Compliance minimisation problems under a fully discrete and Riemannian framework
Karen Estefania Loayza Romero (Imperial)

Abstract: Compliance minimization problems aim to find as stiff structures as possible for prescribed volumes, which can be used in designing bridges, offshore structures, turbomachinery, or biomedical devices. Usually, they are treated as topology optimization problems, which, unfortunately, are highly ill-posed and often exhibit solutions with microstructures that are difficult to treat numerically and, more importantly, to manufacture. One way to guarantee the existence of solutions to this problem is to fix the topology (the number of holes) of the structure, which transforms the problem into a shape optimization problem. However, even by treating them as shape optimization problems, many drawbacks remain. For example, there are discrepancies in the existence of solutions between the continuous and discretized problems, and there is an increasing mesh degeneracy along the optimization process.

In this talk, we focus on developing a framework that allows us to solve fully discrete shape optimization problems based on the assumption that discrete shapes are triangular meshes and that they form a smooth manifold. This manifold will be endowed with a particular complete Riemannian metric that takes care of the mesh degeneracy and helps us to provide regularization to guarantee the existence of solutions.

Main Speaker (3pm)

Approximating the inverse Hessian in 4D-Var data assimilation
Alison Ramage (U. Strathclyde)

Abstract: Large-scale variational data assimilation problems are commonly
found in applications like numerical weather prediction and
oceanographic modelling. The 4D-Var method is frequently used to
calculate a forecast model trajectory that best fits the available
observations to within the observational error over a period of time.
One key challenge is that the state vectors used in realistic
applications typically contain a very large number of unknowns so,
due to memory limitations, in practice it is often impossible
to assemble, store or manipulate the matrices involved explicitly. In
this talk we present a limited memory approximation to the inverse
Hessian, computed using the Lanczos method, based on a multilevel approach. We illustrate one use of this approximation by showing its potential effectiveness as a preconditioner within a Gauss-Newton iteration.