Towards a Lagrange-Newton approach for PDE constrained shape optimization

The novel Riemannian view on shape optimization developed in [Schulz, FoCM, 2014] is extended to a Lagrange-Newton approach for PDE constrained shape optimization problems. The extension is based on optimization on Riemannian vector space bundles and exemplified for a simple numerical example.Comment: 16 pages, 4 figures, 1 tabl

Ensemble Kalman filter for neural network based one-shot inversion

We study the use of novel techniques arising in machine learning for inverse problems. Our approach replaces the complex forward model by a neural network, which is trained simultaneously in a one-shot sense when estimating the unknown parameters from data, i.e. the neural network is trained only for the unknown parameter. By establishing a link to the Bayesian approach to inverse problems, an algorithmic framework is developed which ensures the feasibility of the parameter estimate w.r. to the forward model. We propose an efficient, derivative-free optimization method based on variants of the ensemble Kalman inversion. Numerical experiments show that the ensemble Kalman filter for neural network based one-shot inversion is a promising direction combining optimization and machine learning techniques for inverse problems

Optimal Control of Active Nematics

In this work we present the first systematic framework to sculpt active nematics systems, using optimal control theory and a hydrodynamic model of active nematics. We demonstrate the use of two different control fields, (1) applied vorticity and (2) activity strength, to shape the dynamics of an extensile active nematic that is confined to a disk. In the absence of control inputs, the system exhibits two attractors, clockwise and counterclockwise circulating states characterized by two co-rotating topological $+\frac{1}{2}$ defects. We specifically seek spatiotemporal inputs that switch the system from one attractor to the other; we also examine phase-shifting perturbations. We identify control inputs by optimizing a penalty functional with three contributions: total control effort, spatial gradients in the control, and deviations from the desired trajectory. This work demonstrates that optimal control theory can be used to calculate non-trivial inputs capable of restructuring active nematics in a manner that is economical, smooth, and rapid, and therefore will serve as a guide to experimental efforts to control active matter

Functional a posteriori error estimates for time-periodic parabolic optimal control problems

This paper is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds