Article
Mathematics
Teng Wang, Mei Feng, Xiang Wang, Hongjia Chen
Summary: In this paper, the authors investigate the relationship between Rayleigh-Ritz projection and linearization and establish bounds for the backward errors of approximate eigenpairs. Numerical experiments are conducted to support the predictions of the backward error analysis.
LINEAR & MULTILINEAR ALGEBRA
(2022)
Article
Mathematics, Applied
Tom Lewis, Aaron Rapp, Yi Zhang
Summary: This paper further analyzes the dual-wind discontinuous Galerkin (DWDG) method for approximating Poisson's problem by examining the relationship between the Laplacian and the discrete Laplacian. The DWDG methods are derived from the DG differential calculus framework, which replaces continuous differential operators with discrete ones. We establish error estimates and explore the relationship between the DWDG approximation and the Ritz projection. Numerical experiments are conducted to validate the theoretical results.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Engineering, Multidisciplinary
Carla Manni, Espen Sande, Hendrik Speleers
Summary: We prove that Galerkin discretizations of eigenvalue problems related to the Laplace operator, subject to any standard type of homogeneous boundary conditions, have no outliers in certain optimal spline subspaces. These optimal subspaces are obtained by imposing specific additional boundary conditions on the full spline space defined on certain uniform knot sequences.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Akira Imakura, Keiichi Morikuni, Akitoshi Takayasu
Summary: We propose a method to compute and verify eigenvalues and corresponding eigenvectors of generalized Hermitian eigenvalue problems in a region. The method uses complex moments and the Rayleigh-Ritz procedure to extract the desired eigencomponents and project the eigenvalue problem into a reduced form. The proposed method enables faster computation and verification of eigenvectors, even for nearly singular matrix pencils and in the presence of multiple and nearly multiple eigenvalues.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Kensuke Aishima
Summary: Dynamic mode decomposition (DMD) is an efficient analysis method for time series data. This study introduces an adaptive averaging technique to DMD for strong convergence in the statistical sense under a newly constructed noise model. The averaging technique can be generally applied to the new noise model as a preconditioning step, and it is proven that the estimator of the original DMD is inconsistent in the statistical sense.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Yeon-Ho Jeong, Seung-Hwan Boo, Solomon C. Yim
Summary: In this manuscript, a new effective method for eigenpair reanalysis of large-scale finite element (FE) models is proposed. This method utilizes the matrix block-partitioning algorithm in the Rayleigh-Ritz approach and expresses the Ritz basis matrix using thousands of block matrices of very small size. A new formulation is derived to avoid significant computational costs from the projection procedure. An algorithm is presented to recognize which blocks are changed in the modified FE model to achieve computational cost savings when computing new eigenpairs. The performance of the proposed method is demonstrated by solving several practical engineering problems and comparing the results with other methods.
JOURNAL OF COMPUTATIONAL DESIGN AND ENGINEERING
(2023)
Article
Mathematics, Applied
Pedro Massey, Demetrio Stojanoff, Sebastian Zarate
Summary: This paper discusses the relationship between the singular values of a complex self-adjoint matrix and subspaces, obtaining sharp upper bounds and partially confirming conjectures by Knyazev and Argentati.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Xuefeng Liu, Tomas Vejchodsky
Summary: A fully computable guaranteed error bound in the L2 norm sense is proposed for conforming finite element approximations of the Laplacian eigenfunctions. The bound is based on the a priori error estimate for the Galerkin projection of the conforming finite element method and has an optimal speed of convergence for the eigenfunctions with the worst regularity. The resulting error estimate is robust and illustrated by numerical examples.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics
Zengfeng Liu, Lingsheng Meng
Summary: New multiplicative perturbation bounds for orthogonal projection in a general unitarily invariant norm, the Q-norm and the spectral norm are derived in this paper. These bounds always improve the existing bounds. An example is given to demonstrate the optimality of these new bounds.
LINEAR & MULTILINEAR ALGEBRA
(2023)
Article
Computer Science, Artificial Intelligence
Xueying Zhao, Minru Bai, Defeng Sun, Libin Zheng
Summary: This paper focuses on the bound constrained robust low-rank tensor completion (RTC) problem and proposes a nonconvex model with novel nonconvex terms to handle biased solutions. A proximal majorization-minimization (PMM) algorithm is developed to solve the proposed model, achieving global convergence and lower recovery error bounds.
SIAM JOURNAL ON IMAGING SCIENCES
(2022)
Article
Mathematics, Applied
Robert J. Webber, Erik H. Thiede, Douglas Dow, Aaron R. Dinner, Jonathan Weare
Summary: In this paper, the error of a dynamical spectral estimation method called VAC is analyzed by bounding the approximation error and estimation error. The analysis establishes the convergence properties of VAC and suggests new strategies for tuning VAC to improve accuracy.
SIAM JOURNAL ON MATHEMATICS OF DATA SCIENCE
(2021)
Article
Engineering, Mechanical
Allison Kaminski, J. Gregory McDaniel
Summary: This study aims to address the need for knowing natural frequencies in system design by proposing a scalar perturbed eigenvalue expression and developing reference plots to predict associated errors. It avoids the expensive calculation of matrix-vector products and effectively predicts errors in approximated natural frequencies.
JOURNAL OF VIBRATION ENGINEERING & TECHNOLOGIES
(2023)
Article
Mathematics, Applied
Riya Jain, Amiya K. Pani, Sangita Yadav
Summary: This paper develops the hybridizable discontinuous Galerkin (HDG) method for a linear parabolic integro-differential equation and analyzes uniform in time a priori error bounds. An extended Ritz-Volterra projection is introduced to handle the integral term and achieve optimal order convergence. Element-by-element post-processing is proposed to achieve higher convergence order. Numerical examples in two-dimensional domains are provided to verify the results.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Gyurhan H. Nedzhibov
Summary: In this study, we present another representation of a relatively new method for solving the eigenvalue problem of the Frobenius companion matrix and establish its equivalence to the Weierstrass method. By utilizing this dependence, we provide new proofs for some known results and uncover new theoretical properties for the Weierstrass method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Automation & Control Systems
Rahul Moghe, Maruthi R. Akella
Summary: This paper introduces a projection scheme to handle eigenvalue bounds for adaptive control with uncertain symmetric matrix parameters. Conventional parameter projection techniques are generally unable to handle explicit eigenvalue bounds. The continuous projection scheme presented here maintains the closed-loop stability properties for adaptive controllers while simultaneously satisfying a priori available eigenvalue bounds of the uncertain symmetric matrix valued parameters. The new projection shows improved performance in numerical simulations of rigid body attitude tracking control and trajectory tracking of robotic manipulators with unknown inertia parameters.
IEEE TRANSACTIONS ON AUTOMATIC CONTROL
(2023)
Article
Computer Science, Theory & Methods
Merico E. Argentati, Andrew V. Knyazev, Klaus Neymeyr, Evgueni E. Ovtchinnikov, Ming Zhou
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2017)
Article
Mathematics, Applied
Eugene Vecharynski, Andrew Knyazev
LINEAR ALGEBRA AND ITS APPLICATIONS
(2016)
Article
Mathematics, Applied
Peizhen Zhu, Andrew V. Knyazev
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2017)
Article
Chemistry, Physical
Grigory Kolesov, Chungwei Lin, Andrew Knyazev, Keisuke Kojima, Joseph Katz, Koichi Akiyama, Eiji Nakai, Hiroyuki Kawahara
PHYSICAL CHEMISTRY CHEMICAL PHYSICS
(2019)
Proceedings Paper
Automation & Control Systems
Rien Quirynen, Andrew Knyazev, Stefano Di Cairano
2018 EUROPEAN CONTROL CONFERENCE (ECC)
(2018)
Proceedings Paper
Automation & Control Systems
Alexander Malyshev, Rien Quirynen, Andrew Knyazev, Stefano Di Cairano
2018 EUROPEAN CONTROL CONFERENCE (ECC)
(2018)
Proceedings Paper
Computer Science, Hardware & Architecture
Alyson Fox, Geoffrey Sanders, Andrew Knyazev
2018 IEEE HIGH PERFORMANCE EXTREME COMPUTING CONFERENCE (HPEC)
(2018)
Proceedings Paper
Automation & Control Systems
Alexander Malyshev, Rien Quirynen, Andrew Knyazev
Proceedings Paper
Automation & Control Systems
Alexander Malyshev, Rien Quirynen, Andrew Knyazev
Proceedings Paper
Automation & Control Systems
Rien Quirynen, Andrew Knyazev, Stefano Di Cairano
Proceedings Paper
Automation & Control Systems
Andrew Knyazev, Alexander Malyshev
2017 AMERICAN CONTROL CONFERENCE (ACC)
(2017)
Proceedings Paper
Computer Science, Hardware & Architecture
David Zhuzhunashvili, Andrew Knyazev
2017 IEEE HIGH PERFORMANCE EXTREME COMPUTING CONFERENCE (HPEC)
(2017)
Proceedings Paper
Computer Science, Theory & Methods
Andrew Knyazev, Hassan Mansour, Dong Tian, Akshay Gadde
2017 INTERNATIONAL CONFERENCE ON SAMPLING THEORY AND APPLICATIONS (SAMPTA)
(2017)
Proceedings Paper
Computer Science, Theory & Methods
Andrew Knyazev, Alexander Malyshev
2017 INTERNATIONAL CONFERENCE ON SAMPLING THEORY AND APPLICATIONS (SAMPTA)
(2017)
Proceedings Paper
Automation & Control Systems
Arvind U. Raghunathan, Andrew V. Knyazev
2016 AMERICAN CONTROL CONFERENCE (ACC)
(2016)