Hybrid Numerical Asymptotic Methods.

In the first part of this thesis, we develop a quasi analytic numerical method, for computing the smooth solution of a variety of singularly perturbed, elliptic, second order differential equations. Our scheme is an iterative algorithm in which an approximate operator is used at each step to improve on the current iterate. The approximate operator is obtained by a scheme that approximately factors the original operator into two first order operators. Hence, inverting the approximate operator amounts to the solution of two first order equations and has lower complexity than inverting the original elliptic operator. Our factorization scheme is based on ideas and techniques from asymptotic analysis. We consider the application of this methodology to the convection diffusion equation and to the exact amplitude equations of geometrical optics. Furthermore, we use this methodology in the context of long range wave propagation in the ocean. We propose a modification to the parabolic equation method which accounts for backscattering. A detailed analysis of both analytical and numerical aspects of this scheme is presented for a one dimensional model problem. In the second part of this thesis, we use asymptotic methods to analyze problems modeled by linear recurrence equations. We concentrate on applications in the analysis of algorithms. First, we obtain the asymptotic behavior of the Eulerian Numbers A(n, k), directly from the partial difference equation that they satisfy. The distribution of A(n, k) for large n arises in the average case analysis of sorting algorithms. The second problem we study, is the tradeoff between inner and outer iterations for the Chebyshev iterative algorithm, for solving linear systems. Specifically, we consider this algorithm when the sequence of tolerance values for the inner iteration, varies from one iteration to the other. We determine the asymptotic convergence rate of the algorithm and find the sequence of tolerances that yields the lowest cost. There are two underlying themes in this thesis. In all problems, we use the same mathematical tools: the W.K.B. method or its multidimensional counterpart, the ray method. Furthermore, in all problems there is an interplay between discrete/numerical methods and analytic/asymptotic methods.