Optimal Control, Numerics, and Applications of Fractional PDEs
Abstract
This article provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems.
Keywords
Cite
@article{arxiv.2106.13289,
title = {Optimal Control, Numerics, and Applications of Fractional PDEs},
author = {Harbir Antil and Thomas S. Brown and Ratna Khatri and Akwum Onwunta and Deepanshu Verma and Mahamadi Warma},
journal= {arXiv preprint arXiv:2106.13289},
year = {2021}
}