You are here

Comparison of Different Noise Forcings, Regularization of Noise, and Optimal Control for the Stochastic Navier-Stokes Equations

Title: Comparison of Different Noise Forcings, Regularization of Noise, and Optimal Control for the Stochastic Navier-Stokes Equations.
23 views
13 downloads
Name(s): Zhao, Wenju, author
Gunzburger, Max D., professor directing dissertation
Sussman, Mark, university representative
Peterson, Janet S., committee member
Quaife, Bryan, committee member
Huang, Chen (Professor of Scientific Computing), committee member
Florida State University, degree granting institution
College of Arts and Sciences, degree granting college
Department of Scientific Computing, degree granting department
Type of Resource: text
Genre: Text
Doctoral Thesis
Issuance: monographic
Date Issued: 2017
Publisher: Florida State University
Place of Publication: Tallahassee, Florida
Physical Form: computer
online resource
Extent: 1 online resource (144 pages)
Language(s): English
Abstract/Description: Stochastic Navier-Stokes equations have been widely applied in various computational fluid dynamics (CFD) fields in recent years. It can be considered as another milestone in CFD. Our work focuses on exploring some theoretical and numerical properties of the stochastic Navier-Stokes equations and related optimal control problems. In particular, we consider: a numerical comparison of solutions of the stochastic Navier-Stokes equations perturbed by a large range of random noises in time and space; effective Martingale regularized methods for the stochastic Navier-Stokes equations with additive noises; and the stochastic Navier-Stokes equations constrained stochastic boundary optimal control problems. We systemically provide numerical simulation methods for the stochastic Navier-Stokes equations with different types of noises. The noises are classified as colored or white based on their autocovariance functions. For each type of noise, we construct a representation and a simulation method. Numerical examples are provided to illustrate our schemes. Comparisons of the influence of different noises on the solution of the Navier-Stokes system are presented. To improve the simulation accuracy, we impose a Martingale correction regularized method for the stochastic Navier-Stokes equations with additive noise. The original systems are split into two parts, a linear stochastic Stokes equations with Martingale solution and a stochastic modified Navier-Stokes equations with smoother noise. In addition, a negative fractional Laplace operator is introduced to regularize the noise term. Stability and convergence of the path-wise modified Navier-Stokes equations are proved. Numerical simulations are provided to illustrate our scheme. Comparisons of non-regularized and regularized noises for the Navier-Stokes system are presented to further demonstrate the efficiency of our numerical scheme. As a consequence of the above work, we consider a stochastic optimal control problem constrained by the Navier-Stokes equations with stochastic Dirichlet boundary conditions. Control is applied only on the boundary and is associated with reduced regularity, compared to interior controls. To ensure the existence of a solution and the efficiency of numerical simulations, the stochastic boundary conditions are required to belong almost surely to H¹(∂D). To simulate the system, state solutions are approximated using the stochastic collocation finite element approach, and sparse grid techniques are applied to the boundary random field. One-shot optimality systems are derived from Lagrangian functionals. Numerical simulations are then made, using a combination of Monte Carlo methods and sparse grid methods, which demonstrate the efficiency of the algorithm.
Identifier: FSU_SUMMER2017_Zhao_fsu_0071E_14002 (IID)
Submitted Note: A Dissertation submitted to the Department of Scientific Computing in partial fulfillment of the requirements for the degree of Doctor of Philosophy.
Degree Awarded: Summer Semester 2017.
Date of Defense: July 13, 2017.
Keywords: Stochastic Navier-Stokes equations, Stochastic optimal control
Bibliography Note: Includes bibliographical references.
Advisory Committee: Max Gunzburger, Professor Directing Dissertation; Mark Sussman, University Representative; Janet Peterson, Committee Member; Bryan Quaife, Committee Member; Chen Huang, Committee Member.
Subject(s): Mathematics
Persistent Link to This Record: http://purl.flvc.org/fsu/fd/FSU_SUMMER2017_Zhao_fsu_0071E_14002
Owner Institution: FSU

Choose the citation style.
Zhao, W. (2017). Comparison of Different Noise Forcings, Regularization of Noise, and Optimal Control for the Stochastic Navier-Stokes Equations. Retrieved from http://purl.flvc.org/fsu/fd/FSU_SUMMER2017_Zhao_fsu_0071E_14002