Obstacle Problem Using Random Minimizing Movement

This repository contains the code used for the numerical simulations presented in our paper Mini-Batch Descent in Semiflows. The code implements the Random Minimizing Movement scheme applied to solving the obstacle problem.

Problem Description

In our paper, we explore the application of Random Minimizing Movement, a method where in each iteration of the minimizing movement, a randomly selected sub-functional of the total functional is minimized rather than following the full functional. This approach is illustrated using the obstacle problem, comparing the obtained solutions with real solutions.

Repository Structure

• utils.py: Contains several functions. Among them are:

  • smoothmax : Used to approximate the max function
  • two_mountains : 2d function considered as the obstacle. It has the shape of two mountains
  • obs_1d : 1d function considered as the obstacle for the 1d case.
  • make_gif : Generate gifs
  • batches_gen : Generate a set that contains sets randomly generated with a certain probability.
  • unity_partition_1d : Partition of the unity 1d, defined in the interval [-1,1]
  • unity_partition_2d : Partition of the unity 2d, defined in the interval [-1,1]x[-1,1]

• solvers.py: Contain the solvers. Namely,

  • solver_op : Obstacle problem solver (inspired by DyCon-blog)
  • sol_one_realization : Solve one realization of the random minimizing movement
  • solver_avg_mb_op : Compute the average of a given number of realizations of the random minimizing movement

• mini_batch_obstacle_problem.ipynb: Jupyter Notebooks with examples demonstrating the code.

References

For more details on the theory and numerical results, please refer to Section 4.3 of our paper on arXiv: Mini-Batch Descent in Semiflows.