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UG 4: A novel flexible software system for simulating PDE based models on high performance computers

In this paper we describe the concept of the renewed software package UG, that is used as a flexible simulation framework for the solution of partial differential equations. A general overview of the concepts of the new implementation is given: The modularization of the software package into several libraries libGrid, libAlgebra, libDiscretization and pcl is described and all major modules are discussed in detail. User backends through scripting and visual editing are briefly considered and examples show the new features of the current implementation.

Software Framework ug4: Parallel Multigrid on the Hermit Supercomputer

The modeling of physical phenomena in a variety of fields of scientific interest lead to a formulation in terms of partial differential equations. Especially when complex geometries as the domain of definition are involved, a direct and exact solution is not accessible, but numerical schemes are used to compute an approximate discrete solution. In this report, we focus on elliptic and parabolic types of equations that include spatial operators of second order. When discretizing such problems using commonly known discretization schemes such as finite element methods or finite volume methods, large systems of linear equations arise naturally. Their solution takes the largest amount of the overall computing time.

10,000 Performance Models per Minute – Scalability of the UG4 Simulation Framework

Numerically addressing scientific questions such as simulating drug diffusion through the human stratum corneum is a challenging task requiring complex codes and plenty of computational resources. The UG4 framework is used for such simulations, and though empirical tests have shown good scalability so far, its sheer size precludes analytical modeling of the entire code. We have developed a process which combines the power of our automated performance modeling method and the workflow manager JUBE to create insightful models for entire UG4 simulations. Examining three typical use cases, we identified and resolved a previously unknown latent scalability bottleneck. In collaboration with the code developers, we validated the performance expectations in each of the use cases, creating over 10,000 models in less than a minute, a feat previously impossible without our automation techniques.

Scalable shape optimization methods for structured inverse modeling in 3D diffusive processes

In this work we consider inverse modeling of the shape of cells in the outermost layer of human skin. We propose a novel algorithm that combines mathematical shape optimization with high-performance computing. Our aim is to fit a parabolic model for drug diffusion through the skin to data measurements. The degree of freedom is not the permeability itself, but the shape that distinguishes regions of high and low diffusivity. These are the cells and the space in between. The key part of the method is the computation of shape gradients, which are then applied as deformations to the finite element mesh, in order to minimize a tracking type objective function. Fine structures in the skin require a very high resolution in the computational model. We therefor investigate the scalability of our algorithm up to millions of discretization elements.

Multigrid analysis of spatially resolved hepatitis C virus protein simulations

Viruses are a major challenge to human health and prosperity. This holds true for various viruses which are either threatening Europe (like Dengue and Yellow fever) or which are currently causing big health problems like the hepatitis C virus (HCV). HCV causes chronic liver diseases like cirrhosis and cancer and is the main reason for liver transplantations. Exploring biophysical properties of virus-encoded components and viral life cycle is an exciting new area of current virological research. In this context, spatial resolution is an aspect that has not yet been received much attention despite strong biological evidence suggesting that intracellular spatial dependence is a crucial factor in the viral replication process. We are developing first spatio-temporal resolved models which mimic the behavior of the important components of virus replication within single liver cells. HCV replication is strongly associated to the intracellular Endoplasmatic Reticulum (ER) network. Here, we present the computational basis for the estimation of the diffusion constant of a central component of HCV genome (viral RNA) replication, namely the NS5a protein, on the surface of realistic reconstructed ER geometries. The basic surface partial differential equation (sPDE) evaluations are performed with UG4 using fast massively parallel multigrid solvers. The numerics of the simulations are studied in detail. Integrated concentrations within special subdomains correspond to experimental FRAP time series. In particular, we analyze the refinement stability in time and space for these integrated concentrations based on diffusion sPDEs upon large unstructured surface grids using heuristic values for the NS5a diffusion constant. This builds up a solid basis for future research not included in this presentation. e.g. the presented refinement stability analysis of the single sPDEs allows for parameter estimations for the NS5a diffusion constant. Our advanced Finite Volume/multigrid techniques also could be applied for studying life cycles of other viruses.

Automated Performance Modeling of the UG4 Simulation Framework

Many scientific research questions such as the drug diffusion through the upper part of the human skin are formulated in terms of partial differential equations and their solution is numerically addressed using grid based finite element methods. For detailed and more realistic physical models this computational task becomes challenging and thus complex numerical codes with good scaling properties up to millions of computing cores are required. Employing empirical tests we presented very good scaling properties for the geometric multigrid solver in Reiter et al. (Comput Vis Sci 16(4):151–164, 2013) using the UG4 framework that is used to address such problems. In order to further validate the scalability of the code we applied automated performance modeling to UG4 simulations and presented how performance bottlenecks can be detected and resolved in Vogel et al. (10,000 performance models per minute—scalability of the UG4 simulation framework. In: Traff JL, Hunold S, Versaci F (eds) Euro-Par 2015: Parallel processing, theoretical computer science and general issues, vol 9233. Springer, Springer, Heidelberg, pp 519–531, 2015). In this paper we provide an overview on the obtained results, present a more detailed analysis via performance models for the components of the geometric multigrid solver and comment on how the performance models coincide with our expectations.

A Massively Parallel Multigrid Method with Level Dependent Smoothers for Problems with High Anisotropies

Anisotropic layers, as often seen in biological and geological domains, impose difficulties to several aspects of numerical simulations. In this article we examine how the highly scalable approach to massively parallel geometric multigrid solvers presented in Reiter et al. (Comput Vis Sci 16(4):151–164, 2013) can be extended to problem domains featuring such anisotropies. Considering the real world problem of drug diffusion through the human skin we combine hierarchically distributed multigrids, anisotropic refinement, and level dependent smoothing strategies to create a robust and highly scalable multigrid solver for anisotropic domains.

A Concept for Quantitative Comparison of Mathematical and Natural Language and its possible Effect on Learning

Starting with the question whether there is a connection between the mathematical capabilities of a person and his or her mother tongue, we introduce a new modeling approach to quantitatively compare natural languages with mathematical language. The question arises from educational assessment studies that indicate such a relation. Texts written in natural languages can be deconstructed into a dependence graph, in simple cases a dependence tree. The same kind of deconstruction is also possible for mathematical texts. This gives an idea of how to quantitatively compare mathematical and natural language. To that end, we develop algorithms to define the distance between graphs. In this paper, we restrict the structure to trees. In order to measure the distance between trees, we use algorithms based on previous work measuring the distance of neurons using the constrained tree edit distance. Once a distance matrix has been computed, this matrix can be used to perform a cluster analysis.

Massively Parallel Multigrid for the Simulation of Skin Permeation on Anisotropic Tetrakaidecahedral Cell Geometries

Numerical simulation based on mathematical models is an important pillar for enhancing the understanding of permeation processes in the skin. To adequately resolve the complex geometrical structure of the skin, special models based on tetrakaidecahedral cells have been suggested. While these models preserve many of the desirable properties of the underlying geometry, they impose challenges regarding mesh generation and solver robustness.

Towards the Implementation of a New Multigrid Solver in the DNS Code FS3D for Simulations of Shear-Thinning Jet Break-Up at Higher Reynolds Numbers

Liquid jet break-up appears in many technical applications, as well as in nature. It consists of complex physical processes, which happen on very small scales in space and time. This makes them hard to capture by experimental methods; and therefore a prime subject for numerical investigations. The state-of-the-art approach combines the Volume of Fluid (VOF) method with Direct Numerical Simulations (DNS) as employed in the ITLR in-house code Free Surface 3D (FS3D). The simulation of these jets is dependent on very fine grids, with most of the computational costs incurred by solving the Pressure Poisson Equation. In order to simulate larger computational domains, we tried to improve the performance of FS3D by the implementation of a new multigrid solver. For this we selected the solver contained in the UG4 package developed by the Goethe Center for Scientific Computing at the University of Frankfurt. We will show simulations of the primary break-up of shear-thinning liquid jets and explain why larger computational domains are necessary. Results are preliminary. We demonstrate that the implementation of UG4 into FS3D provides a noticeable increase in weak scaling performance, while the change in strong scaling is yet detrimental. We will then discuss ways to further improve these results.