Peter Bastian's research while affiliated with Universität Heidelberg and other places
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Publications (143)
Ill-conditioned and multiscale partial differential equations (PDEs) arise in many fields. It is a very challenging problem to compute a resolved, fine-scale solution or to find a robust low-dimensional approximation. In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite ae...
Numerical models of complex real-world phenomena often necessitate High Performance Computing (HPC). Uncertainties increase problem dimensionality further and pose even greater challenges. We present a parallelization strategy for multilevel Markov chain Monte Carlo, a state-of-the-art, algorithmically scalable Uncertainty Quantification (UQ) algor...
Highly heterogeneous, anisotropic coefficients, e.g. in the simulation of carbon-fibre composite components, can lead to extremely challenging finite element systems. Direct solvers for the resulting large and sparse linear systems suffer from severe memory requirements and limited parallel scalability, while iterative solvers in general lack robus...
Nonlinear algebraic systems of equations resulting from Discontinuous Galerkin (DG) discretisation of partial differential equations are typically solved by Newton’s method. In this study, we propose a nonlinear preconditioner for Newton’s method for solving the system of equations which is the modification of RASPEN (Restricted Additive Schwarz Pr...
SIMD vectorization has lately become a key challenge in high-performance computing. However, hand-written explicitly vectorized code often poses a threat to the software’s sustainability. In this publication, we solve this sustainability and performance portability issue by enriching the simulation framework dune-pdelab with a code generation appro...
Coordination of fate transition and cell division is crucial to maintain the plant architecture and to achieve efficient production of plant organs. In this paper, we analysed the stem cell dynamics at the shoot apical meristem (SAM) that is one of the plant stem cells locations. We designed a mathematical model to elucidate the impact of hormonal...
![Fig. 6 Model of a neural network for matrix inversion, cf. [53]](publication/343327374/figure/fig3/AS:938719947288581@1600819684879/Model-of-a-neural-network-for-matrix-inversion-cf-53_Q320.jpg)
In the Exa-Dune project we have developed, implemented and optimised numerical algorithms and software for the scalable solution of partial differential equations (PDEs) on future exascale systems exhibiting a heterogeneous massively parallel architecture. In order to cope with the increased probability of hardware failures, one aim of the project...
This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first Dune version in 2007 and the main papers describing that state Bastian etal. (2008a, 2008b). This discussion is accompanied with a de...
The accurate numerical simulation of high Reynolds number incompressible flows is a challenging topic in computational fluid dynamics. Classical inf-sup stable methods like the Taylor-Hood element or only $L^2$-conforming discontinuous Galerkin (DG) methods relax the divergence constraint in the variational formulation. However, unlike divergence-f...
In the Exa-Dune project we have developed, implemented and optimised numerical algorithms and software for the scalable solution of partial differential equations (PDEs) on future exascale systems exhibiting a heterogeneous massively parallel architecture. In order to cope with the increased probability of hardware failures, one aim of the project...
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Coordination of fate transition and cell division is crucial to maintain the plant architecture and to achieve efficient production of plant organs. In this paper, we analysed the stem cell dynamics at the shoot apical meristem (SAM) that is one of the plant stem cells locations. We designed a mathematical model to...
This paper presents the basic concepts and the module structure of the Distributed and Unified Numerics Environment and reflects on recent developments and general changes that happened since the release of the first Dune version in 2007 and the main papers describing that state [1, 2]. This discussion is accompanied with a description of various a...
Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering, including for example computational fluid dynamics, nuclear reactor simulations and combustion models. To achieve optimal performance, solvers have to exhibit...
SIMD vectorization has lately become a key challenge in high-performance computing. However, hand-written explicitly vectorized code often poses a threat to the software's sustainability. In this publication we solve this sustainability and performance portability issue by enriching the simulation framework dune-pdelab with a code generation approa...
Efficient and suitably preconditioned iterative solvers for elliptic partial differential equations (PDEs) of the convection-diffusion type are used in all fields of science and engineering. To achieve optimal performance, solvers have to exhibit high arithmetic intensity and need to exploit every form of parallelism available in modern manycore CP...
![FIGURE 2 FEM solutions on uniform grids [top row] and its L 2 − error...](profile/Abdullah-Shah-6/publication/323364425/figure/fig1/AS:598270552317953@1519650223050/FEM-solutions-on-uniform-grids-top-row-and-its-L-2-error-distributions-bottom-row_Q320.jpg)
![FIGURE 3 FEM solutions on adaptive grids [top row] and its L 2 − error...](profile/Abdullah-Shah-6/publication/323364425/figure/fig2/AS:598270552313857@1519650223728/FEM-solutions-on-adaptive-grids-top-row-and-its-L-2-error-distributions-bottom-row_Q320.jpg)
This article presents an efficient and robust algorithm for the numerical solution of the Allen-Cahn equation, which represents the motion of antiphase boundaries. The proposed algorithm is based on the diagonally implicit fractional-step θ− scheme for time discretization and the conforming finite element method for space discretization. For the st...
Achieving a substantial part of peak performance on todays and future high-performance computing systems is a major challenge for simulation codes. In this paper we address this question in the context of the numerical solution of partial differential equations with finite element methods, in particular the discontinuous Galerkin method applied to...
In this paper we consider discontinuous Galerkin (DG) methods for the incompressible Navier-Stokes equations in the framework of projection methods. In particular we employ symmetric interior penalty DG methods within the second-order rotational incremental pressure correction scheme. The major focus of the paper is threefold: i) We propose a modif...
In the geostatistical inverse problem of subsurface hydrology, continuous hydraulic parameter fields, in most cases hydraulic conductivity, are estimated from measurements of dependent variables, such as hydraulic heads, under the assumption that the parameter fields are autocorrelated random space functions. Upon discretization, the continuous fie...
In this article, we present an adaptive time-stepping technique for numerical simulation of dendritic crystal growth model. The diagonally implicit fractional step θ-scheme for time discretisation and conforming Q1 finite-element method for space discretisation are used. The performance of the scheme is illustrated by simulating two-dimensional den...
We present advances concerning efficient finite element assembly and linear solvers on current and upcoming HPC architectures obtained in the frame of the Exa-Dune project, part of the DFG priority program 1648 Software for Exascale Computing (SPPEXA). In this project, we aim at the development of both flexible and efficient hardware-aware software...
In this contribution we present advances concerning efficient parallel multiscale methods and uncertainty quantification that have been obtained in the frame of the DFG priority program 1648 Software for Exascale Computing (SPPEXA) within the funded project Exa-Dune. This project aims at the development of flexible but nevertheless hardware-specifi...
Uncertainty quantification (UQ) for porous media flow is of great importance for many societal, environmental and industrial problems. An obstacle for progress in this area is the extreme computational effort needed for solving realistic problems. It is expected that exa-scale computers will open the door for a significant progress in this area. We...
A discontinuous Galerkin scheme was implemented in the DUNE framework to solve the compressible, inviscid Euler equations in a multi-dimensional Cartesian grid. It uses a HLLC Riemann solver for the numerical fluxes in the interfaces, a total variation bounded limiter to handle discontinuities, a positivity preserving limiter for near vacuum condit...
In this study, we consider the simulation of subsurface flow and solute
transport processes in the stationary limit. In the convection-dominant case,
the numerical solution of the transport problem may exhibit non-physical
diffusion and under- and overshoots. For an interior penalty discontinuous
Galerkin (DG) discretization, we present a $h$-adapt...
A multicomponent multiphase reactive transport simulator has been developed
to facilitate the investigation of a large variety of phenomena in porous media
including component transport, diffusion, microbiological growth and decay,
cell attachment and detachment and phase exchange. The coupled problem is
solved using operator splitting. This approa...
In the EXA-DUNE project we strive to (i) develop and implement numerical algorithms for solving PDE problems efficiently on heterogeneous architectures, (ii) provide corresponding domain-specific abstractions that allow application scientists to effectively use these methods, and (iii) demonstrate performance on porous media flow problems. In this...
Laboratory batch experiments have been employed to develop a mathematical
model for the growth of Escherichia coli (HB101 K12 pGLO) in the presence or
absence of dissolved organic carbon and oxygen. The proposed model based on
modified double Contois kinetics is able to predict the cell densities, organic
carbon utilization and oxygen transfer and...
A numerical scheme based on artificial compressibility formulation for simulating 3D two-phase incompressible flows. The coupled nonlinear systems composing of the incompressible Navier-Stokes equations and volume preserving Allen-Cahn type phase-field equation are recast into conservative forms with source terms, which are suited for implementing...
In the quality assurance of scientific frameworks, both the special characteristics of scientific software and the large variability in frameworks must be accounted for. In previous research, the authors developed a process for handling the variability of a framework using software product line variability modeling. They described the design of a q...
The extraction of groundwater for drinking water purposes is one of the most important uses of the natural subsurface. Sustainable management of groundwater resources requires detailed knowledge of the hydraulic properties within the subsurface. Typically, these properties are not directly accessible. The evaluation of hydraulic properties therefor...
In this paper we formulate and test numerically a fully-coupled discontinuous
Galerkin (DG) method for incompressible two-phase flow with discontinuous
capillary pressure. The spatial discretization uses the symmetric interior
penalty DG formulation with weighted averages and is based on a wetting-phase
potential / capillary potential formulation o...
In neurophysiology, extracellular signals-as measured by local field potentials (LFP) or electroencephalography-are of great significance. Their exact biophysical basis is, however, still not fully understood. We present a three-dimensional model exploiting the cylinder symmetry of a single axon in extracellular fluid based on the Poisson-Nernst-Pl...
The testing of scientific frameworks is a challenging task. The special characteristics of scientific software e.g. missing test oracle, the need for high performance parallel computing, and high priority of non-functional requirements, need to be accounted for as well as the large variability in a framework. In our previous research, we have shown...
The quasi-linear geostatistical approach can be used to estimate the
spatial distribution of heterogeneous hydraulic conductivity fields
based on point-wise measurements of observable quantities such as the
hydraulic head or the concentration of a tracer. The accurate and
efficient solution of steady-state groundwater flow and solute transport
prob...
Carbon Capture and storage is simulated with a two-phase two-component
flow model employing a special set of primary variables (capillary
pressure / phase pressure) to deal with the (dis-)appearance of the
non-wetting phase. The implementation is based on DUNE PDElab. Numerical
results of massive parallel simulations for test cases with millions of...
The development of better macro scale models for multi-phase flow in
porous media is still impeded by the lack of suitable methods for the
simulation of such flow regimes on the pore scale. The highly
complicated geometry of natural porous media imposes requirements with
regard to stability and computational efficiency which current numerical
metho...
The capillary fringe (CF) is a highly dynamic soil zone, which is
located above the groundwater level. It results from the capillary water
rise into the unsaturated soil zone and therewith offers a broad range
of growth conditions for microorganisms. These conditions change from
aerobic (good oxygen supply) at the top of the CF to anaerobic (no
ava...
SUMMARYA discontinuous Galerkin method for the solution of the immiscible and incompressible two‐phase flow problem based on the nonsymmetric interior penalty method is presented. Therefore, the incompressible Navier–Stokes equation is solved for a domain decomposed into two subdomains with different values of viscosity and density as well as a sin...
Carbon Capture and Storage (CCS) is a recently discussed new technology,
aimed at allowing an ongoing use of fossil fuels while preventing the produced
CO2 to be released to the atmosphere. CSS can be modeled with two components
(water and CO2) in two phases (liquid and CO2). To simulate the process, a
multiphase flow equation with equilibrium phas...
This paper describes a massively parallel algebraic multigrid method based on
non-smoothed aggregation. It is especially suited for solving heterogeneous
elliptic problems as it uses a greedy heuristic algorithm for the aggregation
that detects changes in the coefficients and prevents aggregation across them.
Using decoupled aggregation on each pro...
Hydraulic conductivity is a key parameter for the simulation of groundwater flow and transport. Typically, it is highly variable in space and difficult to determine by direct methods. The most common approach is to infer hydraulic-conductivity values from measurements of dependent quantities, such as hydraulic head and concentration. In geostatisti...
The quasi-linear geostatistical approach is an inversion scheme that can
be used to estimate the spatial distribution of a heterogeneous
hydraulic conductivity field. The estimated parameter field is
considered to be a random variable that varies continuously in space,
meets the measurements of dependent quantities (such as the hydraulic
head, the...
Pore scale simulations of multi phase flow in porous media present a
promising approach in the development and verification of continuum
scale models as well as in the understanding of the underlying processes
of flow phenomena like hysteresis or the peculiarities of the capillary
fringe. As typical pore geometries involve complicated geometries wi...
In this contribution, we motivate the use of simulation studies in
virtual soil systems using high performance computing systems. We focus
on the scale of a field or a field plot, which is the unit cell of land
management and of hydrologic simulation models. Processes in the vadose
zone of such a unit cell are strongly dependent on the spatial
vari...
The assessment of hydraulic aquifer parameters is important for the
evaluation of anthropogenic impacts on groundwater resources. The
distribution of these parameters determines flow paths and solute travel
times and is therefore critical for the successful design and deployment
of remediation schemes at contaminated sites. The geostatistical
appro...
The capillary fringe (CF) is a highly dynamic zone in a porous media at
the interface between water-saturated aquifer and vadose zone, where
steep biogeochemical gradients and thus high bioactivities are expected.
In recent years, considerable effort has been undertaken to deepen the
understanding of the physical (flow, diffusion, dispersion), geoc...
The simulation of ground-penetrating radar (GPR) measurements requires
the solution of Maxwell's equations. While finite-differences
time-domain (FDTD) solvers are faster on structured grids,
finite-element time-domain (FETD) and discontinuous Galerkin time-domain
(DGTD) allow to resolve complicated structures and avoid staircase
approximations. So...
We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin (DG) discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections, and the first coarse level space is the subspace spanned by continuous linear basis functions. The linear sys...
Testing a scientific framework is a challenging task, given the framework's large variability. The approach taken here is to apply software product line engineering, using variability modeling to support the selection of test applications and test cases. Specifically, regression testing is used for a complex scientific framework called the Distribu...
This paper presents recent results of a network project aiming at the modelling and simulation of coupled surface and subsurface flows. In particular, a discontinuous Galerkin method for the shallow water equations has been devel-oped which includes a special treatment of wetting and drying. A robust solver for saturated–unsaturated groundwater flo...
We present a Dune-Grid extension that enhances existing Dune grids with the ability to designate arbitrary subsets of their leaf entity complex as subdomains and present them as new grid objects. We describe the functionality of this module, which is called Dune-Multidomaingrid and available as free software, and outline its implementation. In part...
Estimating the hydraulic conductivity of an aquifer is an important task
of groundwater inverse modeling. Typically, the number of available
measurement locations is limited. Using different types of measurements
helps improving the estimate because different types of measurements
have different sensitivity pattern. In this study, we consider the m...
For the simulation of transport processes in porous media effective parameters for the physical processes on the target scale are required. Numerical upscaling, as well as multiscale approaches can help where experiments are not possible, or hard to conduct.
In 2009, Bastian and Engwer proposed an Unfitted Discontinuous Galerkin (UDG) method for so...
Testing scientific software involves dealing with special challenges like missing test oracle and different possible sources of a problem. When testing scientific frameworks, additionally a large variety of mathematical algorithms and possible applications for the framework has to be handled. We propose to use concepts of software product line engi...
Consider the stationary diffusion equation with Dirichlet boundary conditions
$$-\nabla.(K \nabla u)=f\qquad \qquad {\rm in}\ \Omega,$$ (1a)
$$\qquad{u = g}\qquad\qquad \qquad\qquad\quad{\rm on}\partial \Omega.$$ (1b)
In the two-dimensional discretisation benchmark session at the FVCA5 conference, we participated with a Mimetic Finite Difference (MFD) method. In this paper, we present results for the three-dimensional case using the same method. Since the previous conference, the equivalence of MFD, Hybrid Finite Volume and Mixed Finite Volume methods has been d...
We present a new algebraic multigrid (AMG) algorithm for the solution of linear systems arising from discontinuous Galerkin discretizations of heterogeneous elliptic problems. The algorithm is based on the idea of subspace corrections and the first coarse level space is the subspace spanned by continuous linear basis functions. The linear system as...
We describe an abstract interface for the geometric coupling of finite element grids. The scope of the interface encompasses
a wide range of domain decomposition techniques in use today, including nonconforming grids and grids of different dimensions.
The couplings are described as sets of remote intersections, which encapsulate the relationships b...
Pore scale simulations of multi phase flow in porous media present a promising approach in the development and verification of continuum scale models as well as in the understanding of the underlying processes of flow phenomena like hysteresis or the peculiarities of the capillary fringe. However, the applicability of modern numerical models is sti...
We present a model for multiphase reactive flow in the capillary fringe including hysteretical effects and phase transfer. The numerical approach is based on a pressure-pressure formulation and cell-centered finite volumes. Operator splitting into three substeps is used to handle the coupled system: i) transport of the two phases ii) transport of t...
The simulation of two-phase flow in porous media has a wide variety of applications. The equations governing these flows are inherently nonlinear. In applications like CO2 sequestration in geological formations and petroleum engineering, we often have to cope with unstructured geometries and highly heterogeneous media, which raise additional diffic...
We describe PDELab, an extensible C++ template library for finite element methods based on the Distributed and Unified Numerics Environment (Dune). PDELab considerably simplifies the implementation of discretization schemes for systems of partial differential equations (PDEs) by setting up global functions and operators from a simple element-local...
Large-scale parallel codes require the data to be decomposed between the set of processes active in the computation. This data decomposition implies recurring communication schemes. The paper introduces generic template classes in C++ for describing the data decomposition. The aim is to store the data in arbitrary existent efficient sequential data...
In this paper we present a new approach to simulations on complex-shaped domains. The method is based on a discontinuous Galerkin (DG) method, using trial and test functions defined on a structured grid. Essential boundary conditions are imposed weakly via the DG formulation. This method offers a discretization where the number of unknowns is indep...
Models for multiphase flow in porous media on the continuum scale have to provide descriptions for phenomena like hysteresis and immobile residual water saturation. These are essentially determined by the pore geometries of the media. Hence, numerical simulations of flow processes on the pore scale are of fundamental importance for model understand...
When performing solute transport in porous media one often observes macroscopic tailing, even with non-sorbing solutes. This is typically traced back to structural heterogeneity and heuristically represented by the mobile-immobile model (MIM). However, tailing has also been observed in an almost perfect uniform medium. In this presentation we show...
We give a mathematically rigorous definition of a grid for algorithms solving partial differential equations. Unlike previous
approaches (Benger 2005, PhD thesis; Berti 2000, PhD thesis), our grids have a hierarchical structure. This makes them suitable
for geometric multigrid algorithms and hierarchical local grid refinement. The description is al...
Hairy roots are plants genetically transformed by Agrobacterium rhizogenes, which do not produce shoots and are composed mainly by roots. Hairy roots of Ophiorrhiza mungos Linn. are currently gaining interest of pharmacologists, since a secondary product of their metabolism, camptothecin, is used in chemotherapy. To optimize the production of valua...
European Geosciences Union General Assembly 2008 Vienna, Austria, 13 – 18 April 2008
The simulation of realistic multiphase flow problems in porous media requires efficient solution methods and means for handling
the complicated structure of the media. The numerical toolbox UG has been developed to be able to use the approaches-multigrid,
unstructured grids, adaptivity, and parallel computing-for a wide range of applications. An ex...
In a companion paper (Bastian et al. 2007, this issue) we introduced an abstract definition of a parallel and adaptive hierarchical
grid for scientific computing. Based on this definition we derive an efficient interface specification as a set of C++ classes.
This interface separates the applications from the grid data structures. Thus, user implem...
The numerical solution of partial differential equations frequently requires solving large and sparse linear systems. When using the Finite Element Method these systems exhibit a natural block structure that is exploited for efficiency in the "Iterative Solver Template Library" (ISTL). Based on existing sequential preconditioned iterative solvers w...
A multiscale approach is presented to model growth of hairy roots. On the macroscopic scale, a continuous model is derived,
which includes growth and nutrient transport. Water transport is considered on the microscopic scale. A Discontinuous Galerkin
scheme for complex geometries is used to compute the permeability of root bulks. This permeability...
In this paper we present a new approach to simulations on complex shaped domains. The method is based on a Discontinuous Galerkin method, using trial and test functions defined on a structured grid. Essential Boundary conditions are imposed weakly via the Discontinuous Galerkin formulation. This method offers a discretization where the number of un...
We develop a low Mach number reactive flow solver to compute laminar steady flames with detailed chemical kinetics and transport
properties. Convergence to the steady solution is obtained by time marching. The fluid dynamical and thermo-reactive balanced
equation are advanced in time one after another, but each of them with an implicit scheme. The...
In this paper we develop higher-order Discontinuous Galerkin finite element methods for the simulation of two-phase flow in
heterogeneous porous media and experimental methods to determine structure and saturation in two-dimensional Hele-Shaw cells
with high resolution in space and time. Together with additional measurements of governing parameters...
The calculation of the relative hydraulic conductivity function based on water retention data is an attractive and widely used approach, since direct measurements of unsaturated conductivities are difficult. We show theoretically under which conditions an air-entry value for water retention data is definitely required when using the statistical app...