Physical Sciences and Mathematics Commons^™

Improving Efficiency Of Rational Krylov Subspace Methods, Shengjie Xu

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This thesis studies two classes of numerical linear algebra problems, approximating the product of a function of a matrix with a vector, and solving the linear eigenvalue problem $Av=\lambda Bv$ for a small number of eigenvalues. These problems are solved by rational Krylov subspace methods (RKSM). We present several improvements in two directions: pole selection and applying inexact methods.

In Chapter 3, a flexible extended Krylov subspace method ($\mathcal{F}$-EKSM) is considered for numerical approximation of the action of a matrix function $f(A)$ to a vector $b$, where the function $f$ is of Markov type. $\mathcal{F}$-EKSM has the same framework as …

On Variants Of Sliding And Frank-Wolfe Type Methods And Their Applications In Video Co-Localization, Seyed Hamid Nazari

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In this dissertation, our main focus is to design and analyze first-order methods for computing approximate solutions to convex, smooth optimization problems over certain feasible sets. Specifically, our goal in this dissertation is to explore some variants of sliding and Frank-Wolfe (FW) type algorithms, analyze their convergence complexity, and examine their performance in numerical experiments. We achieve three accomplishments in our research results throughout this dissertation. First, we incorporate a linesearch technique to a well-known projection-free sliding algorithm, namely the conditional gradient sliding (CGS) method. Our proposed algorithm, called the conditional gradient sliding with linesearch (CGSls), does not require the …

Managing Risk For Power System Operations And Planning: Applications Of Conditional Value-At-Risk And Uncertainty Quantification To Optimal Power Flow And Distributed Energy Resources Investment, Thanh To

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Renewable energy sources are indispensable components of sustainable electrical systems that reduce human dependence on fossil fuels. However, due to their intermittent nature, there are issues that need to be addressed to ensure the security and resiliency of these power systems. This dissertation formulates several practical problems, from an optimization perspective, stemming from the increasing penetration of intermittent renewable energy to power systems. A number of Optimal Power Flow (OPF) formulations are investigated and new formulations are proposed to control both operations and planning risks by utilizing the Conditional Value–at–Risk (CVaR) measure. Our formulations provide system operators and investors analysis …

Advancements In Gaussian Process Learning For Uncertainty Quantification, John C. Nicholson

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Gaussian processes are among the most useful tools in modeling continuous processes in machine learning and statistics. The research presented provides advancements in uncertainty quantification using Gaussian processes from two distinct perspectives. The first provides a more fundamental means of constructing Gaussian processes which take on arbitrary linear operator constraints in much more general framework than its predecessors, and the other from the perspective of calibration of state-aware parameters in computer models. If the value of a process is known at a finite collection of points, one may use Gaussian processes to construct a surface which interpolates these values to …