MODIFICATION OF THE IMPLEMENTATION OF THE JACOBI METHOD IN SIMULATING SUPERDIFFUSION OF RADON ON RECONFIGURABLE COMPUTER SYSTEMS
Abstract
When studying natural objects, the problem of modeling complex systems with a structure that cannot be described by means of Euclidean geometry tools often arises, therefore, fractal geometry and the corresponding mathematical apparatus are used to represent them. So the model of radon transport in an inhomogeneous medium, using superdiffusion, displays real data more accurately than the classical one. An increase in the concentration of radon in the air is one of the signs of an approaching earthquake, which makes it necessary to simulate the propagation of this radioactive inert gas in real time. Reconfigurable computing systems have great potential for solv-ing problems in real time, but the currently existing means for solving systems of linear equations have low efficiency due to the irregular structure of matrices obtained by discretizing the radon superdiffusion model using adaptive grids. The basic subgraph of the Jacobi method is trans-formed as follows: the input data is vectorized, the structure of the frame in which the value of one unknown is calculated is divided into several microframes, parallelizing the calculations in the first microframe, where the sum of the products of the matrix coefficients and the values of the unknowns from the previous iteration is performed. The results obtained are buffered for subse-quent delivery to the second microframe, where the final processing and output of the iteration result takes place. The described approach allows to reduce equipment downtime when solving a system of linear equations with sparse irregular matrices, and gives a speed gain by 5–15 times in comparison with existing methods for solving linear system on reconfigurable computing systems
References
1. Parovik R.I. Model' nestatsionarnoy diffuzii-advektsii radona sisteme grunt-atmosfera [Model of unsteady diffusion-advection of radon in the soil-atmosphere system], Vestnik KRAUNTS. Fiz.-mat. nauki [Bulletin KRASEC. Physical and Mathematical Sciences], 2010, Issue 1 (1), pp. 39-45.
2. Parovik R.I., Shevtsov B.M. Protsessy perenosa radona v sredakh s fraktal'noy strukturoy [Ra-don transfer processes in media with fractal structure], Matematicheskoe modelirovanie [Mathematical modeling], 2009, Vol. 21, No. 8, pp. 30-36.
3. Terekhov K.M., Vassilevski Yu.V. Two-phase water flooding simulations on dynamic adaptive octree grids with two-point nonlinear fluxes, Russian Journal of Numerical Analysis and Mathematical Modelling, 2013, Vol. 28, No. 3, pp. 267-288.
4. Afendikov A.L., Men'shov I.S., Merkulov K.D., Pavlukhin P.V. Metod adaptivnykh dekartovykh setok dlya resheniya zadach gazovoy dinamiki [The method of adaptive cartesian grids for solv-ing problems of gas dynamics]. Moscow: Rossiyskaya akademiya nauk, 2017, 63 p.
5. Sukhinov A.A. Postroenie dekartovykh setok s dinamicheskoy adaptatsiey k resheniyu [Con-struction of Cartesian grids with dynamic adaptation to the solution], Matematicheskoe modelirovanie [Mathematical modeling], 2010, Vol. 22, No. 1, pp. 86-98.
6. Levin I.I., Dordopulo A.I., Pelipets A.V. Realizatsiya iteratsionnykh metodov resheniya sistem lineynykh uravneniy v zadachakh matematicheskoy fiziki na rekonfiguriruemykh vychislitel'nykh sistemakh [Implementation of iterative methods for solving systems of linear equations in problems of mathematical physics on reconfigurable computing systems], Vestnik YuUrGU. Seriya: Vychislitel'naya matematika i informatika [Bulletin of SUSU. Series: Com-putational Mathematics and Computer Science], 2016, Vol. 5, No. 4, pp. 5-18. DOI: 10.14529/cmse160401.
7. Pelipets A.V. Rasparallelivanie iteratsionnykh metodov resheniya sistem lineynykh algebraicheskikh uravneniy na rekonfiguriruemykh vychislitel'nykh sistemakh [Parallelization of iterative methods for solving systems of linear algebraic equations on reconfigurable com-puting systems], Super-komp'yuternye tekhnologii (SKT-2016): Mater. 4-y Vserossiyskoy nauchno-tekhnicheskoy konferentsii [Super-computer Technologies (SKT-2016): Materials of the 4th All-Russian Scientific and Technical Conference], 2016, pp. 194-198.
8. Levin I.I., Pelipets A.V. Effektivnaya realizatsiya rasparallelivaniya na rekonfiguriruemykh sistemakh [Effective implementation of parallelization on reconfigurable systems], Vestnik komp'yuternykh i informatsionnykh tekhnologiy [Bulletin of Computer and Information Tech-nologies], 2018, No. 8, pp. 11-16.
9. Levin I.I., Dordopulo A.I., Sorokin D.A., Kalyaev Z.V., Doronchenko Yu.I. Rekonfiguriruemye komp'yutery na osnove plis Xilinx Virtex Ultrascale [Reconfigurable computers based on FPGA Xilinx Virtex Ultrascale], Parallel'nye vychisli-tel'nye tekhnologii (PaVT'2019): Korotkie stat'i i opisaniya plakatov XIII Mezhduna-rodnoy nauchnoy konferentsii [Parallel Computing Technologies (PaVT'2019): Short articles and poster descriptions of the XIII Inter-national Scientific Conference], 2019, pp. 288-298.
10. Dordopulo A.I., Levin I.I. Metody reduktsii vychisleniy dlya programmirovaniya gibridnykh rekonfiguriruemykh vychislitel'nykh sistem [Methods of reduction of calculations for pro-gramming hybrid reconfigurable computing systems], XII mul'tikonferentsiya po problemam upravleniya (MKPU-2019): Mater. XII mul'tikonferentsii [XII Multi-conference on manage-ment problems (MCPU-2019): Materials of the multi-conference]: in 4 vol., 2019, pp. 78-82.
11. Levin I.I., Pelipets A.V., Sorokin D.A. Reshenie zadachi LU-dekompozitsii na rekonfiguriruemykh vychislitel'nykh sistemakh: otsenka i perspektivy [Solving the LU-decomposition problem on recon-figurable computing systems: assessment and prospects], Izvestiya YuFU. Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2015, No. 7 (168), pp. 62-70.
12. Guzik V.F., Kalyaev I.A., Levin I.I. Rekonfiguriruemye vychislitel'nye sistemy [Reconfigurable computing systems], under the general ed. of. I.A. Kalyaeva. Rostov-on-Don: Izd-vo YuFU, 2016, 472 p.
13. Samarskiy A.A., Nikolaev E.S. Metody resheniya setochnykh uravneniy [Methods for solving grid equations]. Moscow: Nauka, 1978, 592 p.
14. Kalyaev I.A., Levin I.I., Semernikov E.A., Shmoilov V.I. Reconfigurable multipipeline compu-ting structures. New York: Nova Science Publishers, 2012, 330 р.
15. Mandel'brot B. Fraktal'naya geometriya prirody [Fractal geometry of nature]. Moscow: Institut komp'yuternykh issledovaniy, 2002, 656 p.
16. Kasarkin A.V. Metod resheniya grafovykh NP-polnykh zadach na rekonfiguriruemykh vy-chislitel'nykh sistemakh na osnove printsipa rasparallelivaniya po iteratsiyam [A method for solving graph NP-complete problems on reconfigurable numerical systems based on the prin-ciple of parallelization by iterations], Izvestiya YuFU. Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2020, No. 7 (217), pp. 121-129.
17. Nakhushev A.M. Drobnoe ischislenie i ego primenenie [Fractional calculus and its application]. Moscow: Fizmatlit, 2003, 272 p.
18. Nakhusheva V.A. Differentsial'nye uravneniya matematicheskikh modeley nelokal'nykh protsessov [Differential equations of mathematical models of nonlocal processes]. Moscow: Nauka, 2006, 173 p.
19. Zaslavsky G.M. Chaos, fractional kinetics, and anomalous transport, Physics Reports, 2002, Vol. 371, pp. 461-580.
20. Krylov S.S., Bobrov N.Yu. Fraktaly v geofizike [Fractals in geophysics]. Saint Petersburg: Izd-vo S-Pb. universiteta, 2004, 138 p.
