Nonlocal operators in geometry and nonlocal geometry
Mariel Sáez-Trumper (Pontificia Universidad Católica de Chile, Chile)
In this course, we will discuss a number of geometric notions that can naturally be associated with nonlocal operators. In the first instance, we will study nonlocal notions of area and perimeter. In the second part, we will concentrate on notions of conformal geometry, in which nonlocal operators appear through extension problems (in the style of Cafarelli-Silvestre's work). Finally, we will study possible connections between the two principles.
Numerical methods for nonlocal problems
Abner Salgado (University of Tennessee, USA)
The grand goal of this series of lectures is to, starting from a minimal background, acquaint the audience with the emerging field of numerical analysis for nonlocal problems, both when the nonlocality is spatial as well as temporal. To set the stage, we will begin with a quick review of finite element methods for the approximation of local, elliptic, and second-order equations. To deal with time-dependent problems, we will discuss either the method of lines or Rothe's method, and illustrate it in the case of the heat equation. Local problems, and their analysis, will serve as a roadmap for the treatment of nonlocal ones.
Next, we will deal with the case of the so-called "spectral fractional Laplacian". We will consider several ideas that "localize" this problem. In other words, the numerical approximation of this problem reduces to the (numerical) solution of a series of local problems. Among these approaches, we will mention rational approximations, the Balakrishnan formula, and the Caffarelli-Silvestre extension.
Another class of spatial nonlocal operators is exemplified by the "integral fractional Laplacian", which will be the next topic. As the name suggests, this is a nonlocal operator which is defined by means of a singular integral operator. Its direct numerical approximation, one by means of Fourier transforms, and one by means of regularization, are discussed. Finally, we embark on the study of nonlocal, in time, operators, i.e., fractional derivatives. We discuss the possible definitions of a derivative of fractional order and settle, for the sake of illustration, to study a time-fractional heat equation, where the fractional derivative is of Caputo type. We discuss several approaches for its discretization over uniform and time-varying temporal grids, as well as their error analysis. A recurring theme of these lectures will be the fine interplay between the regularity and the rate of approximation that our numerical schemes are able to afford. We show illustrative simulations, applications, and mention challenging open questions.