- Title Information
- Title
- The Holy Trinity of Stochastic Modeling and Uncertainty Quantification: High-Dimensionality, Sparsity and Rare Events
- Name:
Personal
- Name Part
- Yang, Xiu
- Role
- Role Term:
Text
- creator
- Origin Information
- Copyright Date
- 2014
- Physical Description
- Extent
- 23, 179 p.
- digitalOrigin
- born digital
- Note
- Thesis (Ph.D. -- Brown University (2014)
- Name:
Personal
- Name Part
- George , Karniadakis
- Role
- Role Term:
Text
- Director
- Name:
Personal
- Name Part
- Boris, Rozovsky
- Role
- Role Term:
Text
- Reader
- Name:
Personal
- Name Part
- Xiaoliang, Wan
- Role
- Role Term:
Text
- Reader
- Name:
Corporate
- Name Part
- Brown University. Applied Mathematics
- Role
- Role Term:
Text
- sponsor
- Genre (aat)
- theses
- Abstract
- In this thesis, we present new developments to the three most important issues in stochastic modeling and uncertainty quantification (UQ): high-dimensionality, sparsity and rare events. With respect to high-dimensionality, we present an adaptive Analysis Of VAriance (ANOVA) method to construct the generalized polynomial chaos (gPC) expansion of the system hierarchically and apply this method to study compressible supersonic flow over rough surface in aerodynamics and a ring oscillator in circuit simulations. We demonstrate that for both examples even a draconian truncation of the ANVOA expansion leads to accurate solutions in terms of mean and standard deviation. With respect to sparsity, we consider the gPC coefficients of the system as a "signal". If this signal is sparse, we employ compressive sensing method to "recover" it with high accuracy but low computational cost. We propose a method combining the reweighted l1 minimization and Chebyshev sampling strategy to compute the gPC expansion of stochastic partial differential equations (PDE) efficiently. This method is able to exploit information from a limited number of realizations, hence it is suitable for problems with expensive deterministic solver. Furthermore, we apply this method to construct the response surface of particle based simulation models and use this response surface in a Bayesian inference framework to identify parameters of molecular dynamics models. This approach provides a general framework for mescoscale model calibration, and it leads to parametric compression and dimensionality reduction. With respect to rare events, we firstly consider the problem of narrow escape of a Brownian particle in a boundary domain. A potential is imposed to trap the particle in it. We use PDE constrained optimization method to optimize the shape of the potential, hence to maximize the mean first passage time. We also consider a stochastic problem with disparate correlations length in adjacent domains. Instead of solving the problem globally, we use domain decomposition method to reduce the complexity of the problem in each subdomain and use PDE constrained optimization to obtain the solution. This method does not require an explicit expression of the information transferred across the interface of subdomains and can lead to accurate results.
- Subject
- Topic
- uncertainty quantification
- Subject
- Topic
- high-dimensionality
- Subject
- Topic
- sparsity
- Subject
- Topic
- rare events
- Subject
- Topic
- generalized polynomial chaos
- Record Information
- Record Content Source (marcorg)
- RPB
- Record Creation Date
(encoding="iso8601")
- 20141006
- Language
- Language Term:
Code (ISO639-2B)
- eng
- Language Term:
Text
- English
- Identifier:
DOI
- 10.7301/Z0R78CKH
- Access Condition:
rights statement
(href="http://rightsstatements.org/vocab/InC/1.0/")
- In Copyright
- Access Condition:
restriction on access
- Collection is open for research.
- Type of Resource (primo)
- dissertations