arXiv subjects

Peter Bastian

Publications and source records attributed to Peter Bastian.

At least 19 recordsLinked to original sources

arXiv2025

A non-negativity-preserving cut-cell discontinuous Galerkin method for the diffusive wave equation

A non-negativity-preserving cut-cell discontinuous Galerkin method for the degenerate parabolic diffusive wave approximation of the shallow water equation is presented. The method can handle continuous and discontinuous bathymmetry as well as general triangular meshes. It is complemented by a finite volume method on Delauney triangulations which is also shown to be non-negativity preserving. Both methods feature an upwind flux and can handle Manning's and Chezy's friction law. By numerical experiment we demonstrate the discontinuous Galerkin method to be fully second-order accurate for the Barenblatt analytical solution on an inclined plane. In constrast, the finite volume method is only first-order accurate. Further numerical experiments show that three to four mesh refinements are needed for the finite volume method to match the solution of the discontinuous Galerkin method.

math.NA↗

arXiv2025

Novel bidomain partitioned strategies for the simulation of ventricular fibrillation dynamics

The numerical tools to simulate the bidomain model in cardiac electrophysiology are constantly developing due to the great clinical interest and scientific advances in mathematical models and computational power. The bidomain model consists of an elliptic partial differential equation (PDE) and a non-linear parabolic PDE of reaction-diffusion type, where the reaction term is described by a set of ordinary differential equations (ODEs). We propose and analyze a suite of numerical strategies for the efficient and accurate simulation of cardiac electrophysiology, with a particular focus on ventricular fibrillation in realistic geometries. Specifically, we develop and compare a fully coupled strategy, a traditional decoupled strategy, and a novel partitioned strategy. The centerpiece of this work is a bidomain partitioned strategy enhanced with spectral deferred correction, designed to balance numerical stability and computational efficiency. To address the substantial memory requirements posed by biophysically detailed ionic models, we adopt a compile-time memory-efficient sparse matrix technique. This enables the efficient solution of the coupled nonlinear parabolic PDE and the associated large systems of ODEs that govern ionic gating and concentration dynamics. We perform comprehensive numerical experiments using the Luo-Rudy and Ten Tusscher cell models in both two- and three-dimensional geometries. In addition, we demonstrate the applicability of our approach to bidomain-bath coupling scenarios. The results confirm that the proposed partitioned strategy achieves high accuracy and efficiency compared to standard decoupled strategies. Our findings support the use of advanced partitioned strategies for large-scale simulations in cardiac electrophysiology and highlight their potential for future investigations into cardiac arrhythmias and other pathological conditions.

math.NA↗

arXiv2025

A GenEO-type coarse space with smaller eigenproblems

Coarse spaces are essential to ensure robustness w.r.t. the number of subdomains in two-level overlapping Schwarz methods. Robustness with respect to the coefficients of the underlying partial differential equation (PDE) can be achieved by adaptive (or spectral) coarse spaces involving the solution of local eigenproblems. The solution of these eigenproblems, although scalable, entails a large setup cost which may exceed the cost for the iteration phase. In this paper we present and analyse a new variant of the GenEO (Generalised Eigenproblems in the Overlap) coarse space which involves solving eigenproblems only in a strip connected to the boundary of the subdomain. This leads to a significant reduction of the setup cost while the method satisfies a similar coefficient-robust condition number estimate as the original method, albeit with a possibly larger coarse space.

math.NA↗

arXiv2025

Robust spectral preconditioning for high-P\'{e}clet number convection-diffusion

We introduce a two-level hybrid restricted additive Schwarz (RAS) preconditioner for heterogeneous steady-state convection-diffusion equations at high P\'{e}clet numbers. Our construction builds on the multiscale spectral generalized finite element method (MS-GFEM), wherein the coarse space is spanned by locally optimal basis functions obtained from local generalized eigenproblems on operator-harmonic spaces. Extending the theory of Ma (2025) to convection-diffusion problems in conservation form, we establish exponential convergence of the MS-GFEM approximation with respect to the dimension of the local approximation space. Rewriting MS-GFEM as a RAS-type iteration, we show for coercive problems that this exponential convergence property is inherited by the RAS-type iterative method (at least in the continuous setting). Employed as a preconditioner within the generalized minimal residual method (GMRES), the resulting method requires only a few iterations for high accuracy even with low-dimensional coarse spaces. Through extensive numerical experiments on problems with high-contrast diffusion and non-divergence-free, rotating velocity fields, we demonstrate robustness with respect to the grid P\'{e}clet number and the number of subdomains (tested up to $10^5$ subdomains), while coarse-space dimensions remain small as grid P\'{e}clet numbers increase. By adapting the coarse space and oversampling size, we are able to achieve arbitrarily fast convergence of preconditioned GMRES. As an extension, for which we do not have theory yet, we show effectiveness of the method even for indefinite problems and in the vanishing-diffusion limit.

math.NA↗

arXiv2025

Optimal Spectral Approximation in the Overlaps for Generalized Finite Element Methods

In this paper, we study a generalized finite element method for solving second-order elliptic partial differential equations with rough coefficients. The method uses local approximation spaces computed by solving eigenvalue problems on rings around the boundary of local subdomains. Compared to the corresponding method that solves eigenvalue problems on the whole subdomains, the problem size and the bandwidth of the resulting system matrices are substantially reduced, resulting in faster spectral computations. We prove a nearly exponential a priori decay result for the local approximation errors of the proposed method, which implies the nearly exponential decay of the overall approximation error of the method. The proposed method can also be used as a preconditioner, and only a slight adaptation of our theory is necessary to prove the optimal convergence of the preconditioned iteration. Numerical experiments are presented to support the effectiveness of the proposed method and to investigate its coefficient robustness.

math.NA↗

arXiv2024

The positioning of stress fibers in contractile cells minimizes internal mechanical stress

The mechanics of animal cells is strongly determined by stress fibers, which are contractile filament bundles that form dynamically in response to extracellular cues. Stress fibers allow the cell to adapt its mechanics to environmental conditions and to protect it from structural damage. While the physical description of single stress fibers is well-developed, much less is known about their spatial distribution on the level of whole cells. Here, we combine a finite element method for one-dimensional fibers embedded in an elastic bulk medium with dynamical rules for stress fiber formation based on genetic algorithms. We postulate that their main goal is to achieve minimal mechanical stress in the bulk material with as few fibers as possible. The fiber positions and configurations resulting from this optimization task alone are in good agreement with those found in experiments where cells in 3D-scaffolds were mechanically strained at one attachment point. For optimized configurations, we find that stress fibers typically run through the cell in a diagonal fashion, similar to reinforcement strategies used for composite material.

physics.bio-ph↗

arXiv2023

Scalable multiscale-spectral GFEM with an application to composite aero-structures

In this paper, the first large-scale application of multiscale-spectral generalized finite element methods (MS-GFEM) to composite aero-structures is presented. The crucial novelty lies in the introduction of A-harmonicity in the local approximation spaces, which in contrast to [Babuska, Lipton, Multiscale Model. Simul. 9, 2011] is enforced more efficiently via a constraint in the local eigenproblems. This significant modification leads to excellent approximation properties, which turn out to be essential to capture accurately material strains and stresses with a low dimensional approximation space, hence maximising model order reduction. The implementation of the framework in the DUNE software package, as well as a detailed description of all components of the method are presented and exemplified on a composite laminated beam under compressive loading. The excellent parallel scalability of the method, as well as its superior performance compared to the related, previously introduced GenEO method are demonstrated on two realistic application cases, including a C-shaped wing spar with complex geometry. Further, by allowing low-cost approximate solves for closely related models or geometries this efficient, novel technology provides the basis for future applications in optimisation or uncertainty quantification on challenging problems in composite aero-structures.

math.NA↗

arXiv2022

The Sweep Method for radiative Transfer in Arepo

We introduce the radiative transfer code Sweep for the cosmological simulation suite Arepo. Sweep is a discrete ordinates method in which the radiative transfer equation is solved under the infinite speed of light, steady state assumption by a transport sweep across the entire computational grid. Since Arepo is based on an adaptive, unstructured grid, the dependency graph induced by the sweep dependencies of the grid cells is non-trivial. In order to solve the topological sorting problem in a distributed manner, we employ a task-based-parallelism approach. The main advantage of the sweep method is that the computational cost scales only with the size of the grid, and is independent of the number of sources or the distribution of sources in the computational domain, which is an advantage for radiative transfer in cosmological simulations, where there are large numbers of sparsely distributed sources. We successfully apply the code to a number of physical tests such as the expansion of HII regions, the formation of shadows behind dense objects, the scattering of light, as well as its behavior in the presence of periodic boundary conditions. In addition, we measure its computational performance with a focus on highly parallel, large-scale simulations.

astro-ph.IM↗

arXiv2021

High Performance Uncertainty Quantification with Parallelized Multilevel Markov Chain Monte Carlo

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) algorithm for Bayesian inverse problems, and a new software framework allowing for large-scale parallelism across forward model evaluations and the UQ algorithms themselves. The main scalability challenge presents itself in the form of strong data dependencies introduced by the MLMCMC method, prohibiting trivial parallelization. Our software is released as part of the modular and open-source MIT UQ Library (MUQ), and can easily be coupled with arbitrary user codes. We demonstrate it using the DUNE and the ExaHyPE Engine. The latter provides a realistic, large-scale tsunami model in which identify the source of a tsunami from buoy-elevation data.

cs.MS↗

arXiv2021

Multilevel Spectral Domain Decomposition

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 robustness. Two-level spectral domain decomposition methods can provide such robustness for symmetric positive definite linear systems, by using coarse spaces based on independent generalized eigenproblems in the subdomains. Rigorous condition number bounds are independent of mesh size, number of subdomains, as well as coefficient contrast. However, their parallel scalability is still limited by the fact that (in order to guarantee robustness) the coarse problem is solved via a direct method. In this paper, we introduce a multilevel variant in the context of subspace correction methods and provide a general convergence theory for its robust convergence for abstract, elliptic variational problems. Assumptions of the theory are verified for conforming, as well as for discontinuous Galerkin methods applied to a scalar diffusion problem. Numerical results illustrate the performance of the method for two- and three-dimensional problems and for various discretization schemes, in the context of scalar diffusion and linear elasticity.

math.NA↗

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↗

arXiv2019

A high-order discontinuous Galerkin pressure robust splitting scheme for incompressible flows

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-free methods, this relaxation leads to a pressure-dependent contribution in the velocity error which is proportional to the inverse of the viscosity, thus resulting in methods that lack pressure robustness and have difficulties in preserving structures at high Reynolds numbers. The present paper addresses the discretization of the incompressible Navier-Stokes equations with high-order DG methods in the framework of projection methods. The major focus in this article is threefold: i) We present a novel postprocessing technique in the projection step of the splitting scheme that reconstructs the Helmholtz flux in $H(\text{div})$. In contrast to the previously introduced $H(\text{div})$ postprocessing technique, the resulting velocity field is pointwise divergence-free in addition to satisfying the discrete continuity equation. ii) Based on this Helmholtz flux $H(\text{div})$ reconstruction, we obtain a high order in space, pressure robust splitting scheme as numerical experiments in this paper demonstrate. iii) With this pressure robust splitting scheme, we demonstrate that a robust DG method for underresolved turbulent incompressible flows can be realized.

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

Matrix-free multigrid block-preconditioners for higher order Discontinuous Galerkin discretisations

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 CPUs. The computationally most expensive components of the solver are the repeated applications of the linear operator and the preconditioner. For discretisations based on higher-order Discontinuous Galerkin methods, sum-factorisation results in a dramatic reduction of the computational complexity of the operator application while, at the same time, the matrix-free implementation can run at a significant fraction of the theoretical peak floating point performance. Multigrid methods for high order methods often rely on block-smoothers to reduce high-frequency error components within one grid cell. Traditionally, this requires the assembly and expensive dense matrix solve in each grid cell, which counteracts any improvements achieved in the fast matrix-free operator application. To overcome this issue, we present a new matrix-free implementation of block-smoothers. Inverting the block matrices iteratively avoids storage and factorisation of the matrix and makes it is possible to harness the full power of the CPU. We implemented a hybrid multigrid algorithm with matrix-free block-smoothers in the high order DG space combined with a low order coarse grid correction using algebraic multigrid where only low order components are explicitly assembled. The effectiveness of this approach is demonstrated by solving a set of representative elliptic PDEs of increasing complexity, including a convection dominated problem and the stationary SPE10 benchmark.

math.NA↗

arXiv2018

Automatic Code Generation for High-Performance Discontinuous Galerkin Methods on Modern Architectures

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 approach. The approach is based on the well-known domain-specific language UFL, but combines it with loopy, a more powerful intermediate representation for the computational kernel. Given this flexible tool, we present and implement a new class of vectorization strategies for the assembly of Discontinuous Galerkin methods on hexahedral meshes exploiting the finite element's tensor product structure. The optimal variant from this class is chosen by the code generator through an autotuning approach. The implementation is done within the open source PDE software framework Dune and the discretization module dune-pdelab. The strength of the proposed approach is illustrated with performance measurements for DG schemes for a scalar diffusion reaction equation and the Stokes equation. In our measurements, we utilize both the AVX2 and the AVX512 instruction set, achieving 40\% to 60\% of the machine's theoretical peak performance for one matrix-free application of the operator.

math.NA↗

arXiv2017

High-performance Implementation of Matrix-free High-order Discontinuous Galerkin Methods

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 a convection-diffusion-reaction model problem. Assuming tensor product structure of basis functions and quadrature on cuboid meshes in a matrix-free approach a substantial reduction in computational complexity can be achieved for operator application compared to a matrix-based implementation while at the same time enabling SIMD vectorization and the use of fused-multiply-add. Close to 60\% of peak performance are obtained for a full operator evaluation on a Xeon Haswell CPU with 16 cores and speedups of several hundred (with respect to matrix-based computation) are achieved for polynomial degree seven. Excellent weak scalability on a single node as well as the roofline model demonstrate that the algorithm is fully compute-bound with a high flop per byte ratio. Excellent scalability is also demonstrated on up to 6144 cores using message passing.

math.NA↗

arXiv2016

A Stable and High-Order Accurate Discontinuous Galerkin Based Splitting Method for the Incompressible Navier-Stokes Equations

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 modified upwind scheme based on the Vijayasundaram numerical flux that has favourable properties in the context of DG. ii) We present a novel postprocessing technique in the Helmholtz projection step based on $H(\text{div})$ reconstruction of the pressure correction that is computed locally, is a projection in the discrete setting and ensures that the projected velocity satisfies the discrete continuity equation exactly. As a consequence it also provides local mass conservation of the projected velocity. iii) Numerical results demonstrate the properties of the scheme for different polynomial degrees applied to two-dimensional problems with known solution as well as large-scale three-dimensional problems. In particular we address second-order convergence in time of the splitting scheme as well as its long-time stability.

math.NA↗

arXiv2014

Application of reactive transport modelling to growth and transport of microorganisms in the capillary fringe

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 approach allows a flexible adaptation of the solution strategy to the concrete problem. Moreover, the individual submodels were optimised to be able to describe behaviour of Escherichia coli (HB101 K12 pGLO) in the capillary fringe in the presence or absence of dissolved organic carbon and oxygen under steady-state and flow conditions. Steady-state and flow through experiments in a Hele-Shaw cell, filled with quartz sand, were conducted to study eutrophic bacterial growth and transport in both saturated and unsaturated porous media. As E. coli cells can form the green fluorescent protein (GFP), the cell densities, calculated by evaluation of measured fluorescence intensities (in situ detection) were compared with the cell densities computed by numerical simulation. The comparison showed the laboratory experiments can be well described by our mathematical model.

cs.CE↗