Numerical Analysis

Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud \ud In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud \ud With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright

Parallel Factorizations in Numerical Analysis

In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.Comment: 15 pages, 5 figure

Numerical Analysis of Parallel Replica Dynamics

Parallel replica dynamics is a method for accelerating the computation of processes characterized by a sequence of infrequent events. In this work, the processes are governed by the overdamped Langevin equation. Such processes spend much of their time about the minima of the underlying potential, occasionally transitioning into different basins of attraction. The essential idea of parallel replica dynamics is that the exit time distribution from a given well for a single process can be approximated by the minimum of the exit time distributions of $N$ independent identical processes, each run for only 1/N-th the amount of time. While promising, this leads to a series of numerical analysis questions about the accuracy of the exit distributions. Building upon the recent work in Le Bris et al., we prove a unified error estimate on the exit distributions of the algorithm against an unaccelerated process. Furthermore, we study a dephasing mechanism, and prove that it will successfully complete.Comment: 37 pages, 4 figures, revised and new estimates from the previous versio

Numerical Analysis of Black Hole Evaporation

Black hole formation/evaporation in two-dimensional dilaton gravity can be described, in the limit where the number $N$ of matter fields becomes large, by a set of second-order partial differential equations. In this paper we solve these equations numerically. It is shown that, contrary to some previous suggestions, black holes evaporate completely a finite time after formation. A boundary condition is required to evolve the system beyond the naked singularity at the evaporation endpoint. It is argued that this may be naturally chosen so as to restore the system to the vacuum. The analysis also applies to the low-energy scattering of $S$-wave fermions by four-dimensional extremal, magnetic, dilatonic black holes.Comment: 10 pages, 9 figures in separate uuencoded fil

Theoretical and Analytical Investigation of Electromagnetic Problems Using Dispersive Material and the Kramers-Kronig Transformations

In reality, there is no material with constant permittivity, permeability, and conductivity values over the entire frequency spectrum. The variation in these parameters is well-known as the dispersion phenomenon, which can be analytically interpreted using the Kramers-Kronig relations. Through this thesis, we extensively explain how to take advantage of the dispersion in these parameters to numerically investigate some electromagnetic problems. Among these is a practical problem of separating the electric conductive losses from the dielectric losses of any dispersive lossy material. On the other hand, the performance of particular electrically small antennas is improved by exploiting the frequency dispersion in the dielectric and magnetic material. Electrically small antennas have great attention in many applications due to their drastically reduced size. However, the size reduction comes at the expense of performance degradation, such as increasing the internally stored electromagnetic energy, narrowing the operating bandwidth, decreasing radiation efficiency, and poor matching to surrounding media, particularly when the antenna element has direct contact with a lossy medium like biomedical tissues. Therefore, we try to numerically investigate the performance of these antennas by coating them with dispersive dielectric or magnetic material. In this thesis, some artificially synthesized material with frequency-dependent permittivity, permeability, and electric conductivity values over a wide range of frequencies are suggested to represent the dispersive lossy material. The Kramers-Kronig (KK) relations are employed as a mathematical solution to interrelate the real and imaginary parts of the suggested frequency-dependent relative permittivity and permeability values of the artificial material. Finally, the solution methods are verified by applying them to real-world material found in the literature

Numerical analysis of the master equation

Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme that remains stable with much larger time increments than can be used in standard methods. When only the stationary distribution is required, a direct iteration method is even more rapid; this method may be extended to construct the quasi-stationary distribution of a process with an absorbing state. Applications to birth-and-death processes reveal gains in efficiency of two or more orders of magnitude.Comment: 7 pages 3 figure