arXiv subjects

Christian Engwer

Publications and source records attributed to Christian Engwer.

At least 37 records · Page 2Linked to original sources

arXiv2021

DUNEuro -- A software toolbox for forward modeling in bioelectromagnetism

Accurate and efficient source analysis in electro- and magnetoencephalography using sophisticated realistic head geometries requires advanced numerical approaches. This paper presents DUNEuro, a free and open source C++ software toolbox for forward modeling in bioelectromagnetism. Building upon the DUNE framework, it provides implementations of modern fitted and unfitted finite element methods to efficiently solve the forward problems in electro- and magnetoencephalography. The user can choose between a variety of different source models that are implemented. The software's aim is to provide interfaces that are extendible and easy-to-use. In order to enable a closer integration into existing analysis pipelines, interfaces to Python and Matlab are provided. The practical use is demonstrated by a source analysis example of somatosensory evoked potentials using a realistic six compartment head model.

cs.MS↗

arXiv2020

Realistic head modeling of electromagnetic brain activity: An integrated Brainstorm pipeline from MRI data to the FEM solution

Human brain activity generates scalp potentials (electroencephalography EEG), intracranial potentials (iEEG), and external magnetic fields (magnetoencephalography MEG), all capable of being recorded, often simultaneously, for use in research and clinical applications. The so-called forward problem is the modeling of these fields at their sensors for a given putative neural source configuration. While early generations modeled the head as a simple set of isotropic spheres, today s ubiquitous magnetic resonance imaging (MRI) data allows detailed descriptions of head compartments with assigned isotropic and anisotropic conductivities. In this paper, we present a complete pipeline, integrated into the Brainstorm software, that allows users to generate an individual and accurate head model from the MRI and then calculate the electromagnetic forward solution using the finite element method (FEM). The head model generation is performed by the integration of the latest tools for MRI segmentation and FEM mesh generation. The final head model is divided into five main compartments: white matter, grey matter, CSF, skull, and scalp. For the isotropic compartments, widely-used default conductivity values are assigned. For the brain tissues, we use the process of the effective medium approach (EMA) to estimate anisotropic conductivity tensors from diffusion-weighted imaging (DWI) data. The FEM electromagnetic calculations are performed by the DUNEuro library, integrated into Brainstorm and accessible with a user-friendly graphical interface. This integrated pipeline, with full tutorials and example data sets freely available on the Brainstorm website, gives the neuroscience community easy access to advanced tools for electromagnetic modeling using FEM.

physics.med-ph↗

arXiv2020

The DUNE Framework: Basic Concepts and Recent Developments

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 advanced features, such as coupling of domains and cut cells, grid modifications such as adaptation and moving domains, high order discretizations and node level performance, non-smooth multigrid methods, and multiscale methods. A brief discussion on current and future development directions of the framework concludes the paper.

cs.MS↗

arXiv2020

Estimating the extent of glioblastoma invasion

Glioblastoma Multiforme is a malignant brain tumor with poor prognosis. There have been numerous attempts to model the invasion of tumorous glioma cells via partial differential equations in the form of advection-diffusion-reaction equations. The patient-wise parametrisation of these models, and their validation via experimental data has been found to be difficult, as time sequence measurements are generally missing. Also the clinical interest lies in the actual (invisible) tumor extent for a particular MRI/DTI scan and not in a predictive estimate. Therefore we propose a stationalised approach to estimate the extent of glioblastoma (GBM) invasion at the time of a given MRI/DTI scan. The underlying dynamics can be derived from an instationary GBM model, falling into the wide class of advection-diffusion-reaction equations. The stationalisation is introduced via an analytical solution of the Fisher-KPP equation, the simplest model in the considered model class. We investigate the applicability in 1D and 2D, in the presence of inhomogeneous diffusion coefficients and on a real 3D DTI-dataset.

q-bio.CB↗

arXiv2019

Strategies for the vectorized Block Conjugate Gradients method

Block Krylov methods have recently gained a lot of attraction. Due to their increased arithmetic intensity they offer a promising way to improve performance on modern hardware. Recently Frommer et al. presented a block Krylov framework that combines the advantages of block Krylov methods and data parallel methods. We review this framework and apply it on the Block Conjugate Gradients method,to solve linear systems with multiple right hand sides. In this course we consider challenges that occur on modern hardware, like a limited memory bandwidth, the use of SIMD instructions and the communication overhead. We present a performance model to predict the efficiency of different Block CG variants and compare these with experimental numerical results.

math.NA↗

arXiv2019

Monotonicity considerations for stabilized DG cut cell schemes for the unsteady advection equation

For solving unsteady hyperbolic conservation laws on cut cell meshes, the so called small cell problem is a big issue: one would like to use a time step that is chosen with respect to the background mesh and use the same time step on the potentially arbitrarily small cut cells as well. For explicit time stepping schemes this leads to instabilities. In a recent preprint [arXiv:1906.05642], we propose penalty terms for stabilizing a DG space discretization to overcome this issue for the unsteady linear advection equation. The usage of the proposed stabilization terms results in stable schemes of first and second order in one and two space dimensions. In one dimension, for piecewise constant data in space and explicit Euler in time, the stabilized scheme can even be shown to be monotone. In this contribution, we will examine the conditions for monotonicity in more detail.

math.NA↗

arXiv2019

Dynamic and weighted stabilizations of the $L$-scheme applied to a phase-field model for fracture propagation

We consider a phase-field fracture propagation model, which consists of two (nonlinear) coupled partial differential equations. The first equation describes the displacement evolution, and the second is a smoothed indicator variable, describing the crack position. We propose an iterative scheme, the so-called $L$-scheme, with a dynamic update of the stabilization parameters during the iterations. Our algorithmic improvements are substantiated with two numerical tests. The dynamic adjustments of the stabilization parameters lead to a significant reduction of iteration numbers in comparison to constant stabilization values.

math.NA↗

arXiv2019

Exa-Dune -- Flexible PDE Solvers, Numerical Methods and Applications

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 was to add flexible, application-oriented resilience capabilities into the framework. Continuous improvement of the underlying hardware-oriented numerical methods have included GPU-based sparse approximate inverses, matrix-free sum-factorisation for high-order discontinuous Galerkin discretisations as well as partially matrix-free preconditioners. On top of that, additional scalability is facilitated by exploiting massive coarse grained parallelism offered by multiscale and uncertainty quantification methods where we have focused on the adaptive choice of the coarse/fine scale and the overlap region as well as the combination of local reduced basis multiscale methods and the multilevel Monte-Carlo algorithm. Finally, some of the concepts are applied in a land-surface model including subsurface flow and surface runoff.

cs.MS↗

arXiv2019

A stabilized DG cut cell method for discretizing the linear transport equation

We present new stabilization terms for solving the linear transport equation on a cut cell mesh using the discontinuous Galerkin (DG) method in two dimensions with piecewise linear polynomials. The goal is to allow for explicit time stepping schemes, despite the presence of cut cells. Using a method of lines approach, we start with a standard upwind DG discretization for the background mesh and add penalty terms that stabilize the solution on small cut cells in a conservative way. Then, one can use explicit time stepping, even on cut cells, with a time step length that is appropriate for the background mesh. In one dimension, we show monotonicity of the proposed scheme for piecewise constant polynomials and total variation diminishing in the means stability for piecewise linear polynomials. We also present numerical results in one and two dimensions that support our theoretical findings.

math.NA↗

arXiv2019

Efficient implementation of the Localized Orthogonal Decomposition method

In this paper we present algorithms for an efficient implementation of the Localized Orthogonal Decomposition method (LOD). The LOD is a multiscale method for the numerical simulation of partial differential equations with a continuum of inseparable scales. We show how the method can be implemented in a fairly standard Finite Element framework and discuss its realization for different types of problems, such as linear elliptic problems with rough coefficients and linear eigenvalue problems.

math.NA↗

arXiv2018

Function space bases in the dune-functions module

The dune-functions Dune module provides interfaces for functions and function space bases. It forms one abstraction level above grids, shape functions, and linear algebra, and provides infrastructure for full discretization frameworks like dune-pdelab and dune-fem. This document describes the function space bases provided by dune-functions. These are based on an abstract description of bases for product spaces as trees of simpler bases. From this description, many different numberings of degrees of freedom by multi-indices can be derived in a natural way. We describe the abstract concepts, document the programmer interface, and give a complete example program that solves the stationary Stokes equation using Taylor-Hood elements.

cs.MS↗

arXiv2018

A high-level C++ approach to manage local errors, asynchrony and faults in an MPI application

C++ advocates exceptions as the preferred way to handle unexpected behaviour of an implementation in the code. This does not integrate well with the error handling of MPI, which more or less always results in program termination in case of MPI failures. In particular, a local C++ exception can currently lead to a deadlock due to unfinished communication requests on remote hosts. At the same time, future MPI implementations are expected to include an API to continue computations even after a hard fault (node loss), i.e. the worst possible unexpected behaviour. In this paper we present an approach that adds extended exception propagation support to C++ MPI programs. Our technique allows to propagate local exceptions to remote hosts to avoid deadlocks, and to map MPI failures on remote hosts to local exceptions. A use case of particular interest are asynchronous 'local failure local recovery' resilience approaches. Our prototype implementation uses MPI-3.0 features only. In addition we present a dedicated implementation, which integrates seamlessly with MPI-ULFM, i.e. the most prominent proposal for extending MPI towards fault tolerance. Our implementation is available at https://gitlab.dune-project.org/christi/test-mpi-exceptions .

cs.DC↗

arXiv2017

A Mixed Finite Element Method to Solve the EEG Forward Problem

Finite element methods have been shown to achieve high accuracies in numerically solving the EEG forward problem and they enable the realistic modeling of complex geometries and important conductive features such as anisotropic conductivities. To date, most of the presented approaches rely on the same underlying formulation, the continuous Galerkin (CG)-FEM. In this article, a novel approach to solve the EEG forward problem based on a mixed finite element method (Mixed-FEM) is introduced. To obtain the Mixed-FEM formulation, the electric current is introduced as an additional unknown besides the electric potential. As a consequence of this derivation, the Mixed-FEM is, by construction, current preserving, in contrast to the CG-FEM. Consequently, a higher simulation accuracy can be achieved in certain scenarios, e.g., when the diameter of thin insulating structures, such as the skull, is in the range of the mesh resolution. A theoretical derivation of the Mixed-FEM approach for EEG forward simulations is presented, and the algorithms implemented for solving the resulting equation systems are described. Subsequently, first evaluations in both sphere and realistic head models are presented, and the results are compared to previously introduced CG-FEM approaches. Additional visualizations are shown to illustrate the current preserving property of the Mixed-FEM. Based on these results, it is concluded that the newly presented Mixed-FEM can at least complement and in some scenarios even outperform the established CG-FEM approaches, which motivates a further evaluation of the Mixed-FEM for applications in bioelectromagnetism.

math.NA↗

arXiv2016

ArbiLoMod: Local Solution Spaces by Random Training in Electrodynamics

The simulation method ArbiLoMod has the goal to provide users of Finite Element based simulation software with quick re-simulation after localized changes to the model under consideration. It generates a Reduced Order Model (ROM) for the full model without ever solving the full model. To this end, a localized variant of the Reduced Basis method is employed, solving only small localized problems in the generation of the reduced basis. The key to quick re-simulation lies in recycling most of the localized basis vectors after a localized model change. In this publication, ArbiLoMod's local training algorithm is analyzed numerically for the non-coercive problem of time harmonic Maxwell's equations in 2D, formulated in H(curl).

math.NA↗

arXiv2016

ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications

Engineers manually optimizing a structure using Finite Element based simulation software often employ an iterative approach where in each iteration they change the structure slightly and resimulate. Standard Finite Element based simulation software is usually not well suited for this workflow, as it restarts in each iteration, even for tiny changes. In settings with complex local microstructure, where a fine mesh is required to capture the geometric detail, localized model reduction can improve this workflow. To this end, we introduce ArbiLoMod, a method which allows fast recomputation after arbitrary local modifications. It employs a domain decomposition and a localized form of the Reduced Basis Method for model order reduction. It assumes that the reduced basis on many of the unchanged domains can be reused after a localized change. The reduced model is adapted when necessary, steered by a localized error indicator. The global error introduced by the model order reduction is controlled by a robust and efficient localized a posteriori error estimator, certifying the quality of the result. We demonstrate ArbiLoMod for a coercive, parameterized example with changing structure.

math.NA↗

arXiv2016

A discontinuous Galerkin Method for the EEG Forward Problem using the Subtraction Approach

In order to perform electroencephalography (EEG) source reconstruction, i.e., to localize the sources underlying a measured EEG, the electric potential distribution at the electrodes generated by a dipolar current source in the brain has to be simulated, which is the so-called EEG forward problem. To solve it accurately, it is necessary to apply numerical methods that are able to take the individual geometry and conductivity distribution of the subject's head into account. In this context, the finite element method (FEM) has shown high numerical accuracy with the possibility to model complex geometries and conductive features, e.g., white matter conductivity anisotropy. In this article, we introduce and analyze the application of a discontinuous Galerkin (DG) method, a finite element method that includes features of the finite volume framework, to the EEG forward problem. The DG-FEM approach fulfills the conservation property of electric charge also in the discrete case, making it attractive for a variety of applications. Furthermore, as we show, this approach can alleviate modeling inaccuracies that might occur in head geometries when using classical FE methods, e.g., so-called "skull leakage effects", which may occur in areas where the thickness of the skull is in the range of the mesh resolution. Therefore, we derive a DG formulation of the FEM subtraction approach for the EEG forward problem and present numerical results that highlight the advantageous features and the potential benefits of the proposed approach.

cs.CE↗

arXiv2016

The Unfitted Discontinuous Galerkin Method for Solving the EEG Forward Problem

Objective: The purpose of this study is to introduce and evaluate the unfitted discontinuous Galerkin finite element method (UDG-FEM) for solving the electroencephalography (EEG) forward problem. Methods: This new approach for source analysis does not use a geometry conforming volume triangulation, but instead uses a structured mesh that does not resolve the geometry. The geometry is described using level set functions and is incorporated implicitly in its mathematical formulation. As no triangulation is necessary, the complexity of a simulation pipeline and the need for manual interaction for patient specific simulations can be reduced and is comparable with that of the FEM for hexahedral meshes. In addition, it maintains conservation laws on a discrete level. Here, we present the theory for UDG-FEM forward modeling, its verification using quasi-analytical solutions in multi-layer sphere models and an evaluation in a comparison with a discontinuous Galerkin (DG-FEM) method on hexahedral and on conforming tetrahedral meshes. We furthermore apply the UDG-FEM forward approach in a realistic head model simulation study. Results: The given results show convergence and indicate a good overall accuracy of the UDG-FEM approach. UDG-FEM performs comparable or even better than DG-FEM on a conforming tetrahedral mesh while providing a less complex simulation pipeline. When compared to DG-FEM on hexahedral meshes, an overall better accuracy is achieved. Conclusion: The UDG-FEM approach is an accurate, flexible and promising method to solve the EEG forward problem. Significance: This study shows the first application of the UDG-FEM approach to the EEG forward problem.

cs.CE↗

arXiv2016

An unfitted discontinuous Galerkin scheme for conservation laws on evolving surfaces

Motivated by considering partial differential equations arising from conservation laws posed on evolving surfaces, a new numerical method for an advection problem is developed and simple numerical tests are performed. The method is based on an unfitted discontinuous Galerkin approach where the surface is not explicitly tracked by the mesh which means the method is extremely flexible with respect to geometry. Furthermore, the discontinuous Galerkin approach is well-suited to capture the advection driven by the evolution of the surface without the need for a space-time formulation, back-tracking trajectories or streamline diffusion. The method is illustrated by a one-dimensional example and numerical results are presented that show good convergence properties for a simple test problem.

math.NA↗