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Anova for Parameter Dependent Nonlinear PDEs and Numerical Methods for the Stochastic Stokes Equations

Title: Anova for Parameter Dependent Nonlinear PDEs and Numerical Methods for the Stochastic Stokes Equations.
Name(s): Chen, Zheng, 1965-, author
Gunzburger, Max, professor directing dissertation
Huffer, Fred, outside committee member
Peterson, Janet, committee member
Wang, Xiaoqiang, committee member
Department of Mathematics, degree granting department
Florida State University, degree granting institution
Type of Resource: text
Genre: Text
Issuance: monographic
Date Issued: 2007
Publisher: Florida State University
Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource
Language(s): English
Abstract/Description: This dissertation includes the application of analysis-of-variance (ANOVA) expansions to analyze solutions of parameter dependent partial differential equations and the analysis and finite element approximations of the Stokes equations with stochastic forcing terms. In the first part of the dissertation, the impact of parameter dependent boundary conditions on the solutions of a class of nonlinear PDEs is considered. Based on the ANOVA expansions of functionals of the solutions, the effects of different parameter sampling methods on the accuracy of surrogate optimization approaches to PDE constrained optimization is considered. The effects of the smoothness of the functional and the nonlinearity in the PDE on the decay of the higher-order ANOVA terms are studied. The concept of effective dimensions is used to determine the accuracy of the ANOVA expansions. Demonstrations are given to show that whenever truncated ANOVA expansions of functionals provide accurate approximations, optimizers found through a simple surrogate optimization strategy are also relatively accurate. The effects of several parameter sampling strategies on the accuracy of the surrogate optimization method are also considered; it is found that for this sparse sampling application, the Latin hypercube sampling method has advantages over other well-known sampling methods. Although most of the results are presented and discussed in the context of surrogate optimization problems, they also apply to other settings such as stochastic ensemble methods and reduced-order modeling for nonlinear PDEs. In the second part of the dissertation, we study the numerical analysis of the Stokes equations driven by a stochastic process. The random processes we use are white noise, colored noise and the homogeneous Gaussian process. When the process is white noise, we deal with the singularity of matrix Green's functions in the form of mild solutions with the aid of the theory of distributions. We develop finite element methods to solve the stochastic Stokes equations. In the 2D and 3D cases, we derive error estimates for the approximate solutions. The results of numerical experiments are provided in the 2D case that demonstrate the algorithm and convergence rates. On the other hand, the singularity of the matrix Green's functions necessitates the use of the homogeneous Gaussian process. In the framework of theory of abstract Wiener spaces, the stochastic integrals with respect to the homogeneous Gaussian process can be defined on a larger space than L2 . With some conditions on the density function in the definition of the homogeneous Gaussian process, the matrix Green's functions have well defined integrals. We have studied the probability properties of this kind of integral and simulated discretized colored noise.
Identifier: FSU_migr_etd-3851 (IID)
Submitted Note: A Dissertation submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester, 2007.
Date of Defense: July 2, 2007.
Keywords: Optimization, Stochastic Stokes Equations, Finite Element Methods, White Noise, Colored Noise, ANOVA, Simulation
Bibliography Note: Includes bibliographical references.
Advisory Committee: Max Gunzburger, Professor Directing Dissertation; Fred Huffer, Outside Committee Member; Janet Peterson, Committee Member; Xiaoqiang Wang, Committee Member.
Subject(s): Mathematics
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Host Institution: FSU

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Chen, Z. (2007). Anova for Parameter Dependent Nonlinear PDEs and Numerical Methods for the Stochastic Stokes Equations. Retrieved from