Two stochastic algorithms for solving elastostatics problems governed by the Lamé equation

References

[1] B. V. Budaev and D. B. Bogy, Probabilistic approach to the Lamé equations of linear elastostatics, Internat. J. Solids Structures 40 (2003), no. 23, 6285–6306. 10.1016/S0020-7683(03)00364-0Search in Google Scholar

[2] E. B. Dynkin, Theory of Markov Processes, Prentice-Hall, Englewood Cliffs, 1961. Search in Google Scholar

[3] A. I. Koshelev, Application of the universal iterative process to some problems of mechanics, Vestnik St. Petersburg Univ. Math. 1 (2008), no. 2, 47–55. 10.3103/S1063454108020064Search in Google Scholar

[4] M. E. Muller, Some continuous Monte Carlo methods for the Dirichlet problem, Ann. Math. Statist. 27 (1956), 569–589. 10.1214/aoms/1177728169Search in Google Scholar

[5] D. P. O’Leary, G. W. Stewart and J. S. Vandergraft, Estimating the largest eigenvalue of a positive definite matrix, Math. Comp. 33 (1979), no. 148, 1289–1292. 10.1090/S0025-5718-1979-0537973-XSearch in Google Scholar

[6] A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC, Boca Raton, 2002. 10.1201/9781420035322Search in Google Scholar

[7] S. Roux, Generalized Brownian motion and elasticity, J. Stat. Phys. 48 (1987), no. 1–2, 201–213. 10.1007/BF01010406Search in Google Scholar

[8] Y. Saad, Iterative Methods for Sparse Linear Systems, PWS Publishing, New York, 1996. Search in Google Scholar

[9] Y. Saad and H. A. van der Vorst, Iterative solution of linear systems in the 20th century, J. Comput. Appl. Math. 123 (2000), 1–33. 10.1016/S0377-0427(00)00412-XSearch in Google Scholar

[10] K. K. Sabelfeld, Monte Carlo Methods in Boundary Value Problems, Springer Ser. Comput. Phys., Springer, Berlin, 1991. 10.1007/978-3-642-75977-2Search in Google Scholar

[11] K. K. Sabelfeld, Random walk on spheres method for solving drift-diffusion problems, Monte Carlo Methods Appl. 22 (2016), no. 4, 265–275. 10.1515/mcma-2016-0118Search in Google Scholar

[12] K. K. Sabelfeld, Random walk on spheres algorithm for solving transient drift-diffusion-reaction problems, Monte Carlo Methods Appl. 23 (2017), no. 3, 189–212. 10.1515/mcma-2017-0113Search in Google Scholar

[13] K. K. Sabelfeld, A new randomized vector algorithm for iterative solution of large linear systems, Appl. Math. Lett. 126 (2022), Paper No. 107830. 10.1016/j.aml.2021.107830Search in Google Scholar

[14] K. K. Sabelfeld and A. Kireeva, A new global random walk algorithm for calculation of the solution and its derivatives of elliptic equations with constant coefficients in an arbitrary set of points, Appl. Math. Lett. 107 (2020), Paper No. 106466. 10.1016/j.aml.2020.106466Search in Google Scholar

[15] K. K. Sabelfeld and A. Kireeva, A global random walk on spheres algorithm for calculating the solution and its derivatives of the drift-diffusion-reaction equations, Math. Methods Appl. Sci. 45 (2022), no. 3, 1420–1431. 10.1002/mma.7861Search in Google Scholar

[16] K. K. Sabelfeld and I. A. Shalimova, Spherical and Plane Integral Operators for PDEs, De Gruyter, Berlin, 2013, 10.1515/9783110315332Search in Google Scholar

[17] K. K. Sabelfeld and D. Smirnov, A global random walk on grid algorithm for second order elliptic equations, Monte Carlo Methods Appl. 27 (2021), no. 3, 211–225. 10.1515/mcma-2021-2092Search in Google Scholar

[18] K. K. Sabelfeld and D. Talay, Integral formulation of the boundary value problems and the method of random walk on spheres, Monte Carlo Methods Appl. 1 (1995), no. 1, 1–34. 10.1515/mcma.1995.1.1.1Search in Google Scholar

[19] I. Shalimova and K. K. Sabelfeld, Development and implementation of branching random walk on spheres algorithms for solving the 2D elastostatics Lamé equation, Monte Carlo Methods Appl. 29 (2023), no. 1, 79–93. 10.1515/mcma-2022-2131Search in Google Scholar

[20] A. V. Starchenko and V. N. Bercun, Methods of parallel computations (in Russian), Izdatel’stvo Tomskogo Universiteta, Tomsk, 2013. Search in Google Scholar

[21] A. J. Walker, New fast method for generating discrete random numbers with arbitrary frequency distributions, Electr. Letters 10 (1974), 127–128. 10.1049/el:19740097Search in Google Scholar

[22] The Siberian Branch of the Russian Academy of Sciences Siberian Supercomputer Center, http://www.sscc.icmmg.nsc.ru, accessed April 27, 2023. Search in Google Scholar