Local Algorithms for Nonlocal Problems in Wave Theory

Although many mathematical models in wave theory lead to hyperbolic initial-boundary value problems which are inherently local due to the finite domain-of-dependence of the solution at any point on past values, there are also examples where nonlocal operators, in particular space-time integral op- erators, arise. A primary example is the radiation boundary condition needed to truncate an unbounded domain to a finite one to enable numerical solu- tions, as well as closely-related operators for unidirectional propagation. In this work we show how to leverage results from rational function approxima- tion theory to construct spectrally-convergent local algorithms for evaluating such operators. For an important class of problems - systems equivalent to the scalar wave equation, such as acoustics or Maxwell’s equations in ho- mogeneous, isotropic media, we will explain the construction, analysis, and implementation of our complete radiation boundary conditions, which are in a certain sense optimal. We will also discuss other applications of these approximation methods to inverse problems and fractional differential equa- tions, as well as the fundamental barriers to extending the successful methods to other systems.

Contatto:
luca.formaggia@polimi.it