Efficient and parallelizable numerics for optimal control and nonlinear partial differential equations

Abstract

The analysis and optimal control of differential equations are of central interest in various fields of research as well as industrial applications. Given the limitations of analytical methods for solving related problems, one must often devise potent numerical approximations. In practice, attaining accurate solutions will necessitate large-scale approximations that require tailored methods to facilitate efficient computation. This thesis focuses on three topics related to the overarching theme of numerical methods for large-scale optimal control problems and the numerical analysis of complex nonlinear dynamical systems.

In the first part, we derive a new parallel-in-time approach for solving large-scale optimization problems constrained by time-dependent partial differential equations arising from fluid dynamics. The solver consists of a preconditioner used within a flexible GMRES iteration. The preconditioner involves the use of a block circulant approximation of the original matrices, enabling parallelization-in-time via the use of fast Fourier transforms. We devise bespoke matrix approximations which may be applied within this framework. These make use of block row and column operations, saddle-point approximations, commutator arguments for divergence and gradient terms, as well as inner solvers such as the Uzawa method, Chebyshev semi-iteration, and multigrid.

Theoretical results underpin our strategy of applying a block circulant strategy, and numerical experiments demonstrate the effectiveness and robustness of our approach on Stokes and Oseen problems. Notably, satisfying results for the strong and weak scaling of our methods are provided within a fully parallel architecture. To our knowledge, this is the first parallelizable preconditioner for fluid flow problems with such a strong theoretical foundation.

While the first part of this thesis addresses optimal control for linear systems, which are well understood, the analysis and computation of nonlinear problems pose significantly greater challenges. Linearization methods for nonlinear systems can provide a partial solution by converting the problem into a linear system. This allows for the application of techniques designed for linear systems to be utilized for nonlinear systems as well. However, linearization methods currently lack a solid mathematical framework when applied to partial differential equations and in the context of their optimal control. In the second part of this thesis, we explore how the Carleman linearization---one particular class of linearization methods---can be extended to dynamical systems on infinite-dimensional Hilbert spaces with quadratic nonlinearities. We demonstrate the well-posedness and convergence of the truncated Carleman linearization under suitable assumptions on the dynamical system, which encompass many common parabolic semi-linear partial differential equations. Upon discretization, we show that the total approximation error of the linearization decomposes into two independent components: the discretization error and the linearization error. This decomposition yields convergence bounds of the linearization independent of the discretization. Furthermore, it motivates the use of non-standard structure-exploiting numerical methods. Finally, we verify the theoretical convergence results with numerical experiments.

Lastly, we consider the solution of saddle-point systems with a tree-based block structure. Such problems stem from optimal control problems that consist of small, distributed control problems linked by coupling on a limited number of degrees of freedom with a specific graph structure. As our key contribution, we propose several structure-exploiting preconditioners to be used during applications of the GMRES algorithm and analyze their properties. We adapt several concepts originating in the field of multigrid methods, obtaining a variety of adapted multi-level methods. We analyze the complexity of all algorithms, and derive a number of results on eigenvalues of the preconditioned system and convergence of iterative methods. We validate our theoretical findings through a range of numerical experiments, demonstrating the convergence and efficacy of the developed preconditioners.