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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

URL: https://docs.lib.purdue.edu/open_access_dissertations/744

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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

URL: https://docs.lib.purdue.edu/open_access_dissertations/242

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>L_{p}</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

Record Details Similar Records

❌

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

APA (6^{th} 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 (16^{th} 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 (7^{th} 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. 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

❌

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 October 25, 2020. https://docs.lib.purdue.edu/open_access_dissertations/1507.

MLA Handbook (7^{th} Edition):

Qi, Xin. “UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS.” 2014. Web. 25 Oct 2020.

Vancouver:

Qi X. 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/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