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Development of stability-preserving time-limited model reduction framework for 2-D and 1-D models with error bound

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Control Theory and Technology 2022年第3期20卷 371-381页

作者： Muhammad Imran Syeda Hira Ambreen Syed Nooh Hamdani Muhammad Imran Muhammad Ejaz Naveed Maria Siddiqui Department of Electrical Engineering Military College of Signals(MCS)National University of Sciences and Technology(NUST)Islamabad43000Pakistan Department of Electrical Engineering Sir Syed(Center For Advanced Studies)Institute of Technology(SSCASE)Islamabad43000Pakistan College of Science and Humanities Rumah Majmaah UniversityRumahKingdom of Saudi Arabia

Due to their complex structure,2-D models are challenging to work with;additionally,simulation,analysis,design,and control get increasingly difficult as the order of the model ***,in particular time intervals,Gawronski and Juang’s time-limited model reduction schemes produce an unstable reduced-order model for the 2-D and 1-D *** revealed some stability preservation solutions to address this key flaw which ensure the stability of 1-D reduced-order systems;nevertheless,these strategies result in large approximation ***,to the best of the authors’knowledge,there is no literature available for the stability preserving time-limited-interval Gramian-based model reduction framework for the 2-D discrete-time *** this article,2-D models are decomposed into two separate sub-models(i.e.,two cascaded 1-D models)using the condition of minimal *** reduction procedures are conducted on these obtained two 1-D sub-models using limited-time *** suggested methodology works for both 2-D and 1-D ***,the suggested methodology gives the stability of the reduced model as well as a priori error-bound expressions for the 2-D and 1-D *** results and comparisons between existing and suggested methodologies are provided to demonstrate the effectiveness of the suggested methodology.

关键词： Model reduction Weighted Gramian Limited interval Gramian Minimal realization Approximation error error bound

error bounds of Two Smoothing Approximations for Semi-infinite Minimax Problems

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Acta Mathematicae Applicatae Sinica 2009年第4期25卷 685-696页

作者： Hong-xia Yin Department of Mathematics and Statistics Minnesota State University Mankato 273 Wissink Hall Mankato MN 56001 USA

In the paper we investigate smoothing method for solving semi-infinite minimax problems. Not like most of the literature in semi-infinite minimax problems which are concerned with the continuous time version（i.e., the one dimensional semi-infinite minimax problems）, the primary focus of this paper is on multi- dimensional semi-infinite minimax problems. The global error bounds of two smoothing approximations for the objective function are given and compared. It is proved that the smoothing approximation given in this paper can provide a better error bound than the existing one in literature.

关键词： Semi-infinite minimax problem smoothing method aggregate function error bound polynomial interpolation

error bounds for Uniform Asymptotic Expansions-Modified Bessel Function of Purely Imaginary Order

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Chinese Annals of Mathematics,Series B 2010年第5期31卷 759-780页

作者： Wei SHI Roderick WONG Wuhan Institute of Physics and Mathematics The Chinese Academy of Sciences Wuhan 430071 China. Department of Mathematics City University of Hong Kong Tat Chee Avenue Kowloon Hong Kong China.

The authors modify a method of Olde Daalhuis and Temme for representing the remainder and coefficients in Airy-type expansions of *** using a class of rational functions,they express these quantities in terms of Cauchy-type integrals;these expressions are natural generalizations of integral representations of the coe?cients and the remainders in the Taylor expansions of analytic *** using the new representation,a computable error bound for the remainder in the uniform asymptotic expansion of the modified Bessel function of purely imaginary order is derived.

关键词： Modified Bessel function of purely imaginary order Airy function Uniform asymptotic expansion error bound

error bounds of Lanczos approach for trust-region subproblem

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Frontiers of Mathematics in China 2018年第2期13卷 459-481页

作者： Leihong ZHANG Weihong YANG Chungen SHEN Jiang FENG School of Mathematics Shanghai University of Finance and Economics Shanghai 200433 China Shanghai Key Laboratory of Financial Information Technology Shanghai University of Finance and Economics Shanghai 200433 China School of Mathematical Sciences Fudan University Shanghai 200433 China College of Science University of Shanghai for Science and Technology Shanghai 200093China

Because of its vital role of the trust-region subproblem （TRS） in various applications, for example, in optimization and in ill-posed problems, there are several factorization-free algorithms for solving the large-scale sparse TRS. The truncated Lanczos approach proposed by N. I. M. Gould, S. Lucidi, M. Roma, and P. L. Toint [SIAM J. Optim., 1999, 9： 504-525] is a natural extension of the classical Lanczos method for the symmetric linear system and eigenvalue problem and, indeed follows the classical Rayleigh-Ritz procedure for eigenvalue computations. It consists of 1） projecting the original TRS to the Krylov subspa^es to yield smaller size TRS＇s and then 2） solving the resulted TRS＇s to get the approximates of the original TRS. This paper presents a posterior error bounds for both the global optimal value and the optimal solution between the original TRS and their projected counterparts. Our error bounds mainly rely on the factors from the Lanczos process as well as the data of the original TRS and, could be helpful in designing certain stopping criteria for the truncated Lanczos approach.

关键词： Trust-region method trust-region subproblem (TRS) Lanczos method Steihaug-Toint conjugate-gradient iteration error bound

Sufcient Conditions for error bounds and Linear Regularity in Banach Spaces

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Acta Mathematica Sinica,English Series 2014年第3期30卷 423-436页

作者： Zhou WEI Qing Hai HE Department of Mathematics Yunnan University

In this paper, we study error bounds for lower semicontinuous functions defined on Banach space and linear regularity for finitely many closed subset in Banach spaces. By using Clarke＇s subd- ifferentials and Ekeland variational principle, we establish several sufficient conditions ensuring error bounds and linear regularity in Banach spaces.

关键词： error bound linear regularity normal cone Clarke subdifferential Ekeland variationalprinciple

Low-rank spectral estimation algorithm of learning Markov model

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Frontiers of Mathematics in China 2024年第3期19卷 137-155页

作者： Yongye ZHAO Shujun BI Department of Basic Courses Guangzhou Maritime UniversityGuangzhou 510725China Department of Mathematics South China University of TechnologyGuangzhou 510640China

This paper proposes a low-rank spectral estimation algorithm of learning Markov ***,an approximate projection algorithm for the rank-constrained frequency matrix set is proposed,and thereafter its local Lipschitzian error bound ***,we propose a low-rank spectral estimation algorithm for estimating the state transition frequency matrix and the probability matrix of Markov model by applying the approximate projection algorithm to correct the maximum likelihood estimation of the frequency matrix,and prove that there is only a multiplying constant difference in estimation errors between the low-rank spectral estimation and the maximum likelihood estimation under appropriate ***,numerical comparisons with the prevailing maximum likelihood estimation,spectral estimation,and rank-constrained maxi-mum likelihood estimation show that the low-rank spectral estimation algorithm is effective.

关键词： Markov model low-rank spectral estimation error bound approximate projection

New Upper bounds for the Inverse of H-Matrices Including S-SDD Matrices and Linear Complementarity Problems

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Journal of Mathematical Research with Applications 2024年第2期44卷 170-186页

作者： Yebo XIONG Department of Mathematics and Statistics Hunan First Normal University

A partition reduction method is used to obtain new upper bounds for the inverses of H-matrices and S-strictly diagonally dominant(S-SDD) matrices. The estimates are expressed via the determinants of third order matrices. Numerical experiments with various random matrices show that they are stable and better than the estimates presented in literatures. We use these upper bounds to improve known error estimates for linear complementarity problems with H-matrices and S-SDD matrices.

关键词： linear complementarity problem error bound upper bound S-SDD matrices H-matrice

Structured backward error for palindromic polynomial eigenvalue problems~ II:Approximate eigentriplets

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Frontiers of Mathematics in China 2018年第6期13卷 1397-1426页

作者： Changli LIU Ren-Cang LI School of Mathematics Sichuan UniversityChengdu 610065China Department of Mathematics University of Texas at ArlingtonArlingtonTX 76019USA

A detailed structured backward error analysis for four kinds of palindromic polynomial eigenvalue problems (PPEPs) P(λ)≡(∑ℓ=0dAℓλℓ)x=0,Ad−ℓ=εAℓ*,ℓ=0,1...,⌊d2⌋, for an approximate eigentriplet is performed, where * is one of the two actions: transpose and conjugate transpose, and ε ∈ {±1} The analysis is concerned with estimating the smallest perturbation to P(λ); while preserving the respective palindromic structure, such that the given approximate eigentriplet is an exact eigentriplet of the perturbed PPEP. Previously, R. Li, W. Lin, and C. Wang [Numer. Math., 2010, 116(1): 95[122] had only considered the case of an approximate eigenpair for PPEP but commented that attempt for an approximate eigentriplet was unsuccessful. Indeed, the latter case is much more complicated. We provide computable upper bounds for the structured backward errors. Our main results in this paper are several informative and very sharp upper bounds that are capable of revealing distinctive features of PPEP from general polynomial eigenvalue problems (PEPs). In particular, they reveal the critical cases in which there is no structured backward perturbation such that the given approximate eigentriplet becomes an exact one of any perturbed PPEP, unless further additional conditions are imposed. These critical cases turn out to the same as those from the earlier studies on an approximate eigenpair. [ABSTRACT FROM AUTHOR]

关键词： 65F15 65G99 eigentriplet error bound Palindromic polynomial eigenvalue problem (PPEP) structured backward error

THE CRUCIAL CONSTANTS IN THE EXPONENTIAL-TYPE error ESTIMATES FOR GAUSSIAN INTERPOLATION

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Analysis in Theory and Applications 2008年第2期24卷 183-194页

作者： Lintian Luh Providence University Shalu Taichung Taiwan

It＇s well-known that there is a very powerful error bound for Gaussians put forward by Madych and Nelson in 1992. It＇s of the form｜f（x） - s（x）｜≤（Cd）c/d｜｜f｜｜h where C, c are constants, h is the Gaussian function, s is the interpolating function, and d is called fill distance which, roughly speaking, measures the spacing of the points at which interpolation occurs. This error bound gets small very fast as d → 0. The constants C and c are very sensitive. A slight change of them will result in a huge change of the error bound. The number c can be calculated as shown in [9]. However, C cannot be calculated, or even approximated. This is a famous question in the theory of radial basis functions. The purpose of this paper is to answer this question.

关键词： radial basis function interpolation Gaussian error bound