This python package reproduces numerical experiments from the paper
J. Calder, N. Drenska, D. Mosaphir Numerical solution of a PDE arising from prediction with expert advice, arXiv preprint, 2024.
The code numerically solves a degenerate elliptic PDE arising in the problem of prediction from expert advise in relatively high dimensions. There is code for solving the PDE on a regular dense grid, which works for low dimensions, and on a sparse grid that works in relatively high dimensions.
The code for the regular dense grid is mainly contained in solvers.py. The scripts test_2d.py, test_3d.py, and test_4d.py show how to run the solvers and generate the figures from the paper for the 2,3 and 4 expert problems (the n expert problem involves solving a PDE in n-1 dimensions). The convergence_rates.py script generates plots showing the convergence rates in these cases, where the solution of the PDE is known. The solutions are stored in the folder solutions/ for several different dimensions and grid resolutions. These will be loaded automatically by the code, instead of computing them, when available.
Note that many of these solutions take up far more space than the 100MB limit in github, so they have not been included in the repository. You can reference the .gitignore file for a list of files that are not included. The authors are happy to provide any of these files directly to anyone upon request.
The code for the sparse sector grid solvers is mainly contained in sparse_solvers.py. The script sparse_test.py shows how to run the solver and generate the figures from the paper. The script num_grid_points.py prints out the number of grid points used by the sparse solver, compared to what the dense solver would use. The file plots.py contains plotting functionality used by many scripts.
The folder meshes contains the sparse sector meshes for many different dimensions and grid resolutions, since they are computationally intensive to compute. The folder sparse_solutions contains many sparse solutions as well as the stencils for computing directional second derivatives, since they are very computationally intensive to compute. All stored files are automatically loaded by the code when available.
As before, there are many sparse solutions that take up to 10GB to store and cannot be included in the repository. See the .gitignore file for a list of additional files that are available upon request.
Any questions should be sent to Jeff Calder (jwcalder@umn.edu).