Adjusted Sparse Tensor Product Spectral Galerkin Method for Solving Pseudodifferential Equations on the Sphere with Random Input Data

Abstract

An adjusted sparse tensor product spectral Galerkin approximation method based on spherical harmonics is introduced and analyzed for solving pseudodifferential equations on the sphere with random input data. These equations arise from geodesy where the sphere is taken as a model of the earth. Numerical solutions to the corresponding \(k\)-th order statistical moment equations are found in adjusted sparse tensor approximation spaces which are accordingly designed to the regularity of the data and the equation. Established convergence theorem shows that the adjusted sparse tensor Galerkin discretization is superior not only to the full tensor product but also to the standard sparse tensor counterpart when the statistical moments of the data are of mixed unequal regularity. Numerical experiments illustrate our theoretical results.

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References

  1. 1.

    Aubin, J.-P.: Applied Functional Analysis, 2nd edn. Pure and Applied Mathematics (New York). Wiley–Interscience, New York (2000), with exercises by Bernard Cornet and Jean-Michel Lasry, Translated from the French by Carole Labrousse

    Google Scholar 

  2. 2.

    Chernov, A.: Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading. Appl. Numer. Math. 62, 360–377 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Chernov, A., Dũng, D.: New explicit-in-dimension estimates for the cardinality of high-dimensional hyperbolic crosses and approximation of functions having mixed smoothness. J. Complex. 32, 92–121 (2016)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Chernov, A., Pham, T.D.: Sparse tensor product spectral Galerkin BEM for elliptic problems with random data on a spheroid. Adv. Comput. Math. 41, 77–104 (2015)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Chernov, A., Schwab, C.: Sparse \(p\)-version BEM for first kind boundary integral equations with random loading. Appl. Numer. Math. 59, 2698–2712 (2009)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Claessens, S.: Solutions to ellipsoidal boundary value problems for gravity field modelling. PhD thesis, Curtin University of Technology, Perth (2005)

  7. 7.

    Cohen, A., DeVore, R., Schwab, C.: Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math. 10, 615–646 (2010)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Dũng, D., Griebel, M.: Hyperbolic cross approximation in infinite dimensions. J. Complex. 33, 55–88 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Dũng, D., Temlyakov, V., Ullrich, T.: Hyperbolic Cross Approximation. Advanced Courses in Mathematics—CRM Barcelona. Birkhäuser, Basel (2018)

    Google Scholar 

  10. 10.

    Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere with Applications to Geomathematics. Oxford University Press, Oxford (1998)

    Google Scholar 

  11. 11.

    Freeden, W., Windheuser, U.: Combined spherical harmonic and wavelet expansion—a future concept in Earth’s gravitational determination. Appl. Comput. Harmon. Anal. 4, 1–37 (1997)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Grafarend, E.W., Krumm, F.W., Schwarze, V.S. (eds.): Geodesy: The Challenge of the 3rd Millennium. Springer, Berlin (2003)

    Google Scholar 

  13. 13.

    Hörmander, L.: Pseudodifferential operators. Commun. Pure Appl. Math. 18, 501–517 (1965)

    Article  Google Scholar 

  14. 14.

    Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations. Applied Mathematical Sciences, vol. 164. Springer, Berlin (2008)

    Google Scholar 

  15. 15.

    Huang, H.-Y., Yu, D.-H.: Natural boundary element method for three dimensional exterior harmonic problem with an inner prolate spheroid boundary. J. Comput. Math. 24, 193–208 (2006)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Khoromskij, B.N., Oseledets, I.: Quantics-TT collocation approximation of parameter-dependent and stochastic elliptic PDEs. Comput. Methods Appl. Math. 10, 376–394 (2010)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Khoromskij, B.N., Schwab, C.: Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. SIAM J. Sci. Comput. 33, 364–385 (2011)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Kohn, J.J., Nirenberg, L.: On the algebra of pseudodifferential operators. Commun. Pure Appl. Math. 18, 269–305 (1965)

    Article  Google Scholar 

  19. 19.

    Langel, R.A., Hinze, W.J.: The Magnetic Field of the Earth’s Lithosphere, the Satellite Perspective. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  20. 20.

    Morton, T.M., Neamtu, M.: Error bounds for solving pseudodifferential equations on spheres. J. Approx. Theory 114, 242–268 (2002)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Müller, C.: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, Berlin (1966)

    Google Scholar 

  22. 22.

    Nédélec, J.-C.: Acoustic and Electromagnetic Equations. Springer, New York (2000)

    Google Scholar 

  23. 23.

    Nitsche, P.-A.: Best \(N\) term approximation spaces for tensor product wavelet bases. Constr. Approx. 24, 49–70 (2006)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Petersen, B.E.: Introduction to the Fourier transform & pseudodifferential operators. Monographs and Studies in Mathematics, vol. 19. Pitman, Boston (1983)

    Google Scholar 

  25. 25.

    Pham, T.D., Tran, T.: Strongly elliptic pseudodifferential equations on the sphere with radial basis functions. Numer. Math. 128, 589–614 (2014)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Pham, T.D., Tran, T.: Solving non-strongly elliptic pseudodifferential equations on a sphere using radial basis functions. Comput. Math. Appl. 70, 1970–1983 (2015)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Pham, T.D., Tran, T., Chernov, A.: Pseudodifferential equations on the sphere with spherical splines. Math. Models Methods Appl. Sci. 21, 1933–1959 (2011)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Pinchon, D., Hoggan, P.E.: Rotation matrices for real spherical harmonics: general rotations of atomic orbitals in space-fixed axes. J. Phys. A 40, 1597–1610 (2007)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Schwab, C., Gittelson, C.J.: Sparse tensor discretizations of high-dimensional parametric and stochastic PDEs. Acta Numer. 20, 291–467 (2011)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Schwab, C., Todor, R.-A.: Sparse finite elements for elliptic problems with stochastic loading. Numer. Math. 95, 707–734 (2003)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Schwab, C., Todor, R.A.: Sparse finite elements for stochastic elliptic problems—higher order moments. Computing 71, 43–63 (2003)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Schwab, C., Todor, R.A.: Karhunen–Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys. 217, 100–122 (2006)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Stephan, E.P.: Boundary integral equations for screen problems in \({{\mathbb{R}}}^{3}\). Integral Equ. Oper. Theory 10, 236–257 (1987)

    Article  Google Scholar 

  34. 34.

    Svensson, S.L.: Pseudodifferential operators—a new approach to the boundary problems of physical geodesy. Manuscr. Geod. 8, 1–40 (1983)

    MATH  Google Scholar 

  35. 35.

    Tran, T., Le Gia, Q.T., Sloan, I.H., Stephan, E.P.: Boundary integral equations on the sphere with radial basis functions: error analysis. Appl. Numer. Math. 59, 2857–2871 (2009)

    MathSciNet  Article  Google Scholar 

  36. 36.

    von Petersdorff, T., Schwab, C.: Sparse finite element methods for operator equations with stochastic data. Appl. Math. 51, 145–180 (2006)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Wendland, W.L., Stephan, E.P.: A hypersingular boundary integral method for two-dimensional screen and crack problems. Arch. Ration. Mech. Anal. 112, 363–390 (1990)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This research was funded by the Department of Science and Technology–Ho Chi Minh City (HCMC-DOST), and the Institute for Computational Science and Technology (ICST) at Ho Chi Minh city, Vietnam under Contract 21/2017/HD-KHCNTT on 21/09/2017. A part of this paper was done when the authors were visiting the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors thank VIASM for providing a fruitful research environment and working condition.

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Correspondence to Duong Thanh Pham.

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Pham, D.T., Dũng, D. Adjusted Sparse Tensor Product Spectral Galerkin Method for Solving Pseudodifferential Equations on the Sphere with Random Input Data. Acta Appl Math 166, 187–214 (2020). https://doi.org/10.1007/s10440-019-00262-4

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Keywords

  • Stochastic pseudodifferential equations
  • Statistical moments
  • Hyperbolic cross spectral methods
  • Spheres

Mathematics Subject Classification

  • 65N30
  • 65N15
  • 35R60
  • 41A25