Reduced basis methods applied to obstacle problems and parametrized distributed optimal control problems with control and state constraints

• Anwendung der Reduzierte-Basis-Methode auf Hindernisprobleme und parametrisierte optimale Steuerungsprobleme mit Kontroll- und Zustandsbeschränkung

Bader, Eduard; Veroy-Grepl, Karen Paula (Thesis advisor); Herty, Michael Matthias (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2016

Abstract

In this thesis, we address new reduced basis approaches for parametrized variational inequalities and parametrized linear-quadratic optimal control problems with constraints on both the control and the state. These problems are challenging because of their nonlinear nature and, after a discretization, usually large number of degrees of freedom. We address the latter difficulty by reducing the complexity of solving suchproblems by applying the reduced basis method, resulting in very small approximate problems. These small problem cannot only be inexpensively solved but also provide inexpensive error estimators/bounds that allow us to directly control the approximation error.Many models in computation engineering science require numerous evaluations of input-output systems that are described by parameter-dependent partial differential equations. Inputs might determine system properties, e.g., the conductivity coefficients, whereas the system's outputs are then, e.g., the heat distribution. Evaluating these input-output relationships usually requires expensive and iterative computations, which makes the need for rapid and yet reliable reduction methods apparent.