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Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem

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Abstract

Probabilistic domain decomposition is proposed as a novel method for solving the two-dimensional Maxwell’s equations as used in the magnetotelluric method. The domain is split into non-overlapping sub-domains and the solution on the sub-domain boundaries is obtained by evaluating the stochastic form of the exact solution of Maxwell’s equations by a Monte-Carlo approach. These sub-domains can be naturally chosen by splitting the sub-surface domain into regions of constant (or at least continuous) conductivity. The solution over each sub-domain is obtained by solving Maxwell’s equations in the strong form. The sub-domain solver used for this purpose is a meshless method resting on radial basis function-based finite differences. The method is demonstrated by solving a number of classical magnetotelluric problems, including the quarter-space problem, the block-in-half-space problem and the triangle-in-half-space problem.

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References

  1. Acebrón, J.A., Busico, M.P., Lanucara, P., Spigler, R.: Domain decomposition solution of elliptic boundary-value problems via Monte Carlo and quasi-Monte Carlo methods. SIAM J. Sci. Comput. 27, 440–457 (2005)

    Article  Google Scholar 

  2. Acebrón, J.A., Rodríguez-Rozas, Á., Spigler, R.: Efficient parallel solution of nonlinear parabolic partial differential equations by a probabilistic domain decomposition. J. Sci. Comput. 43, 135–157 (2010)

    Article  Google Scholar 

  3. Acebrón, J.A., Spigler, R.: A new probabilistic approach to the domain decomposition method. In: Domain Decomposition Methods in Science and Engineering XVI, pp 473–480. Springer (2007)

  4. Bihlo, A., Haynes, R.D.: Parallel stochastic methods for PDE based grid generation. Comput. Math. Appl. 68, 804–820 (2014)

    Article  Google Scholar 

  5. Bihlo, A., Haynes, R.D., Walsh, E.J.: Stochastic domain decomposition for time dependent adaptive mesh generation. J. Math. Study 48, 106–124 (2015)

    Article  Google Scholar 

  6. Bossy, M., Champagnat, N., Leman, H., Maire, S., Violeau, L., Yvinec, M.: Monte Carlo methods for linear and non-linear Poisson–Boltzmann equation. ESAIM: Proc. Surv. 48, 420–446 (2015)

    Article  Google Scholar 

  7. Buchmann, F.M.: Simulation of stopped diffusions. J. Comput. Phys. 202, 446–462 (2005)

    Article  Google Scholar 

  8. Chave, A.D., Jones, A.G.: Introduction to the magnetotelluric method. In: Chave, A.D., Jones, A.G. (eds.) The Magnetotelluric Method: Theory and Practice, pp 1–18. Cambridge University Press (2012)

  9. Ding, H., Shu, C., Yeo, K., Xu, D.: Development of least-square-based two-dimensional finite-difference schemes and their application to simulate natural convection in a cavity. Comput. Fluids 33, 137–154 (2004)

    Article  Google Scholar 

  10. Dolean, V., Gander, M.J., Veneros, E.: Schwarz methods for second order Maxwell equations in 3D with coefficient jumps, Domain Decomposition Methods in Science and Engineering XXII, pp. 471–479. Springer (2016)

  11. Farquharson, C.G.: Constructing piecewise-constant models in multidimensional minimum-structure inversions. Geophysics 73, K1–K9 (2007)

    Article  Google Scholar 

  12. Fischer, G., Schnegg, P.A.: The magnetotelluric dispersion relations over 2-D structures. Geophys. J. Int. 115, 1119–1123 (1993)

  13. Fischer, G., Szarka, L., Adam, A.,Weaver, J.: The magnetotelluric phase over 2-D structures. Geophys. J. Int. 108, 778–786 (1992)

  14. Fornberg, B., Flyer, N.: A Primer on Radial Basis Functions with Applications to the Geosciences, vol. 3529. SIAM Press, Philadelphia (2015)

    Book  Google Scholar 

  15. Fornberg, B., Larsson, E., Flyer, N.: Stable computations with Gaussian radial basis functions. SIAM J. Sci. Comput. 33, 869–892 (2011)

    Article  Google Scholar 

  16. Fornberg, B., Lehto, E.: Stabilization of RBF-generated finite difference methods for convective PDEs. J. Comput. Phys. 230, 2270–2285 (2011)

    Article  Google Scholar 

  17. Fornberg, B., Lehto, E., Powell, C.: Stable calculation of Gaussian-based RBF-FD stencils. Comput. Math. Appl. 65, 627–637 (2013)

    Article  Google Scholar 

  18. Gobet, E.: Weak approximation of killed diffusion using Euler schemes. Stochastic Process. Appl. 87, 167–197 (2000)

    Article  Google Scholar 

  19. Karatzas, I., Shreve, S.E.: Brownian motion and stochastic calculus, vol. 113 of Graduate Texts in Mathematics. Springer, New York (1991)

    Google Scholar 

  20. Lejay, A.: Simulation of a stochastic process in a discontinuous layered medium. Electron. Comm. Probab. 16, 764–774 (2011)

    Article  Google Scholar 

  21. Lejay, A., Maire, S.: Simulating diffusions with piecewise constant coefficients using a kinetic approximation. Comput. Methods Appl. Mech. Eng. 199, 2014–2023 (2010)

    Article  Google Scholar 

  22. Lejay, A., Maire, S.: New Monte Carlo schemes for simulating diffusions in discontinuous media. J. Comput. Appl. Math. 245, 97–116 (2013)

    Article  Google Scholar 

  23. Lejay, A., Pichot, G.: Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps. J. Comput. Phys. 231, 7299–7314 (2012)

    Article  Google Scholar 

  24. Maire, S., Nguyen, G.: Stochastic finite differences for elliptic diffusion equations in stratified domains, hal-00809203 (2013)

  25. Mascagni, M., Simonov, N.A.: Monte Carlo methods for calculating some physical properties of large molecules. SIAM J. Sci. Comput. 26, 339–357 (2004)

    Article  Google Scholar 

  26. Milewski, S.: Meshless finite difference method with higher order approximation—applications in mechanics. Arch. Comput. Methods Eng. 19, 1–49 (2012)

    Article  Google Scholar 

  27. Nguyen, V.P., Rabczuk, T., Bordas, S., Duflot, M.: Meshless methods: a review and computer implementation aspects. Math. Comput. Simul. 79, 763–813 (2008)

    Article  Google Scholar 

  28. Preis, T., Virnau, P., Paul, W., Schneider, J.J.: GPU accelerated Monte Carlo simulation of the 2D and 3D Ising model. J. Comput. Phys. 228, 4468–4477 (2009)

    Article  Google Scholar 

  29. Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes: the art of scientific computing, 3rd edn. Cambridge University Press, Cambridge (2007)

    Google Scholar 

  30. Quarteroni, A., Valli, A.: Domain decomposition methods for partial differential equations. Oxford University Press, Oxford (1999)

    Google Scholar 

  31. Tupper, P.F., Yang, X.: A paradox of state-dependent diffusion and how to resolve it. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 468, 3864–3881 (2012)

    Article  Google Scholar 

  32. van Meel, J.A., Arnold, A., Frenkel, D., Portegies Zwart, S.F., Belleman, R.G.: Harvesting graphics power for MD simulations. Mol. Simul. 34, 259–266 (2008)

    Article  Google Scholar 

  33. Vozoff, K.: The magnetotelluric method. In: Nabighian, M. (ed.) Electromagnetic methods in applied geophysics, pp 641–712. Society of Exploration Geophysicists (1991)

  34. Weaver, J.T.: Mathematical methods for geo-electromagnetic induction, vol. 7. Research Studies Press, Baldock (1994)

    Google Scholar 

  35. Weiss, C.: The two- and three-dimensional forward problems. In: Chave, A.D., Jones, A.G. (eds.) The Magnetotelluric Method: Theory and Practice, pp 303–346. Cambridge University Press (2012)

  36. Wittke, J., Tezkan, B.: Meshfree magnetotelluric modelling. Geophysical J. Int. 198, 1255–1268 (2014)

    Article  Google Scholar 

  37. Wright, G.B., Fornberg, B.: Scattered node compact finite difference-type formulas generated from radial basis functions. J. Comput. Phys. 212, 99–123 (2006)

    Article  Google Scholar 

  38. Zhdanov, M.S., Varentsov, I.M., Weaver, J.T., Golubev, N.G., Krylov, V.A.: Methods for modelling electromagnetic fields results from COMMEMI—the international project on the comparison of modelling methods for electromagnetic induction. J. Appl. Geophys. 37, 133–271 (1997)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s (NL), A1C 5S7, Canada

    Alexander Bihlo, Ronald D. Haynes & J Concepción Loredo-Osti

  2. Department of Earth Sciences, Memorial University of Newfoundland, St. John’s (NL), A1B 3X5, Canada

    Colin G. Farquharson

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  1. Alexander Bihlo
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  2. Colin G. Farquharson
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  3. Ronald D. Haynes
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  4. J Concepción Loredo-Osti
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Correspondence to Alexander Bihlo.

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Bihlo, A., Farquharson, C.G., Haynes, R.D. et al. Probabilistic domain decomposition for the solution of the two-dimensional magnetotelluric problem. Comput Geosci 21, 117–129 (2017). https://doi.org/10.1007/s10596-016-9598-8

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  • DOI: https://doi.org/10.1007/s10596-016-9598-8

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