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RETRIEVAL OF INITIAL CONDITION FOR BURGERS' EQUATION USING REDUCED-ORDER ENKF VIA POD-BASED SPARSE OBSERVATIONS

JOURNAL OF INDUSTRIAL AND MANAGEMENT OPTIMIZATION 2023年第6期19卷 4222-4232页

作者： Li, Jie Wang, Yuepeng Ren, Zhigang NUIST Sch Math & Stat Nanjing 210044 Peoples R China Guangdong Univ Technol GDUT Sch Automat Guangzhou 510006 Peoples R China

The data assimilation (DA) is a popular method to solve uncertainty quantification problem which has attracted more and more attention in disaster assessment and climate change research. However, the DA usually faces the computational problem associated with "the curse of dimensionality" when the high-dimensional inverse problem is involved. For this, the usual strategy is to combine the dimensionality reduction method with the DA to mitigate the computational demand in practical application. In this paper, we develop a reduced-order model (ROM)-based EnKF used for retrieving the initial condition, where the prediction model and the observation model are all POD-dependent. In addition to the reduced-order model (ROM) being prediction model, the idea of compressed sensing motivates the acquisition of the observation model that allows an efficient combination of the EnKF with qd algorithm to obtain the optimal sparse observation locations. In this way, computational acceleration is gained in retrieving the initial condition. The effectiveness of this algorithm is demonstrated through retrieving the initial condition of an one-dimensional (1-D) Burgers' equation. Experimental results show that a satisfactory retrieval result is obtained using the current ROM-based EnKF with the computational (CPU) time, 42.64s, nearly 25 times faster, and the error result 7.47 x 10(-3), two order of magnitude O(10(2)) more accurate, than the implementation of the traditional full-order model (FOM)-based EnKF where the computational instability is observed at each iteration though in the same condition as set in the ROM-based EnKF, leading to a poor result (the CPU time 1057.97s, and the error accuracy 4.06 x 10(-1)) The present study extends the application of ROM to efficiently dealing with ill-posed problem as a regularization strategy through replacing the FOM with the ROM such that the low-order truncation of the POD basis leads to the ROM with as few degrees of freedom as possible

关键词： Data assimilation sparse observations ensemble Kalman filter qd algorithm Burgers' equation

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Hessenberg and Tridiagonal Matrices

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丛书名： Other Titles in Applied Mathematics

2025年

作者： Gérard Meurant

Publishing June 2025This is the only book devoted exclusively to Hessenberg and tridiagonal matrices. Hessenberg matrices are involved in Krylov methods for solving linear systems or computing eigenvalues and eigenvec... 详细信息

ISBN: (数字)9781611978452

ISBN: (纸本)9781611978445

Publishing June 2025

This is the only book devoted exclusively to Hessenberg and tridiagonal matrices. Hessenberg matrices are involved in Krylov methods for solving linear systems or computing eigenvalues and eigenvectors, the QR algorithm for computing eigenvalues, and in many other areas of scientific computing (for instance, control theory). Matrices that are both upper and lower Hessenberg are tridiagonal. Their entries are zero except for the main diagonal and the subdiagonal and updiagonal next to it.

Hessenberg and Tridiagonal Matrices: Theory and Examples

presents known and new results;
describes the theoretical properties of the matrices, their determinants, LU factorizations, inverses, and eigenvalues;
illustrates the theoretical properties with applications and examples as well as numerical experiments; and
considers unitary Hessenberg matrices, inverse eigenvalue problems, and Toeplitz tridiagonal matrices.

关键词： Hessenberg matrices tridiagonal matrices LU factorization eigenvalues inverse eigenvalue problems Toeplitz matrices determinants Krylov methods qd algorithm Gauss quadrature

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A Preliminary Study of Multi-task MAP-Elites with Knowledge Transfer for Robotic Arm Design

A Preliminary Study of Multi-task MAP-Elites with Knowledge ...

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IEEE Congress on Evolutionary Computation (CEC)

作者： Liu, Peng Guo, Zheng Yu, Hua Han Linghu Li, Yijing Hou, Yaqing Ge, Hongwei Zhang, Qiang Dalian Univ Technol Sch Comp Sci & Technol Dalian Peoples R China Chinese Univ Hong Kong Shenzhen Peoples R China Dalian Maritime Univ Dalian Peoples R China

ISBN: (数字)9781665467087

ISBN: (纸本)9781665467087

The structure design of robotic arms is of great importance on completing industrial tasks successfully. This is a typical multi-task optimization problem when considering different constraints as different tasks. However, mainstream methods for multi-task optimization such as evolutionary multitasking and Multi-task MAP-Elites algorithms tend to encounter problems such as high computational cost and slow convergence when solving large-scale robotic arm tasks. To this end, this paper proposes a new framework based on the MAP-Elites algorithms for solving large-scale robot arm design tasks, called Multi-task MAP-Elites with Knowledge Transfer (MMKT). Specifically, this paper designs the group-based knowledge transfer process for large-scale task optimization in which all tasks are classified into different groups according to their similarity to generate multiple knowledge transfer areas;and knowledge transfer strategies are designed to enhance the quality of solutions with low fitness value. We test the effectiveness of the MMKT framework in planar robotic arm experiments (2000, 5000, and 10,000 tasks;10, 15-dimensional search space). The experimental results prove that the MMKT outperforms the MME, CMA-ES, and classical ES algorithms.

关键词： Robotic Arm Design Multi-task Optimization MFEA qd algorithm Knowledge Transfer

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Computing recursive orthogonal polynomial with Schur complements

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2020年第0期373卷 112406-000页

作者： Sidki, S. Sadaka, R. Benazzouz, A. Mohammed V Univ Rabat Ecole Normale Super Ave Mohammed Belhassan El OuazzaniBP 5118 Rabat Morocco

In the present paper, we give the expression of orthogonal polynomials as Schur complements. We use the Schur complement and its properties to get a recurrence relation between two families of orthogonal polynomials. We derive two recursive algorithms for computing the orthogonal polynomials by using the qd algorithm and the well known recurrence relation of Hankel determinants. We also give a useful recurrence relation between the orthogonal polynomials of the same family and auxiliary polynomials, which allows us to build a third algorithm for computing recursively the orthogonal polynomials. (C) 2019 Elsevier B.V. All rights reserved.

关键词： Orthogonal polynomials Schur complement Sylvester's identity Recursive orthogonal polynomial algorithm qd algorithm

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Accurate quotient-difference algorithm: Error analysis, improvements and applications

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APPLIED MATHEMATICS AND COMPUTATION 2017年 309卷 245-271页

作者： Du, Peibing Barrio, Roberto Jiang, Hao Cheng, Lizhi Natl Univ Def Technol Sch Sci Changsha 410073 Hunan Peoples R China Univ Zaragoza Dept Matemat Aplicada E-50009 Zaragoza Spain Univ Zaragoza IUMA E-50009 Zaragoza Spain Natl Univ Def Technol Coll Comp Changsha 410073 Hunan Peoples R China Natl Univ Def Technol State Key Lab High Performance Computat Changsha 410073 Hunan Peoples R China

The compensated quotient-difference (Compqd) algorithm is proposed along with some applications. The main motivation is based on the fact that the standard quotient difference (qd) algorithm can be numerically unstable. The Compqd algorithm is obtained by applying error-free transformations to improve the traditional qd algorithm. We study in detail the error analysis of the qd and Compqd algorithms and we introduce new condition numbers so that the relative forward rounding error bounds can be derived directly. Our numerical experiments illustrate that the Compqd algorithm is much more accurate than the qd algorithm, relegating the influence of the condition numbers up to second order in the rounding unit of the computer. Three applications of the new algorithm in the obtention of continued fractions and in pole and zero detection are shown. (C) 2017 Elsevier Inc. All rights reserved.

关键词： qd algorithm Compensated qd algorithm Error-free transformation Rounding error Continued fractions Pole detection

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An application of the discrete-time Toda lattice to the progressive algorithm by Lanczos and related problems

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JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL 2018年第17期51卷 174001-174001页

作者： Nakamura, Yoshimasa Sekido, Hiroto Kyoto Univ Grad Sch Informat Dept Appl Math & Phys Kyoto 6068501 Japan

The finite or the semi-infinite discrete-time Toda lattice has many applications to various areas in applied mathematics. The purpose of this paper is to review how the Toda lattice appears in the Lanczos algorithm through the quotient-difference algorithm and its progressive form (pqd). Then a multistep progressive algorithm (MPA) for solving linear systems is presented. The extended Lanczos parameters can be given not by computing inner products of the extended Lanczos vectors but by using the pqd algorithm with highly relative accuracy in a lower cost. The asymptotic behavior of the pqd algorithm brings us some applications of MPA related to eigenvectors.

关键词： discrete-time Toda lattice qd algorithm Lanczos algorithm multistep progressive algorithm

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Generalized qd algorithm and Markov-Bernstein inequalities for Jacobi weight

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NUMERICAL algorithmS 2009年第4期51卷 429-447页

作者： Draux, Andr Moalla, Borhane Sadik, Mohamed INSA Rouen Dept Genie Math F-76131 Mont St Aignan France Acad Mil Fondouk Jedid 8012 Tunisia

The Markov-Bernstein inequalities for the Jacobi measure remained to be studied in detail. Indeed the tools used for obtaining lower and upper bounds of the constant which appear in these inequalities, did not work, since it is linked with the smallest eigenvalue of a five diagonal positive definite symmetric matrix. The aim of this paper is to generalize the qd algorithm for positive definite symmetric band matrices and to give the mean to expand the determinant of a five diagonal symmetric matrix. After that these new tools are applied to the problem to produce effective lower and upper bounds of the Markov-Bernstein constant in the Jacobi case. In the last part we com pare, in the particular case of the Gegenbauer measure, the lower and upper bounds which can be deduced from this paper, with those given in Draux and Elhami (Comput J Appl Math 106:203-243, 1999) and Draux (Numer Algor 24:31-58, 2000).

关键词： Orthogonal polynomials Jacobi polynomials Gegenbauer polynomials Markov-Bernstein inequalities qd algorithm LR algorithm

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BLOCK FACTORIZATIONS AND qd-TYPE TRANSFORMATIONS FOR THE MR³ algorithm

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2011年 38卷 363-400页

作者： Willems, Paul R. Lang, Bruno Univ Wuppertal Fac Math & Nat Sci D-42097 Wuppertal Germany

Factorizing symmetric tridiagonal matrices and propagating the factorizations to shifted matrices are central tasks in the MR3 algorithm for computing partial eigensystems. In this paper we propose block bidiagonal factorizations LDL* with 1 x 1 and 2 x 2 blocks in D as an alternative to the bidiagonal and twisted factorizations used hitherto. With block factorizations, the element growth can be reduced (or avoided altogether), which is essential for the success of the MR3 algorithm, in particular, if the latter is used to determine the singular value decomposition of bidiagonal matrices. We show that the qd algorithm used for shifting bidiagonal factorizations, e.g., LDL* - tau I -: L+D+(L+)* can be extended to work with blocks in a mixed stable way, including criteria for determining a suitable block structure dynamically.

关键词： symmetric tridiagonal matrix eigensystem MRRR algorithm block bidiagonal factorizations qd algorithm theory and implementation

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From qd to LR, or, how were the qd and LR algorithms discovered?

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IMA JOURNAL OF NUMERICAL ANALYSIS 2011年第3期31卷 741-754页

作者： Gutknecht, Martin H. Parlett, Beresford N. ETH Seminar Appl Math CH-8092 Zurich Switzerland Univ Calif Berkeley Dept Math Berkeley CA 94720 USA

Perhaps, the most astonishing idea in eigenvalue computation is Rutishauser's idea of applying the LR transform to a matrix for generating a sequence of similar matrices that become more and more triangular. The same idea is the foundation of the ubiquitous QR algorithm. It is well known that this idea originated in Rutishauser's qd algorithm, which precedes the LR algorithm and can be understood as applying LR to a tridiagonal matrix. But how did Rutishauser discover qd and when did he find the qd-LR connection? We checked some of the early sources and have come up with an explanation.

关键词： matrix eigenvalues qd algorithm LR algorithm QR algorithm Rutishauser

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Markov-Bernstein inequalities for generalized Gegenbauer weight

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APPLIED NUMERICAL MATHEMATICS 2011年第12期61卷 1301-1321页

作者： Draux, Andre Sadik, Mohamed Moalla, Borhane INSA Rouen Dept Genie Math F-76801 St Etienne France Acad Mil Fondouk Jedid 8012 Tunisia

The Markov-Bernstein inequalities for generalized Gegenbatter weight are studied. A special basis of the vector space P-n of real polynomials in one variable of degree at most equal to n is proposed. It is produced by quasi-orthogonal polynomials with respect to this generalized Gegenbauer measure. Thanks to this basis the problem to find the Markov-Bernstein constant is separated in two eigenvalue problems. The first has a classical form and we are able to give lower and upper bounds of the Markov-Bernstein constant by using the Newton method and the classical qd algorithm applied to a sequence of orthogonal polynomials. The second is a generalized eigenvalue problem with a five diagonal matrix and a tridiagonal matrix. A lower bound is obtained by using the Newton method applied to the six term recurrence relation produced by the expansion of the characteristic determinant. The asymptotic behavior of an upper bound is studied. Finally, the asymptotic behavior of the Markov-Bernstein constant is O(n(2)) in both cases. (C) 2011 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Markov-Bernstein inequalities Gegenbauer polynomials Generalized Gegenbauer polynomials Quasi-orthogonal polynomials Eigenvalue problem Generalized eigenvalue problem qd algorithm

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