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Adaptive step size numerical integration for stochastic differential equations with discontinuous drift and diffusion

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numerical algorithms 2021年第2期87卷 849-872页

作者： Malik, Avinash Univ Auckland Dept Elect Comp & Software Engn Auckland New Zealand

Stochastic hybrid systems (SHSs) are a modelling framework for a cyber-physical system (CPS), used to simulate, validate, and verify safety critical controllers under uncertainty. Popular simulation tools can miss detecting discontinuities when simulating SHS, thereby producing incorrect outputs during simulation. We propose a novel adaptive step size simulation/integration technique for a subset of SHS-stochastic differential equations (SDEs) with discontinuous drift and diffusion coefficients. Each integration step, of the Euler-Maruyama numerical solution of the SDEs, is made dependent upon the values of the continuous variables inducing the discontinuity. This in turn guarantees convergence of the system trajectory towards the discontinuity without missing it. A thorough analysis and extensive benchmarking of the proposed integration technique shows the efficacy of the approach when simulating complex SHSs.

关键词： Stochastic hybrid systems Stochastic differential equations numerical integration algorithms Convergence analysis

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Questioning numerical integration methods for microsphere (and microplane) constitutive equations

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MECHANICS OF MATERIALS 2015年第Oct.期89卷 216-228页

作者： Verron, Erwan Ecole Cent Nantes Inst Rech Genie Civil & Mecan GeM UMR CNRS 6183 F-44321 Nantes 3 France

In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated. Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress-strain prediction are illustrated with a directional version of the classical Mooney-Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations. (C) 2015 Elsevier Ltd. All rights reserved.

关键词： Constitutive equations Microsphere models numerical integration algorithms

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CNN based hole filler template design using numerical integration techniques

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17th International Conference on Artificial Neural Networks (ICANN 2007)

作者： Murugesan, K. Elango, P. Natl Inst Technol Dept Comp Applicat Tiruchirappalli 620015 Tamil Nadu India Natl Inst Technol Dept Math Tiruchirappalli 620015 Tamil Nadu India

ISBN: (纸本)9783540746898

This paper presents, a design method for the template of a hole-fliler used to improve the pe rformance of the handwritten character recognition using numerical integration algorithms, based on the dynamic analysis of a cellular neural network (CNN). This is done by analyzing the features of the hole-filter template and the dynamic process of CNN using popular numerical integration algorithms to obtain a set of inequalities satisfying its output characteristics as well as the parameter range of the hole-filler template. Simulation results are presented for Enter, Modified Euler and RK methods and compared. It was found that RK Method performs well in terms of settling time and computation time for all step sizes.

关键词： cellular neural network hole-filler template numerical integration algorithms Euler method Modified-Euler method RK-method

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Efficiency of numerical integration algorithms related to divisor theory in cyclotomic fields

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MATHEMATICAL NOTES 1997年第1-2期61卷 242-245页

作者： Temirgaliev, N AL FARIBI KAZAKH STATE NATL UNIV ALMA ATAKAZAKHSTAN

For function classes with dominant mixed derivative and bounded mixed difference in the metric of L-q (1 < q less than or equal to 2), quadrature formulas are constructed so that the following properties are achieved simultaneously: the grid is simple, the algorithm is efficient and close to the optimal algorithm for constructing the grid, and the order of the error on the power scale cannot be further improved. The case q = 2 was studied earlier.

关键词： numerical integration algorithms quadrature formulas divisor theory cyclotomic fields

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Computational integration

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丛书名： Other Titles in Applied Mathematics

1998年

作者： Arnold R. Krommer Christoph W. Ueberhuber

ISBN: (数字)9781611971460

ISBN: (纸本)9780898713749

This survey covers a wide range of topics fundamental to calculating integrals on computer systems and discusses both the theoretical and computational aspects of numerical and symbolic methods. It includes extensive sections on one- and multidimensional integration formulas, like polynomial, number-theoretic, and pseudorandom formulas, and deals with issues concerning the construction of numerical integration algorithms.

关键词： Computational integration integration formula numerical integration algorithms parallel integrational algorithms numerical integration software numerial methods

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SIMULATION 1984年第4期43卷 169-174页

作者： KOHN, W [o_1]Lockheed Engineering & Management Services Company Inc. 1830 NASA Road 1 Houston Texas 77058

The rigid and flexible dynamic behavior of many systems in aerospace structures can be modeled by a set of coupled non linear second-order differential equations. In these systems, the mass matrix function of the system, which contains the masses of the different modules in the structure, its moment of inertia tensors, and its bending moment coefficients (finite number of modes), exhibits eigenvalue dispersion between four and nine orders of magnitude. In addition, due to small gravity gradient and aerodynamic forces and moments, these systems are acted on by small or negligible damping forces and moments. The two characteristics alluded to above imply that the differential equa tions representing these systems have poor numerical stability properties. This study is concerned with the development of a similarity transformation on the variables of integration which results in an equivalent system with higher damping and conse quently better numerical stability than the original one.

关键词： nonlinear second-order differential equations numerical integration algorithms numerical sta bility piecewise-linearization real-time simula tion similarity transformation system dynamics

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numerical-integration algorithms IN POWER SYSTEM DYNAMIC ANALYSIS

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PROCEEDINGS OF THE INSTITUTION OF ELECTRICAL ENGINEERS-LONDON 1974年第6期121卷 467-473页

作者： HUMPAGE, WD WONG, KP LEE, YW UNIV MANCHESTER INST SCI & TECHNOL POWER SYST LAB POB 88 MANCHESTER M60 1QD ENGLAND UNIV BRISTOL DEPT ELECT ENGN BRISTOL ENGLAND

The outcome of recent test applications of a range of integration algorithms to models representative of those now widely encountered in large- scale power- system dynamic analysis is reported. The paper collects together the main requirements of practical dynamic analysis, and it gives the principal schemes and solution strategies that might be used in meeting these requirements. Plant equation sets are summarised in a form that is of general application and interpretation in large analysis schemes, and, for these sets, comparative studies are made of the performance of explicit and implicit methods of single- and multiple-step sequences of numerical integration. In total, some ten different methods have been included in the evaluations, and, for most of these, two, and, in some cases, three, alternative modes of implementation are studied and compared. Methods of decomposition are invoked to allow different step lengths, and, where required, different integration methods, to be associated with individual subsets of significantly different response rates. Finally, the likely role of automatic algorithm control in this area of applied analysis in power systems is discussed

关键词： numerical integration algorithms numerical analysis numerical integration and differentiation dynamic analysis Power engineering computing Power systems electrical engineering applications of computers computer aided analysis power systems integration

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