Research Highlight
Discrete-Least-Squares Reduced Basis Method for Parametric PDEs
Achievement
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. 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.
Figure 1: Offline cost comparison. The error versus the number of finite element solves for the standard DLS and reduced-basis DLS.
Figure 2: Online cost comparison. The error versus the CPU time for the standard DLS and reduced-basis DLS.
Publication:
M. Gunzburger, M. Schneier, C. Webster, and G. Zhang, An improved discrete least-squares/reduced-basis method for parameterized elliptic PDEs, Journal of Scientific Computing, 81 (1), pp. 76-91, 2019.
Q. Guan, M. Gunzburger, C. Webster and G. Zhang, Reduced basis methods for nonlocal diffusion problems with random input data, Computer Methods in Applied Mechanics and Engineering, 317 (2017), pp. 746-770.