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Application of the Fast Automatic Differentiation Technique for Solving Inverse Coefficient Problems

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2020年第1期60卷 15-25页

作者： Albu, A. F. Evtushenko, Yu G. Zubov, V., I Russian Acad Sci Fed Res Ctr Comp Sci & Control Dorodnicyn Comp Ctr Moscow 119333 Russia Moscow Inst Phys & Technol Dolgoprudnyi 141701 Moscow Oblast Russia

Results obtained by the authors in solving inverse coefficient problems are overviewed. The inverse problem under consideration is to determine a temperature-dependent thermal conductivity coefficient from experimental observations of the temperature field in the studied substance and (or) the heat flux on the surface of the object. The study is based on the Dirichlet boundary value problem for the nonstationary heat equation stated in the general -dimensional formulation. For this general case, an analytical expression for the cost functional gradient is obtained. The features of solving the inverse problem and the difficulties encountered in the solution process are discussed.

关键词： heat conduction inverse coefficient problems gradient heat equation numerical algorithm

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HITTING TIME FOR BESSEL PROCESSES-WALK ON MOVING SPHERES algorithm (WOMS)

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ANNALS OF APPLIED PROBABILITY 2013年第6期23卷 2259-2289页

作者： Deaconu, Madalina Herrmann, Samuel Univ Lorraine INRIA Nancy Grand Est IECN UMR 7502 F-54506 Vandoeuvre Les Nancy France Univ Lorraine INRIA Nancy Grand Est Project Team TOSCA F-54506 Vandoeuvre Les Nancy France Univ Bourgogne IMB UMR 5584 F-21078 Dijon France

In this article we investigate the hitting time of some given boundaries for Bessel processes. The main motivation comes from mathematical finance when dealing with volatility models, but the results can also be used in optimal control problems. The aim here is to construct a new and efficient algorithm in order to approach this hitting time. As an application we will consider the hitting time of a given level for the Cox-Ingersoll-Ross process. The main tools we use are on one side, an adaptation of the method of images to this particular situation and on the other side, the connection that exists between Cox-Ingersoll-Ross processes and Bessel processes.

关键词： Bessel processes Cox-Ingersoll-Ross processes hitting time method of images numerical algorithm

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Walk On Spheres algorithm for Helmholtz and Yukawa Equations via Duffin Correspondence

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METHODOLOGY AND COMPUTING IN APPLIED PROBABILITY 2017年第2期19卷 589-602页

作者： Yang, Xuxin Rasila, Antti Sottinen, Tommi Hunan First Normal Univ Dept Math Changsha 410205 Hunan Peoples R China Aalto Univ Dept Math & Syst Anal POB 11100 Espoo 00076 Finland Univ Vaasa Dept Math & Stat Fac Technol POB 700 Vaasa 65101 Finland

We show that a constant-potential time-independent Schrodinger equation with Dirichlet boundary data can be reformulated as a Laplace equation with Dirichlet boundary data. With this reformulation, which we call the Duffin correspondence, we provide a classical Walk On Spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the boundary value problem. We compare the obtained Duffin WOS algorithm with existing modified WOS algorithms.

关键词： Brownian motion Helmholtz equation Linearized Poisson-Boltzmann equation Monte Carlo simulation numerical algorithm Walk On Spheres algorithm Yukawa equation

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Systems of quasilinear conservation laws and algorithmization of variational principles

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COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS 2015年第9期55卷 1554-1566页

作者： Rykov, Yu. G. Feodoritova, O. B. Russian Acad Sci Keldysh Inst Appl Math Moscow 125047 Russia

A previously formulated new approach to the consideration of systems of quasilinear hyperbolic equations on the basis of variational principles is described in more detail in the case of special systems of three equations. It is shown that each field of characteristics can be represented as a solution of a variational problem. Moreover, the Rankine-Hugoniot relations at the corner points of the characteristics or at the intersections of the characteristics of a single family hold automatically. In the simplest case of the Hopf equation, a numerical algorithm is constructed on the basis of a variational principle.

关键词： hyperbolic system gas dynamics equations characteristics variational problem discontinuous solutions Rankine-Hugoniot relations numerical algorithm

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An improved technique to determine the controlling unstable equilibrium point in a power system

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I-FUNDAMENTAL THEORY AND APPLICATIONS 1996年第4期43卷 313-323页

作者： Treinen, RT Vittal, V Kliemann, W IOWA STATE UNIV SCI & TECHNOL DEPT MATHAMESIA 50011

The accuracy of stability assessment provided by the transient energy function (TEF) method depends on the determination of the controlling unstable equilibrium point (UEP), The technique that currently determines the controlling UEP in the TEF method is based on the so-called exit point method and has also been recently labeled the BCU method, The exit point method consists of two basic steps, First, the exit point is approximated by the point theta(egsa), where the first maximum of the potential energy along the fault-on trajectory is encountered, Second, the minimum gradient point theta(mgp) along the trajectory from theta(egsa) is computed, The controlling UEP is then obtained by solving a system of nonlinear algebraic equations with theta(mgp) as an initial guess, It has been observed that this method lacks robustness in the sense that the following two problems may occur, 1) There may be no detection of the minimum gradient point theta(mgp) and hence, no determination of the controlling UEP, 2) if theta(mgp) is found, then based on the definition of the controlling UEP, it may not be in the domain of convergence of the controlling UEP for the particular solving algorithm used, Hence, another equilibrium point, possibly a stable equilibrium paint, not the controlling UEP will be located, This results in a flawed transient stability assessment, The result of this research has been the development of a new numerical technique for determining the controlling UEP, With an initial starting point that is close to the exit point this technique efficiently produces a sequence of points, An analytical foundation for this method is given which shows that under certain assumptions this sequence will converge to the controlling UEP, Hence this new technique exhibits a substantial improvement over the exit point method because of the following reasons: (1) the technique does not attempt to detect the point theta(mgp), (2) the technique can produce a point that is close to t

关键词： power system analysis computing power system stability power system transients BCU method IEEE 50-generator controlling unstable equilibrium point convergence exit point minimum gradient point nonlinear algebraic equations numerical algorithm power syste differential algebraic equations Power system analysis computing Power system stability Riccati equations exit point power systems power system transients Converge

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Application of Residual Power Series Method for the Solution of Time-fractional Schrodinger Equations in One-dimensional Space

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FUNDAMENTA INFORMATICAE 2019年第2期166卷 87-110页

作者： Abu Arqub, Omar Univ Jordan Fac Sci Dept Math Amman 11942 Jordan

The object of this article is to present the computational solution of the time-fractional Schrodinger equation subject to given constraint condition based on the generalized Taylor series formula in the Caputo sense. The algorithm methodology is based on construct a multiple fractional power series solution in the form of a rabidly convergent series with minimum size of calculations using symbolic computation software. The proposed technique is fully compatible with the complexity of this problem and obtained results are highly encouraging. Efficacious computational experiments are provided to guarantee the procedure and to illustrate the theoretical statements of the present algorithm in order to show its potentiality, generality, and superiority for solving such fractional equation. Graphical results and numerical comparisons are presented and discussed quantitatively to illustrate the solution.

关键词： Fractional Schrodinger equation Multiple fractional power series Residual power series method numerical algorithm Symbolic computations

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An algorithm for maximizing a convex function over a simple set

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JOURNAL OF GLOBAL OPTIMIZATION 1996年第4期8卷 379-391页

作者： Enhbat, R MONGOLIAN NATL UNIV DEPT MATH ULAANBAATAR MONGOLIA

The problem of maximizing a convex function on a so-called simple set is considered. Based on the optimality conditions [19], an algorithm for solving the problem is proposed. This numerical algorithm is shown to be c... 详细信息

关键词： global optimization optimality condition convex function numerical algorithm simple set

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Short Communication: Optimal Insurance to Maximize Exponential Utility When Premium Is Computed by a Convex Functional

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SIAM JOURNAL ON FINANCIAL MATHEMATICS 2024年第1期15卷 SC15-SC27页

作者： Caot, Jingyi Li, Dongchen Young, Virginia R. Zou, Bin York Univ Dept Math & Stat Toronto ON M3J 1P3 Canada Univ Michigan Dept Math Ann Arbor MI 48109 USA Univ Connecticut Dept Math Storrs CT 06269 USA

We find the optimal indemnity to maximize the expected utility of terminal wealth of a buyer of insurance whose preferences are modeled by an exponential utility. The insurance premium is computed by a convex functional. We obtain a necessary condition for the optimal indemnity;then, because the candidate optimal indemnity is given implicitly, we use that necessary condition to develop a numerical algorithm to compute it. We prove that the numerical algorithm converges to a unique indemnity that, indeed, equals the optimal policy. We also illustrate our results with numerical examples.

关键词： optimal insurance numerical algorithm convex premium functional

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numerical Simulation of Vibration of Composite Pipelines Conveying Pulsating Fluid

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INTERNATIONAL JOURNAL OF APPLIED MECHANICS 2019年第9期11卷

作者： Khudayarov, B. A. Komilova, Kh M. Turaev, F. Zh Tashkent Inst Irrigat & Agr Mechanizat Engineers Dept Higher Math Tashkent 100000 Uzbekistan

Vibration problems of pipelines made of composite materials conveying pulsating flow of gas and fluid are investigated in the paper. A dynamic model of motion of pipelines conveying pulsating fluid flow supported by a Hetenyi's base is developed taking into account the viscosity properties of the structure material, axial forces, internal pressure and Winkler's viscoelastic base. To describe the processes of viscoelastic material strain, the Boltzmann-Volterra integral model with weakly singular hereditary kernels is used. Using the Bubnov-Galerkin method, the problem is reduced to the study of a system of ordinary integro-differential equations (IDE). A computational algorithm is developed based on the elimination of the features of IDE with weakly singular kernels, followed by the use of quadrature formulas. The effect of rheological parameters of the pipeline material, flow rate and base parameters on the vibration of a viscoelastic pipeline conveying pulsating fluid is analyzed. The convergence analysis of the approximate solution of the Bubnov-Galerkin method is carried out. It was revealed that the viscosity parameters of the material and the pipeline base lead to a significant change in the critical flow rate. It was stated that an increase in excitation coefficient of pulsating flow and the parameter of internal pressure leads to a decrease in the critical flow rate. It is shown that an increase in the singularity parameter, the Winkler base parameter, the rigidity parameter of the continuous base layer and the Reynolds number increases the critical flow rate.

关键词： Mathematical model viscoelasticity pipeline numerical algorithm vibration fluid

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Analysis of pandemic with game methodology and numerical approximation

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ARCHIVES OF CONTROL SCIENCES 2023年第3期33卷 651-680页

作者： Matusik, Radoslaw Nowakowski, Andrzej Univ Lodz Fac Math & Comp Sci Banacha 22 Lodz Poland

We build a mathematical game model of pandemic transmission, including vaccinations of population and budget costs of different acting to eliminate pandemic. We assume the interactions among different groups: vaccinated, susceptible, exposed, infectious, super-spreaders, hospitalized and fatality, defining a system of ordinary differential equations, which describes compartment model of disease and costs of the treatment. The goal of the game is to describe the development disease under different types of treatment, but including costs of them and social restrictions, during the shortest time period. To this effect we construct a dual dynamic programming method to describe open-loop Nash equilibrium for treatment, a group of people having antibodies and budget costs. Next, we calculate numerically an approximate open-loop Nash

关键词： COVID-19 game model of pandemic approximate dual dynamic program-ming sufficient approximate optimality conditions for Nash equilibrium numerical algorithm

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