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High-Precision algorithms for Constructing Optimal Trajectories via Solving Hamiltonian Systems *

IFAC Proceedings Volumes 2009年第2期42卷 70-75页

作者： Andrey A. Krasovskii Alexander M. Tarasyev Institute of Mathematics and Mechanics Ural Branch of RAS S. Kovalevskoi Str. 16 620219 Ekaterinburg Russia and Vienna Institute of Demography Austrian Academy of Sciences Wohllebengasse 12-14 1040 Vienna Austria Institute of Mathematics and Mechanics UrB RAS S. Kovalevskoi Str. 16 620219 Ekaterinburg Russia

The paper deals with analysis of optimal control problems arising in models of economic growth. The Pontryagin maximum principle is applied for analysis of the optimal investment problem. Specifically, the research is based on existence results and necessary conditions of optimality in problems with infinite horizon. Properties of Hamiltonian systems are examined for different regimes of optimal control. The existence and uniqueness result is proved for a steady state of the Hamiltonian system. Analysis of properties of eigenvalues and eigenvectors is completed for the linearized system in a neighborhood of the steady state. Description of behavior of the nonlinear Hamiltonian system is provided on the basis of results of the qualitative theory of differential equations. This analysis allows us to outline proportions of the main economic factors and trends of optimal growth in the model. A numerical algorithm for construction of optimal trajectories of economic growth is elaborated on the basis of constructions of backward procedures and conjugation of an approximation linear dynamics with the nonlinear Hamiltonian dynamics. High order precision estimates are obtained for the proposed algorithm. These estimates establish connection between precision parameters in the phase space and precision parameters for functional indices.

关键词： Nonlinear control systems numerical algorithms Precision measurements Economic design

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Comparison of numerical algorithms for bivariate sequential tests based on marginal criteria

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COMPUTATIONAL STATISTICS & DATA ANALYSIS 2005年第3期49卷 631-641页

作者： Zhao, YL Grambsch, PM Neaton, JD Univ Minnesota Sch Publ Hlth Div Biostat Minneapolis MN 55455 USA

Group sequential tests are widely used for interim analyses in randomized clinical trials. These tests have been extended to bivariate outcomes, but calculation of boundary values is computationally intensive. In this paper, we compare 3 algorithms: (1) quasi-Monte Carlo;(2) Schervish's MULNOR;and (3) a bivariate version of Armitage's recursive formula proposed by Jennison and Turnbull, for calculating boundary values for the L-region bivariate design which takes account of the correlation between the outcomes. Up to 5 interim analyses (stages) and a range of values for the correlation between the two endpoints were considered. The three algorithms produced very similar boundary values, but the quasi-Monte Carlo approach was much faster than the other two. The quasi-Monte Carlo method is an attractive approach for studying higher dimensions-designs with more stages and designs with more than two outcomes. The increased speed of this algorithm may permit the development of more flexible monitoring guidelines for multiple endpoints for clinical trials. (c) 2004 Elsevier B.V. All rights reserved.

关键词： group sequential monitoring bivariate responses numerical algorithms

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numerical ALGORITHM FOR CALCULATING THE GENERALIZED MITTAG-LEFFLER FUNCTION

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SIAM JOURNAL ON numerical ANALYSIS 2008年第1期47卷 69-88页

作者： Seybold, Hansjoerg Hilfer, Rudolf ETH IfB CH-8093 Zurich Switzerland Univ Stuttgart ICP D-70569 Stuttgart Germany Johannes Gutenberg Univ Mainz Inst Phys D-55099 Mainz Germany

A numerical algorithm for calculating the generalized Mittag-Leffler E alpha, beta(z) function for arbitrary complex argument z and real parameters alpha > 0 and beta is an element of R is presented. The algorithm uses the Taylor series, the exponentially improved asymptotic series, and integral representations to obtain optimal stability and accuracy of the algorithm. Special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm.

关键词： special functions of mathematical physics fractional calculus generalized Mittag-Leffler functions numerical algorithms

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A numerical algorithm for simulation of the Q-switched fiber laser using the travelling wave model

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LITHUANIAN MATHEMATICAL JOURNAL 2008年第3期48卷 270-281页

作者： Ciegis, R. Dement'ev, A. Laukaityte, I. Vilnius Gediminas Tech Univ LT-10223 Vilnius Lithuania Inst Phys LT-2600 Vilnius Lithuania

We develop a finite-difference scheme for approximation of a system of nonlinear PDEs describing the Q-switching process. We construct it by using staggered grids. The transport equations are approximated along characteristics, and quadratic nonlinear functions are linearized using a special selection of staggered grids. The stability analysis proves that a connection between time and space steps arises only due to approximation requirements in order to follow exactly the directions of characteristics. The convergence analysis of this scheme is done in two steps. First, some estimates of the uniform boundedness of the discrete solution are proved. This part of the analysis is done locally, in some neighborhood of the exact solution. Second, on the basis of the obtained estimates, the main stability inequality is proved. The second-order convergence rate with respect to the space and time coordinates follows from this stability estimate. Using the obtained convergence result, we prove that the local stability analysis in the selected neighborhood of the exact solution is sufficient.

关键词： Q-switch nonlinear problems numerical algorithms convergence analysis staggered grids

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numerical bounds for semi-Markovian quantities and application to reliability

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METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 2008年第2期10卷 179-198页

作者： Mercier, Sophie Univ Paris Est Lab Anal & Math Appl CNRS UMR 8050 F-77454 Marne La Vallee France

We propose new easily computable bounds for different quantities which are solutions of Markov renewal equations linked to some continuous-time semi-Markov process (SMP). The idea is to construct two new discrete-time SMP which bound the initial SMP in some sense. The solution of a Markov renewal equation linked to the initial SMP is then shown to be bounded by solutions of Markov renewal equations linked to the two discrete time SMP. Also, the bounds are proved to converge. To illustrate the results, numerical bounds are provided for two quantities from the reliability field: mean sojourn times and probability transitions.

关键词： continuous and discrete time homogeneous semi-Markov processes Markov renewal equations numerical algorithms

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Theoretical formulation and numerical implementation of electromagnetic and thermomechanical loading processes in solids

Theoretical formulation and numerical implementation of elec...

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作者： Thomas, Jesse D. University of Michigan

学位级别：Ph.D.

The electromagnetic-thermal-mechanical process of electromagnetic forming (EMF) is a high velocity manufacturing technique that uses electromagnetic (Lorentz) body forces to shape sheet metal parts. One of the several advantages of this technique is the considerable ductility increase observed in several metals, particularly aluminum. In this research two important aspects of this process are addressed: the influence of the sheet metal's constitutive response on ductility and the consistent formulation of the fully coupled electromagnetic-mechanical problem. The first part of the work addresses the necking localization of a metal sheet during an electromagnetic process. A "weak band" model is used to predict the onset of necking of a thin sheet under plane stress and in-plane electric currents, an idealization of the local conditions in unconstrained electromagnetic loading. This work finds that EMF increases ductility over quasistatic techniques, due to the material's strain-rate sensitivity, and details how the material constitutive response and process characteristics affect ductility. The general theory is subsequently applied to freely expanding electromagnetically loaded aluminum tubes. Necking strains are measured in tubes of various geometries that are loaded by different forming coils and electric currents. The experimental results show reasonable agreement with the corresponding theoretical forming limit predictions, which indicate a two to three fold increase in the forming limits with respect to the quasistatic case. The second part of the work pertains to predictive modeling of EMF processes. Recent works do not consistently account for the coupling between electromagnetic fields and finite deformations. Typically, separate solutions to the electromagnetic and finite strain mechanical problems are combined in lock-step. The present work employs a fully coupled Lagrangian (reference configuration) least-action variational principle. This principle forms

关键词： Electromagnetic-thermal-Mechanical Processes Continuum Mechanics High Strain Rate Ductility numerical algorithms Variational Calculus Thesis

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Finite deformation pseudo-elasticity of shape memory alloys - Constitutive modelling and. finite element implementation

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INTERNATIONAL JOURNAL OF PLASTICITY 2008年第3期24卷 455-482页

作者： Reese, Stefanie Christ, Daniel Tech Univ Carolo Wilhelmina Braunschweig Inst Solid Mech D-38106 Braunschweig Germany

In this paper we suggest a new phenomenological material model for shape memory alloys. In contrast to many earlier concepts of this kind the present approach includes arbitrarily large deformations. The work is motivated by the requirement, also expressed by regulatory agencies, to carry out finite element simulations of NiTi stents. Depending on the quality of the numerical results it is possible to circumvent, at least partially, expensive experimental investigations. Stent structures are usually designed to significantly reduce their diameter during the insertion into a catheter. Thereby large rotations combined with moderate and large strains occur. In this process an agreement of numerical and experimental results is often hard to achieve. One of the reasons for this discrepancy is the use of unrealistic material models which mostly rely on the assumption of small strains. In the present paper we derive a new constitutive model which is no longer limited in this way. Further its efficient implementation into a finite element formulation is shown. One of the key issues in this regard is to fulfil "inelastic" incompressibility in each time increment. Here we suggest a new kind of exponential map where the exponential function is suitably computed by means of the spectral decomposition. A series expansion is completely avoided. Finite element simulations of stent structures show that the new concept is well appropriate to demanding finite element analyses as they occur in practically relevant problems. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： A. phase transformation B. constitutive behaviour finite strain C. finite elements numerical algorithms

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The connected-moments polynomial approach for Hamiltonian eigenvalues calculation and its application to the one-particle systems

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INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY 2008年第2期108卷 272-278页

作者： Bartashevich, Igor FESU Dept Phys St Petersburg 197378 Russia

The new connected-moments polynomial approach (CMP) is developed for evaluation of Hamiltonian eigenvalues. It is based on properties of specially designed polynomial and does not use any basis set and variational procedure. Like all the methods based on hamiltonain moments knowledge, the CMP is conceptually simple but is less tedious and is usually convergent even for very "crude" trial functions. This method is applicable not only to the ground state energy calculation but also to the excited states. The formalism is presented in two modifications: non-local (integral) and local (integral-free) ones. An accuracy of both versions is illustrated by numerical examples of Hamiltonian eigenvalues calculations for harmonic and anharmonic oscillators. (C) 2007 Wiley Periodicals, Inc.

关键词： connected-moments polynomial Hamiltonian moments one-dimensional Schrodinger equation numerical algorithms

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Converting to and from dilated integers

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IEEE TRANSACTIONS ON COMPUTERS 2008年第4期57卷 567-573页

作者： Raman, Rajeev Wise, David S. Dept Comp Sci Leicester LEQ 7RH Leics England Indiana Univ Dept Comp Sci Bloomington IN 47405 USA

Dilated integers form an ordered group of the Cartesian indices into a d-dimensional array represented in the Morton order. Efficient implementations of its operations can be found elsewhere. This paper offers efficient casting (type)conversions to and from an ordinary integer representation. As the Morton order representation for 2D and 3D arrays attracts more users because of its excellent block locality, the efficiency of these conversions becomes important. They are essential for programmers who would use Cartesian indexing there. Two algorithms for each casting conversion are presented here, including to-and-from dilated integers for both d = 2 and d = 3. They fall into two families. One family uses newly compact table lookup, so the cache capacity is better preserved, The other generalizes better to all d, using processor-local arithmetic that is newly presented as abstract d-ary and (d-1)-ary recurrences. Test results for two and three dimensions generally favor the former.

关键词： analysis programming techniques numerical algorithms design performance memory hierarchy caches paging compilers classes quadtrees octrees dilation contraction.

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Stress field tomography based on 3D photoelasticity

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JOURNAL OF THE MECHANICS AND PHYSICS OF SOLIDS 2008年第3期56卷 1065-1085页

作者： Wijerathne, M. L. L. Oguni, Kenji Hori, Muneo Univ Tokyo Earthquake Res Inst Bunkyo Ku Tokyo 11130032 Japan Japan Agcy Marine Earth Sci & Technol Inst Res Earth Evolut Kanazawa Ku Yokohama Kanagawa 2360001 Japan

tomographic method for identification of stress fields based on 3D photoelasticity has been developed. A second order tensor field tomographic method based on the general inverse problem of 3D photoelasticity, previously developed by the authors, is found to be highly sensitive to errors in photoelastic observations. In this study a new tomographic method for stress field with fairly high robustness to errors in photoelastic observations has been developed by introducing both equilibrium condition and linear elasticity to the previously developed general tensor field tomographic method. This new stress field tomographic method expands unknown 3D stress distributions as a linear combination of independent set of basis functions and a new inverse problem is posed: identify the amplitudes of basis functions based on photoelastic observations. Just as the inverse problem of 3D photoelasticity, this newly posed inverse problem is also nonlinear and ill posed. Unlike conventional approaches to 3D photoelasticity, both these nonlinearity and ill-posedness are properly treated using a load incremental approach. Load incremental approach chops the nonlinear solution space into segments with unique Solutions by conducting photoelastic observations at sufficiently small increments in external load. Validating both numerically and experimentally, it is shown that this new stress field tomographic method has sufficient robustness against errors in photoelastic observations and is applicable to experimental stress measurements. (C) 2007 Elsevier Ltd. All rights reserved.

关键词： 3D stress field tomography nondestructive evaluation mechanical testing numerical algorithms optical interferometry

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