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Purdue University

1. Kloster, Kyle. Graph diffusions and matrix functions: fast algorithms and localization results.

Degree: PhD, Mathematics, 2016, Purdue University

URL: https://docs.lib.purdue.edu/open_access_dissertations/1404

Network analysis provides tools for addressing fundamental applications in graphs such as webpage ranking, protein-function prediction, and product categorization and recommendation. As real-world networks grow to have millions of nodes and billions of edges, the scalability of network analysis algorithms becomes increasingly important. Whereas many standard graph algorithms rely on matrix-vector operations that require exploring the entire graph, this thesis is concerned with graph algorithms that are local (that explore only the graph region near the nodes of interest) as well as the localized behavior of global algorithms. We prove that two well-studied matrix functions for graph analysis, PageRank and the matrix exponential, stay localized on networks that have a skewed degree sequence related to the power-law degree distribution common to many real-world networks. Our results give the first theoretical explanation of a localization phenomenon that has long been observed in real-world networks. We prove our novel method for the matrix exponential converges in sublinear work on graphs with the specified degree sequence, and we adapt our method to produce the first deterministic algorithm for computing the related heat kernel diffusion in constant-time. Finally, we generalize this framework to compute any graph diffusion in constant time.
*Advisors/Committee Members: David F Gleich, Jianlin Xia, Greg Buzzard, Jie Shen.*

Subjects/Keywords: community detection; graph diffusions; knapsack problem; localization; matrix exponential; PageRank

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Kloster, K. (2016). Graph diffusions and matrix functions: fast algorithms and localization results. (Doctoral Dissertation). Purdue University. Retrieved from https://docs.lib.purdue.edu/open_access_dissertations/1404

Chicago Manual of Style (16^{th} Edition):

Kloster, Kyle. “Graph diffusions and matrix functions: fast algorithms and localization results.” 2016. Doctoral Dissertation, Purdue University. Accessed September 19, 2020. https://docs.lib.purdue.edu/open_access_dissertations/1404.

MLA Handbook (7^{th} Edition):

Kloster, Kyle. “Graph diffusions and matrix functions: fast algorithms and localization results.” 2016. Web. 19 Sep 2020.

Vancouver:

Kloster K. Graph diffusions and matrix functions: fast algorithms and localization results. [Internet] [Doctoral dissertation]. Purdue University; 2016. [cited 2020 Sep 19]. Available from: https://docs.lib.purdue.edu/open_access_dissertations/1404.

Council of Science Editors:

Kloster K. Graph diffusions and matrix functions: fast algorithms and localization results. [Doctoral Dissertation]. Purdue University; 2016. Available from: https://docs.lib.purdue.edu/open_access_dissertations/1404

Purdue University

2. Qi, Xin. UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS.

Degree: PhD, Mathematics, 2014, Purdue University

URL: https://docs.lib.purdue.edu/open_access_dissertations/1507

Uncertainties widely exist in physical, finance, and many other areas. Some uncertainties are determined by the nature of the research subject, such as random variable and stochastic process. However, in many problems uncertainty is a result of lack of knowledge and may not be modeled as random variables/processes because of the lack of probability information. This is often referred to as epistemic uncertainty, and the traditional probabilistic approaches cannot be readily employed. First two parts of this work study epistemic uncertainties in the forward problems. A method to compute upper and lower bounds for the quantity of interest of problems whose uncertain inputs are of epistemic type is presented. Relative entropy is an important measure to study the distance between multiple probabilities. Its properties have motivated many important existing inequalities for quantifying epistemic uncertainties. Based on these works, we extend the inequalities to a large family of functions, the integrable functions, which play an important role in engineering and research. To be more specific, we provide upper and lower bounds for the statistics such as statistical moments of the quantities of our interest under the existence of epistemic uncertainty. We present the theoretical derivation of the bounds, along with numerical examples to illustrate their computations. Based on derived analytical lower and upper bounds, a procedure to compute numerical bounds of when the underlying system is subject to epistemic uncertainty is discussed. In particular, we consider the case where the uncertain inputs to the system take the form of parameters, physical and/or hyperparameters, and with unknown probability distributions. Our goal is to compute the lower and upper bounds of the statistical moments of quantity-of-interest of the system response. We discuss exclusively the numerical algorithms for computing such bounds. More importantly, we established the properties of such numerical bounds and analyzed their accuracy compared to the analytical bounds.
*Advisors/Committee Members: Dongbin Xiu, Suchuan Dong, Greg Buzzard, Jianlin Xia.*

Subjects/Keywords: Bayesian; Epistemic Uncertainty; Model discrepancy; Uncertainty Quantification

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Qi, X. (2014). UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS. (Doctoral Dissertation). Purdue University. Retrieved from https://docs.lib.purdue.edu/open_access_dissertations/1507

Chicago Manual of Style (16^{th} Edition):

Qi, Xin. “UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS.” 2014. Doctoral Dissertation, Purdue University. Accessed September 19, 2020. https://docs.lib.purdue.edu/open_access_dissertations/1507.

MLA Handbook (7^{th} Edition):

Qi, Xin. “UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS.” 2014. Web. 19 Sep 2020.

Vancouver:

Qi X. UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS. [Internet] [Doctoral dissertation]. Purdue University; 2014. [cited 2020 Sep 19]. Available from: https://docs.lib.purdue.edu/open_access_dissertations/1507.

Council of Science Editors:

Qi X. UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS. [Doctoral Dissertation]. Purdue University; 2014. Available from: https://docs.lib.purdue.edu/open_access_dissertations/1507

3. Imberti, David Michael. Methods For Increasing Domains Of Convergence In Iterative Linear System Solvers.

Degree: PhD, Mathematics, 2013, Purdue University

URL: https://docs.lib.purdue.edu/open_access_dissertations/127

In this thesis, we introduce and improve various methods for increasing the domains of convergence for iterative linear system solvers. We rely on the following three approaches: making the iteration adaptive, or nesting an inner iteration inside of a previously determined outer iteration; using deflation and projections to manipulate the spectra inherent to the iteration; and/or focusing on reordering schemes. We will analyze a specific combination of these three strategies. In particular, we propose to examine the influence of nesting a Flexible Generalized Minimum Residual algorithm together with an inner Recursive Projection Method using a banded preconditioner resulting from the Fiedler reordering.
*Advisors/Committee Members: Ahmed Sameh, Jianlin Xia, Ahmed Sameh, Jianlin Xia, Zhiqiang Cai, Bradley Lucier.*

Subjects/Keywords: pure sciences; applied sciences; flexible generalized minimum residual algorithm; fiedler; generalized minimum residual algorithm; nested convergence; recursive projection method; Computer Sciences; Mathematics

…Imberti, David M. Ph.D., *Purdue* *University*, December 2013. Methods for Increasing
Domains of… …Convergence in Iterative Linear System Solvers.
Major Professors:
Ahmed Sameh and *Jianlin* *Xia*.
In…

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Imberti, D. M. (2013). Methods For Increasing Domains Of Convergence In Iterative Linear System Solvers. (Doctoral Dissertation). Purdue University. Retrieved from https://docs.lib.purdue.edu/open_access_dissertations/127

Chicago Manual of Style (16^{th} Edition):

Imberti, David Michael. “Methods For Increasing Domains Of Convergence In Iterative Linear System Solvers.” 2013. Doctoral Dissertation, Purdue University. Accessed September 19, 2020. https://docs.lib.purdue.edu/open_access_dissertations/127.

MLA Handbook (7^{th} Edition):

Imberti, David Michael. “Methods For Increasing Domains Of Convergence In Iterative Linear System Solvers.” 2013. Web. 19 Sep 2020.

Vancouver:

Imberti DM. Methods For Increasing Domains Of Convergence In Iterative Linear System Solvers. [Internet] [Doctoral dissertation]. Purdue University; 2013. [cited 2020 Sep 19]. Available from: https://docs.lib.purdue.edu/open_access_dissertations/127.

Council of Science Editors:

Imberti DM. Methods For Increasing Domains Of Convergence In Iterative Linear System Solvers. [Doctoral Dissertation]. Purdue University; 2013. Available from: https://docs.lib.purdue.edu/open_access_dissertations/127