After successful completion of the course, students are able to recognize Volterra and Fredholm equations of first and second type, understand how the spectral properties of integral operators can lead to well-conditioned systems, understand how the Fast Multipole Method can perform a matrix-vector multiply within nearly linear complexity, understand the foundations of hierarchical matrices.

The main objective of this class is twofold: on one hand, to establish a connection between boundary value problems and boundary integral equations; on the other, to study discretization techniques for boundary integral operators. We shall begin by covering some classical results of potential theory (single and double layer potential, jump conditions) and see how they can transform a boundary value problem into an integral equation; we will restrict our attention to the Laplace and Helmholtz problems. We will discuss numerical discretization techniques, and focus on fast multipole methods; finally, we will see how those methods have lead to the concept of hierarchical matrices. Since this is an advanced class at the forefront of research, a tailored set of notes by the instructor will be provided.

The course will be mostly taught through chalk lectures. Exercises will be assigned during class and discussed during exercise sessions.

Oral examination.

Basic concepts of functional analysis, Lebesgue integration theory, and numerical linear algebra.