Model reduction with the reduced basis method and sparse grids

Benjamin Peherstorfer, Stefan Zimmer, Hans Joachim Bungartz

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

The reduced basis (RB) method has become increasingly popular for problems where PDEs have to be solved for varying parameters in order to evaluate a parameter-dependent output function. The idea of the RB method is to compute the solution of the PDE for varying parameters in a problem-specific low-dimensional subspace X N of the high-dimensional finite element space. We will discuss how sparse grids can be employed within the RB method or to circumvent the RB method altogether. One drawback of the RB method is that the solvers of the governing equations have to be modified and tailored to the reduced basis. This is a severe limitation of the RB method. Our approach interpolates the output function s on a sparse grid. Thus, we compute the respond to a new parameter with a simple function evaluation. No modification or in-depth knowledge of the governing equations and its solver are necessary. We present numerical examples to show that we obtain not only competitive results with the interpolation on sparse grids but that we can even be better than the RB approximation if we are only interested in a rough but very fast approximation.

Original languageEnglish (US)
Title of host publicationSparse Grids and Applications
EditorsJochen Garcke, Michael Griebel
Pages223-242
Number of pages20
DOIs
StatePublished - Mar 5 2013

Publication series

NameLecture Notes in Computational Science and Engineering
Volume88
ISSN (Print)1439-7358

Fingerprint

Reduced Basis Methods
Sparse Grids
Model Reduction
Function evaluation
Interpolation
Governing equation
Interpolate
Output
Evaluation Function
Approximation
Rough
High-dimensional
Subspace
Finite Element
Numerical Examples
Necessary
Dependent
Evaluate

ASJC Scopus subject areas

  • Modeling and Simulation
  • Engineering(all)
  • Discrete Mathematics and Combinatorics
  • Control and Optimization
  • Computational Mathematics

Cite this

Peherstorfer, B., Zimmer, S., & Bungartz, H. J. (2013). Model reduction with the reduced basis method and sparse grids. In J. Garcke, & M. Griebel (Eds.), Sparse Grids and Applications (pp. 223-242). (Lecture Notes in Computational Science and Engineering; Vol. 88). https://doi.org/10.1007/978-3-642-31703-3-11

Model reduction with the reduced basis method and sparse grids. / Peherstorfer, Benjamin; Zimmer, Stefan; Bungartz, Hans Joachim.

Sparse Grids and Applications. ed. / Jochen Garcke; Michael Griebel. 2013. p. 223-242 (Lecture Notes in Computational Science and Engineering; Vol. 88).

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Peherstorfer, B, Zimmer, S & Bungartz, HJ 2013, Model reduction with the reduced basis method and sparse grids. in J Garcke & M Griebel (eds), Sparse Grids and Applications. Lecture Notes in Computational Science and Engineering, vol. 88, pp. 223-242. https://doi.org/10.1007/978-3-642-31703-3-11
Peherstorfer B, Zimmer S, Bungartz HJ. Model reduction with the reduced basis method and sparse grids. In Garcke J, Griebel M, editors, Sparse Grids and Applications. 2013. p. 223-242. (Lecture Notes in Computational Science and Engineering). https://doi.org/10.1007/978-3-642-31703-3-11
Peherstorfer, Benjamin ; Zimmer, Stefan ; Bungartz, Hans Joachim. / Model reduction with the reduced basis method and sparse grids. Sparse Grids and Applications. editor / Jochen Garcke ; Michael Griebel. 2013. pp. 223-242 (Lecture Notes in Computational Science and Engineering).
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