Optimization in PDE computation

Often in times in practice, the partial differential equations people encounter present multi-scale structure that are numerically hard to capture, or have more dimensions than one could afford to solve. When those happen, the traditional numerical PDE methods, including finite difference, finite element or spectral types of method always fail to behave to its optimal. On the other hand, the advancements took place over the years in optimization and compress sensing provide alternative means to handle data. In this talk, we give three examples on utilizing optimization techniques for PDE computation: 1. low energy + sparsity decomposition for hyperbolic equations; 2. dealing with high dimensional randomness in the media for elliptic equations; 3. coupling radiative transport equation (high dimensional problem) with heat equation (one dimensional problem).