A shape optimization algorithm for cellular composites

We propose and investigate a mesh deformation technique for PDE constrained shape optimization. Introducing a gradient penalization to the inner product for linearized shape spaces, mesh degeneration can be prevented within the optimization iteration allowing for the scalability of employed solvers. We illustrate the approach by a shape optimization for cellular composites with respect to linear elastic energy under tension. The influence of the gradient penalization is evaluated and the parallel scalability of the approach demonstrated employing a geometric multigrid solver on hierarchically distributed meshes.

On outflow boundary conditions in finite element simulations of non-Newtonian internal flows

The matter of appropriate boundary conditions for open or truncated outflow regions in internal flow is still focus of discussion and research. In most practical applications, one can at best estimate mean pressure values or flow rates at such outlets. In the context of finite element methods, it is known that enforcing mean pressures through the pseudo-tractions arising from the Laplacian Navier-Stokes formulation yields accurate, physically consistent solutions. Nevertheless, when generalised Newtonian fluid models are considered, the resulting non-uniform viscosity fields render the classical Laplacian formulation inadequate. Thus, it is common practice to use the socalled stress-divergence formulation with natural boundary conditions known for causing nonphysical outflow behaviour. In order to overcome such a limitation, this work presents a novel mixed variational formulation that can be seen as a generalisation of the Laplacian Navier-Stokes form to fluids with shear-rate-dependent viscosity, as appearing in hemodynamic and polymeric flows. By appropriately manipulating the viscous terms in the variational formulation and employing a simple projection of the constitutive law, it is possible to devise a formulation with the desired natural boundary conditions and low computational complexity. Several numerical examples are presented to showcase the potential of our method, revealing improved accuracy and robustness in comparison with the state of the art.

Optimization with Partial Differential Equation Constraints for Bio-imaging Applications

Molecular transport and interaction are of funda- mental importance in biology and medicine. The spatiotemporal diffusion map can reflect the regulation of molecular interactions and their intracellular functions. To construct subcellular diffusion maps based on bio-imaging data, we explore a general optimization framework with diffusion equation constraints (OPT-PDE). For the solution of the spatially piecewise constant and anisotropic diffusion.

A Semi-Algebraic Two Level Solver

We develop a simple semi-algebraic 2-level solver built on traditional multigrid ideas. It is designed to be easily incorporated into existing simulation software. It exhibits good convergence for many classes of challenging problems including discontinuous diffusion, convection- diffusion, and Helmholtz equations. It has built-in structure that makes it simple to generalize in several interesting directions.

A Subdivision-based Geometric Multigrid Method

In this paper we introduce a smooth subdi- vision theory-based geometric multigrid method. While theory and efficiency of geometric multigrid methods rely on grid regularity, this requirement is often not directly fulfilled in applications where partial differential equations are defined on complex geometries. Instead of generating multigrid hierarchies with classical linear refinement, we here propose the use of smooth subdivision theory for automatic grid hierarchy regularization within a geometric multigrid solver. This subdivi- sion multigrid method is compared to the classical geometric multigrid method for two benchmark problems. Numerical tests show significant improvement factors for iteration numbers and solve times when comparing subdivision to classical multigrid. A second study fo- cusses on the regularizing effects of surface subdivision refinement, using the Poisson-Nernst-Planck equations. Subdivision multigrid is demonstrated to outperform classical multigrid.

Multilevel Initialization for Layer-Parallel Deep Neural Network Training

This paper investigates multilevel initializa- tion strategies for training very deep neural networks with a layer-parallel multigrid solver. The scheme is based on a continuous interpretation of the training problem as an optimal control problem, in which neu- ral networks are represented as discretizations of time- dependent ordinary differential equations. A key goal is to develop a method able to intelligently initialize the network parameters for the very deep networks en- abled by scalable layer-parallel training. To do this, we apply a uniform refinement strategy across the time domain, that is equivalent to refining in the layer di- mension. This refinement algorithm builds good ini- tializations for deep networks with network parameters coming from the coarser trained networks. The effec- tiveness of multilevel strategies (called nested iteration) for training is investigated using the Peaks and Indian Pines classification data sets. In both cases, the vali- dation accuracy achieved by nested iteration is higher than non-nested training. Moreover, run time to achieve the same validation accuracy is reduced. For instance, the Indian Pines example takes around 25% less time to train with the nested iteration algorithm. Finally, using the Peaks problem, we present preliminary anec- dotal evidence that the initialization strategy provides a regularizing effect on the training process, reducing sensitivity to hyperparameters and randomness in ini- tial network parameters.