Optimal control problems constrained to nonlinear partial differential equations appear in many applications. Because the discretization of these problems leads to large-scale systems, a direct approach is not always viable. We consider the aggressive space-mapping approach using surrogate models derived with the help of model order reduction techniques in order to efficiently solve such problems, while maintaining an acceptable accuracy. Numerical results based on a medical application (laser-induced thermotherapy) are used to evaluate the effectiveness of the proposed approach.In many applications, the direct solution of optimal control problems is not always viable, due to the associated computational costs. Therefore, it is often beneficial to use surrogate models in order to improve efficiency. These models can be drawn from a model hierarchy. Leaning on the classification from [1], model hierarchies can be categorized into three broad types: physical-based, algebraic-based and grid-based. For example, projection-based model order reduction techniques deliver an algebraic-based model hierarchy. While one could simply optimize with respect to a coarse (cheap) model in order to reduce the computational effort at the cost of accuracy, the space-mapping approach offers a more elegant solution by aligning the optimization of a fine (accurate) model with a coarse one.Assuming that a fine and coarse model f and c, respectively, are available, then the space-mapping function p : U f → U c ,

In this paper, we develop a nonlinear reduction framework based on our recently introduced extended group finite element method. By interpolating nonlinearities onto approximation spaces defined with the help of finite elements, the extended group finite element formulation achieves a noticeable reduction in the computational overhead associated with nonlinear finite element problems. However, the problem's size still leads to long solution times in most applications. Aiming to make real-time and/or many-query applications viable, we apply model order reduction and complexity reduction techniques in order to reduce the problem size and efficiently handle the reduced nonlinear terms, respectively. For this work, we focus on the proper orthogonal decomposition and discrete empirical interpolation methods. While similar approaches based on the group finite element method only focus on semilinear problems, our proposed framework is also compatible with quasilinear problems. Compared to existing methods, our reduced models prove to be superior in many different aspects as demonstrated in three numerical benchmark problems.

We investigate the planning of minimally invasive tumor treatments via laser-induced thermotherapy. The goal is to control the laser in order to obtain an optimal treatment, e.g. eradicating the tumor, while leaving as much healthy tissue unharmed as possible. To this end, we define a PDE-constrained optimal control problem. As these problems are usually computationally expensive, we propose a simplified modeling approach using reduced-order models. Numerical results illustrate the viability of our approach.

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Interpolation methods for nonlinear finite element discretizations are commonly used to eliminate the computational costs associated with the repeated assembly of the nonlinear systems. While the group finite element formulation interpolates nonlinear terms onto the finite element approximation space, we propose the use of a separate approximation space that is tailored to the nonlinearity. In many cases, this allows for the exact reformulation of the discrete nonlinear problem into a quadratic problem with algebraic constraints. Furthermore, the substitution of the nonlinear terms often shifts general nonlinear forms into trilinear forms, which can easily be described by third-order tensors. The numerical benefits as well as the advantages in comparison to the original group finite element method are studied using a wide variety of academic benchmark problems.