Abstract
We present a quasi-Monte Carlo spectral method for a class of elliptic partial differential equations (PDEs) with random coefficients defined on the unit sphere. The random coefficients are parametrised by the Karhunen-Loève expansion, while the exact solution is approximated by the spherical harmonics. The expectation of the solution is approximated by a quasi-Monte Carlo integration rule. A method for obtaining error estimates between the exact and the approximate solution is also proposed. Some numerical experiments are provided in the last section.
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Acknowledgements
The author acknowledges the support of the Australian Research Council under its Centre of Excellence program and helpful conversations with Prof. Ian Sloan, Prof. Christoph Schwab, Dr. Josef Dick and Dr. Frances Kuo.
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Gia, Q.T.L. (2013). A QMC-Spectral Method for Elliptic PDEs with Random Coefficients on the Unit Sphere. In: Dick, J., Kuo, F., Peters, G., Sloan, I. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2012. Springer Proceedings in Mathematics & Statistics, vol 65. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41095-6_26
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DOI: https://doi.org/10.1007/978-3-642-41095-6_26
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