# SIAM Colloquium Summer Term 2018

In the summer term 2018 the first SIAM Colloquium took place. The goal of this lecture series was, to give an insight in current local research. We invited four researchers from the Mathematikon to present their current work. At the last date, we had an informal get together with pizza and beer.

# Talks

**Mechanochemical pattern formation in biological tissue**

**Felix Brinkmann**

In this talk a finite element method for mechanochemical pattern formation will be presented. A biological application of this prototypical model is embryonic development of fertilized cells. We model biological tissues using the simple, hyperelastic Saint Venant-Kirchhoff model. The growth processes are modeled by splitting the deformation gradient into an active part and an elastic response. The active part depends on the concentration of signaling molecules, which are modeled by an reaction-diffusion equation. Evolving patterns are reinforced by a feedback mechanism which depends on mechanical cues, e.g. stress, compression or strain and which is robust to changes in the parameters or in the initial conditions. Finally, implementation details such as parallelization will be addressed. All problems, in particular in 3D, are solved with the software library Gascoigne 3D.

**Estimation of Parameters in Nonlinear Dynamic Models - The Multiple Shooting Approach**

**Dr. Johannes P. Schlöder**

**Finite Element Simulation of Drugs Spreading in the Vitreous Body**

**Simon Dörsam**

The injection of a drug into the vitreous body of a human eye for the treatment of retinal diseases is the most common form of medical intervention worldwide. In the worst case, the treatment prevents that the patient loses his eyesight. The aim of an optimal therapy is that the drug operates locally around the area of the macula as long as possible. We present numerical simulations of the drug distribution, which are generated by using the Finite Element method. Therefore, the mathematical model is a Darcy equation combined with a transport-anisotropic-diffusion equation. The numerical grid is constructed with the help of parameter estimation methods, which fit measurement data from different patients. The discretization is realized by using the Crank-Nicolson scheme in time, the Raviart-Thomas elements for the velocity, discontinuous zero-order elements for the pressure and Lagrange elements for the concentration. Finally, we investigate the influence of the position of the injection on the drug distribution. This is realized by introducing specific output functionals, which measure the mean or relative amount of the drug in the vitreous and in the area of action. Our simulations show that the injections should be located in the center of the vitreous body for a more efficient therapy.

**Using differential equations to solve nonlinear algebraic and nonlinear optimization problems**

**Dr. Andreas Potschka**

Damped Newton-type and Gauß-Newton methods for the solution of nonlinear algebraic equations or nonlinear least squares problems can be interpreted as explicit Euler time-stepping methods for a generalized Newton-flow, which is an autonomous differential equation that can be formally obtained by considering infinitesimally small step sizes for the Newton-type method. We present how this viewpoint can be exploited to construct efficient algorithms with guaranteed convergence to the closest solution in the sense of the generalized Newton flow under reasonable assumptions. We then consider a nonsmooth gradient flow for nonlinear optimization problems with equality and inequality constraints. A projected implicit Euler time-stepping method on this flow results in a novel sequential homotopy algorithm. This approach can be used to globalize the convergence of arbitrary locally convergent optimization algorithms, including algorithms of inexact sequential quadratic programming type suitable for large scale problems. We highlight the beneficial properties of this approach for the resulting quadratic programming subproblems: Guaranteed feasibility, strict convexity, and the strongest possible constraint qualifications independently of the original nonlinear problem and transition to fast local convergence.