Abstract
We focus on the analysis of variance (ANOVA) method for high dimensional approximations employing the Dirac measure. This anchored-ANOVA representation converges exponentially fast for certain classes of functions but the error depends strongly on the anchor points. We employ the concept of “weights per dimension” to construct a theory that leads to the optimal anchor points. We then present examples of a function approximation as well as numerical solutions of the stochastic advection equation up to 500 dimensions using a combination of anchored-ANOVA and polynomial chaos expansions.
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
Preview
Unable to display preview. Download preview PDF.
References
Bieri, M. and Schwab, C.: Sparse high order FEM for elliptic sPDEs, Tech. Report 22, ETH, Switzerland, May 2008
Caflisch, R.E., Morokoff, W. and Owen, A.: Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance, 1, 27–46 (1997)
Dick, J., Sloan, I. H., Wang, X. and Wozniakowski, H.: Liberating the weights, Journal of Complexity, 20, 593–623 (2004)
Genz, A.: A package for testing multiple integration subroutines, Numerical Integration: Recent developments, software and applications, Ed. Reidel, pp. 337–340 (1987)
Griebel, M.: Sparse grids and related approximation schemes for higher dimensional problems, In: Foundations of Computational Mathematics (FoCM05), Santander, L. Pardo, A. Pinkus, E. Suli, and M. Todd, eds., Cambridge University Press, Cambridge, 106–161 (2006)
Griebel, M. and Holtz, M.: Dimension-wise integration of high-dimensional functions with applications to finance, INS Preprint No. 0809, University of Bonn, Germany (2009)
Jardak, M., Su, C.H. and Karniadakis, G.E.: Spectral polynomial chaos solutions of the stochastic advection equation, Journal of Scientific Computing, 17, 319–338 (2002)
Larcher, G.,Leobacher, G. and Scheicher, K.: On the tractability of the Brownian bridge algorithm, Journal of Complexity, 19, 511–528 (2003)
Larcher, G.,Leobacher, G. and Scheicher, K.: The dimension distribution and quadrature test functions, Statistica Sinica, 13, 1–17 (2003)
Paskov, S.H. and Traub, J.F.: Faster valuation of financial derivatives, Journal of Portfolio Management, 22, 113–120 (1995)
Sloan, I.H. and Wozniakowski, H.: When are Quasi-Monte Carlo algorithms efficient for high dimensional integrals? Journal of Complexity, 14, 1–33 (1998)
Wang, X. and Fang, K.-T.: The effective dimension and quasi-Monte Carlo integration, Journal of Complexity, 19, 101–124 (2003)
Acknowledgements
This work was supported by an OSD/AFOSR MURI grant and also by a DOE Computational Mathematics grant.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer Berlin Heidelberg
About this paper
Cite this paper
Zhang, Z., Choi, M., Karniadakis, G.E. (2011). Anchor Points Matter in ANOVA Decomposition. In: Hesthaven, J., Rønquist, E. (eds) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol 76. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15337-2_32
Download citation
DOI: https://doi.org/10.1007/978-3-642-15337-2_32
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15336-5
Online ISBN: 978-3-642-15337-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)