Numerical experiments with MG continuation algorithms

This paper presents a systematic investigation of the numerical continuation algorithms for bifurcation problems (simple turning points and Hopf bifurcation points) of 2D nonlinear elliptic equations. The continuation algorithms employed are based only on iterative methods (Preconditioned Generalized Conjugate Gradient, PGCG, and Multigrid, MG). PGCG is mainly used as coarse grid solver in the MG cycle. Numerical experiments were made with the MG continuation algorithms developed by Hackbusch [W. Hackbusch, Multi-Grid Solution of Continuation Problems, Lecture Notes in Math., vol. 953, Springer, Berlin, 1982], Meis et al. [T.F. Meiss, H. Lehman, H. Michael, Application of the Multigrid Method to a Nonlinear Indefinite Problem, Lecture Notes in Math., vol. 960, Springer, Berlin, 1982], and Mittelmann and Weber [H.D. Mittelmann, H. Weber, Multi-grid solution of bifurcation problems, SIAM J. Sci. Statist. Comput. 6 (1985) 49]. The mathematical models selected, as test problems, are well-known diffusion–reaction systems; non-isothermal catalyst pellet and Lengyel–Epstein model of the CIMA reaction. The numerical methods proved to be efficient and reliable so that computations with fine grids can easily be performed.     

Numerical experiments with various optimal estimators

Numerical experiments are performed to measure the relative accuracy and computational efficiency of various estimators for the parameters and state of a linear dynamic system with forcing using a finite sequence of measurements containing noise. This nonlinear estimation problem is treated by estimators based on least-squares and maximumlikelihood criteria and, for the linearized problem, two mechanizations of the Kalman-Bucy estimator are applied. A digital computer simulation of an example problem is performed, and a Monte Carlo technique is used to generate statistics of the errors in the estimates empirically. This process is carried out for a range ofa priori error statistics.This work was supported by the National Science Foundation, Systems Grant No. GU-1153.     

Numerical experiments with a multistep Radau method

This paper describes an implementation of multistep collocation methods, which are applicable to stiff differential problems, singular perturbation problems, and D.A.E.s of index 1 and 2.These methods generalize one-step implicit Runge-Kutta methods as well as multistep one-stage BDF methods. We give numerical comparisons of our code with two representative codes for these methods, RADAU5 and LSODE.     

Numerical experiments with two approximate inverse preconditioners

We present the results of numerical experiments aimed at comparing two recently proposed sparse approximate inverse preconditioners from the point of view of robustness, cost, and effectiveness. Results for a standard ILU preconditioner are also included. The numerical experiments were carried out on a Cray C98 vector processor. This work was partially supported by the GA AS CR under grant 2030706 and by the grant GA CR 205/96/0921.     

Numerical experiments with an inexact Jacobian trust-region algorithm

We present the adaptation and implementation of a composite-step trust region algorithm, developed in (Walther, SIAM J. Optim. 19(1):307–325, 2008), that incorporates the approximation of the Jacobian of the equality constraints with a specialized quasi-Newton method. The forming and/or factoring of the exact Jacobian in each optimization step is avoided. Hence, the presented approach is especially well suited for equality constrained optimization problems where the Jacobian of the constraints is dense.     

Numerical experiments in semi-infinite programming

A quasi-Newton algorithm for semi-infinite programming using an L exact penalty function is described, and numerical results are presented. Comparisons with three Newton algorithms and one other quasi-Newton algorithm show that the algorithm is very promising in practice.     

Numerical experiments with LP formulations of the maximum clique problem

Central European Journal of Operations Research - The maximum clique problems calls for determining the size of the largest clique in a given graph. This graph problem affords a number of zero-one...     

Numerical experiments with the inhomogeneity indicator in positron emission tomography

By means of numerical methods, the question is studied of applicability of the inhomogeneity indicator in positron emission tomography. The signal registered by the tomograph is described in terms of an imitation model using the Monte Carlo method. The possibility is demonstrated of the effective use of the inhomogeneity indicator for solving the problem under consideration. Some numerical results are presented in graphical form for reconstructing the boundaries of unknown activity sources.     

Divergent fourier series: Numerical experiments

An open problem in the theory of Fourier series is whether there are functions f L ₁ such that the partial sums S _n(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly bad functions Kahane proved that the rate of divergence is faster than o(log log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums S _n(f, x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense optimal, and conclude that, under this assumption, ...(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for L ₁ functions.     

Numerical experiments with hemodynamics of the arterial part of the myocardium

A hydraulic model of blood supply to the arterial part of the myocardium is considered. The model is described by a system of linear algebraic equations whose matrix is determined by the geometry and the size of the blood vessels in the circulation system. Simulation results are reported for blood flow supplying different parts of the myocardium. Translated from Obratnye Zadachi Estestvoznaniya, Published by Moscow University, Moscow, 1997, pp. 159–168.     

Numerical experiments with universal barrier functions for cones of Chebyshev systems

Based on previous explicit computations of universal barrier functions, we describe numerical experiments for solving certain classes of convex optimization problems. The comparison is given of the performance of the classical affine-scaling algorithm with the similar algorithm built upon the universal barrier function.     

Numerical experiments with a method of successive approximation for conformal mapping

A method of successive approximation for the conformal mapping of simply connected regions onto the unit disc and numerical experiments with it are described. The experiments gave unexpectedly good results.

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Zusammenfassung Es werden ein Schmiegungsverfahren für die konforme Abbildung einfach zusammenhängender Gebiete auf den Einheitskreis und numerische Experimente damit beschrieben. Letztere ergaben unerwartet gute Resultate.

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Born December 6, 1942; killed August 31, 1979, while climbing in the mountains of Canada.     

Regularities in data from factorial experiments

This article documents a meta‐analysis of 113 data sets from published factorial experiments. The study quantifies regularities observed among factor effects and multifactor interactions. Such regularities are known to be critical to efficient planning and analysis of experiments and to robust design of engineering systems. Three previously observed properties are analyzed: effect sparsity, hierarchy, and heredity. A new regularity is introduced and shown to be statistically significant. It is shown that a preponderance of active two‐factor interaction effects are synergistic, meaning that when main effects are used to increase the system response, the interaction provides an additional increase and that when main effects are used to decrease the response, the interactions generally counteract the main effects. © 2006 Wiley Periodicals, Inc. Complexity 11: 32–45, 2006     

Numerical experiments with variations of the Gauss-Newton algorithm for nonlinear least squares

In this paper, the classical Gauss-Newton method for the unconstrained least squares problem is modified by introducing a quasi-Newton approximation to the second-order term of the Hessian. Various quasi-Newton formulas are considered, and numerical experiments show that most of them are more efficient on large residual problems than the Gauss-Newton method and a general purpose minimization algorithm based upon the BFGS formula. A particular quasi-Newton formula is shown numerically to be superior. Further improvements are obtained by using a line search that exploits the special form of the function.     

Numerical experiments with ABS algorithms for linear systems on a parallel machine

Numerical results are obtained on sequential and parallel versions of ABS algorithms for linear systems for both full matrices andq-band matrices. The results using the sequential algorithm on full matrices indicate the superiority of a particular implementation of the symmetric algorithm. The condensed form of the algorithm is well suited for implementation in a parallel environment, and results obtained on the IBM 4381 system favor a synchronous implementation over the asynchronous one. Results are obtained from sequential implementations of theLU, Cholesky, and symmetric algorithms of the ABS class forq-band matrices able to reduce memory storage. A simple parallelization of do-loops for calculating components gives interesting performances.This work has been developed in the framework of a collaboration between IBM-ECSEC, Rome, Italy, and the Department of Mathematics of the University of Bergamo, Bergamo, Italy.The author is grateful to Prof. J. Abaffy (University of Economics, Budapest), Prof. L. Dixon (Hatfield Polytechnic), and Prof. E. Spedicato (Department of Mathematics, University of Bergamo) for useful suggestions.     

Numerical experiments in problems of asymptotic representation theory

The article presents the results of numerical computations of statistics related to Young diagrams, including estimates on the maximum and average (with respect to the Plancherel distribution) dimension of irreducible representations of the symmetric group S _n . The computed limit shapes of two-dimensional and three-dimensional diagrams distributed according to the Richardson statistics are also presented. Bibliography: 14 titles.     

Numerical experiments on a flexible variant of GMRES-DR

The Flexible GMRES (FGMRES [1]) and the GMRES with deflated restarting (GMRES-DR [2]) methods are two algorithms derived from GMRES [3], that are considered as powerful when solving large non hermitian systems of linear equations. GMRES-DR is a variant of GMRES with an improved restarting technique that maintains in the Krylov subspace harmonic Ritz vector from the previous restart. In situations where the convergence of restarted GMRES is slow and where the matrix has few eigenvalues close to the origin, this technique has proved very efficient. The new method that we propose is the Flexible GMRES with deflated restarting (FGMRES-DR [6]), which combines the two above mentioned algorithms in order to yield better performance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical experiments on dual matrix algorithms for function minimization

In Ref. 2, four algorithms of dual matrices for function minimization were introduced. These algorithms are characterized by the simultaneous use of two matrices and by the property that the one-dimensional search for the optimal stepsize is not needed for convergence. For a quadratic function, these algorithms lead to the solution in at mostn+1 iterations, wheren is the number of variables in the function. Since the one-dimensional search is not needed, the total number of gradient evaluations for convergence is at mostn+2. In this paper, the above-mentioned algorithms are tested numerically by using five nonquadratic functions. In order to investigate the effects of the stepsize on the performances of these algorithms, four schemes for the stepsize factor are employed, two corresponding to small-step processes and two corresponding to large-step processes. The numerical results show that, in spite of the wide range employed in the choice of the stepsize factor, all algorithms exhibit satisfactory convergence properties and compare favorably with the corresponding quadratically convergent algorithms using one-dimensional searches for optimal stepsizes.     

Numerical experiments on flow control of near wall turbulence

In this paper we demonstrate by means of direct numerical simulations (DNS) of channel flow with surface structuring or modified boundary conditions how turbulence can be controlled by influencing the anisotropy state of near wall turbulence. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)