Numerical simulations are ubiquitous in the world, but sometimes it can be hard to understand the presence of numerical artifacts or the validity of the model.
In this talk, we discuss two recent examples of using calculus of variations techniques to build provable numerical simulations/models.
The first example is from optimal transport. I will discuss joint work with Hicham Kouhkouh (Graz) on a fast algorithm for computing the Wasserstein Distance inspired from the Jacobs-Leger Back and Forth Algorithm.
While the Back and Forth Algorithm has no convergence proof, we are able to give a convergence proof for our algorithm under some general conditions.
The second example is from the study of nematic liquid crystals. I will discuss some joint work with Amit Acharya (Carnegie Mellon), Irene Fonseca (Carnegie Mellon), and Kerrek Stinson (Uni Bonn) on a new director field model for liquid crystal defects. We will compare this model to the more standard Q-tensor models and show that under suitable conditions the model is a computable regularization of defects in the liquid crystal.
Finally, I will discuss the many open questions and ongoing work related to these problems