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Advances in Numerical Linear Algebra and Its Applications
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Advances in Numerical Linear Algebra and Its Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (30 April 2024) | Viewed by 5911

Special Issue Editors

Dr. Fei Xue

E-Mail Website
Guest Editor
School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA
Interests: numerical analysis; scientific computing; numerical linear algebra
Dr. Qifeng Liao

E-Mail Website
Guest Editor
School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China
Interests: model order reduction; uncertainty quantification; numerical methods for partial differential equations; finite element methods
Dr. Yuanzhe Xi

E-Mail Website
Guest Editor
Department of Mathematics, Emory University, Atlanta, GA 30322, USA
Interests: numerical linear algebra; optimization; deep learning; high performance computing

Special Issue Information

Dear Colleagues,

Recent advancements in numerical linear algebra have been brought about by the development of exciting collaborations between traditional and new fields in computational mathematics. Novel iterative methods, such as preconditioning techniques and Anderson acceleration, have seen considerable new developments and applications, while approximation theory and randomized methods have played increasingly important roles in speeding up the rate of convergence and lowering computational costs in various ways. Tools and language from numerical linear algebra (such as tensor train/ring and their decompositions) have been critical for the development and analysis of numerical algorithms for uncertainty quantification, machine learning and data science.

For this Special Issue, we are interested in papers exploring numerical linear algebra, with a primary focus on iterative and acceleration methods for linear/nonlinear systems and functions of matrices (such as polynomial/rational Krylov subspace methods and Anderson acceleration), preconditioning techniques, approximation theory and practice, parameter-dependent, randomized and stochastic methods, machine learning and artificial intelligence. We welcome submissions in these and related fields, and hope that this Special Issue succeeds as an international forum for researchers to summarize and share their most recent developments.

Dr. Fei Xue
Dr. Qifeng Liao
Dr. Yuanzhe Xi
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Krylov subspace methods
  • preconditioning
  • acceleration methods
  • approximation theory
  • randomized sketching
  • tensor decomposition
  • machine learning
  • data science

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Published Papers (4 papers)

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Research

18 pages, 706 KiB  
Article
ANOVA-GP Modeling for High-Dimensional Bayesian Inverse Problems
by Xiaoyu Shi, Hanyu Zhang and Guanjie Wang
Mathematics 2024, 12(2), 301; https://doi.org/10.3390/math12020301 - 17 Jan 2024
Viewed by 1310
Abstract
Markov chain Monte Carlo (MCMC) stands out as an effective method for tackling Bayesian inverse problems. However, when dealing with computationally expensive forward models and high-dimensional parameter spaces, the challenge of repeated sampling becomes pronounced. A common strategy to address this challenge is [...] Read more.
Markov chain Monte Carlo (MCMC) stands out as an effective method for tackling Bayesian inverse problems. However, when dealing with computationally expensive forward models and high-dimensional parameter spaces, the challenge of repeated sampling becomes pronounced. A common strategy to address this challenge is to construct an inexpensive surrogate of the forward model, which cuts the computational cost of individual samples. While the Gaussian process (GP) is widely used as a surrogate modeling strategy, its applicability can be limited when dealing with high-dimensional input or output spaces. This paper presents a novel approach that combines the analysis of variance (ANOVA) decomposition method with Gaussian process regression to handle high-dimensional Bayesian inverse problems. Initially, the ANOVA method is employed to reduce the dimension of the parameter space, which decomposes the original high-dimensional problem into several low-dimensional sub-problems. Subsequently, principal component analysis (PCA) is utilized to reduce the dimension of the output space on each sub-problem. Finally, a Gaussian process model with a low-dimensional input and output is constructed for each sub-problem. In addition to this methodology, an adaptive ANOVA-GP-MCMC algorithm is proposed, which further enhances the adaptability and efficiency of the method in the Bayesian inversion setting. The accuracy and computational efficiency of the proposed approach are validated through numerical experiments. This innovative integration of ANOVA and Gaussian processes provides a promising solution to address challenges associated with high-dimensional parameter spaces and computationally expensive forward models in Bayesian inference. Full article
(This article belongs to the Special Issue Advances in Numerical Linear Algebra and Its Applications)
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29 pages, 2099 KiB  
Article
A Flexible Extended Krylov Subspace Method for Approximating Markov Functions of Matrices
by Shengjie Xu and Fei Xue
Mathematics 2023, 11(20), 4341; https://doi.org/10.3390/math11204341 - 19 Oct 2023
Viewed by 1194
Abstract
A flexible extended Krylov subspace method (F-EKSM) is considered for numerical approximation of the action of a matrix function f(A) to a vector b, where the function f is of Markov type. F-EKSM has the same [...] Read more.
A flexible extended Krylov subspace method (F-EKSM) is considered for numerical approximation of the action of a matrix function f(A) to a vector b, where the function f is of Markov type. F-EKSM has the same framework as the extended Krylov subspace method (EKSM), replacing the zero pole in EKSM with a properly chosen fixed nonzero pole. For symmetric positive definite matrices, the optimal fixed pole is derived for F-EKSM to achieve the lowest possible upper bound on the asymptotic convergence factor, which is lower than that of EKSM. The analysis is based on properties of Faber polynomials of A and (I−A/s)−1. For large and sparse matrices that can be handled efficiently by LU factorizations, numerical experiments show that F-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime. Full article
(This article belongs to the Special Issue Advances in Numerical Linear Algebra and Its Applications)
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10 pages, 424 KiB  
Article
Distribution of Eigenvalues and Upper Bounds of the Spread of Interval Matrices
by Wenshi Liao and Pujun Long
Mathematics 2023, 11(19), 4032; https://doi.org/10.3390/math11194032 - 22 Sep 2023
Viewed by 915
Abstract
The distribution of eigenvalues and the upper bounds for the spread of interval matrices are significant in various fields of mathematics and applied sciences, including linear algebra, numerical analysis, control theory, and combinatorial optimization. We present the distribution of eigenvalues within interval matrices [...] Read more.
The distribution of eigenvalues and the upper bounds for the spread of interval matrices are significant in various fields of mathematics and applied sciences, including linear algebra, numerical analysis, control theory, and combinatorial optimization. We present the distribution of eigenvalues within interval matrices and determine upper bounds for their spread using Geršgorin’s theorem. Specifically, through an equality for the variance of a discrete random variable, we derive upper bounds for the spread of symmetric interval matrices. Finally, we give three numerical examples to illustrate the effectiveness of our results. Full article
(This article belongs to the Special Issue Advances in Numerical Linear Algebra and Its Applications)
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15 pages, 665 KiB  
Article
Estimating Failure Probability with Neural Operator Hybrid Approach
by Mujing Li, Yani Feng and Guanjie Wang
Mathematics 2023, 11(12), 2762; https://doi.org/10.3390/math11122762 - 18 Jun 2023
Cited by 2 | Viewed by 1367
Abstract
Evaluating failure probability for complex engineering systems is a computationally intensive task. While the Monte Carlo method is easy to implement, it converges slowly and, hence, requires numerous repeated simulations of a complex system to generate sufficient samples. To improve the efficiency, methods [...] Read more.
Evaluating failure probability for complex engineering systems is a computationally intensive task. While the Monte Carlo method is easy to implement, it converges slowly and, hence, requires numerous repeated simulations of a complex system to generate sufficient samples. To improve the efficiency, methods based on surrogate models are proposed to approximate the limit state function. In this work, we reframe the approximation of the limit state function as an operator learning problem and utilize the DeepONet framework with a hybrid approach to estimate the failure probability. The numerical results show that our proposed method outperforms the prior neural hybrid method. Full article
(This article belongs to the Special Issue Advances in Numerical Linear Algebra and Its Applications)
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