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Numerical solution of stochastic Nash games with state-dependent noise for weakly coupled large-scale systems

APPLIED MATHEMATICS AND COMPUTATION 2008年第2期197卷 844-857页

作者： Sagara, Muneomi Mukaidani, Hiroaki Yamamoto, Toru Hiroshima Univ Grad Sch Educ Higashihiroshima 7398524 Japan

This paper discusses the infinite horizon stochastic Nash games with state-dependent noise. After establishing the asymptotic structure along with the positive semidefiniteness for the solutions of the cross-coupled stochastic algebraic Riccati equations (CSAREs), a new algorithm that combines Newton's method with two fixed point algorithms for solving the CSAREs is derived. As a result, it is shown that the proposed algorithm attains quadratic convergence and the reduced-order computations for sufficiently small parameter e. As another important feature, the high-order approximate strategy that is based on the iterative solutions is proposed. Using such strategy, the degradation of the cost functional is investigated. Finally, in order to demonstrate the efficiency of the proposed algorithms, computational examples are provided. (C) 2007 Elsevier Inc. All rights reserved.

关键词： stochastic Nash games cross-coupled stochastic algebraic Riccati equations (CSAREs) Newton's method Newton-Kantorovich theorem fixed point algorithm

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Numerical computation for solving algebraic Riccati equations of weakly coupled systems

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ELECTRICAL ENGINEERING IN JAPAN 2007年第1期160卷 39-48页

作者： Mukaidani, Hiroaki Xu, Hua Monden, Yoshimi Hiroshima Univ Hiroshima Japan Univ Tsukuba Tsukuba Ibaraki 305 Japan

In this paper, an algorithm for solving the algebraic Riccati equation (ARE) that has an indefinite sign quadratic term related to weakly coupled large-scale systems is investigated. A novel contribution is that a new iterative algorithm is derived by combining Newton's method and the fixed point algorithm. As a result, for sufficiently small F, we can obtain an ARE solution with a quadratic convergence rate. Moreover, it is possible to calculate the ARE solution for the same dimension of each subsystem. As another important feature, an algorithm for solving the filter ARE is also discussed. Finally, in order to demonstrate the efficiency of the proposed algorithm, a numerical example is given. (C) 2007 Wiley Periodicals, Inc.

关键词： weakly coupled systems algebraic Riccati equation Newton's method fixed point algorithm

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A numerical algorithm for finding solution of sign-indefinite algebraic Riccati equations for general multiparameter singularly perturbed systems

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APPLIED MATHEMATICS AND COMPUTATION 2007年第1期189卷 255-270页

作者： Mukaidani, Hiroaki Hiroshima Univ Grad Sch Educ Higashihiroshima 7398524 Japan

In this paper, a computational algorithm for solving sign-indefinite general multiparameter algebraic Riccati equation (SIGMARE) that arises in the H-infinity filtering problem is investigated. After establishing the asymptotic structure of the solution of the SIGMARE, in order to solve the SIGMARE, Newton's method and two fixed point algorithms are combined. As a result, the new iterative algorithm achieves the quadratic convergence property and succeeds in reducing the computing workspace dramatically. As another important feature, the convergence criteria for small parameters epsilon(i) is derived for the first time. Moreover, it is shown that the uniqueness and positive semidefiniteness of the convergence solutions are guaranteed in the neighborhood of the initial conditions. (c) 2006 Elsevier Inc. All rights reserved.

关键词： general multiparameter singularly perturbed system (GMSPS) sign-indefinite general multiparameter algebraic Riccati equation (SIGMARE) H-infinity filtering problem Newton's method fixed point algorithm

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The iteratively reweighted estimating equation in minimum distance problems

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2004年第2期45卷 105-124页

作者： Basu, A Lindsay, BG Indian Stat Inst Appl Stat Unit Kolkata 700108 W Bengal India Penn State Univ Dept Stat University Pk PA 16802 USA

The class of density based minimum distance estimators provide attractive alternatives to the maximum likelihood estimator because several members of this class have nice robustness properties while being first-order efficient under the assumed model. A helpful computational technique-similar to the iteratively reweighted least squares used in robust regression-is introduced which makes these estimators computationally much more feasible. This technique is much simpler than the Newton-Raphson (NR) method to implement. The loss suffered in the rate of convergence compared to the NR method can be made to vanish in some exponential family situations by a little modification in the weight function-in which case the performance is comparable to the NR method. For a large number of parameters the performance of this modified version is actually expected to be better than the NR method. In view of the widespread interest in density based robust procedures, this modification appears to be of great practical value. (C) 2002 Elsevier B.V. All rights reserved.

关键词： disparity robustness hellinger distance iteratively reweighted least squares iteratively reweighted estimating equation Newton-Raphson algorithm fixed point algorithm

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ABSORBING BOUNDARY CONDITIONS FOR GENERAL NONLINEAR SCHRODINGER EQUATIONS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2011年第2期33卷 1008-1033页

作者： Antoine, Xavier Besse, Christophe Klein, Pauline Nancy Univ Inst Elie Cartan Nancy INRIA CORIDA Team CNRS UMR 7502 F-54506 Vandoeuvre Les Nancy France Univ Lille 1 Sci & Technol Univ Lille Nord France INRIA SIMPAF Team CNRS UMR 8524Lab Paul Painleve F-59655 Villeneuve Dascq France

This paper addresses the construction of different families of absorbing boundary conditions for the one- and two-dimensional Schrodinger equations with a general variable nonlinear potential. Various semidiscrete time schemes are built for the associated initial boundary value problems. Finally, some numerical simulations give a comparison of the various absorbing boundary conditions and associated schemes to analyze their accuracy and efficiency.

关键词： absorbing boundary conditions pseudodifferential operators nonlinear Schrodinger equation with potential stable semidiscrete schemes fixed point algorithm relaxation scheme

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Parameter Estimation For Multivariate Generalized Gaussian Distributions

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IEEE TRANSACTIONS ON SIGNAL PROCESSING 2013年第23期61卷 5960-5971页

作者： Pascal, Frederic Bombrun, Lionel Tourneret, Jean-Yves Berthoumieu, Yannick Supelec SONDRA F-91192 Gif Sur Yvette France Univ Bordeaux IPB ENSEIRB Matmeca Lab IMS F-33400 Talence France Univ Toulouse IRIT INP ENSEEIHT F-31071 Toulouse France

Due to its heavy-tailed and fully parametric form, the multivariate generalized Gaussian distribution (MGGD) has been receiving much attention in signal and image processing applications. Considering the estimation issue of the MGGD parameters, the main contribution of this paper is to prove that the maximum likelihood estimator (MLE) of the scatter matrix exists and is unique up to a scalar factor, for a given shape parameter beta is an element of (0, 1). Moreover, an estimation algorithm based on a Newton-Raphson recursion is proposed for computing the MLE of MGGD parameters. Various experiments conducted on synthetic and real data are presented to illustrate the theoretical derivations in terms of number of iterations and number of samples for different values of the shape parameter. The main conclusion of this work is that the parameters of MGGDs can be estimated using the maximum likelihood principle with good performance.

关键词： Covariance matrix estimation fixed point algorithm multivariate generalized Gaussian distribution

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Newton's method for solving cross-coupled sign-indefinite algebraic Riccati equations for weakly coupled large-scale systems

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APPLIED MATHEMATICS AND COMPUTATION 2007年第1期188卷 103-115页

作者： Mukaidani, Hiroaki Hiroshima Univ Grad Sch Educ Higashihiroshima 7398524 Japan

In this paper, a new algorithm for solving cross-coupled sign-indefinite algebraic Riccati equations (CSAREs) for weakly coupled large-scale systems is proposed. It is shown that since the proposed algorithm is based on the Newton's method, the quadratic convergence is attained. Moreover, the local uniqueness of the convergence solutions for the CSAREs is investigated. Finally, in order to overcome the computation of large- and sparse-matrix related to the Newton's method, the fixed point algorithm and the alternating direction implicit (ADI) method are combined. (C) 2006 Elsevier Inc. All rights reserved.

关键词： weakly coupled large-scale systems cross-coupled sign-indefinite algebraic Riccati equations (CSAREs) Newton's method Newton-Kantorovich theorem fixed point algorithm fixed point theorem alternating direction implicit (ADI) method

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Krylov subspace split Bregman methods

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APPLIED NUMERICAL MATHEMATICS 2023年 184卷 371-390页

作者： Alotaibi, Majed Buccini, Alessandro Reichel, Lothar Kent State Univ Dept Math Sci Kent OH 44242 USA Univ Cagliari Dipartimento Matemat & Informat I-09124 Cagliari Italy

Split Bregman methods are popular iterative methods for the solution of large-scale minimization problems that arise in image restoration and basis pursuit. This paper investigates the possibility of projecting large-scale problems into a Krylov subspace of fairly small dimension and solving the minimization problem in the latter subspace by a split Bregman algorithm. We are concerned with the restoration of images that have been contaminated by blur and Gaussian or impulse noise. Computed examples illustrate that the projected split Bregman methods described are fast and give computed solutions of high quality.(c) 2022 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Split Bregman method Krylov method Golub-Kahan bidiagonalization fixed point algorithm Cross validation

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Complete Dictionary Learning via l⁴-Norm Maximization over the Orthogonal Group

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JOURNAL OF MACHINE LEARNING RESEARCH 2020年第1期21卷 1-68页

作者： Zhai, Yuexiang Yang, Zitong Liao, Zhenyu Wright, John Ma, Yi Univ Calif Berkeley Dept Elect Engn & Comp Sci Berkeley CA 94720 USA Kuaishou Technol Beijing Peoples R China Columbia Univ Dept Elect Engn New York NY 10027 USA

This paper considers the fundamental problem of learning a complete (orthogonal) dictionary from samples of sparsely generated signals. Most existing methods solve the dictionary (and sparse representations) based on heuristic algorithms, usually without theoretical guarantees for either optimality or complexity. The recent l(1)-minimization based methods do provide such guarantees but the associated algorithms recover the dictionary one column at a time. In this work, we propose a new formulation that maximizes the l(4)-norm over the orthogonal group, to learn the entire dictionary. We prove that under a random data model, with nearly minimum sample complexity, the global optima of the l(4)-norm are very close to signed permutations of the ground truth. Inspired by this observation, we give a conceptually simple and yet effective algorithm based on "matching, stretching, and projection" (MSP). The algorithm provably converges locally and cost per iteration is merely an SVD. In addition to strong theoretical guarantees, experiments show that the new algorithm is significantly more efficient and effective than existing methods, including KSVD and l(1)-based methods. Preliminary experimental results on mixed real imagery data clearly demonstrate advantages of so learned dictionary over classic PCA bases.

关键词： sparse dictionary learning l(4)-norm maximization orthogonal group measure concentration fixed point algorithm

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Two-grid method for regularized frictional elastostatics problems

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ENGINEERING COMPUTATIONS 1995年第7期12卷 657-664页

作者： Lebon, F Laboratoire de Mécanique et Génie Civil Université Montpellier 2 Pl. E. Bataillon 34095 Montpellier Cedex 5 France

This paper is the description of a new two-grid algorithm to solve frictional contact problems. A regularized formulation is introduced and the discretized problem is solved using an internal non linear two-grid technique coupled with a diagonal fixed point algorithm. Mathematical background is given, and superconvergence is obtained.

关键词： two-grid method contact problems fixed point algorithm

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