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An Improved Discrete Least-Squares/Reduced-Basis Method for Parameterized Elliptic PDEs

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Abstract

It is shown that the computational efficiency of the discrete least-squares (DLS) approximation of solutions of stochastic elliptic PDEs is improved by incorporating a reduced-basis method into the DLS framework. In particular, we consider stochastic elliptic PDEs with an affine dependence on the random variable. The goal is to recover the entire solution map from the parameter space to the finite element space. To this end, first, a reduced-basis solution using a weak greedy algorithm is constructed, then a DLS approximation is determined by evaluating the reduced-basis approximation instead of the full finite element approximation. The main advantage of the new approach is that one only need apply the DLS operator to the coefficients of the reduced-basis expansion, resulting in huge savings in both the storage of the DLS coefficients and the online cost of evaluating the DLS approximation. In addition, the recently developed quasi-optimal polynomial space is also adopted in the new approach, resulting in superior convergence rates for a wider class of problems than previous analyzed. Numerical experiments are provided that illustrate the theoretical results.

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Authors and Affiliations

  1. Department of Scientific Computing, Florida State University, Tallahassee, FL, 32306-4120, USA

    Max Gunzburger & Michael Schneier

  2. Computational and Applied Mathematics Division, Oak Ridge National Laboratory, Oak Ridge, TN, 37831-6164, USA

    Clayton Webster & Guannan Zhang

Authors
  1. Max Gunzburger
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  2. Michael Schneier
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  3. Clayton Webster
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  4. Guannan Zhang
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Corresponding author

Correspondence to Max Gunzburger.

Additional information

This material is based upon work supported in part by the U.S. Air Force of Scientific Research under Grants 1854-V521-12 and FA9550-15-1-0001; the U.S. Defense Advanced Research Projects Agency, Defense Sciences Office under contract and Award HR0011619523 and 1868-A017-15; the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under contracts and Awards ERKJ259, ERKJ320, DE-SC0009324, and DE-SC0010678; by the National Science Foundation under the Award Numbers 1620027 and 1620280.

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Gunzburger, M., Schneier, M., Webster, C. et al. An Improved Discrete Least-Squares/Reduced-Basis Method for Parameterized Elliptic PDEs. J Sci Comput 81, 76–91 (2019). https://doi.org/10.1007/s10915-018-0661-6

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  • DOI: https://doi.org/10.1007/s10915-018-0661-6

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