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Continuous Problem (continuous + problem)
Selected AbstractsA dynamic approach for evaluating parameters in a numerical methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 1 2005A. A. Oberai Abstract A new methodology for evaluating unknown parameters in a numerical method for solving a partial differential equation is developed. The main result is the identification of a functional form for the parameters which is derived by requiring the numerical method to yield ,optimal' solutions over a set of finite-dimensional function spaces. The functional depends upon the numerical solution, the forcing function, the set of function spaces, and the definition of the optimal solution. It does not require exact or approximate analytical solutions of the continuous problem, and is derived from an extension of the variational Germano identity. This methodology is applied to the one-dimensional, linear advection,diffusion problem to yield a non-linear dynamic diffusivity method. It is found that this method yields results that are commensurate to the SUPG method. The same methodology is then used to evaluate the Smagorinsky eddy viscosity for the large eddy simulation of the decay of homogeneous isotropic turbulence in three dimensions. In this case the resulting method is found to be more accurate than the constant-coefficient and the traditional dynamic versions of the Smagorinsky model. Copyright © 2004 John Wiley & Sons, Ltd. [source] Solving singularly perturbed advection,reaction equations via non-standard finite difference methodsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 14 2007Jean M.-S. Abstract We design and implement two non-standard finite difference methods (NSFDMs) to solve singularly perturbed advection,reaction equations (SPARE). Our methods constitute a big plus to the class of those ,rare' fitted operator methods, which can be extended to singularly perturbed partial differential equations. Unlike the standard finite difference methods (SFDMs), the NSFDMs designed in this paper allow the time and the space step sizes to vary independently of one another and of the parameter , in the SPARE under consideration. The NSFDMs replicate the linear stability properties of the fixed points of the continuous problem. Furthermore, these methods preserve the positivity and boundedness properties of the exact solution. Numerical simulations that confirm the theoretical results are presented. Copyright © 2007 John Wiley & Sons, Ltd. [source] On the convergence of the finite integration technique for the anisotropic boundary value problem of magnetic tomographyMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 9 2003Roland Potthast The reconstruction of a current distribution from measurements of the magnetic field is an important problem of current research in inverse problems. Here, we study an appropriate solution to the forward problem, i.e. the calculation of a current distribution given some resistance or conductivity distribution, respectively, and prescribed boundary currents. We briefly describe the well-known solution of the continuous problem, then employ the finite integration technique as developed by Weiland et al. since 1977 for the solution of the problem. Since this method can be physically realized it offers the possibility to develop special tests in the area of inverse problems. Our main point is to provide a new and rigorous study of convergence for the boundary value problem under consideration. In particular, we will show how the arguments which are used in the proof of the continuous case can be carried over to study the finite-dimensional numerical scheme. Finally, we will describe a program package which has been developed for the numerical implementation of the scheme using Matlab. Copyright © 2003 John Wiley & Sons, Ltd. [source] A specially structured nonlinear integer resource allocation problemNAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 7 2003Kurt M. Bretthauer Abstract We present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush-Kuhn-Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770,792, 2003. [source] Numerical analysis of the stochastic Stokes equations of Wick typeNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2007H. Manouzi Abstract We propose a finite element method for the numerical solution of the stochastic Stokes equations of the Wick type. We give existence and uniqueness results for the continuous problem and its approximation. Optimal error estimates are derived and algorithmic aspects of the method are discussed. Our method will reduce the problem of solving stochastic Stokes equations to solving a set of deterministic ones. Moreover, one can reconstruct particular realizations of the solution directly from Wiener chaos expansions once the coefficients are available. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] Remarks on Duality in Graph Spaces of First-Order Linear OperatorsPROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2006Max JensenArticle first published online: 4 DEC 200 Graph spaces provide a setting alternative to Sobolev spaces and BV spaces, which is suitable for the analysis of first-order linear boundary value problems such as Friedrichs systems. Besides investigations of the well-posedness of the continuous problem there is also an increasing interest in the error analysis of finite element methods within a graph space framework. In this text we elucidate various methods for an explicit representation of dual spaces of graph spaces. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] A specially structured nonlinear integer resource allocation problemNAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 7 2003Kurt M. Bretthauer Abstract We present an algorithm for solving a specially structured nonlinear integer resource allocation problem. This problem was motivated by a capacity planning study done at a large Health Maintenance Organization in Texas. Specifically, we focus on a class of nonlinear resource allocation problems that involve the minimization of a convex function over one general convex constraint, a set of block diagonal convex constraints, and bounds on the integer variables. The continuous variable problem is also considered. The continuous problem is solved by taking advantage of the structure of the Karush-Kuhn-Tucker (KKT) conditions. This method for solving the continuous problem is then incorporated in a branch and bound algorithm to solve the integer problem. Various reoptimization results, multiplier bounding results, and heuristics are used to improve the efficiency of the algorithms. We show how the algorithms can be extended to obtain a globally optimal solution to the nonconvex version of the problem. We further show that the methods can be applied to problems in production planning and financial optimization. Extensive computational testing of the algorithms is reported for a variety of applications on continuous problems with up to 1,000,000 variables and integer problems with up to 1000 variables. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 770,792, 2003. [source] | |||||||||||||