| Author: | Luo, Jianfeng |
| Title: | Convergence analysis and uncertainty quantification of differential complementarity systems with uncertainty |
| Advisors: | Chen, Xiaojun (AMA) |
| Degree: | Ph.D. |
| Year: | 2021 |
| Subject: | Programming (Mathematics) Mathematical optimization Hong Kong Polytechnic University -- Dissertations |
| Department: | Department of Applied Mathematics |
| Pages: | xx, 111 pages : color illustrations |
| Language: | English |
| Abstract: | In this thesis, our research focuses on the differential complementarity systems. We extend deterministic differential complementarity systems to random ones to obtain three types of differential complementarity systems with uncertainty parameters. We are devoted to analyzing the quantification and convergence of the numerical methods of the above three systems, which consist of two initial value systems and a boundary value system. Firstly, we study a differential linear stochastic complementarity system consisting of a deterministic ordinary differential equation and a linear stochastic complementarity problem. We investigate the existence of solutions of the above system under two cases where the coefficient matrix of the linear stochastic complementarity problem is a P-matrix and a positive semi-definite matrix, respectively. For the discrete version of our system under the first case, we adopt the independent identically distributed samples as the outer scheme and the time-stepping method as the inner one. The corresponding convergence analysis is conducted. We also introduce a regularization parameter to the second case to guarantee the uniqueness of solutions. The convergence is carried out as the parameter decreases to zero. Numerical examples verify our theoretical results. We then study a differential complementarity system which composes of an ordinary differential equation with uncertain parameters and a nonlinear stochastic complementarity problem. We first adopt the time-stepping method to discretize this system. The mean square convergence and error bound of the time-stepping method are established. For uncertainty quantification, we introduce a multilevel Monte Carlo approach and compare it with the standard Monte Carlo approach. The error analysis illustrates that the multilevel Monte Carlo approach can save more computational costs. Numerical experiments are also conducted to confirm our results. At last, we apply a differential nonlinear complementarity system with a boundary condition to study an optimal control problem with parameter uncertainty. The differential complementarity system can be derived from a necessary condition of the optimal control problem. It consists of a random ordinary differential equation and an expected stochastic complementarity problem. We establish the well-posedness of the complementarity system under some mild conditions. Furthermore, we adopt the explicit Euler time-stepping method as the outer scheme and the independent identically distributed samples as the inner one. Moreover, we analyze the corresponding convergence properties of two discrete approximation schemes. Finally, a numerical example of a linear-quadratic optimal control is presented to confirm the convergence theorem. |
| Rights: | All rights reserved |
| Access: | open access |
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