Stochastic modelling of diffusion equations on a parallel machine. (English) Zbl 0854.65128
Summary: We present the parallelization of the code “MIXAGE” 3D on the \(T\)-Node tandem of JRC-Ispra. This code solves numerically parabolic systems of partial differentiation equations. These equations, which govern many physical, chemical or biological phenomena, describe time-dependent diffusion in heterogeneous media. We mainly use stochastic differential equations associated to the equation \(\partial f/\partial t = \nabla (D\nabla f)\). Moreover, we define the evolution operators corresponding to the different physical phenomena. By a process that we call “mixing”, we construct the general solution considering simultaneously all the physical phenomena.
In view of the implementation of the code “MIXAGE” 3D on the \(T\)-Node, we have chosen geometric parallelization. Using a matrix \(7 \times 7\) processor, the CPU time reached with the \(T\)-Node is of the same order as with the CRAY II machine.
In view of the implementation of the code “MIXAGE” 3D on the \(T\)-Node, we have chosen geometric parallelization. Using a matrix \(7 \times 7\) processor, the CPU time reached with the \(T\)-Node is of the same order as with the CRAY II machine.
MSC:
| 65C99 | Probabilistic methods, stochastic differential equations |
| 65Y05 | Parallel numerical computation |
| 35K55 | Nonlinear parabolic equations |
| 65M99 | Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems |
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
Keywords:
parallel computation; code “MIXAGE” 3D on the \(T\)-Node tandem of JRC-Ispra; parabolic systems of partial differentiation equations; stochastic differential equations; geometric parallelizationSoftware:
MIXAGEReferences:
| [1] | Azencott, T., Formule de Taylor Stochastique et Développements Asymptotiques d’Intégrales de Feynman, (Lecture Notes No. 921 (1983), Springer: Springer Berlin) · Zbl 0484.60064 |
| [2] | Huard, A.; Laigle, P.; Mastrangelo, V.; Talbi, M.; Xhemalce, S., A method based on a stochastic approach for space dependent nuclear reactor kinetics in one dimension, Comput. Phys. Commun., 46, 351 (1987) · Zbl 0664.65119 |
| [3] | Mastrangelo, V., Stochastic modelisation and parallel computing, invited lecturer Architecture, Programming environment and application of the supernode network of transputers, Euro Courses — Joint Research Centre Ispra (4-8 November 1991) |
| [4] | Huard, A.; Talbi, M.; Xhemalce, S., Solution approchée d’une équation aux dérivées partielles parabolique par une méthode stochastique, C.R. Acad. Sc. Paris, t. 3C2, No. 9 (1986), Série 5 · Zbl 0609.60071 |
| [5] | Mastrangelo, M.; Mastrangelo, V., Stochastic resolution of space-time nuclear reactor kinetics in multigroup diffusion theory, Transport Theor. Stat. Phys., 13, 533 (1984) · Zbl 0547.60062 |
| [6] | Laigle, P.; Mastrangelo, V.; Xhemalce, S., Résolution stochastique de systèmes d’équations aux dérivées partielles du type parabolique affine et applications physiques, EDF/Bulletin de la Direction des Etudes et Recherches, No. 4, 17 (1990), Série C · Zbl 0725.65105 |
| [7] | Revue “La Recherche”. Revue “La Recherche”, Les nouveaux ordinateurs (November 1988), 2nd ed. |
| [8] | Pierre, V., Nouvelles architectures d’ordinateurs, processeurs et systèmes d’exploitation, ediTest (1989) |
| [9] | Telmat Informatique. Telmat Informatique, T-Node User Manuel (1990), 2nd ed. |
| [10] | Pountain, D.; May, D., A tutorial introduction to Occam programming (1988), PSP/Professional Books |
| [11] | INMOS Limited, OCCAM 2 reference Manual, ((1989), Prentice Hall: Prentice Hall Englewood Cliffs, NJ), CAR Hoare Series · Zbl 0777.68003 |
| [12] | 3L Ltd parallel Fortran User Guide (1988), 3L Ltd. |
| [13] | Perihelion Software Ltd., The Helios operating system (1989), Prentice Hall: Prentice Hall Englewood Cliffs, NJ · Zbl 0797.68036 |
| [14] | Pountain, D., Virtual channels: the next generation of transputers (April 1990), Byte |
| [15] | Hoare, C. A.R., Communicating Sequential processes (1985), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0637.68007 |
| [16] | CRAY, SN-2088 Autotasking User’s Guide (1988) |
| [17] | Heidrich, D., Implémentation des langages C, Fortran et Pascal, Paralèles 3L sur une machine MIMD à réseau reconfigurable: supernode (October 1990), Rapport de DEA Université de Mulhouse |
| [18] | Dongarra, J. J., Overview of current high-performance computers, supercomputing Europe ’89 (1989) |
| [19] | Hockney, R. W.; Jesshope, C. R., Parallel Computers (1981), Adam Hilger: Adam Hilger Bristol · Zbl 0523.68004 |
| [20] | INMOS, Transputer development system (1988), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ |
| [21] | Gibbons, A.; Rytter, W., Efficient parallel algorithms (1988), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0771.68015 |
| [22] | Mastrangelo, V., Modélisation stochastique et calcul parallèle, JRC-Ispra/CEC, Technical Note No. I.91.58 (April 1991) |
| [23] | Topexpress Ltd., Mathematical procedure library reference manual.; Topexpress Ltd., Mathematical procedure library reference manual. |
| [24] | Topexpress Ltd., Vector library reference manual.; Topexpress Ltd., Vector library reference manual. |
| [25] | N.A. Software Ltd., Liverpool parallel transputer mathematical library.; N.A. Software Ltd., Liverpool parallel transputer mathematical library. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
