Alexander Heinlein

Alexander Heinlein
Delft University of Technology | TU · Department of Applied Mathematics

Dr. rer. nat.
Scientific computing, domain decomposition methods, high-performance computing, scientific machine learning

About

65
Publications
2,777
Reads
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330
Citations
Citations since 2017
58 Research Items
311 Citations
2017201820192020202120222023020406080100
2017201820192020202120222023020406080100
2017201820192020202120222023020406080100
2017201820192020202120222023020406080100
Additional affiliations
July 2019 - September 2019
Sandia National Laboratories
Position
  • Researcher
February 2018 - present
University of Cologne
Position
  • Managing Coordinator
July 2017 - October 2017
Sandia National Laboratories
Position
  • Researcher
Education
October 2006 - November 2011
University of Duisburg-Essen
Field of study
  • Mathematics

Publications

Publications (65)
Preprint
Physics-informed neural networks (PINNs) [4, 10] are an approach for solving boundary value problems based on differential equations (PDEs). The key idea of PINNs is to use a neural network to approximate the solution to the PDE and to incorporate the residual of the PDE as well as boundary conditions into its loss function when training it. This p...
Article
Full-text available
We show that the concept of topology optimization for metallization grid patterns of thin-film solar devices can be applied to monolithically integrated solar cells. Different irradiation intensities favor different topological grid designs as well as a different thickness of the transparent conductive oxide (TCO) layer. For standard laboratory eff...
Preprint
Full-text available
Two-level domain decomposition preconditioners lead to fast convergence and scalability of iterative solvers. However, for highly heterogeneous problems, where the coefficient function is varying rapidly on several possibly non-separated scales, the condition number of the preconditioned system generally depends on the contrast of the coefficient f...
Preprint
In order to make the numerical simulation of atherosclerotic plaque growth feasible, a temporal homogenization approach is employed. The resulting macro-scale problem for the plaque growth can be further accelerated by using parallel time integration schemes, such as the parareal algorithm. However, the parallel scalability is dominated by the comp...
Article
A convolution neural network (CNN)-based approach for the construction of reduced order surrogate models for computational fluid dynamics (CFD) simulations is introduced; it is inspired by the approach of Guo, Li, and Iori [X. Guo, W. Li, and F. Iorio, Convolutional neural networks for steady flow approximation, in Proceedings of the 22nd ACM SIGKD...
Preprint
The numerical simulation of atherosclerotic plaque growth is computationally prohibitive, since it involves a complex cardiovascular fluid-structure interaction (FSI) problem with a characteristic time scale of milliseconds to seconds, as well as a plaque growth process governed by reaction-diffusion equations, which takes place over several months...
Article
Full-text available
A temporal homogenization approach for the numerical simulation of atherosclerotic plaque growth is extended to fully coupled fluid-structure interaction (FSI) simulations. The numerical results indicate that the two-scale approach yields significantly different results compared to a simple heuristic averaging, where only stationary long-scale FSI...
Preprint
Full-text available
Different graph partitioning methods, i.e., linear partioning, parallel hypergraph (PHG) partioning, and two approaches using ParMETIS, are considered to generate an unstructured decomposition of the second-level coarse operator of three-level FROSch (Fast and Robust Overlapping Schwarz) preconditioners in the Trilinos software library. In our cont...
Article
The course of an epidemic can often be successfully described mathematically using compartment models. These models result in a system of ordinary differential equations. Two well-known examples are the SIR S I R and the SEIR S E I R models. The transition rates between the different compartments are defined by certain parameters that are specific...
Preprint
Full-text available
The parallel performance of the three-level Fast and Robust Overlapping Schwarz (FROSch) preconditioners is investigated for linear elasticity. The FROSch framework is part of the Trilinos software library and contains a parallel implementation of different preconditioners with energy minimizing coarse spaces of GDSW (Gen-eralized Dryja-Smith-Widlu...
Preprint
Full-text available
A temporal homogenization approach for the numerical simulation of atherosclerotic plaque growth is extended to fully coupled fluid-structure interaction (FSI) simulations. The numerical results indicate that the two-scale approach yields significantly different results compared to a simple heuristic averaging, where only stationary long-scale FSI...
Article
Full-text available
Scientific machine learning (SciML), an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within SciML given by the combination of domain decomposition methods (DDMs) with machine learning techn...
Chapter
The convergence rate of classic domain decomposition methods in general deteriorates severely for large discontinuities in the coefficient functions of the considered partial differential equation. To retain the robustness for such highly heterogeneous problems, the coarse space can be enriched by additional coarse basis functions. These can be obt...
Technical Report
A convolution neural network (CNN)-based approach for the construction of reduced order surrogate models for computational fluid dynamics (CFD) simulations is introduced; it is inspired by the approach of Guo, Li, and Iori [X. Guo, W. Li, and F. Iorio, Convolutional neural networks for steady flow approximation, in Proceedings of the 22nd ACM SIGKD...
Chapter
The convergence rate of classical domain decomposition methods for diffusion or elasticity problems usually deteriorates when large coefficient jumps occur along or across the interface between subdomains. In fact, the constant in the classical condition number bounds [11, 12] will depend on the coefficient jump.
Conference Paper
We compare the spectra of local generalized eigenvalue problems in different adaptive coarse spaces for overlapping and nonoverlapping domain decomposition methods. In particular, we compare the AGDSW (Adaptive Generalized Dryja--Smith--Widlund), the OS-ACMS (Overlapping Schwarz-Approximate Component Mode Synthesis), and the SHEM (Spectral Harmonic...
Technical Report
Scientific machine learning, an area of research where techniques from machine learning and scientific computing are combined, has become of increasing importance and receives growing attention. Here, our focus is on a very specific area within scientific machine learning given by the combination of domain decomposition methods with machine learning te...
Chapter
To solve (1), we consider nonlinear domain decomposition methods of the Schwarz type, e.g., ASPIN (Additive Schwarz Preconditioned Inexact Newton) [1, 10] or RASPEN (Restricted Additive Schwarz Preconditioned Exact Newton) [3].
Chapter
For second order elliptic partial differential equations, such as diffusion or elasticity, with arbitrary and high coefficient jumps, the convergence rate of domain decomposition methods with classical coarse spaces typically deteriorates. One remedy is the use of adaptive coarse spaces, which use eigenfunctions computed from local generalized eige...
Chapter
Full-text available
The GDSW (Generalized Dryja–Smith–Widlund) preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner [23] with exact local solvers [5, 4].
Chapter
Full-text available
This article describes a parallel implementation of a two-level overlapping Schwarz preconditioner with the GDSW (Generalized Dryja–Smith–Widlund) coarse space described in previous work [12, 10, 15] into the Trilinos framework; cf. [16]. The software is a significant improvement of a previous implementation [12]; see Sec. 4 for results on the impr...
Technical Report
The course of an epidemic can be often successfully described mathematically using compartment models. These models result in a system of ordinary differential equations. Two well-known examples are the SIR and the SEIR models. The transition rates between the different compartments are defined by certain parameters which are specific for the respecti...
Technical Report
Full-text available
A new reduced dimension adaptive GDSW (Generalized Dryja-Smith-Widlund) overlapping Schwarz method for linear second-order elliptic problems in three dimensions is introduced. It is robust with respect to large contrasts of the coefficients of the partial differential equations. The condition number bound of the new method is shown to be independen...
Chapter
Computational Fluid Dynamics (CFD) simulations are a numerical tool to model and analyze the behavior of fluid flow. However, accurate simulations are generally very costly because they require high grid resolutions. In this paper, an alternative approach for computing flow predictions using Convolutional Neural Networks (CNNs) is described; in par...
Chapter
Different parallel two-level overlapping Schwarz preconditioners with Generalized Dryja–Smith–Widlund (GDSW) and Reduced dimension GDSW (RGDSW) coarse spaces for elasticity problems are considered. GDSW type coarse spaces can be constructed from the fully assembled system matrix, but they additionally need the index set of the interface of the corr...
Technical Report
The hybrid ML-FETI-DP algorithm combines the advantages of adaptive coarse spaces in domain decomposition methods and certain supervised machine learning techniques. Adaptive coarse spaces ensure robustness of highly scalable domain decomposition solvers, even for highly heterogeneous coefficient distributions with arbitrary coefficient jumps. Howe...
Article
Full-text available
Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared to preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this paper, several techniques to further im...
Technical Report
Computational Fluid Dynamics (CFD) simulations are a numerical tool to model and analyze the behavior of fluid flow. However, accurate simulations are generally very costly because they require high grid resolutions. In this paper, an alternative approach for computing flow predictions using Convolutional Neural Networks (CNNs) is described; in par...
Technical Report
The convergence rate of classic domain decomposition methods in general deteriorates severely for large discontinuities in the coefficient functions of the considered partial differential equation. To retain the robustness for such highly heterogeneous problems, the coarse space can be enriched by additional coarse basis functions. These can be obt...
Article
Domain decomposition methods are robust and parallel scalable, preconditioned iterative algorithms for the solution of the large linear systems arising in the discretization of elliptic partial differential equations by finite elements. The convergence rate of these methods is generally determined by the eigenvalues of the preconditioned system. Fo...
Article
Full-text available
A benchmark for the comparison of MRI (Magnetic Resonance Imaging) measurements and CFD (Computational Fluid Dynamics) simulations for blood flow in intracranial aneurysms is presented. The benchmark setting is designed to allow for CFD simulations that are completely independent of the MRI measurements. This facilitates a fair comparison of both m...
Article
Full-text available
MRI (Magnetic Resonance Imaging) measurements and CFD (Computational Fluid Dynamics) simulations for blood flow in intracranial aneurysms are compared for a benchmark problem. In particular, it is shown that noise and other artifacts in the MRI measurements have an influence on certain properties of the flow field, e.g., on the boundary flow and ma...
Article
Full-text available
Parallel computational results for problems in dislocation mechanics are presented using the deal.II adaptive finite element software and the Fast and Robust Overlapping Schwarz (FROSch) Preconditioner.
Article
Full-text available
A robust two-level overlapping Schwarz method for scalar elliptic model problems with highly varying coefficient functions is introduced. While the convergence of standard coarse spaces may depend strongly on the contrast of the coefficient function, the condition number bound of the new method is independent of the coefficient function. Indeed, th...
Chapter
Full-text available
A three-level extension of the GDSW overlapping Schwarz preconditioner in two dimensions is presented, constructed by recursively applying the GDSW preconditioner to the coarse problem. Numerical results, obtained for a parallel implementation using the Trilinos software library, are presented for up to 90,000 cores of the JUQUEEN supercomputer. Th...
Article
In this contribution, Fluid‐Structure‐Interaction (FSI) in blood vessels, in detail the simulation of realistic arterial geometries, where the interaction of the blood flow and the vessel wall is of special interest, is considered. Based on pervious research, cf. [1], our existing framework for FSI‐simulations is extended towards realistic arterial...
Technical Report
We compare the spectra of local generalized eigenvalue problems in different adaptive coarse spaces for overlapping and nonoverlapping domain decomposition methods. In particular, we compare the AGDSW (Adaptive Generalized Dryja-Smith-Widlund), the OS-ACMS (Overlapping Schwarz-Approximate Component Mode Synthesis), and the SHEM (Spectral Harmonical...
Technical Report
Adaptive FETI-DP (Finite Element Tearing and Interconnecting - Dual-Primal) methods are considered for the solution of two-dimensional scalar elliptic model problems with complex coefficient distributions where large coefficient jumps can occur along or across the domain decomposition interface. The adaptive coarse space is obtained by solving cert...
Article
Full-text available
Two-level overlapping Schwarz domain decomposition methods for second-order elliptic problems in two dimensions are proposed using coarse spaces constructed from the Approximate Component Mode Synthesis (ACMS) multiscale discretization approach. These coarse spaces are based on eigenvalue problems using Schur complements on subdomain edges. It is t...
Chapter
We consider a recent overlapping Schwarz method with an energy-minimizing coarse space of reduced size. In numerical experiments for up to 64,000 cores, we show that the parallel efficiency and the total time to solution is improved significantly, compared to our previous overlapping Schwarz method using an alternative energy-minimizing coarse spac...
Chapter
We propose robust coarse spaces for two-level overlapping Schwarz preconditioners, which are extensions of the energy minimizing coarse space known as GDSW (Generalized Dryja, Smith, Widlund). The resulting two-level methods with adaptive coarse spaces are robust for second order elliptic problems in two dimensions, even in presence of a highly het...
Article
In this contribution, results regarding fluid-structure interaction (FSI) simulations for three-dimensional arterial walls are presented. In detail, a benchmark problem for FSI simulations in arteries of sufficient complexity, which combines sophisticated nonlinear models for the fluid and the structure, cf. [1], as well as a short segment from a p...
Chapter
Parallel results obtained with a new implementation of an overlapping Schwarz method using an energy minimizing coarse space are presented. We consider structured and unstructured domain decompositions for scalar elliptic and linear elasticity model problems in two dimensions. In particular, strong and weak parallel scalability studies for up to 10...
Article
We describe a new implementation of a two-level overlapping Schwarz preconditioner with energy-minimizing coarse space (GDSW: generalized Dryja--Smith--Widlund) and show numerical results for an additive and a hybrid additive-multiplicative version. Our parallel implementation makes use of the Trilinos software library and provides a framework for...
Thesis
Full-text available
Accurate simulations of transmural wall stresses in artherosclerotic coronary arteries may help to predict plaque rupture. Therefore, a robust and efficient numerical framework for Fluid-Structure Interaction (FSI) of the blood flow and the arterial wall has to be set up, and suitable material laws for the modeling of the fluid and the structural r...
Chapter
Parallel overlapping Schwarz preconditioners are considered and applied to the structural block in monolithic fluid-structure interaction (FSI). The two-level overlapping Schwarz method uses a coarse level based on energy minimizing functions. Linear elastic as well as nonlinear, anisotropic hyperelastic structural models are considered in an FSI p...
Article
Full-text available
The accurate prediction of transmural stresses in arterial walls requires on the one hand robust and efficient numerical schemes for the solution of boundary value problems including fluid-structure interactions (FSI) and on the other hand the use of a material model for the vessel wall which is able to capture the relevant features of the material...
Article
A Fluid–Structure Interaction (FSI) problem can be reinterpreted as a heterogeneous problem with two subdomains. It is possible to describe the coupled problem at the interface between the fluid and the structure, yielding a nonlinear Steklov–Poincaré problem. The linear system can be linearized by Newton iterations on the interface and the resulti...
Article
A special finite element method based on approximate component mode synthesis (ACMS) was introduced in Hetmaniuk and Lehoucq (2010). ACMS was developed for second order elliptic partial differential equations with rough or highly varying coefficients. Here, a parallel implementation of ACMS is presented and parallel scalability issues are discussed...
Conference Paper
The simulation of the physiological loading situation of arteries with moderate atherosclerotic plaque may provide additional indicators for medical doctors to estimate if the plaque is likely to rupture and if surgical intervention is required. In particular the transmural stresses are important in this context. They depend strongly on the mechani...

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Projects

Projects (4)
Project
Scientific Machine Learning is a rapidly evolving field of research that combines and further develops techniques of scientific computing and machine learning. Topics of interest include but are not restricted to - Physics-informed machine learning - Machine learning and iterative methods/domain decomposition methods - Machine learning enhanced simulations - Hybrid modeling (machine learning + first principle modeling) …
Project
FROSch (Fast and Robust Overlapping Schwarz) is a framework for overlapping Schwarz domain decomposition methods in Trilinos. It is part of the Trilinos package ShyLU.