GMRES for oscillatory matrix-valued differential equations

We investigate the use of Krylov subspace methods to solve linear, oscillatory ODEs. When we apply a Krylov subspace method to a properly formulated equation, we retain the asymptotic accuracy of the asymptotic expansion whilst converging to the exact solution. We will demonstrate the effectiveness of this method by computing Error and Mathieu functions

Fast, numerically stable computation of oscillatory integrals with stationary points

We present a numerically stable way to compute oscillatory integrals of the form $\int{-1}^{1} f(x)e^{i\omega g(x)}dx$. For each additional frequency, only a small, well-conditioned linear system with a Hessenberg matrix must be solved, and the amount of work needed decreases as the frequency increases. Moreover, we can modify the method for computing oscillatory integrals with stationary points. This is the first stable algorithm for oscillatory integrals with stationary points which does not lose accuracy as the frequency increases and does not require deformation into the complex plane

On Krylov Methods in Infinite-dimensional Hilbert Space

This thesis contains the development of key features for the solution to inverse linear problems Af = g on infinite-dimensional Hilbert space H using projection methods. Particular attention is paid to Krylov subspace methods. Intrinsic, key operator-theoretic constructs that guarantee the \u2018Krylov solvability\u2019 of the problem Af = g are developed and investigated for this class of projection methods. This theory is supported by numerous examples, counterexamples, and some numerical tests. Results for both bounded and unbounded operators on general Hilbert spaces are considered, with special attention paid to the Krylov method of conjugate-gradients in the unbounded setting