확률버거즈 방정식의 수치근사와 최적제어
Numerical approximation and distributed control of the stochastic Burgers equation
- 주제(키워드) 버거즈 방정식 , 스파스 그리드 , 유한요소법
- 발행기관 아주대학교
- 지도교수 이형천
- 발행년도 2016
- 학위수여년월 2016. 8
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/ajou/000000022746
- 본문언어 영어
- 저작권 아주대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
The objective of this work is to study an efficient approximation of the solution to stochastic Burgers equation driven by an additive space-time noise and a distributed feedback control problem for the stochastic Burgers equation with a random diffusion coefficient. For these goals, we introduce several stochastic tools in uncertainty quantification, which involve the white noise, the stochastic integrals, the Karhunen-Loeve expansion, and the sparse grid stochastic collocation method. We discuss existence and uniqueness of the stochastic Burgers equation with an additive space-time noise, which is an infinite dimensional random process, through the Orstein-Uhlenbeck (OU) process. To approximate the OU process, we use the Karhunen-Loeve expansion, and sparse grid stochastic collocation method. About spatial discretization of Burgers equation, two separate finite element approximations are presented: the conventional Galerkin method and Galerkin-conservation method. Numerical experiments are provided to demonstrate the efficacy of schemes mentioned above. Meanwhile, we study a control problem for stochastic Burgers equation with a random diffusion coefficient. Numerical schemes for computing the stochastic Burgers equation are developed. We apply the finite element method in spatial discretization and the sparse grid stochastic collocation method in random parameter space. We also use those schemes to compute closed-loop, suboptimal state feedback control. Several numerical experiments are performed to demonstrate the efficiency and plausibility of proposed approximation methods for the stochastic Burgers equation and the related control problem.
more목차
1 Introduction 1
2 Stochastic Tools in Uncertainty Quantification 9
2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.1.1 Brownian motion and white noise . . . . . . . . . . . 11
2.1.2 Stochastic integral . . . . . . . . . . . . . . . . . . . 14
2.2 Karhunen-Loeve exapansion . . . . . . . . . . . . . . . . . . 17
2.3 Sparse grid stochastic collocation method . . . . . . . . . . 25
2.3.1 Stochastic collocation method . . . . . . . . . . . . . 27
2.3.2 Smolyak formula . . . . . . . . . . . . . . . . . . . . 28
2.3.3 Interpolation abscissas . . . . . . . . . . . . . . . . . 31
3 Stochastic Burgers Equation with Additive Noise 35
3.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Ornstein-Uhlenbeck process and Stochastic Burgers equation 37
3.2.1 Stochastic parabolic equation . . . . . . . . . . . . . 38
3.2.2 Stochastic Burgers equation . . . . . . . . . . . . . . 40
3.3 Approximation of the Ornstein-Uhlenbeck process . . . . . 44
3.4 Two finite element methods for the Burgers equation . . . . 47
3.4.1 The Galerkin method . . . . . . . . . . . . . . . . . 47
3.4.2 The Galerkin-conservation method . . . . . . . . . . 48
3.4.3 Application to the stochastic problems . . . . . . . . 50
3.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 52
4 Distributed Control of the Stochastic Burgers Equation
with Random Input Data 65
4.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.3 Finite element approximation . . . . . . . . . . . . . . . . . 69
4.4 Distributed feedback control of the stochastic Burgers equation 73
4.4.1 Linear quadratic regulator design . . . . . . . . . . . 73
4.4.2 Linear feedback controllers with state estimate feedback 75
4.5 Numerical experiments . . . . . . . . . . . . . . . . . . . . . 77
5 Conclusion 89
