Theses/Dissertations - Mathematics
http://hdl.handle.net/2104/4791
2016-02-06T18:43:01ZA multigrid Krylov method for eigenvalue problems.
http://hdl.handle.net/2104/9514
A multigrid Krylov method for eigenvalue problems.
We are interested in computing eigenvalues and eigenvectors of matrices derived from differential equations. They are often large sparse matrices, including both symmetric and non symmetric cases. Restarted Arnoldi methods are iterative methods for eigenvalue problems based on Krylov subspaces. Multigrid methods solve differential equations by taking advantage of the hierarchy of discretizations. A multigrid Krylov method is proposed by combining Arnoldi and multigrid methods. We compare the new approach with other methods, and explore the theory to explain its efficiency.
2015-07-31T00:00:00ZKrylov methods for solving a sequence of large systems of linear equations.
http://hdl.handle.net/2104/9511
Krylov methods for solving a sequence of large systems of linear equations.
Consider solving a sequence of linear systems A_{(i)}x^{(i)}=b^{(i)}, i=1, 2, ... where A₍ᵢ₎ ϵℂⁿᵡⁿ and b⁽ⁱ⁾ϵℂⁿ using some variations of Krylov subspace methods, like GMRES. For a single system Ax=b, it is well-known that the eigenvectors of the coefficient matrix A can be used to speed up the convergence of GMRES by deflating the corresponding eigenvalues. In this dissertation, we propose a deflation-based algorithm that utilizes the eigenvalue and eigenvector information obtained from one system to improve the convergence of GMRES for solving the subsequent systems. When the change in the system is small enough, the algorithm will REUSE the eigenvectors from the previous system to deflate the small eigenvalues from the new system via a projection to speed up convergence. When the change is significant enough that projection loses effectiveness, the algorithm will RECYCLE the eigenvectors from the previous system by adding them to the new Krylov subspace, thus improving them so that they can be suitable candidates for deflation once again. If the system has changed too much, or the new system is completely unrelated to the previous system, the algorithm will REGENERATE a new set of eigenvectors to help with deflation.
2015-07-22T00:00:00ZBoundary condition dependence of spectral zeta functions.
http://hdl.handle.net/2104/9459
Boundary condition dependence of spectral zeta functions.
In this work, we provide the analytic continuation of the spectral zeta function associated with the one-dimensional regular Sturm-Liouville problem and the two-dimensional Laplacian on the annulus. In the one-dimensional setting, we consider general separated and coupled boundary conditions, and on the annulus we restrict our work to Dirichlet-Robin boundary conditions. In both cases, we use our results to calculate the coefficients of the asymptotic expansion of the associated heat kernel. In the one-dimensional case, we additionally use the analytically continued spectral zeta function to compute the determinant of the Sturm-Liouville operator.
2015-07-14T00:00:00ZQuadratic Lyapunov theory for dynamic linear switched systems.
http://hdl.handle.net/2104/8899
Quadratic Lyapunov theory for dynamic linear switched systems.
Eisenbarth, Geoffrey B.
In this work, a special class of time-varying linear systems in the arbitrary time scale setting is examined by considering the qualitative properties of their solutions. Building on the work of J. DaCunha and A. Ramos, Lyapunov's Second (or Direct) Method is utilized to determine when the solutions to a given switched system are asymptotically stable.
Three major classes of switched systems are analyzed which exhibit a convenient containment scheme so as to recover early results as special cases of later, more general results. The stability of switched systems under both arbitrary and particular switching is considered, in addition to design parameters of the time scale domain which also imply stability. A new approach to Lyapunov theory for time scales is then considered for switched systems which do not necessarily belong to any class of systems, contrasting and generalizing previous results. Finally, extensions of the contained theory are considered and a nontrivial generalization of a major result by D. Liberzon and A. Agrachev is investigated and conjectured.
2014-01-28T00:00:00Z