Abstract
In this article, we show that the multilevel Monte Carlo method for elliptic stochastic partial differential equations is a sparse grid approximation. By using this interpretation, the method can straightforwardly be generalized to any given quadrature rule for high dimensional integrals like the quasi Monte Carlo method or the polynomial chaos approach. Besides the multilevel quadrature for approximating the solution’s expectation, a simple and efficient modification of the approach is proposed to compute the stochastic solution’s variance. Numerical results are provided to demonstrate and quantify the approach.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This holds under the assumptions that the dimensions of V j (1) and V j (2) scale like geometric sequences.
- 2.
Error estimates in respectively L 2(D) and L 1(D) are derived by straightforward modifications, yielding the convergence rate 4 − ℓ. Then, the error analysis of the multilevel quadrature can be performed with respect to these norms, provided that the precision of the quadrature (10) is also chosen as \({\epsilon }_{\mathcal{l}} = {4}^{-\mathcal{l}}\).
- 3.
There holds the identity \({V }_{j}^{(1)}\,=\,{P}_{j}\left ({L}_{\rho }^{2}\left (\square ; {H}_{0}^{1}(D)\right )\right )\) where \({P}_{j}\,:\,{L}_{\rho }^{2}\left (\square ; {H}_{0}^{1}(D)\right ) \rightarrow {L}_{\rho }^{2}\left (\square ; {\mathcal{S}}_{j}^{p}(D)\right )\) is the Galerkin projection with respect to the bilinear form \(A : {L}_{\rho }^{2}\left (\square ; {H}_{0}^{1}(D)\right ) \times {L}_{\rho }^{2}\left (\square ; {H}_{0}^{1}(D)\right ) \rightarrow \mathbb{R}\) given by
$$A(v,w) :={ \int }_{\square }{\int }_{D}\alpha (\mathbf{x},\mathbf{y}){\nabla }_{\mathbf{x}}v(\mathbf{x},\mathbf{y}){\nabla }_{\mathbf{x}}w(\mathbf{x},\mathbf{y})\rho (\mathbf{y})\text{ d}\mathbf{x}\text{ d}\mathbf{y}.$$This bilinear form stems from the weak formulation of the parametric diffusion problem (6) in the Bochner space \({L}_{\rho }^{2}\left (\square ; {H}_{0}^{1}(D)\right )\). The equivalence of this weak definition to the pointwise definition (15) follows immediately from the analyticity of the solutions to (6) in the parameter y.
References
A. Barth, C. Schwab, and N. Zollinger. Multi-level monte carlo finite element method for elliptic PDEs with stochastic coefficients. Numer. Math., 119(1):123–161, 2011.
H.-J. Bungartz and M. Griebel. Sparse grids. Acta Numer., 13:147–269, 2004.
E. W. Cheney and W. A. Light. Approximation Theory in Tensor Product Spaces. Springer, Berlin, 1980.
A. Cohen, R. DeVore, and C. Schwab. Convergence rates of best N-term Galerkin approximations for a class of elliptic sPDEs. Found. Comput. Math., 10:615–646, 2010.
J. Diestel and J. J. Uhl, Jr. Vector Measures. American Mathematical Society, Providence, 1979.
P. Frauenfelder, C. Schwab, and R. Todor. Finite elements for elliptic problems with stochastic coefficients. Comput. Methods Appl. Mech. Engrg., 194(2–5):205–228, 2005.
T. Gerstner and M. Griebel. Numerical integration using sparse grids. Numer. Algorithms, 18:209–232, 1998.
T. Gerstner and S. Heinz. Dimension- and time-adaptive multilevel Monte Carlo methods. In J. Garcke and M. Griebel, editors, Sparse Grids and Applications, LNCSE. Springer, 107–120, 2012.
R. Ghanem and P. Spanos. Stochastic finite elements: a spectral approach. Springer-Verlag, New York, 1991.
M. Giles. Multilevel Monte Carlo path simulation. Operations Research, 56(3):607–617, 2008.
M. Giles and B. Waterhouse. Multilevel quasi-Monte Carlo path simulation. Radon Series Comp. Appl. Math., 8:1–18, 2009.
M. Griebel and H. Harbrecht. On the construction of sparse tensor product spaces. Preprint 497, Berichtsreihe des SFB 611, Universität Bonn, 2011. (to appear in Math. Comput.).
H. Harbrecht, M. Peters, and R. Schneider. On the low-rank approximation by the pivoted Cholesky decomposition. Appl. Numer. Math., 62(4):428–440, 2012.
M. Hegland. Adaptive sparse grids. ANZIAM J., 44:C335–C353, 2002.
S. Heinrich. The multilevel method of dependent tests. In Advances in stochastic simulation methods (St. Petersburg, 1998), Stat. Ind. Technol., pages 47–61. Birkhäuser Boston, Boston, MA, 2000.
S. Heinrich. Multilevel Monte Carlo methods. In Lect. Notes in Large Scale Scientific Computing, pages 58–67, London, 2001. Springer-Verlag.
M. Loève. Probability theory. I+II, volume 45 of Graduate Texts in Mathematics. Springer-Verlag, New York, 4th edition, 1977.
H. Niederreiter. Random number generation and quasi-Monte Carlo methods. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992.
P. Protter. Stochastic integration and differential equations: a new approach. Springer, Berlin, 3rd edition, 1995.
C. Schwab and R. Todor. Karhunen-Loève approximation of random fields by generalized fast multipole methods. J. Comput. Phys., 217:100–122, 2006.
R. Todor and C. Schwab. Convergence rates for sparse chaos approximations of elliptic problems with stochastic coefficients. IMA J. Numer. Anal., 27(2):232–261, 2007.
C. Zenger. Sparse grids. In W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations, volume 31 of Notes on Numerical Fluid Mechanics, pages 241–251, Braunschweig/Wiesbaden, 1991. Vieweg-Verlag.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Harbrecht, H., Peters, M., Siebenmorgen, M. (2012). On Multilevel Quadrature for Elliptic Stochastic Partial Differential Equations. In: Garcke, J., Griebel, M. (eds) Sparse Grids and Applications. Lecture Notes in Computational Science and Engineering, vol 88. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31703-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-31703-3_8
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31702-6
Online ISBN: 978-3-642-31703-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)