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The mathematical foundations of dynamic user equilibrium

TRANSPORTATION RESEARCH PART B-METHODOLOGICAL 2019年第Aug.期126卷 309-328页

作者： Friesz, Terry L. Han, Ke Penn State Univ Dept Ind & Mfg Engn University Pk PA 16802 USA Imperial Coll London Dept Civil & Environm Engn London England

This paper is pedagogic in nature, meant to provide researchers a single reference for learning how to apply the emerging literature on differential variational inequalities to the study of dynamic traffic assignment problems that are Cournot-like noncooperative games. The paper is presented in a style that makes it accessible to the widest possible audience. In particular, we apply the theory of differential variational inequalities (DVIs) to the dynamic user equilibrium (DUE) problem. We first show that there is a variational inequality whose necessary conditions describe a DUE. We restate the flow conservation constraint associated with each origin-destination pair as a first-order two-point boundary value problem, thereby leading to a DVI representation of DUE;then we employ Pontryagin-type necessary conditions to show that any DVI solution is a DUE. We also show that the DVI formulation leads directly to a fixed-point algorithm. We explain the fixed-point algorithm by showing the calculations intrinsic to each of its steps when applied to simple examples. (C) 2018 Elsevier Ltd. All rights reserved.

关键词： Dynamic user equilibrium Differential variational inequality Optimal control fixed point algorithm

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A NUMERICAL PERSPECTIVE ON HARTREE-FOCK-BOGOLIUBOV THEORY

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ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS-MODELISATION MATHEMATIQUE ET ANALYSE NUMERIQUE 2014年第1期48卷 53-86页

作者： Lewin, Mathieu Paul, Severine Univ Cergy Pontoise CNRS F-95000 Cergy Pontoise France Univ Cergy Pontoise CNRS Math Lab UMR 8088 F-95000 Cergy Pontoise France

The method of choice for describing attractive quantum systems is Hartree-Fock-Bogoliubov (HFB) theory. This is a nonlinear model which allows for the description of pairing effects, the main explanation for the superconductivity of certain materials at very low temperature. This paper is the first study of Hartree-Fock-Bogoliubov theory from the point of view of numerical analysis. We start by discussing its proper discretization and then analyze the convergence of the simple fixed point (Roothaan) algorithm. Following works by Cances, Le Bris and Levitt for electrons in atoms and molecules, we show that this algorithm either converges to a solution of the equation, or oscillates between two states, none of them being solution to the HFB equations. We also adapt the Optimal Damping algorithm of Cances and Le Bris to the HFB setting and we analyze it. The last part of the paper is devoted to numerical experiments. We consider a purely gravitational system and numerically discover that pairing always occurs. We then examine a simplified model for nucleons, with an effective interaction similar to what is often used in nuclear physics. In both cases we discuss the importance of using a damping algorithm.

关键词： Hartree-Fock-Bogoliubov fixed point algorithm relaxed constraint algorithm nuclear physics

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Field programmable gate arrays-based differential evolution coprocessor: a case study of spectrum allocation in cognitive radio network

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IET COMPUTERS AND DIGITAL TECHNIQUES 2013年第5期7卷 221-234页

作者： Anumandla, Kiran Kumar Peesapati, Rangababu Sabat, Samrat L. Udgata, Siba K. Abraham, Ajith Univ Hyderabad Sch Phys Hyderabad 500046 Andhra Pradesh India Univ Hyderabad Sch Comp & Informat Sci Hyderabad 500046 Andhra Pradesh India Sci Network Innovat & Res Excellence Machine Intelligence Res Labs Auburn WA 98071 USA

In this study, a scalable coprocessor for accelerating the Differential Evolution (DE) algorithm is presented. The coprocessor is interfaced with PowerPC embedded processor of Xilinx Virtex-5 FX70T Field Programmable Gate Array. In the proposed design, the DE algorithm module is tightly coupled with fitness function module to reduce communication and control overhead. The fixed point DE algorithm is implemented in the coprocessor whereas both fixed and floating point DE are implemented in the embedded processor. Performance of the coprocessor is evaluated by optimising benchmark functions of different complexities. The implementation results show that the coprocessor is 73.14-160.2x and 2.19-27.63x faster compared to the software execution time of the floating and fixed point algorithm respectively. As a case study, spectrum allocation problem of cognitive radio network is evaluated with the coprocessor. Results show an acceleration of 76.79-105x and 5.19-6.91x with respect to floating and fixed point DE in embedded processor. It is also observed that the application occupies 56% of BRAM, 54% of DSP48E, 16% of slice LUTs and maximum frequency of operation as 63.55 MHz in a Virtex-5 FPGA. This type of coprocessor is suitable for embedded applications where the fitness function remains unchanged.

关键词： cognitive radio coprocessors embedded systems field programmable gate arrays floating point arithmetic coprocessor development spectrum allocation problem floating point algorithm fixed point algorithm direct software execution embedded processor fitness function evaluation module Xilinx Virtex 5 field programmable gate array hardcore PowerPC embedded processor DE algorithm scalable coprocessor cognitive radio network spectrum allocation differential evolution coprocessor field programmable gate arrays

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algorithmS FOR THE COMPUTATION OF WEIGHTS IN BOUNDED INFLUENCE REGRESSION

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 1988年第3期6卷 251-276页

作者： MARAZZI, A UNIV LAUSANNE INST MED SOCIALE & PREVENT DIV STAT & INFORMAT CH-1005 LAUSANNE SWITZERLAND

A preliminary step for the computation of bounded influence regression is the determination of weights for the points x i in the factor space. This can be achieved through the computation of a robust affinely invariant covariance matrix (with fixed location) and the computation of robust distances of the x i to the origin, in the metric defined by the covariance matrix. For a well known fixed point algorithm convergence has been proved. However, its speed is not very satisfactory in practice. Newton-like and conjugate gradient approaches have been suggested by P.J. Huber. This paper reviews the various proposals, discusses their implementations, some of their convergence properties and compares empirically their performances.

关键词： Robust regression Bounded influence regression Robust covariances fixed point algorithm Newton algorithm Conjugate gradient algorithm

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Fractional elliptic quasi-variational inequalities: Theory and numerics

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INTERFACES AND FREE BOUNDARIES 2018年第1期20卷 1-24页

作者： Antil, Harbir Rautenberg, Carlos N. George Mason Univ Dept Math Sci 4400 Univ DrMS 3F2 Fairfax VA 22030 USA Humboldt Univ Inst Math Unter Linden 6 D-10099 Berlin Germany

This paper introduces an elliptic quasi-variational inequality (QVI) problem class with fractional diffusion of order s is an element of (0, 1), studies existence and uniqueness of solutions and develops a solution algorithm. As the fractional diffusion prohibits the use of standard tools to approximate the QVI, instead we realize it as a Dirichlet-to-Neumann map for a problem posed on a semi-infinite cylinder. We first study existence and uniqueness of solutions for this extended QVI and then transfer the results to the fractional QVI: This introduces a new paradigm in the field of fractional QVIs. Further, we truncate the semi-infinite cylinder and show that the solution to the truncated problem converges to the solution of the extended problem, under fairly mild assumptions, as the truncation parameter r tends to infinity. Since the constraint set changes with the solution, we develop an argument using Mosco convergence. We state an algorithm to solve the truncated problem and show its convergence in function space. Finally, we conclude with several illustrative numerical examples.

关键词： Quasivariational inequality QVI fractional derivatives fractional diffusion free boundary problem Caffarelli-Silvestre and Stinga-Torrea extension weighted Sobolev spaces Mosco convergence fixed point algorithm finite element method

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The Extended Matrix-Variate Beta Probability Distribution on Symmetric Matrices

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METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 2020年第2期22卷 647-676页

作者： Tounsi, Mariem Sfax Univ Lab Probabil & Stat Sfax 3038 Tunisia

The notion of generalized power function in the space of real symmetric matrices is used to introduce a kind of extended matrix-variate beta function. With the aid of this, we define a different versions of extended matrix-variate beta distributions. Some fundamental properties of these distributions are established. We show that using a linear transformation on the extended matrix-variate beta distributions of the first and second kind, we can generalize these distributions. We also show that the distribution of the sum of two independent inverse Riesz matrices introduced by Tounsi and Zine (J Multivar Anal 111:174-182, 2012) can be written in terms of the generalized extended matrix-variate beta function. Finally, using fixed point iterative method, we provide a calculable maximum a posteriori (MAP) estimator for the unknown covariance matrix of a multivariate normal distribution based on the class of the extended matrix-variate beta prior distribution. Additionally, we evaluated the Gaussian finite sample performance by calculating such evaluation criteria as Mean Square Error (MSE) and Hilbert-Schmidt distance (DHS). The obtained results confirm the performance of the proposed prior.

关键词： Random matrices Extended beta function Division algorithm Inverse Riesz distribution Beta-Riesz distribution Covariance estimation MAP estimator fixed point algorithm

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AN EFFICIENT IMPLEMENTATION OF MERRILL'S METHOD FOR SPARSE OR PARTIALLY SEPARABLE SYSTEMS OF NONLINEAR EQUATIONS

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SIAM JOURNAL ON OPTIMIZATION 1991年第2期1卷 206-221页

作者： Bing, Yang Lin, Gao Harbin Shipbuilding Engn Inst Dept Engn Management Harbin 150001 Peoples R China Harbin Shipbuilding Engn Inst Dept Math & Mech Harbin 150001 Peoples R China

When Merrill's method without an extra dimension is applied to solving sparse or partially separable systems of nonlinear equations, the computational efficiency can be further improved, i.e., a larger piece of linearity can be traversed in one step by using a suitable linear system. One of the linear systems is updated and the corresponding technique is shown to update information about the linear system and the large piece in the implementation of the method. Some numerical results of the method support the claim that it is efficient. Also, some mistakes from a previous paper, in which the main technique exploiting sparsity was proposed, are corrected.

关键词： fixed point algorithm systems of nonlinear equations variable dimension algorithm sparsity partial separability

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algorithms with high order convergence speed for blind source extraction

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COMPUTATIONAL OPTIMIZATION AND APPLICATIONS 2007年第1期38卷 123-131页

作者： Georgiev, Pando Pardalos, Panos Cichocki, Andrzej Univ Cincinnati ECECS Dept Cincinnati OH 45221 USA Sofia Univ St K1 Ohridski Sofia 1164 Bulgaria Univ Florida Ctr Appl Optimizat Gainesville FL 32611 USA RIKEN Brain Sci Inst Wako Saitama 3510198 Japan

A rigorous convergence analysis for the fixed point ICA algorithm of Hyvarinen and Oja is provided and a generalization of it involving cumulants of an arbitrary order is presented. We consider a specific optimization problem OP(p), p > 3, integer, arising from a Blind Source Extraction problem (BSE) and prove that every local maximum of OP(p) is a solution of (BSE) in sense that it extracts one source signal from a linear mixture of unknown statistically independent signals. An algorithm for solving OP(p) is constructed, which has a rate of convergence p - 1.

关键词： independent component analysis cumulants fixed point algorithm

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A general optimality criteria algorithm for a class of engineering optimization problems (vol 47, pg 674, 2015)

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ENGINEERING OPTIMIZATION 2015年第5期47卷 718-718页

作者： Belegundu, A. [a] Department of Mechanical and Nuclear Engineering The Pennsylvania State University University Park PA 16802 USA

An optimality criteria (OC)-based algorithm for optimization of a general class of nonlinear programming (NLP) problems is presented. The algorithm is only applicable to problems where the objective and constraint functions satisfy certain monotonicity properties. For multiply constrained problems which satisfy these assumptions, the algorithm is attractive compared with existing NLP methods as well as prevalent OC methods, as the latter involve computationally expensive active set and step-size control strategies. The fixed point algorithm presented here is applicable not only to structural optimization problems but also to certain problems as occur in resource allocation and inventory models. Convergence aspects are discussed. The fixed point update or resizing formula is given physical significance, which brings out a strength and trim feature. The number of function evaluations remains independent of the number of variables, allowing the efficient solution of problems with large number of variables.

关键词： optimality criteria fixed point algorithm re-sizing surrogate multiplier method nonlinear programming

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SOME RESULTS ON THE EXISTENCE, UNIQUENESS, AND COMPUTATION OF THE M-ESTIMATES OF MULTIVARIATE LOCATION AND SCATTER

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SIAM JOURNAL ON SCIENTIFIC AND STATISTICAL COMPUTING 1988年第2期9卷 354-362页

作者： TYLER, DE

The M-estimates of multivariate location and scatter are a class of robust alternatives to the sample mean vector and sample covariance matrix. They also have applications to bounded influence regression. There are a number of problems, though, which need to be resolved before they can become widely applicable. This paper addresses the problems of existence, uniqueness and computation of the M-estimates for finite sample sizes.

关键词： 62H12 62H10 62G05 bounded influence regression covariance matrix finite sample size fixed point algorithm sensitivity bound robust estimation

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