On the convergence of local expansions of layer potentials

In a recently developed quadrature method (quadrature by expansion or QBX), it was demonstrated that weakly singular or singular layer potentials can be evaluated rapidly and accurately on surface by making use of local expansions about carefully chosen off-surface points. In this paper, we derive estimates for the rate of convergence of these local expansions, providing the analytic foundation for the QBX method. The estimates may also be of mathematical interest, particularly for microlocal or asymptotic analysis in potential theory

Debye Sources and the Numerical Solution of the Time Harmonic Maxwell Equations, II

In this paper, we develop a new integral representation for the solution of the time harmonic Maxwell equations in media with piecewise constant dielectric permittivity and magnetic permeability in R^3. This representation leads to a coupled system of Fredholm integral equations of the second kind for four scalar densities supported on the material interface. Like the classical Muller equation, it has no spurious resonances. Unlike the classical approach, however, the representation does not suffer from low frequency breakdown. We illustrate the performance of the method with numerical examples.Comment: 36 pages, 5 figure

The VCG Mechanism for Bayesian Scheduling

We study the problem of scheduling m tasks to n selfish, unrelated machines in order to minimize the makespan, in which the execution times are independent random variables, identical across machines. We show that the VCG mechanism, which myopically allocates each task to its best machine, achieves an approximation ratio of O(ln n&frac; ln ln n). This improves significantly on the previously best known bound of O(m/n) for prior-independent mechanisms, given by Chawla et al. [7] under the additional assumption of Monotone Hazard Rate (MHR) distributions. Although we demonstrate that this is tight in general, if we do maintain the MHR assumption, then we get improved, (small) constant bounds for m ≥ n ln n i.i.d. tasks. We also identify a sufficient condition on the distribution that yields a constant approximation ratio regardless of the number of tasks