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Mixed Precision in Uncertainty Quantification Methods
Martínek, Josef ; Carson, Erin Claire (advisor) ; Hnětynková, Iveta (referee)
This work is concerned with analysing and exploiting mixed precision arithmetic in un- certainty quantification methods with emphasis on the multilevel Monte Carlo (MLMC) method. Although mixed precision can improve performance, it should be used carefully to avoid unwanted effects on the solution accuracy. We provide a rigorous analysis of uncertainty quantification methods in finite precision arithmetic. Based on this analysis, we exploit mixed precision arithmetic in uncertainty quantification methods to improve runtime while preserving the overall error. We begin by stating the model problem, an elliptic PDE with random coefficients and a random right-hand side. Such a problem arises, for example, in uncertainty quantification for groundwater flow. Our focus is on approximating a quantity of interest given as the expected value of a functional of the solution of the PDE problem. To this end, we use the conforming finite element method for approximation in the spatial variable and the MLMC method for approximation of the expected value. We provide a novel rigorous analysis of the MLMC method in finite precision arithmetic and based on this we formulate an adaptive algorithm which determines the optimal precision value on each level of discretisation. To our knowledge, this is a new approach. Our...
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Iterative methods for Tichonov regularization with generalized regularization terms
Kučerová, Andrea ; Hnětynková, Iveta (advisor) ; Carson, Erin Claire (referee)
The aim of this thesis is to study hybrid methods for solving ill-posed linear inverse problems corrupted by white noise. These approaches are based on the combination of iterative Krylov subspace methods and the Tichonov regularization with a general regularization term. We explain the basic properties of ill-posed problems, the idea of regularization, the role of the regularization term to enforce desirable properties to the solution and the theoretical background of Standard and General Tichonov minimization. Then we explain shift invariance of Krylov subspaces. This allows us to describe a hybrid approach where the full size problem is first projected onto a Krylov subspace of a smaller dimension and then the Tichonov minimization is applied to the small projected problem. We focus on the regularization based on the finite difference approximation of derivatives of the solution. The well known regularization terms constructed from forward differences for the first and the second derivative are summarized, then we use the Taylor expansion to construct finite differences of higher orders of precision. We incorporate different variants of boundary conditions. Then the impact of the order of precision of the finite difference schemes on the quality of the solution is studied. In the experiments we use the...
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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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