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Stochastic Collocation Methods: A Survey

  • Dongbin XiuEmail author
Reference work entry

Abstract

Stochastic collocation (SC) has become one of the major computational tools for uncertainty quantification. Its primary advantage lies in its ease of implementation. To carry out SC, one needs only a reliable deterministic simulation code that can be run repetitively at different parameter values. And yet, the modern-day SC methods can retain the high-order accuracy properties enjoyed by most of other methods. This is accomplished by utilizing the large amount of literature in the classical approximation theory. Here we survey the major approaches in SC. In particular, we focus on a few well-established approaches: interpolation, regression, and pseudo projection. We present the basic formulations of these approaches and some of their major variations. Representative examples are also provided to illustrate their major properties.

Keywords

Compressed sensing Interpolation Least squares Stochastic collocation 

References

  1. 1.
    Atkinson, A.C., Donev, A.N., Tobias, R.D.: Optimum Experimental Designs, with SAS. Oxford University Press, Oxford (2007)zbMATHGoogle Scholar
  2. 2.
    Babuska, I., Nobile, F., Tempone, R.: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM J. Numer. Anal. 45(3), 1005–1034 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Barthelmann, V., Novak, E., Ritter, K.: High dimensional polynomial interpolation on sparse grid. Adv. Comput. Math. 12, 273–288 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bungartz, H.-J., Griebel, M.: Sparse grids. Acta Numer. 13, 1–123 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Candès, E.J., Romberg, J.K., Tao, T.: Stable signal recovery from incomplete and inaccurate measurements. Commun. Pure Appl. Math. 59(8), 1207–1223 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Candès, E.J., Tao, T.: Near-optimal signal recovery from random projections: universal encoding strategies? IEEE Trans. Inform. Theory 52(12), 5406–5425 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Cheney, W., Light, W.: A Course in Approximation Theory. Brooks/Cole Publishing Company, Pacific Grove (2000)zbMATHGoogle Scholar
  8. 8.
    Cohen, A., Davenport, M.A., Leviatan, D.: On the stability and accuracy of least squares approximations. Found. Comput. Math. 13(5), 819–834 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cools, R.: Advances in multidimensional integration. J. Comput. Appl. Math. 149, 1–12 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    De Boor, C., Ron, A.: On multivariate polynomial interpolation. Constr. Approx. 6, 287–302 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    De Boor, C., Ron, A.: Computational aspects of polynomial interpolation in several variables. Math. Comput. 58, 705–727 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Donoho, D.L.: Compressed sensing. IEEE Trans. Inform. Theory 52(4), 1289–1306 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Doostan, A., Owhadi, H.: A non-adapted sparse approximation of PDEs with stochastic inputs. J. Comput. Phys. 230(8), 3015–3034 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Engels, H.: Numerical Quadrature and Cubature. Academic, London/New York (1980)zbMATHGoogle Scholar
  15. 15.
    Fedorov, V.V., Leonov, S.L.: Optimal Design for Nonlinear Response Models. CRC Press, Boca Raton (2014)zbMATHGoogle Scholar
  16. 16.
    Ghanem, R.G., Spanos, P.: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991)CrossRefzbMATHGoogle Scholar
  17. 17.
    Mathelin, L., Gallivan K.A.: A compressed sensing approach for partial differential equations with random input data. Commun. Comput. Phys. 12, 919–954 (2012)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Migliorati, G., Nobile, F.: Analysis of discrete least squares on multivariate polynomial spaces with evaluations at low-discrepancy point sets. J. Complex. 31(4), 517–542 (2015)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Migliorati, G., Nobile, F., E. von Schwerin, Tempone, R.: Approximation of quantities of interest in stochastic PDEs by the random discrete L 2 projection on polynomial spaces. SIAM J. Sci. Comput. 35(3), A1440–A1460 (2013)Google Scholar
  20. 20.
    Migliorati, G., Nobile, F., von Schwerin, E., Tempone, R.: Analysis of the discrete L 2 projection on polynomial spaces with random evaluations. Found. Comput. Math. 14(3), 419–456 (2014)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Narayan, A., Xiu, D.: Stochastic collocation methods on unstructured grids in high dimensions via interpolation. SIAM J. Sci. Comput. 34(3), A1729–A1752 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Narayan, A., Xiu, D.: Constructing nested nodal sets for multivariate polynomial interpolation. SIAM J. Sci. Comput. 35(5), A2293–A2315 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Powell, M.J.D.: Approximation Theory and Methods. Cambridge University Press, Cambridge (1981)zbMATHGoogle Scholar
  24. 24.
    Rauhut, H., Ward, R.: Sparse Legendre expansions via 1-minimization. J. Approx. Theory 164, 517–533 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Smolyak, S.A.: Quadrature and interpolation formulas for tensor products of certain classes of functions. Soviet Math. Dokl. 4, 240–243 (1963)zbMATHGoogle Scholar
  26. 26.
    Stroud, A.H.: Approximate Calculation of Multiple Integrals. Prentic-Hall, Englewood Cliffs (1971)zbMATHGoogle Scholar
  27. 27.
    Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM, Philadelphia (2013)zbMATHGoogle Scholar
  28. 28.
    Wasilkowski, G.W., Woźniakowski, H.: Explicit cost bounds of algorithms for multivariate tensor product problems. J. Complex. 11, 1–56 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Xiu, D.: Efficient collocational approach for parametric uncertainty analysis. Commun. Comput. Phys. 2(2), 293–309 (2007)zbMATHMathSciNetGoogle Scholar
  30. 30.
    Xiu, D.: Numerical Methods for Stochastic Computations. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  31. 31.
    Xiu, D., Hesthaven, J.S.: High-order collocation methods for differential equations with random inputs. SIAM J. Sci. Comput. 27(3), 1118–1139 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Xiu, D., Karniadakis, G.E.: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24(2), 619–644 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Yan, L., Guo, L., Xiu, D.: Stochastic collocation algorithms using 1-minimization. Int. J. UQ 2(3), 279–293 (2012)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.Department of Mathematics and Scientific Computing and Imaging InstituteUniversity of UtahSalt Lake CityUSA

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