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Numerical approximation of the time-ordered exponential for spin dynamic simulation
Lazzarino, Lorenzo ; Pozza, Stefano (advisor) ; Congreve, Scott (referee)
We describe, discuss, and compare classes of methods for the numerical solution of non-autonomous linear ODEs using problems coming from Nuclear magnetic resonance (NMR) spectroscopy with Magic-angle spinning (MAS) as case study. The newly intro- duced ⋆-product approach uses a convolution-like product to express the time-ordered exponential to then expand it in a Legendre polynomials basis so to transform the orig- inal ODE problem into a linear algebra problem. The aim is to compare the numerical performance of this method with other commonly used methods. Therefore, we take into account geometric numerical integrators. This group of integrators are frequently used in many different areas of research as, for example, quantum mechanics, molecular dynamics and particle accelerators physics. Their approach can either approximate the solution of the non-autonomous ODE by means of a single time-independent exponential (Magnus Integrators) or by means of a product of time-independent exponentials (Splitting Meth- ods, Commutator-Free Exponential Integrators). Finally, numerical experiments on the NMR MAS case study are performed to test the numerical behaviour of the ⋆-process and compare it with the already established alternatives. 1
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Comparison of iterative matrix methods for information retrieval
Hercík, Jakub ; Carson, Erin Claire (advisor) ; Pozza, Stefano (referee)
This thesis describes the topic of information retrieval and introduces iterative matrix algorithms useful in this context - the Lanczos algorithm used in latent semantic index- ing and the Golub-Kahan-Lanczos bidiagonalization. The efficiency of these techniques is compared in a series of numerical experiments which measure retrieval performance and computation time on collection of real-world datasets. The study of the methods is conducted in both single and double IEEE precision arithmetic, and special attention is payed to the variations. The results of our experiments suggest that in many cases, lower precision can be used without significantly damaging retrieval performance. This finding opens the door to future investigation into the possible use of lower precision in informa- tion retrieval. Provided is also a series of codes in the MATLAB programming language as well as the preprocessed datasets, both of which were utilized in the experiments. 1
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Methods for enforcing non-negativity of solution in Krylov regularization
Hoang, Phuong Thao ; Hnětynková, Iveta (advisor) ; Pozza, Stefano (referee)
The purpose of this thesis is to study how to overcome difficulties one typically encounters when solving non-negative inverse problems by standard Krylov subspace methods. We first give a theoretical background to the non-negative inverse problems. Then we concentrate on selected modifications of Krylov subspace methods known to improve the solution significantly. We describe their properties, provide their implementation and propose an improvement for one of them. After that, numerical experiments are presented giving a comparison of the methods and analyzing the influence of the present parameters on the behavior of the solvers. It is clearly demonstrated, that the methods imposing nonnegativity perform better than the unconstrained methods. Moreover, our improvement leads in some cases to a certain reduction of the number of iterations and consequently to savings of the computational time while preserving a good quality of the approximation.
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