Konrad Simon

Konrad Simon, Jörn Behrens

Long simulation times in climate sciences typically require coarse grids due to computational constraints. Nonetheless, unresolved subscale information significantly influences the prognostic variables and can not be neglected for reliable long term simulations. This is typically done via parametrizations but their coupling to the coarse grid variables often involves simple heuristics. We explore a novel up-scaling approach inspired by multi-scale finite element methods. These methods are well established in porous media applications, where mostly stationary or quasi stationary situations prevail. In advection-dominated problems arising in climate simulations the approach needs to be adjusted. We do so by performing coordinate transforms that make the effect of transport milder in the vicinity of coarse element boundaries. The idea of our method is quite general and we demonstrate it as a proof-of-concept on a one-dimensional passive advection-diffusion equation with oscillatory background velocity and diffusion.

Konrad Simon, Ronen Basri

We consider the problem of matching two shapes assuming these shapes are related by an elastic deformation. Using linearized elasticity theory and the finite element method we seek an elastic deformation that is caused by simple external boundary forces and accounts for the difference between the two shapes. Our main contribution is in proposing a cost function and an optimization procedure to minimize the symmetric difference between the deformed and the target shapes as an alternative to point matches that guide the matching in other techniques. We show how to approximate the nonlinear optimization problem by a sequence of convex problems. We demonstrate the utility of our method in experiments and compare it to an ICP-like matching algorithm.

Konrad Simon, Sameer Sheorey, David Jacobs et al.

We suggest a novel shape matching algorithm for three-dimensional surface meshes of disk or sphere topology. The method is based on the physical theory of nonlinear elasticity and can hence handle large rotations and deformations. Deformation boundary conditions that supplement the underlying equations are usually unknown. Given an initial guess, these are optimized such that the mechanical boundary forces that are responsible for the deformation are of a simple nature. We show a heuristic way to approximate the nonlinear optimization problem by a sequence of convex problems using finite elements. The deformation cost, i.e, the forces, is measured on a coarse scale while ICP-like matching is done on the fine scale. We demonstrate the plausibility of our algorithm on examples taken from different datasets.