Research on multigrid methods for optimization problems is reviewed. Optimization problems considered include shape design, parameter optimization, and optimal control problems governed by partial differential equations of elliptic, parabolic, and hyperbolic type.

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigatio...

Artificial Intelligence (AI) can support diagnostic workflows in oncology by aiding diagnosis and providing biomarkers directly from routine pathology slides. However, AI applications are vulnerable to adversarial attacks. Hence, it is essential to quantify and mitigate this risk before widespread clinical use. Here, we show that convolutional neural networks (CNNs) are highly susceptible to white- and black-box adversarial attacks in clinically relevant weakly-supervised classification tasks. Adversarially robust training and dual batch normalization (DBN) are possible mitigation strategies but require precise knowledge of the type of attack used in the inference. We demonstrate that vision transformers (ViTs) perform equally well compared to CNNs at baseline, but are orders of magnitude more robust to white- and black-box attacks. At a mechanistic level, we show that this is associated with a more robust latent representation of clinically relevant categories in ViTs compared to CNNs. Our results are in line with previous theoretical studies and provide empirical evidence that ViTs are robust learners in computational pathology. This implies that large-scale rollout of AI models in computational pathology should rely on ViTs rather than CNN-based classifiers to provide inherent protection against perturbation of the input data, especially adversarial attacks.

We compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace–Beltrami type metrics are compared with Steklov–Poincaré type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.