http://hdl.handle.net/1721.1/39115 http://hdl.handle.net/1721.1/42456 A generalized precorrected-FFT method for electromagnetic analysis Leibman, Stephen Gerald Boundary Element Methods (BEM) can be ideal approaches for simulating the behavior of physical systems in which the volumes have homogeneous properties. These, especially the so-called "fast" or "accelerated" BEM approaches often have significant computational advantages over other well-known methods which solve partial differential equations on a volume domain. However, the implementation of techniques used to accelerate BEM approaches often comes at a loss of some generality, reducing their applicability to many problems and preventing engineers and researchers from easily building on a common, popular base of code. In this thesis we create a BEM solver which uses the Pre-Corrected FFT technique for accelerating computation, and uses a novel approach which allows users to provide arbitrary basis functions. We demonstrate its utility for both electrostatic and full-wave electromagnetic problems in volumes with homogeneous isotropic permittivity, bounded by arbitrarily complex surface geometries. The code is shown to have performance characteristics similar to the best known approaches for these problems. It also provides an increased level of generality, and is designed in such a way that should allow it to easily be extended by other researchers. Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008. Includes bibliographical references (p. 117-119). http://hdl.handle.net/1721.1/42455 Cheeger sets for unit cube : analytical and numerical solutions for L [infinity] and L² norms Hussain, Mohammad Tariq The Cheeger constant h(Q) of a domain Q is defined as the minimum value of ...... with D varying over all smooth sub-domains of Q. The D that achieves this minimum is called the Cheeger set of Q. We present some analytical and numerical work on the Cheeger set for the unit cube ... using the ...and the ... norms for measuring IIDII. We look at the equivalent max-flow min-cut problem for continuum flows, and use it to get numerical results for the problem. We then use these results to suggest analytical solutions to the problem and optimize these shapes using calculus and numerical methods. Finally we make some observations about the general shapes we get, and how they can be derived using an algorithm similar to the one for finding Cheeger sets for domains in ... Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008. In title on t.p., "L" appears as italic letters and "[infinity]" appears as the symbol. Includes bibliographical references (leaves 47-48). http://hdl.handle.net/1721.1/42454 A preconditioned Newton-Krylov method for computing steady-state pulse solutions of mode-locked lasers Birge, Jonathan R. (Jonathan Richards) We solve the periodic boundary value problem for a mode-locked laser cavity using a specially preconditioned matrix-implicit Newton-Krylov solver. Solutions are obtained at least an order of magnitude faster than with dynamic simulation, the standard method. Our method is demonstrated experimentally on a one-dimensional temporal model of an eight femtosecond mode-locked laser operating in the dispersion-managed soliton regime. Our solver is applicable to finding the steady-state solution of any nonlinear optical cavity with moderate self phase modulation, such as those of solid state lasers, and requires only a model for the round-trip action of the cavity. We conclude by proposing avenues of future work to improve the method's convergence and expand its applicability to lasers with higher degrees of cavity nonlinearity. Our approach can be extended to spatio-temporal cavity models, potentially allowing for the first feasible simulation of the full dynamics of Kerr-lens mode locking. Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008. Includes bibliographical references (p. 47-48). http://hdl.handle.net/1721.1/41737 Computational issues and related mathematics of an exponential annealing homotropy for conic optimization Chen, Jeremy, S.M. Massachusetts Institute of Technology We present a further study and analysis of an exponential annealing based algorithm for convex optimization. We begin by developing a general framework for applying exponential annealing to conic optimization. We analyze the hit-and-run random walk from the perspective of convergence and develop (partially) an intuitive picture that views it as the limit of a sequence of finite state Markov chains. We then establish useful results that guide our sampling. Modifications are proposed that seek to raise the computational practicality of exponential annealing for convex optimization. In particular, inspired by interior-point methods, we propose modifying the hit-and-run random walk to bias iterates away from the boundary of the feasible region and show that this approach yields a substantial reduction in computational cost. We perform computational experiments for linear and semidefinite optimization problems. For linear optimization problems, we verify the correlation of phase count with the Renegar condition measure (described in [13]); for semidefinite optimization, we verify the correlation of phase count with a geometry measure (presented in [4]). Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007. Includes bibliographical references (p. 105-106).