Gregory Beylkin, Department of Applied Mathematics, University of Colorado at Boulder

Fast algorithms for adaptive application of integral operators in high dimensions

Friday January 19th, 4pm, Phillips 383 (refreshments served in Phillips 330 starting at 3:30)

Abstract: The talk will describe an adaptive multiresolution algorithm for applying integral operators with a wide class of radially symmetric kernels in dimensions one, two and three. In dimensions two and three our approach is based on approximating radial kernels (for any finite but arbitrary accuracy) by a sum of Gaussians. Specifically, we will consider operators of the class (-\Delta+\mu^{2}I)^{-\alpha}, where \mu\geq 0 and 0 <\alpha< 3/2, and use the Poisson's and Schrodinger's equations as illustrations.

In physics, chemistry and other applied fields, many important problems may be formulated using integral equations, typically involving Green's functions as their kernels. Often such formulations are preferable to those via partial differential equations (PDEs). For example, evaluating the integral expressing the solution of the Poisson equation in free space (the convolution of the Green's function with the mass or charge density) avoids issues associated with the high condition number of a PDE formulation. Since many physically significant operators depend only on the distance between interacting objects, the operators such as (-\Delta+\mu^{2}I)^{-\alpha} and certain singular operators, such as the projector on divergence-free functions, fall into the class of operators for which our approach yields a fast adaptive algorithm.

We will also briefly discuss an extension of our approach to higher dimensions.