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You searched for +publisher:"Purdue University" +contributor:("Guang Lin"). Showing records 1 – 3 of 3 total matches.

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Purdue University

1. Chen, Yi. Local polynomial chaos expansion method for high dimensional stochastic differential equations.

Degree: PhD, Mathematics, 2016, Purdue University

Polynomial chaos expansion is a widely adopted method to determine evolution of uncertainty in dynamical system with probabilistic uncertainties in parameters. In particular, we focus on linear stochastic problems with high dimensional random inputs. Most of the existing methods enjoyed the efficiency brought by PC expansion compared to sampling-based Monte Carlo experiments, but still suffered from relatively high simulation cost when facing high dimensional random inputs. We propose a localized polynomial chaos expansion method that employs a domain decomposition technique to approximate the stochastic solution locally. In a relatively lower dimensional random space, we are able to solve subdomain problems individually within the accuracy restrictions. Sampling processes are delayed to the last step of the coupling of local solutions to help reduce computational cost in linear systems. We perform a further theoretical analysis on combining a domain decomposition technique with a numerical strategy of epistemic uncertainty to approximate the stochastic solution locally. An establishment is made between Schur complement in traditional domain decomposition setting and the local PCE method at the coupling stage. A further branch of discussion on the topic of decoupling strategy is presented at the end to propose some of the intuitive possibilities of future work. Both the general mathematical framework of the methodology and a collection of numerical examples are presented to demonstrate the validity and efficiency of the method. Advisors/Committee Members: Dongbin Xiu, Suchuan Dong, Dongbin Xiu, Peijun Li, Guang Lin.

Subjects/Keywords: Applied sciences; Generalized polynomial chaos; Stochastic differential equations; Uncertainty quantification; Applied Mathematics

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APA (6th Edition):

Chen, Y. (2016). Local polynomial chaos expansion method for high dimensional stochastic differential equations. (Doctoral Dissertation). Purdue University. Retrieved from https://docs.lib.purdue.edu/open_access_dissertations/744

Chicago Manual of Style (16th Edition):

Chen, Yi. “Local polynomial chaos expansion method for high dimensional stochastic differential equations.” 2016. Doctoral Dissertation, Purdue University. Accessed October 25, 2020. https://docs.lib.purdue.edu/open_access_dissertations/744.

MLA Handbook (7th Edition):

Chen, Yi. “Local polynomial chaos expansion method for high dimensional stochastic differential equations.” 2016. Web. 25 Oct 2020.

Vancouver:

Chen Y. Local polynomial chaos expansion method for high dimensional stochastic differential equations. [Internet] [Doctoral dissertation]. Purdue University; 2016. [cited 2020 Oct 25]. Available from: https://docs.lib.purdue.edu/open_access_dissertations/744.

Council of Science Editors:

Chen Y. Local polynomial chaos expansion method for high dimensional stochastic differential equations. [Doctoral Dissertation]. Purdue University; 2016. Available from: https://docs.lib.purdue.edu/open_access_dissertations/744


Purdue University

2. Chen, Xiaoxiao. Epistemic Uncertainty Quantification in Scientific Models.

Degree: PhD, Mathematics, 2014, Purdue University

In the field of uncertainty quantification (UQ), epistemic uncertainty often refers to the kind of uncertainty whose complete probabilistic description is not available, largely due to our lack of knowledge about the uncertainty. Quantification of the impacts of epistemic uncertainty is naturally difficult, because most of the existing stochastic tools rely on the specification of the probability distributions and thus do not readily apply to epistemic uncertainty. And there have been few studies and methods to deal with epistemic uncertainty. A recent work can be found in [J. Jakeman, M. Eldred, D. Xiu, Numerical approach for quantification of epistemic uncertainty, J. Comput. Phys. 229 (2010) 4648-4663], where a framework for numerical treatment of epistemic uncertainty was proposed. In this paper, firstly, we present a new method, similar to that of Jakeman et al. but significantly extending its capabilities. Most notably, the new method (1) does not require the encapsulation problem to be in a bounded domain such as a hypercube; (2) does not require the solution of the encapsulation problem to converge point-wise. In the current formulation, the encapsulation problem could reside in an unbounded domain, and more importantly, its numerical approximation could be sought in <em>Lp</em> norm. These features thus make the new approach more flexible and amicable to practical implementation. Both the mathematical framework and numerical analysis are presented to demonstrate the effectiveness of the new approach. And then, we apply this methods to work with one of the more restrictive uncertainty models, i.e., the fuzzy logic, where the p-distance, the weighted expected value and variance are defined to assess the accuracy of the solutions. At last, we give a brief introduction to our future work, which is epistemic uncertainty quantification using evidence theory. Advisors/Committee Members: Dongbin Xiu, Dongbin Xiu, Suchuan Dong, Greg Buzzard, Guang Lin.

Subjects/Keywords: Applied Mathematics

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

APA (6th Edition):

Chen, X. (2014). Epistemic Uncertainty Quantification in Scientific Models. (Doctoral Dissertation). Purdue University. Retrieved from https://docs.lib.purdue.edu/open_access_dissertations/242

Chicago Manual of Style (16th Edition):

Chen, Xiaoxiao. “Epistemic Uncertainty Quantification in Scientific Models.” 2014. Doctoral Dissertation, Purdue University. Accessed October 25, 2020. https://docs.lib.purdue.edu/open_access_dissertations/242.

MLA Handbook (7th Edition):

Chen, Xiaoxiao. “Epistemic Uncertainty Quantification in Scientific Models.” 2014. Web. 25 Oct 2020.

Vancouver:

Chen X. Epistemic Uncertainty Quantification in Scientific Models. [Internet] [Doctoral dissertation]. Purdue University; 2014. [cited 2020 Oct 25]. Available from: https://docs.lib.purdue.edu/open_access_dissertations/242.

Council of Science Editors:

Chen X. Epistemic Uncertainty Quantification in Scientific Models. [Doctoral Dissertation]. Purdue University; 2014. Available from: https://docs.lib.purdue.edu/open_access_dissertations/242


Purdue University

3. Dinh, Vu Cao Duy Thien. Probabilistic uncertainty quantification and experiment design for nonlinear models: Applications in systems biology.

Degree: PhD, Mathematics, 2014, Purdue University

Despite the ever-increasing interest in understanding biology at the system level, there are several factors that hinder studies and analyses of biological systems. First, unlike systems from other applied fields whose parameters can be effectively identified, biological systems are usually unidentifiable, even in the ideal case when all possible system outputs are known with high accuracy. Second, the presence of multivariate bifurcations often leads the system to behaviors that are completely different in nature. In such cases, system outputs (as function of parameters/inputs) are usually discontinuous or have sharp transitions across domains with different behaviors. Finally, models from systems biology are usually strongly nonlinear with large numbers of parameters and complex interactions. This results in high computational costs of model simulations that are required to study the systems, an issue that becomes more and more problematic when the dimensionality of the system increases. Similarly, wet-lab experiments to gather information about the biological model of interest are usually strictly constrained by research budget and experimental settings. The choice of experiments/simulations for inference, therefore, needs to be carefully addressed. ^ The work presented in this dissertation develops strategies to address theoretical and practical limitations in uncertainty quantification and experimental design of non-linear mathematical models, applied in the context of systems biology. This work resolves those issues by focusing on three separate but related approaches: (i) the use of probabilistic frameworks for uncertainty quantification in the face of unidentifiability (ii) the use of behavior discrimination algorithms to study systems with discontinuous model responses and (iii) the use of effective sampling schemes and optimal experimental design to reduce the computational/experimental costs. ^ This cumulative work also places strong emphasis on providing theoretical foundations for the use of the proposed framework: theoretical properties of algorithms at each step in the process are investigated carefully to give more insights about how the algorithms perform, and in many cases, to provide feedback to improve the performance of existing approaches. Through the newly developed procedures, we successfully created a general probabilistic framework for uncertainty quantification and experiment design for non-linear models in the face of unidentifiability, sharp model responses with limited number of model simulations, constraints on experimental setting, and even in the absence of data. The proposed methods have strong theoretical foundations and have also proven to be effective in studies of expensive high-dimensional biological systems in various contexts. Advisors/Committee Members: Gregery T. Buzzard, Gregery T. Buzzard, Ann E. Rundell, Zhilan Feng, Guang Lin.

Subjects/Keywords: Biomedical Engineering and Bioengineering; Engineering; Mathematics; Statistics and Probability

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

APA (6th Edition):

Dinh, V. C. D. T. (2014). Probabilistic uncertainty quantification and experiment design for nonlinear models: Applications in systems biology. (Doctoral Dissertation). Purdue University. Retrieved from https://docs.lib.purdue.edu/open_access_dissertations/259

Chicago Manual of Style (16th Edition):

Dinh, Vu Cao Duy Thien. “Probabilistic uncertainty quantification and experiment design for nonlinear models: Applications in systems biology.” 2014. Doctoral Dissertation, Purdue University. Accessed October 25, 2020. https://docs.lib.purdue.edu/open_access_dissertations/259.

MLA Handbook (7th Edition):

Dinh, Vu Cao Duy Thien. “Probabilistic uncertainty quantification and experiment design for nonlinear models: Applications in systems biology.” 2014. Web. 25 Oct 2020.

Vancouver:

Dinh VCDT. Probabilistic uncertainty quantification and experiment design for nonlinear models: Applications in systems biology. [Internet] [Doctoral dissertation]. Purdue University; 2014. [cited 2020 Oct 25]. Available from: https://docs.lib.purdue.edu/open_access_dissertations/259.

Council of Science Editors:

Dinh VCDT. Probabilistic uncertainty quantification and experiment design for nonlinear models: Applications in systems biology. [Doctoral Dissertation]. Purdue University; 2014. Available from: https://docs.lib.purdue.edu/open_access_dissertations/259

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