Topics in quasi-Newton and space-time spectral methods
Date
2019-08-09
Authors
Ebrahim Nataj, Roghayeh
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Abstract
The first part of this thesis focuses on quasi-Newton methods. Broyden's method is a quasi-Newton method which is used to solve a system of nonlinear equations. Almost all convergence theory in the literature assumes existence of a root and bounds on the nonlinear function and its derivative in some neighbourhood of the root. All these conditions cannot be checked in practice. The motivation of this work is to derive a convergence theory where all assumptions can be verified, and the existence of a root and its superlinear rate of convergence are consequences of the theory. The theory is simple in the sense that it contains as few constants as possible. The method of Broyden-Fletcher-Goldfarb-Shanno (BFGS) is also a quasi-Newton method for unconstrained minimization. We generalize a convergence theory where all assumptions are verifiable and existence of a minimizer and superlinear convergence of the iteration are conclusions. In a continuation of this part, we consider Perry nonlinear conjugate gradient (NCG) method and scaled memoryless BFGS method.
In the second part, space-time spectral methods are considered. For time dependent problems, almost all focus has been on low-order finite difference schemes for the time derivative and spectral schemes for spatial derivatives. Spectral methods which converge spectrally in both space and time have appeared recently. In this thesis, it is shown that a Chebyshev spectral collocation method of Tang and Xu for the heat equation converges exponentially when the solution is analytic. We also derive a condition number estimate of the method. Another space-time Chebyshev collocation scheme which is easier to implement is proposed and analyzed. We also present space-time spectral collocation methods for the Schrodinger, wave, Airy and beam equations. In particular, fully spectral convergence and a condition number estimate are shown for Schrodinger and wave equations. Numerical results verify the theoretical results, and demonstrate that the space-time methods also work for some common nonlinear PDEs (Allen-Cahn, viscous Burgers', Sine-Gordon, KdV, Kuramoto--Sivashinsky and Cahn-Hilliard equations).
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Broyden's, BFGS, Perry nonlinear conjugate gradient, scaled memoryless BFGS, unconstrained optimization, quasi-Newton, spectral collocation, Chebyshev collocation, space-time, time dependent partial differential equation