Parallel Computing Toolkit and Numerical Experiments

Petr Girg

University of West Bohemia, Pilsen, Czech Republic

Abstract

Numerous phenomena in physics, chemistry, biology, and economy can be modelled by boundary-value problems for differential equations. These boundary-value problems (BVPs) can often be written in an operator form

L (u) + H (u,  p) = f ,

with L being a  (linear or nonlinear) mapping between some function spaces X and Y, H:X→ Y being an nonlinear operator,  f∈ Y being fixed, and p∈ (ℝ^n) being parameters. Solvability, multiplicity, and bifurcation of solutions are of particular interest for people from praxis. Note that the BVPs depending on parameters describe such important phenomena as resonance, in which the real systems usually undergo drastic changes leading to their collapse (suitable small perturbations in external forcings effect in big changes in their corresponding responses).

With more general nonlinearities involved in the equations, one loses any idea what can be proved or disproved. For that reason, numerical experiments become an indispensable tool of nonlinear analysis nowadays in the following way. In order to get at least a rough idea about the qualitative behavior, one performs a set of numerical experiments by discretizing over the domain of parameters. This approach usually leads to an enormous amount of BVPs. To solve such a number of BVPs would hardly be possible without parallelization. The parallelization with respect to the discretized set of parameters can be considered as a pure data-parallel approach (same code, different data). Parallel Computing Toolkit (PCT for short) brings data-parallel methods into Mathematica efficiently.

In our talk we will concentrate on the use of Parallel Computing Toolkit for performing large-scale numerical experiments. Especially, we will emphasize the possibility of writing parallel programs that can combine numerical and symbolical methods making PCT to be one of the most powerful tools in the market at this time. We will also discuss PCT-based applications running on various platforms in parallel (platforms on nodes-in-use may differ). This property makes PCT a perfect tool for a distributed computing approach if many Mathematica kernels are available (e.g. in campuses with unlimited licence where, for instance, MS Windows, Linux, and Unix versions of Mathematica would be available at the same time).

Numerical experiments referred in this talk have already resulted in discoveries of new phenomena [2,3,7,8,10], or have lead to substantial improvements of previous results [5,6,10]. Some other theoretical results (obtained with the help of numerical experiments) are ready and/or close to being submitted for publication [1,11]. New very interesting open problems were formulated thanks to the numerical experiments [4,9].

At the end of our talk, we would like to introduce our project on computing cluster with a gridMathematica-based computational environment and webMathematica-based interface (a joint project of  Czech Technical University, Prague; University of West Bohemia, Pilsen; and the Czech Academy of Sciences).

References

[1] Benedikt, J., Girg, P., Takáč, P., The Fredholm Alternative for the p-Laplacian at higher eigenvalues in one dimension, in Preparation, 2004. 
[2] Čepička, J., "Numerical Experiments for Nonlinear Problems'',  Ph.D. Thesis, University of West Bohemia, Pilsen, 2001, pp. 97. 
[3] Čepička, J.,  Drábek, P., Girg, P., Quasilinear Boundary Value Problems: Existence and Multiplicity Results, in 
"Variational Methods: Open Problems, Recent Progress, and Numerical Algorithms", John M. Neuberger, Northern Arizona University, Editor, Contemporary Mathematics Series, AMS, 2004, 285 pp., Softcover, ISBN 0-8218-3339-1.
[4] Čepička, J., Drábek, P., Girg, P., Open problems related to the p-Laplacian operator, to appear in  
Bol. Soc. Esp. Mat. Apl. , accepted (2003). 
[5] Del Toro, P., Girg, P.,  Roca, F., Dirichlet Problem with Nonlinearity Depending only on the Derivative,  Appl. Math. Lett. 16(2003), pp. 7-12. 
[6] Drábek, P., Girg, P.,  Roca, F., On the Range Properties of Certain Landesman-Lazer-type Problem, J. Math. Anal. Appl.  257(2001), pp. 131-140.  
[7] Drábek, P.,  Girg, P., Takáč, P., Ulm, M., The Fredholm alternative for the p-Laplacian: bifurcation
from infinity, existence and multiplicity of solutions, Indiana Univ. Math. J., 53 (2) (2004), pp. 433-482.
[8] Drábek, P., Girg, P., Takáč, P., Bounded perturbations of homogeneous quasilinear operators using bifurcations from infinity, to appear in J. Differential Equations,  accepted (2003), electronic version available on-line via www.elsevier.com (2004).
[9] Girg, P., Some Notes on Aeroelastics of Tube Arrays, Technical Report, University of West Bohemia, Pilsen 2004. 
[10] Girg, P., Roca, F., Villegas, S., Semilinear Sturm Liouville Problem with Periodic Nonlinearity, to appear in Nonlinear Analysis T.M.A., accepted (2000).
[11] Girg, P., Takáč, P., Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems,
in Preparation, 2004.

Other References

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