Abstract
We design and analyze algorithms for the efficient sensitivity computation of eigenpairs of parametric elliptic self-adjoint eigenvalue problems on high-dimensional parameter spaces. We quantify the analytic dependence of eigenpairs on the parameters. For the efficient approximate evaluation of parameter sensitivities of isolated eigenpairs on the entire parameter space we propose and analyze a sparse tensor spectral collocation method on an anisotropic sparse grid in the parameter domain. The stable numerical implementation of these methods is discussed and their error analysis is given. Applications to parametric elliptic eigenvalue problems with infinitely many parameters arising from elliptic differential operators with random coefficients are presented.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
I. BABUšKA AND B. Q. GUO, Regularity of the solution of elliptic problems with piecewise analytic data. I. Boundary value problems for linear elliptic equation of second order, SIAM J. Math. Anal., 19 (1988), pp. 172–203.
I. BABUšKA AND J. OSBORN, Eigenvalue problems, in Handb. Numer. Anal., Vol. II, Handb. Numer. Anal., II, North-Holland, Amsterdam, 1991, pp. 641–787.
I. BABUšKA, R. TEMPONE, AND G. E. ZOURARIS, Galerkin finite element approximations of stochastic elliptic partial differential equations, SIAM J. Num. Anal., 42 (2002), pp. 800–825.
I. BABUšKA, F. NOBILE, AND R. TEMPONE, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Num. Anal., 45 (2007), pp. 1005–1034.
J. BäCK, F. NOBILE, L. TAMELLINI, AND R. TEMPONE, Stochastic Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison, Tech. Report 09-33, ICES, 2009.
M. BIERI, A sparse composite collocation finite element method for elliptic sPDEs, Tech. Report 2009-08, Seminar for Applied Mathematics, ETH Zürich, 2009. SIAM Journ. Numer. Anal. (to appear 2011). Available via http://www.sam.math.ethz.ch/reports/2009/08.
M. BIERI, Sparse tensor discretizations of elliptic PDEs with random input data, PhD thesis, ETH Zürich, 2009. Diss ETH No. 18598 Available via http://e-collection.ethbib.ethz.ch/.
M. BIERI, R. ANDREEV, AND CH. SCHWAB, Sparse tensor discretization of elliptic spdes, SIAM J. Sci. Comput., 31 (2009), pp. 4281–4304.
M. BIERI AND CH. SCHWAB, Sparse high order FEM for elliptic sPDEs, Comp. Meth. Appl. Mech. Engrg., 198 (2009), pp. 1149–1170.
D. BRAESS, Finite Elemente, Springer, Berlin, 3rd ed., 2002.
HANS-JOACHIM BUNGARTZ AND MICHAEL GRIEBEL, Sparse grids, Acta Numer., 13 (2004), pp. 147–269.
A. COHEN, R. DEVORE, AND CH. SCHWAB, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs, Tech. Report 2010-03, Seminar for Applied Mathematics, ETH Zürich, 2010.
ALBERT COHEN, RONALD DEVORE, AND CHRISTOPH SCHWAB, Convergence rates of best n-term galerkin approximations for a class of elliptic spdes, Found. Comput. Math., 10 (2010), pp. 615–646. 10.1007/s10208-010-9072-2.
W. DAHMEN, T. ROHWEDDER, R. SCHNEIDER, AND A. ZEISER, Adaptive eigenvalue computation: complexity estimates, Numer. Math., 110 (2008), pp. 277–312.
P. J. DAVIS, Interpolation and approximation, Introductions to higher mathematics, Blaisdell Publishing Company, 1963.
J. FOO, X. WAN, AND G.E. KARNIADAKIS, The multi-element probabilistic collocation method: analysis and simulation, Journal of Computational Physics, (2008), pp. 9572–9595.
P. FRAUENFELDER, CH. SCHWAB, AND R.-A. TODOR, Finite elements for elliptic problems with stochastic coefficients, Comp. Meth. Appl. Mech. Engrg., 194 (2005), pp. 205–228.
W. GAUTSCHI, Orthogonal polynomials. Computation and Approximation, Numer. Math. Sci. Comput., Oxford University Press Inc., 2004.
R. GEUS, The Jacobi-Davidson algorithm for solving large sparse symmetric eigenvalue problems with application to the design of accelerator cavities, PhD thesis, ETH Zürich, 2002. Diss. Nr. 14734.
ROGER G. GHANEM AND POL D. SPANOS, Stochastic Finite Elements, a Spectral Approach, Dover Publications Inc., New York, revised ed., 2003.
CLAUDE JEFFREY GITTELSON, Adaptive Galerkin Methods for Parametric and Stochastic Operator Equations, PhD thesis, ETH Zürich, 2011. ETH Dissertation No. 19533.
CLAUDE JEFFREY GITTELSON, An adaptive stochastic Galerkin method, Tech. Report 2011-11, Seminar for Applied Mathematics, ETH Zürich, 2011.
CLAUDE JEFFREY GITTELSON, Adaptive stochastic Galerkin methods: Beyond the elliptic case, Tech. Report 2011-12, Seminar for Applied Mathematics, ETH Zürich, 2011.
CLAUDE JEFFREY GITTELSON, Stochastic Galerkin approximation of operator equations with infinite dimensional noise, Tech. Report 2011-10, Seminar for Applied Mathematics, ETH Zürich, 2011.
G. GOLUB AND C. VAN LOAN, Matrix computations, The Johns Hopkins University Press, London, 1996.
W. HACKBUSCH, On the computation of approximate eigenvalues and eigenfunctions of elliptic operators by means of a multi-grid method, SIAM J. Numer. Anal., 16 (1979), pp. 201–215.
A. HENROT, Extremum Problems for Eigenvalues of Elliptic Operators, vol. 8 of Frontiers in Mathematics, Birkhäuser Basel, 2006.
M. HERVé, Analyticity in Infinite Dimensional Spaces, vol. 10 of De Gruyter studies in mathematics, Walter de Gruyter, 1989.
VIET-HA HOANG AND CHRISTOPH SCHWAB, Analytic regularity and gpc approximation for parametric and random 2nd order hyperbolic PDEs, Tech. Report 2010-19, Seminar for Applied Mathematics, ETH Zürich, 2010.
VIET-HA HOANG AND CHRISTOPH SCHWAB, Sparse tensor Galerkin discretization for parametric and random parabolic PDEs I: Analytic regularity and gpc-approximation, Tech. Report 2010-11, Seminar for Applied Mathematics, ETH Zürich, 2010.
L. HöRMANDER, An Introduction to Complex Analysis in Several Variables, The University Series in Higher Mathematics, D. van Nostrand Company, 1st ed., 1966.
L. HöRMANDER, An Introduction to Complex Analysis in Several Variables, North Holland Mathematical Library, North Holland, 3rd ed., 1990.
T. KATO, Perturbation theory for linear operators, vol. 132 of Grundlehren Math. Wiss., Springer Berlin, Heidelberg, New-York, 2 ed., 1976.
XIANG MA AND NICHOLAS ZABARAS, An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations, J. Comput. Phys., 228 (2009), pp. 3084–3113.
F. NOBILE, R. TEMPONE, AND C.G. WEBSTER, An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Num. Anal., 46 (2008), pp. 2411–2442.
F. NOBILE, R. TEMPONE, AND C.G. WEBSTER, A sparse grid stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Num. Anal., 46 (2008), pp. 2309–2345.
M. REED AND B. SIMON, Methods of modern mathematical physics. I. Functional analysis, Academic Press, New York, 1972.
M. REED AND B. SIMON, Methods of modern mathematical physics. IV. Analysis of operators, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1978.
F. RELLICH, Perturbation theory of eigenvalue problems, Notes on mathematics and its applications, Gordon and Breach, New York, London, Paris, 1969.
CH. SCHWAB AND R.-A. TODOR, Karhunen-Loève approximation of random fields by generalized fast multipole methods, Journal of Computational Physics, 217 (2006), pp. 100–122.
S.A. SMOLYAK, Quadrature and interpolation formulas for tensor products of certain classes of functions, Sov. Math. Dokl., 4 (1963), pp. 240–243.
D. C. SORENSEN, Numerical methods for large eigenvalue problems, Acta Numer., 11 (2002), pp. 519–584.
R.A. TODOR, Sparse Perturbation algorithms for elliptic PDE’s with stochastic data, PhD thesis, ETH Zürich, 2005. Diss. Nr. 16192.
R.-A. TODOR AND CH. SCHWAB, Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients, IMA J. Num. Anal., 27 (2007), pp. 232–261.
G. W. WASILKOWSKI AND H. WOźNIAKOWSKI, Explicit cost bounds of algorithms for multivariate tensor product problems, J. Complexity, 11 (1995), pp. 1–56.
DONGBIN XIU, Fast numerical methods for stochastic computations: a review, Commun. Comput. Phys., 5 (2009), pp. 242–272.
DONGBIN XIU AND JAN S. HESTHAVEN, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118–1139 (electronic).
DONGBIN XIU AND GEORGE EM KARNIADAKIS, Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos, Comput. Methods Appl. Mech. Engrg., 191 (2002), pp. 4927–4948.
DONGBIN XIU AND GEORGE EM KARNIADAKIS, The Wiener-Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619–644 (electronic).
DONGBIN XIU AND DANIEL M. TARTAKOVSKY, Numerical methods for differential equations in random domains, SIAM J. Sci. Comput., 28 (2006), pp. 1167–1185 (electronic).
Acknowledgements
Supported by SNF grant PDFMP2-127034/1 and by ERC AdG grant STAHDPDE 247277.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Andreev, R., Schwab, C. (2012). Sparse Tensor Approximation of Parametric Eigenvalue Problems. In: Graham, I., Hou, T., Lakkis, O., Scheichl, R. (eds) Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering, vol 83. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22061-6_7
Download citation
DOI: https://doi.org/10.1007/978-3-642-22061-6_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22060-9
Online ISBN: 978-3-642-22061-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)