# Research & development

These are some of the topics of our interests and current work. Don't hesitate to contact us if you need help or are interested in a cooperation in one of these or related topics.

## Analysis and simulation of semiconductors

### Considered semiconductor equations

The mathematical model is given by the van Roosbroeck equations \begin{align*} -\nabla \cdot \varepsilon \nabla w &= C - n - p ,\\ \frac{\partial n}{\partial t} - \nabla \cdot n_i \mu_n e^w \nabla e^{-\phi_n} &= R ,\\ \frac{\partial p}{\partial t} - \nabla \cdot n_i \mu_p e^{-w} \nabla e^{\phi_p} &= R, \end{align*} where $$w$$ is the electrostatic potential, $$n = n_i e^{w - \phi_n}$$ and $$p = n_i e^{\phi_p - w}$$ are the electron and hole densities, $$C$$ is doping, $$\phi_n$$ and $$\phi_p$$ are quasi-Fermi potentials, and $$R = r(\boldsymbol{x},n,p) \left( n_i^2 - np \right)$$ is the recombination/generation rate with $$r(\boldsymbol{x},n,p) \geq 0$$.

For the numerical simulation, it is essential for the discrete system to preserve essential properties of the analytical system, such as the maximum principle, positivity of the carrier concentrations, and current conservation.

### Oskar: semiconductor device simulator

The core idea behind Oskar is the consideration of the physical structure in the numerical solution of the underlying semiconductor equations. Similar structures can be also found in chemical systems and in the image processing (see a phase separation example below). The Voronoi FVM allows to carry over the essential properties of the analytical system to the discretized equations.

Oskar's strength has contributed significantly to the worldwide success of the MPG HLL detectors, including projects for the SLAC National Accelerator Laboratory (example), the European X-ray Free Electron Laser research facility XFEL (example), the High-Energy Accelerator Research Organization KEK, and the Advanced Telescope for High-Energy Astrophysics Athena of the European Space Agency (example).

Oskar was also successfully used in cooperation with the Institute of Applied Photophysics at the TU Dresden to gain a deeper understanding of the mode of operation and major limitations of doped organic semiconductors such as organic light emitting diodes (OLEDs), permeable base transistors (PBTs) and vertical organic field effect transistors (VOFETs).

## Phase separation and image processing

The mathematical description for the non-local phase segregation problem is given by \begin{align} \begin{cases} \frac{\partial u}{\partial t} - \nabla \cdot \left( f (\lvert \nabla v \rvert) \left( \nabla u + \frac {\nabla w}{\phi''(u)} \right) \right) = 0 ~\text{ in $\Omega$} , \\ u(0,\cdot) = u_0(\cdot) , \\ v = \phi'(u) + w , \\ w(t,\boldsymbol{x}) = \int_\Omega \mathcal{K} \left( \lvert \boldsymbol{x}-\boldsymbol{y} \rvert \right) \left(1 - 2 u(t,\boldsymbol{y}) \right) \, \mathrm{d} \boldsymbol{y} , \end{cases} \label{eq:phaseSep} \end{align} where $$\Omega \subset \mathbb{R}^N$$, $$1 \le N \le 3$$ is a bounded Lipschitz domain, $$\phi$$ a convex function, the kernel $$\mathcal{K}$$ represents non-local attracting forces, and $$w$$ and $$v$$ are the interaction and chemical potentials. The initial value $$u_0$$ satisfies $$0 < u_0 < 1$$ and the system is completed with homogeneous Neumann boundary conditions. A short-time version of this model can be used for image segmentation and contrast enhancement.

Phase separation: starting at the smallest scale

Phase separation: the larger reference scale

#### For more details see:

1. H. Gajewski and K. Gärtner:
A dissipative discretization scheme for a nonlocal phase segregation model,
Z. Angew. Math. Mech. 85.11 (2005), pp. 815–822, WIAS Preprint 1004.
2. H. Gajewski and K. Gärtner:
On a nonlocal model of image segmentation,
Z. Angew. Math. Phys. 56 (2005), pp. 572–591, WIAS Preprint 762.

## Graph partitioning and domain separation

Partitioning for $$p = 1.01$$ with 76 cut edges

A non-linear eigenvalue equation \begin{align*} - \nabla \cdot \left( \lvert \nabla u \rvert^{p-2} \nabla u \right) = F_p \lvert u \rvert^{p-2} u \quad \text{ in $\Omega$} \end{align*} with $$\nu \cdot \nabla u = 0$$ on $$\partial \Omega$$ defines an eigenvalue $$F_p$$ with the meaning of the minimal surface halving the volume.

If applied to the graph metric "every edge has size 1", it is equipartitioning the graph with the minimal number of the cut edges.

## Sparse linear solvers

NEC SX Aurora TSUBASA server with 8 Vector Engines
Image by Vectoriser, CC BY-SA 4.0, from Wikimedia Commons

### Direct sparse solvers

Nearly ten years of work went into the parallel direct linear solver PARDISO.

### Solving linear systems on NEC Vector Engine (work in progress)

The new Vector Engine NEC SX Aurora TSUBASA has 8 cores, 24 or 48 GB of HBM2 on chip and can execute 32 double precision floating point operations per cycle and has vector registers for 256 floating point values, resulting in a 16,384 bit vector width and a peak performance of up to 2.45 TFLOPS.

This large vector width requires a redesign of the data structure in order to efficiently utilize the full power of the Vector Engine. Moreover, the high bandwidth of the HBM2 allows effective use of iterative solvers.