Past

Standard finite element or boundary element methods for high frequency scattering problems, with piecewise polynomial approximation spaces, suffer from the limitation that the number of degrees of freedom required to achieve a prescribed level of accuracy grows at least linearly with respect to the frequency. Here we present a new boundary element method for which, by including in the approximation space the products of plane wave basis functions with piecewise polynomials supported on a graded mesh, we can demonstrate a computational cost that grows only logarithmically with respect to the frequency.

The main goal of this talk is to demonstrate connections between the following three big areas of research: the theory of cubature formulas (numerical integration), the discrepancy theory, and nonlinear approximation. First, I will discuss a relation between results on cubature formulas and on discrepancy. In particular, I'll show how standard in the theory of cubature formulas settings can be translated into the discrepancy problem and into a natural generalization of the discrepancy problem. This leads to a concept of the r-discrepancy. Second, I'll present results on a relation between construction of an optimal cubature formula with m knots for a given function class and best nonlinear m-term approximation of a special function determined by the function class. The nonlinear m-term approximation is taken with regard to a redundant dictionary also determined by the function class. Third, I'll give some known results on the lower and the upper estimates of errors of optimal cubature formulas for the class of functions with bounded mixed derivative. One of the important messages (well known in approximation theory) of this talk is that the theory of discrepancy is closely connected with the theory of cubature formulas for the classes of functions with bounded mixed derivative.