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1.
Cho, Heyrim.
High-Dimensional Response-Excitation PDF Methods for
Uncertainty Quantification and Stochastic Modeling.
Degree: PhD, Applied Mathematics, 2015, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:419509/ 
► The probability density approach based on the response-excitation theory is developed for stochastic simulations of non-Markovian systems. This approach provides the complete probabilistic configuration of…
(more)
▼ The probability density approach based on the
response-excitation theory is developed for stochastic simulations
of non-Markovian systems. This approach provides the complete
probabilistic configuration of the solution that enables a
comprehensive study of stochastic systems. By using functional
integral methods we determine a computable evolution equation for
the joint response-excitation probability density function (REPDF)
of stochastic dynamical systems and stochastic partial differential
equations driven by colored noise. We establish its connection to
the classical response approach and its agreement to the
Dostupov-Pugachev equations (Dostupov, 1957) and the
Malakhov-Saichev equations (Gurbatov et al, 1991). An efficient
algorithm has been proposed by using adaptive discontinuous
Galerkin method and probabilistic collocation method combined with
sparse grid. For high-dimensional REPDF systems, we develop the
algorithms concerning high-dimensional numerical approximations,
namely, separated series expansion and the ANOVA approximation.
These methods reduce the computational cost in high-dimensions to
several low-dimensional operations. Alternatively, reduced order
PDF equations are obtained by using the Mori-Zwanzig framework and
conditional moment closures, which establish a preliminary work of
goal-oriented PDF equations. Finally, we demonstrate the
effectiveness of the proposed numerical methods to various
stochastic systems including the tumor cell growth model, chaotic
nonlinear oscillators, advection reaction equation, and Burgers
equation. The second part of the thesis focuses on simulations of
multi-scale stochastic systems. The Karhunen-Loeve expansion is
extended to characterize multiple correlated random processes and
local decomposed random fields. We then propose interface
conditions based on conditional moments and PDE-constrained
optimization that preserve the global statistics while propagating
uncertainty. Finally, the decomposition algorithm is recast to
couple distinct PDF models including the REPDF
system.
Advisors/Committee Members: Karniadakis, George (Director), Rozovsky, Boris (Reader), Venturi, Daniele (Reader), Sapsis, Themistoklis (Reader).
Subjects/Keywords: Uncertainty quantification
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APA (6th Edition):
Cho, H. (2015). High-Dimensional Response-Excitation PDF Methods for
Uncertainty Quantification and Stochastic Modeling. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:419509/
Chicago Manual of Style (16th Edition):
Cho, Heyrim. “High-Dimensional Response-Excitation PDF Methods for
Uncertainty Quantification and Stochastic Modeling.” 2015. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:419509/.
MLA Handbook (7th Edition):
Cho, Heyrim. “High-Dimensional Response-Excitation PDF Methods for
Uncertainty Quantification and Stochastic Modeling.” 2015. Web. 16 Jan 2021.
Vancouver:
Cho H. High-Dimensional Response-Excitation PDF Methods for
Uncertainty Quantification and Stochastic Modeling. [Internet] [Doctoral dissertation]. Brown University; 2015. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:419509/.
Council of Science Editors:
Cho H. High-Dimensional Response-Excitation PDF Methods for
Uncertainty Quantification and Stochastic Modeling. [Doctoral Dissertation]. Brown University; 2015. Available from: https://repository.library.brown.edu/studio/item/bdr:419509/
2.
Choi, Minseok.
Time-dependent Karhunen-Loeve type decomposition methods for
SPDEs.
Degree: PhD, Applied Mathematics, 2014, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:386285/
► A new hybrid methodology for the stochastic partial differential equations (SPDEs) is developed based on the dynamically-orthogonal (DO) and bi-orthogonal (BO) expansions; both approaches are…
(more)
▼ A new hybrid methodology for the stochastic partial
differential equations (SPDEs) is developed based on the
dynamically-orthogonal (DO) and bi-orthogonal (BO) expansions; both
approaches are an extension of the Karhunen-Loeve (KL) expansion.
The original KL expansion provides a low-dimensional representation
for square integrable random processes since it is optimal in the
mean square sense. The solution to SPDEs is represented in a way
that it follows the characteristics of KL expansion on-the-fly at
any given time. To this end, both the spatial and stochastic basis
in the representation are time-dependent unlike the traditional
methods such as polynomial chaos (PC), where only one of them is
time-dependent. In order to overcome the redundancy the DO imposes
the dynamical constraints on the spatial basis while the BO imposes
the static constraints on the spatial and stochastic basis. We
examine the relation of the BO and DO and prove theoretically and
illustrate numerically their equivalence, in the sense that one
method is an exact reformulation of the other by deriving an
invertible and linear transformation matrix governed by a matrix
differential equation that connects the BO and the DO. We also
examine the pathology of the BO that occurs when there is an
eigenvalue crossing leading to the numerical instability. On the
other hand we observe that the DO suffers numerically when there is
a high condition number of the covariance matrix for the stochastic
basis. To this end, we propose a unified hybrid framework of the
two methods by utilizing an invertible and linear transformation
between them. We also present an adaptive algorithm to add or
remove modes to better capture the transient behavior. Several
numerical examples, linear and nonlinear, are presented to
illustrate the DO and BO methods, their equivalence, and adaptive
strategies. It is also shown numerically that two methods converge
exponentially fast with respect to the number of modes giving the
same levels of accuracy, which is comparable with the PC method but
with substantially smaller computational cost compared to
stochastic collocation, especially when the involved parametric
space is high-dimensional.
Advisors/Committee Members: Karniadakis, George (Director), Rozovsky, Boris (Reader), Themistoklis, Sapsis (Reader).
Subjects/Keywords: uncertainty quantification
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Choi, M. (2014). Time-dependent Karhunen-Loeve type decomposition methods for
SPDEs. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:386285/
Chicago Manual of Style (16th Edition):
Choi, Minseok. “Time-dependent Karhunen-Loeve type decomposition methods for
SPDEs.” 2014. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:386285/.
MLA Handbook (7th Edition):
Choi, Minseok. “Time-dependent Karhunen-Loeve type decomposition methods for
SPDEs.” 2014. Web. 16 Jan 2021.
Vancouver:
Choi M. Time-dependent Karhunen-Loeve type decomposition methods for
SPDEs. [Internet] [Doctoral dissertation]. Brown University; 2014. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:386285/.
Council of Science Editors:
Choi M. Time-dependent Karhunen-Loeve type decomposition methods for
SPDEs. [Doctoral Dissertation]. Brown University; 2014. Available from: https://repository.library.brown.edu/studio/item/bdr:386285/
3.
Lee, Chia Ying.
Effective approximations of stochastic partial differential
equations based on Wiener chaos expansions and the Malliavin
calculus.
Degree: PhD, Applied Mathematics, 2011, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11267/
► This thesis studies the application of the Wiener chaos expansion in the analysis of stochastic partial differential equations (SPDEs). Specifically, linear parabolic SPDEs and the…
(more)
▼ This thesis studies the application of the Wiener
chaos expansion in the analysis of stochastic partial differential
equations (SPDEs). Specifically, linear parabolic SPDEs and the
quantized stochastic Navier-Stokes equations are considered, under
the framework of the Malliavin calculus. Especially for these
highly singular SPDEs, the Wiener chaos expansion is a useful tool
for our study of the basic questions of solvability, regularity and
dynamical behaviour, and it enables us to study approximations of
the solutions of SPDEs and to quantify the errors of approximation.
For the quantized stochastic Navier-Stokes equations, we use the
Malliavin calculus to formulate a random perturbation of the
Navier-Stokes equations that is unbiased, and we will show the
existence and uniqueness of steady and time-dependent solutions, as
well as the convergence to steady solution, in a stochastic
weighted space. We also study a stochastic finite element method
for numerical simulation of the solution of linear parabolic SPDEs
and derive error estimates for the numerical solution. Finally, we
show how one basis of the Wiener chaos expansion can be more
efficient than another for approximating the energy of the
solution, so that computational efficiency can be increased when
applied to some physical applications.
Advisors/Committee Members: Rozovsky, Boris (Director), Karniadakis, George (Reader), Ramanan, Kavita (Reader).
Subjects/Keywords: Wiener chaos expansion
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Lee, C. Y. (2011). Effective approximations of stochastic partial differential
equations based on Wiener chaos expansions and the Malliavin
calculus. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11267/
Chicago Manual of Style (16th Edition):
Lee, Chia Ying. “Effective approximations of stochastic partial differential
equations based on Wiener chaos expansions and the Malliavin
calculus.” 2011. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:11267/.
MLA Handbook (7th Edition):
Lee, Chia Ying. “Effective approximations of stochastic partial differential
equations based on Wiener chaos expansions and the Malliavin
calculus.” 2011. Web. 16 Jan 2021.
Vancouver:
Lee CY. Effective approximations of stochastic partial differential
equations based on Wiener chaos expansions and the Malliavin
calculus. [Internet] [Doctoral dissertation]. Brown University; 2011. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:11267/.
Council of Science Editors:
Lee CY. Effective approximations of stochastic partial differential
equations based on Wiener chaos expansions and the Malliavin
calculus. [Doctoral Dissertation]. Brown University; 2011. Available from: https://repository.library.brown.edu/studio/item/bdr:11267/
4.
Papanicolaou, Andrew L.
New Methods in Theory & Applications of Nonlinear
Filtering.
Degree: PhD, Applied Mathematics, 2010, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:11376/
► Hidden Markov models are used in countless signal processing problems, and the associated nonlinear filtering algorithms are used to obtain posterior distributions for the hidden…
(more)
▼ Hidden Markov models are used in countless signal
processing problems, and the associated nonlinear filtering
algorithms are used to obtain posterior distributions for the
hidden states. One reason why posterior distributions are so
important is because they are used to compute estimates which are
optimal given a history of observed data. However, it is often the
case that implementation of these algorithms is near impossible
because of the curse-of-dimensionality which results from testing
every possible hypothesis. This thesis explores new applications of
nonlinear filtering and addresses several issues in algorithm
implementation.
Advisors/Committee Members: Rozovsky, Boris (Director), Dafermos, Constantine (Reader), Karniadakis, George (Reader).
Subjects/Keywords: filtering
Record Details
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APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Papanicolaou, A. L. (2010). New Methods in Theory & Applications of Nonlinear
Filtering. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:11376/
Chicago Manual of Style (16th Edition):
Papanicolaou, Andrew L. “New Methods in Theory & Applications of Nonlinear
Filtering.” 2010. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:11376/.
MLA Handbook (7th Edition):
Papanicolaou, Andrew L. “New Methods in Theory & Applications of Nonlinear
Filtering.” 2010. Web. 16 Jan 2021.
Vancouver:
Papanicolaou AL. New Methods in Theory & Applications of Nonlinear
Filtering. [Internet] [Doctoral dissertation]. Brown University; 2010. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:11376/.
Council of Science Editors:
Papanicolaou AL. New Methods in Theory & Applications of Nonlinear
Filtering. [Doctoral Dissertation]. Brown University; 2010. Available from: https://repository.library.brown.edu/studio/item/bdr:11376/
5.
Srinivasan, Ravi.
Closure and complete integrability in Burgers
turbulence.
Degree: PhD, Applied Mathematics, 2009, Brown University
URL: https://repository.library.brown.edu/studio/item/bdr:158/
► Burgers turbulence (1-D inviscid Burgers equation with random initial data) is a fundamental non-equilibrium model of stochastic coalescence. In this work we demonstrate that at…
(more)
▼ Burgers turbulence (1-D inviscid Burgers equation with
random initial data) is a fundamental non-equilibrium model of
stochastic coalescence. In this work we demonstrate that at the
level of the 2-point correlation function, the entropy solution to
Burgers equation yields a closed, completely integrable system. We
show that the statistical evolution is given by a Lax pair.
Finally, we demonstrate that this equation has an equivalent
kinetic description with a rich family of self-similar solutions,
and in particular admits an explicit solution derived by Groeneboom
in nonparametric statistics. Finally, the closure property and
complete integrability are shown to hold in the general case of 1-D
scalar conservation laws with strictly convex flux.
Advisors/Committee Members: Menon, Govind (director), Menon, Govind (reader), Dafermos, Constantine (reader), Rozovsky, Boris (reader).
Subjects/Keywords: Burgers turbulence
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Record Details
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❌
APA ·
Chicago ·
MLA ·
Vancouver ·
CSE |
Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Srinivasan, R. (2009). Closure and complete integrability in Burgers
turbulence. (Doctoral Dissertation). Brown University. Retrieved from https://repository.library.brown.edu/studio/item/bdr:158/
Chicago Manual of Style (16th Edition):
Srinivasan, Ravi. “Closure and complete integrability in Burgers
turbulence.” 2009. Doctoral Dissertation, Brown University. Accessed January 16, 2021.
https://repository.library.brown.edu/studio/item/bdr:158/.
MLA Handbook (7th Edition):
Srinivasan, Ravi. “Closure and complete integrability in Burgers
turbulence.” 2009. Web. 16 Jan 2021.
Vancouver:
Srinivasan R. Closure and complete integrability in Burgers
turbulence. [Internet] [Doctoral dissertation]. Brown University; 2009. [cited 2021 Jan 16].
Available from: https://repository.library.brown.edu/studio/item/bdr:158/.
Council of Science Editors:
Srinivasan R. Closure and complete integrability in Burgers
turbulence. [Doctoral Dissertation]. Brown University; 2009. Available from: https://repository.library.brown.edu/studio/item/bdr:158/
. 
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