Thomas Ranner

Ana Djurdjevac, Charles M. Elliott, Ralf Kornhuber et al.

In this paper, we introduce and analyse a surface finite element discretization of advection-diffusion equations with uncertain coefficients on evolving hypersurfaces. After stating unique solvability of the resulting semi-discrete problem, we prove optimal error bounds for the semi-discrete solution and Monte Carlo samplings of its expectation in appropriate Bochner spaces. Our theoretical findings are illustrated by numerical experiments in two and three space dimensions.

Till Bretschneider, Cheng-Jin Du, Charles M. Elliott et al.

We present a computational approach for solving reaction-diffusion equations on evolving surfaces which have been obtained from cell image data. It is based on finite element spaces defined on surface triangulations extracted from time series of 3D images. A model for the transport of material between the subsequent surfaces is required where we postulate a velocity in normal direction. We apply the technique to image data obtained from a spreading neutrophil cell. By simulating FRAP experiments we investigate the impact of the evolving geometry on the recovery. We find that for idealised FRAP conditions, changes in membrane geometry, easily account for differences of $\times 10$ in recovery half-times, which shows that experimentalists must take great care when interpreting membrane photobleaching results. We also numerically solve an activator -- depleted substrate system and report on the effect of the membrane movement on the pattern evolution.

Thomas P. Ilett, Omer Yuval, Thomas Ranner et al.

3D shape reconstruction typically requires identifying object features or textures in multiple images of a subject. This approach is not viable when the subject is semi-transparent and moving in and out of focus. Here we overcome these challenges by rendering a candidate shape with adaptive blurring and transparency for comparison with the images. We use the microscopic nematode Caenorhabditis elegans as a case study as it freely explores a 3D complex fluid with constantly changing optical properties. We model the slender worm as a 3D curve using an intrinsic parametrisation that naturally admits biologically-informed constraints and regularisation. To account for the changing optics we develop a novel differentiable renderer to construct images from 2D projections and compare against raw images to generate a pixel-wise error to jointly update the curve, camera and renderer parameters using gradient descent. The method is robust to interference such as bubbles and dirt trapped in the fluid, stays consistent through complex sequences of postures, recovers reliable estimates from blurry images and provides a significant improvement on previous attempts to track C. elegans in 3D. Our results demonstrate the potential of direct approaches to shape estimation in complex physical environments in the absence of ground-truth data.

Christian Engwer, Thomas Ranner, Sebastian Westerheide

Motivated by considering partial differential equations arising from conservation laws posed on evolving surfaces, a new numerical method for an advection problem is developed and simple numerical tests are performed. The method is based on an unfitted discontinuous Galerkin approach where the surface is not explicitly tracked by the mesh which means the method is extremely flexible with respect to geometry. Furthermore, the discontinuous Galerkin approach is well-suited to capture the advection driven by the evolution of the surface without the need for a space-time formulation, back-tracking trajectories or streamline diffusion. The method is illustrated by a one-dimensional example and numerical results are presented that show good convergence properties for a simple test problem.