Some Domain Decomposition and Convex Optimization Algorithms with Applications to Inverse Problems

Domain decomposition and convex optimization play fundamental roles in current computation and analysis in many areas of science and engineering. These methods have been well developed and studied in the past thirty years, but they still require further study and improving not only in mathematics but in actual engineering computation with exponential increase of computational complexity and scale. The main goal of this thesis is to develop some efficient and powerful algorithms based on domain decomposition method and convex optimization. The topicsstudied in this thesis mainly include two classes of convex optimization problems: optimal control problems governed by time-dependent partial differential equations and general structured convex optimization problems. These problems have acquired a wide range of applications in engineering and also demand a very high computational complexity. The main contributions are as follows: In Chapter 2, the relevance of an adequate inner loop starting point (as opposed to a sufficient inner loop stopping rule) is discussed in the context of a numerical optimization algorithm consisting of nested primal-dual proximal-gradient iterations. To study the optimal control problem, we obtain second order domain decomposition methods by combining Crank-Nicolson scheme with implicit Galerkin method in the sub-domains and explicit flux approximation along inner boundaries in Chapter 3. Parallelism can be easily achieved for these explicit/implicit methods. Time step constraints are proved to be less severe than that of fully explicit Galerkin finite element method. Based on the domain decomposition method in Chapter 3, we propose an iterative algorithm to solve an optimal control problem associated with the corresponding partial differential equation with pointwise constraint for the control variable in Chapter 4. In Chapter 5, overlapping domain decomposition methods are designed for the wave equation on account of prediction-correction" strategy. A family of unit decomposition functions allow reasonable residual distribution or corrections. No iteration is needed in each time step. This dissertation also covers convergence analysis from the point of view of mathematics for each algorithm we present. The main discretization strategy we adopt is finite element method. Moreover, numerical results are provided respectivelyto verify the theory in each chapter.Doctorat en Sciences