Skip to main content
×
Home
    • Aa
    • Aa

Error Splitting Preservation for High Order Finite Difference Schemes in the Combination Technique

  • Christian Hendricks (a1), Matthias Ehrhardt (a1) and Michael Günther (a1)
Abstract
Abstract

In this paper we introduce high dimensional tensor product interpolation for the combination technique. In order to compute the sparse grid solution, the discrete numerical subsolutions have to be extended by interpolation. If unsuitable interpolation techniques are used, the rate of convergence is deteriorated. We derive the necessary framework to preserve the error structure of high order finite difference solutions of elliptic partial differential equations within the combination technique framework. This strategy enables us to obtain high order sparse grid solutions on the full grid. As exemplifications for the case of order four we illustrate our theoretical results by two test examples with up to four dimensions.

Copyright
Corresponding author
*Corresponding author. Email addresses: christian.hendricks@gmx.de (C. Hendricks) ehrhardt@math.uni-wuppertal.de (M. Ehrhardt) guenther@math.uni-wuppertal.de (M. Günther)
References
Hide All
[1] De Boor C., A Practical Guide to Splines, Springer, (1978).
[2] Bramble J. H. and Hubbard B. E., New monotone type approximations for elliptic problems, Math. Comp., 18 (1964), pp. 349367.
[3] Bungartz H. J. and Griebel M., Sparse grids, Cambridge University Press, (2004), pp. 1123.
[4] Bungartz H. J. and Griebel M. and Röschke D. and Zenger C., Pointwise convergence of the combination technique for Laplace's equation, East-West J. Numer. Math., 2 (1994), pp. 2145.
[5] Bungartz H. J. and Griebel M. and Rüde U., Extrapolation, combination, and sparse grid techniques for elliptic boundary value problems, Comput. Methods Appl. Mech. Engrg., 116 (1994), pp. 243252.
[6] Ciarlet P. G., Discrete maximum principle for finite-difference operators, Aequationes Mathematicae, 4(3) (1970), pp. 338352.
[7] Gaikwad A. and Toke I. M., GPU based sparse grid technique for solving multidimensional options pricing PDEs, The Workshop on High Performance Computational Finance, (2009), pp. 19.
[8] Garcke J., Sparse Grid Tutorial, Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Mathematical Sciences Institute, (2006).
[9] Griebel M. and Hamaekers J., MSparse grids for the Schrödinger equation, ESAIM-Math. Model. Num., 41(2) (2007), pp. 215247.
[10] Griebel M. and Schneider M. and Zenger C., A combination technique for the solution of sparse grid problems, IMACS Elsevier, Iterative Methods in Linear Algebra, (1992), pp. 263281.
[11] Griebel M. and Thurner V., The efficient solution of fluid dynamics problems by the combination technique, Int. J. Numer. Method H., 5(3) (1995), pp. 251269.
[12] Hall C. A., On Error Bounds for Spline Interpolation, J. Approx. Theory, 1968, pp. 209218.
[13] Hendricks C. and Ehrhardt M., Evaluating the effects of changing market parameters and policy implications in the German electricity market, J. Energy Markets, 7(2) (2014).
[14] Leentvaar C. C. W. and Oosterlee C. W., Pricing multi-asset options with sparse grids and fourth order finite differences, Numerical Mathematics and Advanced Applications, Springer, (2006), pp. 975983.
[15] Reisinger C., Analysis of linear difference schemes in the sparse grid combination technique, IMA J. Numer. Anal., 33(2) (2013), pp. 544581.
[16] Reisinger C., Numerische Methoden für hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben, PhDthesis, Ruprecht-Karls-Universität, 2004.
[17] Zenger C., Sparse grids, Parallel Algorithms for Partial Differential Equations, 31 (1991), pp. 241251.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 16 *
Loading metrics...

Abstract views

Total abstract views: 75 *
Loading metrics...

* Views captured on Cambridge Core between 20th June 2017 - 24th October 2017. This data will be updated every 24 hours.