Fast integral equation methods for the modified Helmholtz equation

We present a collection of integral equation methods for the solution to the two-dimensional, modified Helmholtz equation, u(\x) - \alpha^2 \Delta u(\x) = 0, in bounded or unbounded multiply-connected domains. We consider both Dirichlet and Neumann problems. We derive well-conditioned Fredholm integral equations of the second kind, which are discretized using high-order, hybrid Gauss-trapezoid rules. Our fast multipole-based iterative solution procedure requires only O(N) or $O(N\log N)$ operations, where N is the number of nodes in the discretization of the boundary. We demonstrate the performance of the methods on several numerical examples.Comment: Published in Computers & Mathematics with Application

Slow Convergence in Generalized Central Limit Theorems

We study the central limit theorem in the non-normal domain of attraction to symmetric $\alpha$-stable laws for $0<\alpha\leq2$. We show that for i.i.d. random variables $X_i$, the convergence rate in $L^\infty$ of both the densities and distributions of $\sum_i^n X_i/(n^{1/\alpha}L(n))$ is at best logarithmic if $L$ is a non-trivial slowly varying function. Asymptotic laws for several physical processes have been derived using central limit theorems with $\sqrt{n\log n}$ scaling and Gaussian limiting distributions. Our result implies that such asymptotic laws are accurate only for exponentially large $n$.Comment: To appear in Comptes Rendus de l'Acad\'emie des Sciences, Math\'ematique