## 65K10 Optimization and variational techniques [See also 49Mxx, 93B40]

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#### Keywords

- optimal control (4)
- finite elements (2)
- PDE-constrained optimization (1)
- PFASST (1)
- Trinkwasserversorgung (1)
- Water supply (1)
- adaptive mesh refinement (1)
- adaptive methods (1)
- adjoint gradient computation (1)
- affine conjugate Newton methods (1)

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To solve optimization problems with parabolic PDE constraints, often methods working on the reduced objective functional are used. They are computationally expensive due to the necessity of solving both the state equation and a backward-in-time adjoint equation to evaluate the reduced gradient in each iteration of the optimization method. In this study, we investigate the use of the parallel-in-time method PFASST in the setting of PDE constrained optimization. In order to develop an efficient fully time-parallel algorithm we discuss different options for applying PFASST to adjoint gradient computation, including the possibility of doing PFASST iterations on both the state and adjoint equations simultaneously. We also explore the additional gains in efficiency from reusing information from previous optimization iterations when solving each equation. Numerical results for both a linear and a non-linear reaction-diffusion optimal control problem demonstrate the parallel speedup and efficiency of different approaches.

In gradient-based methods for parabolic optimal control problems, it is necessary to solve both the state equation and a backward-in-time adjoint equation in each iteration of the optimization method. In order to facilitate fully parallel gradient-type and nonlinear conjugate gradient methods for the solution of such optimal control problems, we discuss the application of the parallel-in-time method PFASST to adjoint gradient computation. In addition to enabling time parallelism, PFASST provides high flexibility for handling nonlinear equations, as well as potential extra computational savings from reusing previous solutions in the optimization loop. The approach is demonstrated here for a model reaction-diffusion optimal control problem.

This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.

In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and quasi-Newton methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy coding maintaining an error bound is sufficient.

The paper proposes goal-oriented error estimation and mesh refinement for optimal control problems with elliptic PDE constraints using the value of the reduced cost functional as quantity of interest. Error representation, hierarchical error estimators, and greedy-style error indicators are derived and compared to their counterparts when using the all-at-once cost functional as quantity of interest. Finally, the efficiency of the error estimator and generated meshes are demonstrated on numerical examples.

Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints, otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT's eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete $LBL^T$ factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale 3D PDE-constrained optimal control problems, as well as 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with $150^3$ state variables and $150^2$ control variables, both subject to bound constraints.

Mathematical decision support for operative planning in water supply systems is highly desirable but leads to very difficult optimization problems. We propose a nonlinear programming approach that yields practically satisfactory operating schedules in acceptable computing time even for large networks. Based on a carefully designed model supporting gradient-based optimization algorithms, this approach employs a special initialization strategy for convergence acceleration, special minimum up and down time constraints together with pump aggregation to handle switching decisions, and several network reduction techniques for further speed-up. Results for selected application scenarios at Berliner Wasserbetriebe demonstrate the success of the approach.

The paper extends affine conjugate Newton methods from convex to nonconvex minimization, with particular emphasis on PDE problems originating from compressible hyperelasticity. Based on well-known schemes from finite dimensional nonlinear optimization, three different algorithmic variants are worked out in a function space setting, which permits an adaptive multilevel finite element implementation. These algorithms are tested on two well-known 3D test problems and a real-life example from surgical operation planning.

A new approach to the numerical solution of optimal control problems including control and state constraints is presented. Like hybrid methods, the approach aims at combining the advantages of direct and indirect methods. Unlike hybrid methods, however, our method is directly based on interior-point concepts in function space --- realized via an adaptive multilevel scheme applied to the complementarity formulation and numerical continuation along the central path. Existence of the central path and its continuation towards the solution point is analyzed in some theoretical detail. An adaptive stepsize control with respect to the duality gap parameter is worked out in the framework of affine invariant inexact Newton methods. Finally, the performance of a first version of our new type of algorithm is documented by the successful treatment of the well-known intricate windshear problem.