A meshfree Random Walk on Boundary algorithm with iterative refinement
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Irina Shalimova
und
Karl K. Sabelfeld
Abstract
A hybrid continuous Random Walk on Boundary algorithm and iterative refinement method is constructed. In this method, the density of the double layer boundary integral equation for the Laplace equation is resolved by an isotropic Random Walk on Boundary algorithm and calculated for a set of grid points chosen on the boundary. Then, a residual of the boundary integral equation is calculated deterministically, and the same boundary integral equation is solved where the right-hand side is changed with the residual function. This process is repeated several times until the desired accuracy is achieved. This method is compared against the standard Random Walk on Boundary algorithm in terms of their labor intensity. Simulation experiments have shown that the new method is about 200 times more efficient, and this advantage increases with the increase of the desired accuracy. It is noteworthy that the new hybrid algorithm, unlike the standard Random Walk on Boundary algorithm, solves the Laplace equation efficiently also in non-convex domains.
Funding source: Russian Science Foundation
Funding statement: Support by the Russian Science Foundation under Grant 24-11-00107 is greatly acknowledged.
References
[1] N. M. Günter, Potential Theory and its Applications to Basic Problems of Mathematical Physics, Frederick Ungar, New York, 1967. Suche in Google Scholar
[2] J. H. Halton, Sequential Monte Carlo techniques for the solution of linear systems, J. Sci. Comput. 9 (1994), no. 2, 213–257. 10.1007/BF01578388Suche in Google Scholar
[3] J. H. Halton, Sequential Monte Carlo for linear systems—a practical summary, Monte Carlo Methods Appl. 14 (2008), no. 1, 1–27. 10.1515/MCMA.2008.001Suche in Google Scholar
[4] N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., Society for Industrial and Applied Mathematics, Philadelphia, 2002. 10.1137/1.9780898718027Suche in Google Scholar
[5] O. A. Kurbanmuradov, K. K. Sabelfeld and N. A. Simonov, Random Walk on Boundary Algorithms, Nauka, Novosibirsk, 1989. Suche in Google Scholar
[6] M. Mascagni and N. A. Simonov, The random walk on the boundary method for calculating capacitance, J. Comput. Phys. 195 (2004), no. 2, 465–473. 10.1016/j.jcp.2003.10.005Suche in Google Scholar
[7] C. Moler, Iterative refinement in floating point, J. Assoc. Comput. Mach. 14 (1967), no. 2, 316–321. 10.1145/321386.321394Suche in Google Scholar
[8] K. K. Sabelfeld, Vector algorithms of the Monte Carlo method for solving systems of second-order elliptic equations and the Lamé equation, Dokl. Akad. Nauk SSSR 262 (1982), no. 5, 1076–1080. Suche in Google Scholar
[9] 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.2021.107830Suche in Google Scholar
[10] K. K. Sabelfeld and G. Agarkov, Randomized vector algorithm with iterative refinement for solving boundary integral equations, Monte Carlo Methods Appl. 30 (2024), no. 4, 375–388. 10.1515/mcma-2024-2022Suche in Google Scholar
[11] K. K. Sabelfeld and A. Kireeva, Randomized iterative linear solvers with refinement for large dense matrices, Monte Carlo Methods Appl. 29 (2023), no. 4, 357–378. 10.1515/mcma-2023-2013Suche in Google Scholar
[12] K. K. Sabelfeld and N. A. Simonov, Random Walks on Boundary for Solving PDEs, VSP, Utrecht, 1994. 10.1515/9783110942026Suche in Google Scholar
[13] K. K. Sabelfeld and N. A. Simonov, Stochastic Methods for Boundary Value Problems, De Gruyter, Berlin, 2016. 10.1515/9783110479454Suche in Google Scholar
[14] R. Sugimoto, T. Chen, Y. Jiang, C. Batty and T. Hachisuka, A practical walk-on-boundary method for boundary value problems, ACM Trans. Graphics 42 (2023), no. 4, 1–16. 10.1145/3592109Suche in Google Scholar
[15] J. H. Wilkinson, Rounding Errors in Algebraic Processes, Prentice-Hall, Englewood Cliffs, 1963. 10.2307/2002959Suche in Google Scholar
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Artikel in diesem Heft
- Frontmatter
- Impact of psychrometry on the aerosol distribution pattern in human lungs
- Bayesian inference of traffic intensity in M/M/1 queue under symmetric and asymmetric loss functions
- On the variance of Schatten p-norm estimation with Gaussian sketching matrices
- A meshfree Random Walk on Boundary algorithm with iterative refinement
- Combining randomized and deterministic iterative algorithms for high accuracy solution of large linear systems and boundary integral equations
- Preservation of structural properties of the CIR model by θ-Milstein schemes
Artikel in diesem Heft
- Frontmatter
- Impact of psychrometry on the aerosol distribution pattern in human lungs
- Bayesian inference of traffic intensity in M/M/1 queue under symmetric and asymmetric loss functions
- On the variance of Schatten p-norm estimation with Gaussian sketching matrices
- A meshfree Random Walk on Boundary algorithm with iterative refinement
- Combining randomized and deterministic iterative algorithms for high accuracy solution of large linear systems and boundary integral equations
- Preservation of structural properties of the CIR model by θ-Milstein schemes
