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Walk-on-Spheres Algorithm for Solving Boundary-Value Problems with Continuity Flux Conditions

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Book cover Monte Carlo and Quasi-Monte Carlo Methods 2006

Summary

We consider boundary-value problem for elliptic equations with constant coefficients related through the continuity conditions on the boundary between the domains. To take into account conditions involving the solution’s normal derivative, we apply a new mean-value relation written down at a boundary point. This integral relation is exact and provides a possibility to get rid of the bias caused by usually used finite-difference approximation. Randomization of this mean-value relation makes it possible to continue simulating walk-on-spheres trajectory after it hits the boundary. We prove the convergence of the algorithm and determine its rate. In conclusion, we present the results of some model computations.

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Author information

Authors and Affiliations

  1. Institute of Computational Mathematics and Mathematical Geophysics SB RAS, Novosibirsk, Russia

    Nikolai Simonov

  2. Institute of Molecular Biophysics, Florida State University, Tallahassee, FL, USA

    Nikolai Simonov

Authors
  1. Nikolai Simonov
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Ulm University, Albert-Einstein-Allee 11, 89069, Ulm, Germany

    Alexander Keller

  2. Department of Computer Science, University of Kaiserslautern, 67653, Kaiserslautern, Germany

    Stefan Heinrich

  3. Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore, 117543, Republic of Singapore

    Harald Niederreiter

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Simonov, N. (2008). Walk-on-Spheres Algorithm for Solving Boundary-Value Problems with Continuity Flux Conditions. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_38

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