative models, since their inception, have become increasingly more capable of generating novel and perceptually realistic signals (e.g., images and sound waves). With the emergence of deep models for graph structured data, natura…

A power Schur complement Low-Rank correction preconditioner for general sparse linear systems Open

Qingqing Zheng, Yuanzhe Xi, Yousef Saad · 2020

An effective power based parallel preconditioner is proposed for general large sparse linear systems. The preconditioner combines a power series expansion method with some low-rank correction techniques, where the Sherman-Morrison-Woodbury…

A non-perturbative approach to computing seismic normal modes in rotating planets Open

Jia Shi, Ruipeng Li, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop · 2019

A Continuous Galerkin method-based approach is presented to compute the seismic normal modes of rotating planets. Special care is taken to separate out the essential spectrum in the presence of a fluid outer core using a polynomial filteri…

A Rayleigh-Ritz method based approach to computing seismic normal modes in the presence of an essential spectrum Open

Jia Shi, Ruipeng Li, Yuanzhe Xi, Yousef Saad, Maarten V. de Hoop · 2019

A Rayleigh-Ritz with Continuous Galerkin method based approach is presented to compute the normal modes of a planet in the presence of an essential spectrum. The essential spectrum is associated with a liquid outer core. The presence of a …

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