967 research outputs found

    A Newton Collocation Method for Solving Dynamic Bargaining Games

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    We develop and implement a collocation method to solve for an equilibrium in the dynamic legislative bargaining game of Duggan and Kalandrakis (2008). We formulate the collocation equations in a quasi-discrete version of the model, and we show that the collocation equations are locally Lipchitz continuous and directionally differentiable. In numerical experiments, we successfully implement a globally convergent variant of Broyden's method on a preconditioned version of the collocation equations, and the method economizes on computation cost by more than 50% compared to the value iteration method. We rely on a continuity property of the equilibrium set to obtain increasingly precise approximations of solutions to the continuum model. We showcase these techniques with an illustration of the dynamic core convergence theorem of Duggan and Kalandrakis (2008) in a nine-player, two-dimensional model with negative quadratic preferences.

    Modifications of the Limited Memory BFGS Algorithm for Large-scale Nonlinear Optimization

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    In this paper we present two new numerical methods for unconstrained large-scale optimization. These methods apply update formulae, which are derived by considering different techniques of approximating the objective function. Theoretical analysis is given to show the advantages of using these update formulae. It is observed that these update formulae can be employed within the framework of limited memory strategy with only a modest increase in the linear algebra cost. Comparative results with limited memory BFGS (L-BFGS) method are presented.</p

    Minimal norm constrained interpolation

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    In computational fluid dynamics and in CAD/CAM, a physical boundary is usually known only discreetly and most often must be approximated. An acceptable approximation preserves the salient features of the data such as convexity and concavity. In this dissertation, a smooth interpolant which is locally concave where the data are concave and is locally convex where the data are convex is described. The interpolant is found by posing and solving a minimization problem whose solution is a piecewise cubic polynomial. The problem is solved indirectly by using the Peano Kernal theorem to recast it into an equivalent minimization problem having the second derivative of the interpolant as the solution. This approach leads to the solution of a nonlinear system of equations. It is shown that Newton's method is an exceptionally attractive and efficient method for solving the nonlinear system of equations. Examples of shape-preserving interpolants, as well as convergence results obtained by using Newton's method are also shown. A FORTRAN program to compute these interpolants is listed. The problem of computing the interpolant of minimal norm from a convex cone in a normal dual space is also discussed. An extension of de Boor's work on minimal norm unconstrained interpolation is presented

    Minimal Norm Constrained Interpolation

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    In computational fluid dynamics and in CAD/CAM a physical boundary, usually known only discreetly (say, from a set of measurements), must often be approximated. An acceptable approximation must, of course, preserve the salient features of the data (convexity, concavity, etc.) In this dissertation we compute a smooth interpolant which is locally convex where the data are locally convex and is locally concave where the data are locally concave. Such an interpolant is found by posing and solving a minimization problem. The solution is a piecewise cubic polynomial. We actually solve this problem indirectly by using the Peano kernel theorem to recast this problem into an equivalent minimization problem having the second derivative of the interpolant as the solution. We are then led to solve a nonlinear system of equations. We show that with Newton\u27s method we have an exceptionally attractive and efficient method for solving this nonlinear system of equations. We display examples of such interpolants as well as convergence results obtained by using Newton\u27s method. We list a FORTRAN program to compute these shape-preserving interpolants. Next we consider the problem of computing the interpolant of minimal norm from a convex cone in a normed dual space. This is an extension of de Boor\u27s work on minimal norm unconstrained interpolation

    Domain decomposition methods for the parallel computation of reacting flows

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    Domain decomposition is a natural route to parallel computing for partial differential equation solvers. Subdomains of which the original domain of definition is comprised are assigned to independent processors at the price of periodic coordination between processors to compute global parameters and maintain the requisite degree of continuity of the solution at the subdomain interfaces. In the domain-decomposed solution of steady multidimensional systems of PDEs by finite difference methods using a pseudo-transient version of Newton iteration, the only portion of the computation which generally stands in the way of efficient parallelization is the solution of the large, sparse linear systems arising at each Newton step. For some Jacobian matrices drawn from an actual two-dimensional reacting flow problem, comparisons are made between relaxation-based linear solvers and also preconditioned iterative methods of Conjugate Gradient and Chebyshev type, focusing attention on both iteration count and global inner product count. The generalized minimum residual method with block-ILU preconditioning is judged the best serial method among those considered, and parallel numerical experiments on the Encore Multimax demonstrate for it approximately 10-fold speedup on 16 processors

    A derivative-free bracketing scheme for univariate minimization

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    AbstractA derivative-free scheme for univariate minimization is developed. This scheme has a quadratic convergence rate and requires two function evaluations each iteration

    A multigrid approach for the solution of the 2D semiconductor equations

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    Historical development of the BFGS secant method and its characterization properties

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    The BFGS secant method is the preferred secant method for finite-dimensional unconstrained optimization. The first part of this research consists of recounting the historical development of secant methods in general and the BFGS secant method in particular. Many people believe that the secant method arose from Newton's method using finite difference approximations to the derivative. We compile historical evidence revealing that a special case of the secant method predated Newton's method by more than 3000 years. We trace the evolution of secant methods from 18th-century B.C. Babylonian clay tablets and the Egyptian Rhind Papyrus. Modifications to Newton's method yielding secant methods are discussed and methods we believe influenced and led to the construction of the BFGS secant method are explored. In the second part of our research, we examine the construction of several rank-two secant update classes that had not received much recognition in the literature. Our study of the underlying mathematical principles and characterizations inherent in the updates classes led to theorems and their proofs concerning secant updates. One class of symmetric rank-two updates that we investigate is the Dennis class. We demonstrate how it can be derived from the general rank-one update formula in a purely algebraic manner not utilizing Powell's method of iterated projections as Dennis did it. The literature abounds with update classes; we show how some are related and show containment when possible. We derive the general formula that could be used to represent all symmetric rank-two secant updates. From this, particular parameter choices yielding well-known updates and update classes are presented. We include two derivations of the Davidon class and prove that it is a maximal class. We detail known characterization properties of the BFGS secant method and describe new characterizations of several secant update classes known to contain the BFGS update. Included is a formal proof of the conjecture made by Schnabel in his 1977 Ph.D. thesis that the BFGS update is in some asymptotic sense the average of the DFP update and the Greenstadt update
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