The solution of stochastic PDEs can be approximated using Legendre polynomial expansions. For a class of elliptic PDEs, the coefficient for each polynomial is known to be bounded by an exponentially decaying function. In this project we learn the decay rates and set the coefficients equal to these bounds (or within a sparse deviation) in order to approximate the solution using a minimal number of data-points.
The reference for this work:
Wentz, Jacqueline, and Alireza Doostan. "GenMod: A generative modeling approach for spectral representation of PDEs with random inputs." Journal of Computational Physics 472 (2023): 111691.
To run the code for the examples in this paper run 'scripts/run_paper_optimization.py'. This code is set up to run the first realization for each of the three examples and sample sizes. The output is a .csv file containing the coefficient values. The code can easily be modified to run all realizations presented in the paper. The 'environment.yml' file should be used to set up the correct python environment for runing the code.