Alexander Heinlein

recent publications (full list)

1. PAMM

   First steps towards modeling the interaction of cardiovascular agents and smooth muscle activation in arterial walls

   Ramesh, Sharan Nurani, Uhlmann, Klemens, Saßmannshausen, Lea, Rheinbach, Oliver, Klawonn, Axel, Heinlein, Alexander, and Balzani, Daniel

   In Proc. Appl. Math. Mech.. 2023

   Abstract Bib doi HTML

   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 problems are solved, confirming the importance of incorporating stress variations on small time-scales. In the homogenization approach, a periodic fine-scale problem, which is periodic with respect to the heart beat, has to be solved for each long-scale time step. Even if no exact initial conditions are available, periodicity can be achieved within only 2–3 heart beats by simple time-stepping.

   @inproceedings{Ramesh:2023:FST,
     author = {Ramesh, Sharan Nurani and Uhlmann, Klemens and Sa{\ss}mannshausen, Lea and Rheinbach, Oliver and Klawonn, Axel and Heinlein, Alexander and Balzani, Daniel},
     title = {First steps towards modeling the interaction of cardiovascular agents and smooth muscle activation in arterial walls},
     booktitle = {Proc. Appl. Math. Mech.},
     volume = {22},
     number = {1},
     pages = {e202200133},
     year = {2023},
     doi = {10.1002/pamm.202200133},
     abbr = {PAMM},
     url = {https://onlinelibrary.wiley.com/doi/10.1002/pamm.202200133},
     keywords = {published, recent},
     bibtex_show = {true}
   }

2. Springer LNCSE

   A Three-Level Extension for Fast and Robust Overlapping Schwarz (FROSch) Preconditioners with Reduced Dimensional Coarse Space

   Heinlein, Alexander, Klawonn, Axel, Rheinbach, Oliver, and Röver, Friederike

   In Domain Decomposition Methods in Science and Engineering XXVI. 2023

   Abstract Bib doi HTML Preprint

   The Fast and Robust Overlapping Schwarz (FROSch) preconditioner framework is part of the Trilinos software library and contains parallel implementations of the Generalized-Dryja-Smith-Widlund (GDSW) type overlapping Schwarz domain decomposition preconditioners. It provides implementations of the classical GDSW coarse space as well as of reduced dimensional GDSW coarse spaces, where the coarse problem is smaller compared to the classical approach. To extend the parallel scalability of these approaches, a three-level extension has recently been introduced into the framework. In this paper, we present results, obtained on the SuperMUC-NG supercomputer using up to 85K MPI ranks. The results indicate superior weak parallel scalability of the three-level method compared to the two-level method.

   @inproceedings{Heinlein:2023:TLE,
     author = {Heinlein, Alexander and Klawonn, Axel and Rheinbach, Oliver and R\"over, Friederike},
     title = {A Three-Level Extension for Fast and Robust Overlapping Schwarz (FROSch) Preconditioners with Reduced Dimensional Coarse Space},
     booktitle = {Domain Decomposition Methods in Science and Engineering XXVI},
     pages = {505-513},
     publisher = {Springer International Publishing},
     year = {2023},
     doi = {10.1007/978-3-030-95025-5_54},
     abbr = {Springer LNCSE},
     url = {https://link.springer.com/chapter/10.1007/978-3-030-95025-5_54},
     preprint = {https://tu-freiberg.de/sites/default/files/media/fakultaet-fuer-mathematik-und-informatik-fakultaet-1-9277/prep/2021-02.pdf},
     keywords = {published, reviewed, recent},
     bibtex_show = {true}
   }

3. Springer LNCSE

   Predicting the geometric location of critical edges in adaptive GDSW overlapping domain decomposition methods using deep learning

   Heinlein, Alexander, Klawonn, Axel, Lanser, Martin, and Weber, Janine

   In Domain Decomposition Methods in Science and Engineering XXVI. 2023

   Abstract Bib doi HTML Preprint

   Overlapping GDSW domain decomposition methods are considered for diffusion problems in two dimensions discretized by finite elements. For a diffusion coefficient with high contrast, the condition number is usually dependent on it. A remedy is given by adaptive domain decomposition methods, where the coarse space is enhanced by additional coarse basis functions. These are chosen problem-dependently by solving small local eigenvalue problems. Here, the eigenvalue problems (EVPs) are associated with the edges of the domain decomposition interface; edges, where these EVPs have to be solved are denoted as critical edges. For many applications, not all edges are critical and the solution of the EVPs is not necessary. In an earlier work, a strategy to predict the location of critical edges, based on deep learning, has been proposed for adaptive FETI-DP, a class of nonoverlapping methods. In the present work, this strategy is successfully applied to adaptive GDSW; differences in the classification process for this overlapping method are described. Choosing the classification threshold has been a challenge in all these approaches. Here, for the first time, a heuristic based on the receiver operating characteristic (ROC) curve and the precision-recall graph is discussed. Results for a challenging realistic coefficient function are presented.

   @inproceedings{Heinlein:2023:PGL,
     author = {Heinlein, Alexander and Klawonn, Axel and Lanser, Martin and Weber, Janine},
     title = {Predicting the geometric location of critical edges in adaptive GDSW overlapping domain decomposition methods using deep learning},
     booktitle = {Domain Decomposition Methods in Science and Engineering XXVI},
     pages = {307-315},
     publisher = {Springer International Publishing},
     year = {2023},
     doi = {10.1007/978-3-030-95025-5_32},
     abbr = {Springer LNCSE},
     url = {https://link.springer.com/chapter/10.1007/978-3-030-95025-5_32},
     preprint = {https://kups.ub.uni-koeln.de/36257},
     keywords = {published, reviewed, recent},
     bibtex_show = {true}
   }

4. CM

   Comparison of Arterial Wall Models in Fluid-Structure Interaction Simulations

   Balzani, Daniel, Heinlein, Alexander, Klawonn, Axel, Rheinbach, Oliver, and Schröder, Jörg

   Accepted for publication in Computational Mechanics. March 2023

   Abstract Bib Preprint

   Monolithic fluid-structure interaction (FSI) of blood flow with arterial walls is considered, making use of sophisticated nonlinear wall models. These incorporate the effects of almost incompressibility as well as of the anisotropy caused by embedded collagen fibers. In the literature, relatively simple structural models such as Neo-Hooke are often considered for FSI with arterial walls. Such models lack, both, anisotropy and incompressibility. In this paper, numerical simulations of idealized heart beats in a curved benchmark geometry, using simple and sophisticated arterial wall models, are compared: we consider three different almost incompressible, anisotropic arterial wall models as a reference and, for comparison, a simple, isotropic Neo-Hooke model using four different parameter sets. The simulations show significant quantitative and qualitative differences in the stresses and displacements as well as the lumen cross sections. For the Neo-Hooke models, a significantly larger amplitude in the in- and outflow areas during the heart beat is observed, presumably due to the lack of fiber stiffening. For completeness, we also consider a linear elastic wall using 16 different parameter sets. However, using our benchmark setup, we were not successful in achieving good agreement with our nonlinear reference calculation.

   @techreport{Balzani:2022:CAW,
     author = {Balzani, Daniel and Heinlein, Alexander and Klawonn, Axel and Rheinbach, Oliver and Schr{\"o}der, J{\"o}rg},
     title = {Comparison of Arterial Wall Models in Fluid-Structure Interaction Simulations},
     note = {Accepted for publication in Computational Mechanics. March 2023},
     abbr = {CM},
     preprint = {http://kups.ub.uni-koeln.de/id/eprint/55586},
     keywords = {accepted, reviewed, recent},
     bibtex_show = {true}
   }

5. Springer LNCSE

   Finite basis physics-informed neural networks as a Schwarz domain decomposition method

   Dolean, Victorita, Heinlein, Alexander, Mishra, Siddhartha, and Moseley, Ben

   Accepted for publication in the Domain Decomposition Methods in Science and Engineering XXVII poceedings. February 2023

   Abstract Bib 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 provides a simple and mesh-free approach for solving problems relating to PDEs. However, a key limitation of PINNs is their lack of accuracy and efficiency when solving problems with larger domains and more complex, multi-scale solutions. In a more recent approach, finite basis physics-informed neural networks (FBPINNs) [8] use ideas from domain decomposition to accelerate the learning process of PINNs and improve their accuracy. In this work, we show how Schwarz-like additive, multiplicative, and hybrid iteration methods for training FBPINNs can be developed. We present numerical experiments on the influence of these different training strategies on convergence and accuracy. Furthermore, we propose and evaluate a preliminary implementation of coarse space correction for FBPINNs.

   @techreport{Dolean:2023:FBP,
     author = {Dolean, Victorita and Heinlein, Alexander and Mishra, Siddhartha and Moseley, Ben},
     title = {Finite basis physics-informed neural networks as a Schwarz domain decomposition method},
     note = {Accepted for publication in the Domain Decomposition Methods in Science and Engineering XXVII poceedings. February 2023},
     abbr = {Springer LNCSE},
     preprint = {https://arxiv.org/abs/2211.05560},
     keywords = {accepted, recent},
     bibtex_show = {true}
   }

6. IPDPS

   An Experimental Study of Two-level Schwarz Domain-Decomposition Preconditioners on GPUs

   Yamazaki, Ichitaro, Heinlein, Alexander, and Rajamanickam, Sivasankaran

   Accepted for publication in the IPDPS’23 proceedings. December 2022

   Abstract Bib

   Generalized Dryja–Smith–Widlund (GDSW) preconditioners are two-level overlapping Schwarz domain decomposition (DD) preconditioners that couple a classical one-level overlapping Schwarz preconditioner with an energy-minimizing coarse space. When used together with Krylov subspace methods, GDSW preconditioners are robust and scalable for solving sparse linear systems arising from the discretization of a wide range of partial different equations. However, the solver has a wide variety of algorithmic and implementation choices, which poses several challenges for its GPU implementation, and thus, its GPU implementation has not been presented so far. To address this gap, in this paper, we present FROSch (Fast and Robust Schwarz), a domain decomposition solver package which implements GDSW algorithms for both CPU and GPU clusters. In order to improve the solver performance on GPUs, we use a novel decomposition to run multiple MPI processes on each GPU, reducing both computational and storage costs of the DD solver, and potentially improving the solver convergence rate. This allows the GPU implementation to be competitive or faster than the CPU implementations. We demonstrate the performance of FROSch on the Summit supercomputer with NVIDIA V100 GPUs, where we used NVIDIA Multi-Process Service (MPS) to implement our decomposition strategy. We conduct a thorough experimental study with different solver options including the exact or inexact solution of the local overlapping subdomain problems based on the sparse direct or incomplete factorizations on a GPU, respectively. We also discuss the effect of using the iterative variant of the incomplete LU factorization and sparse-triangular solve as the approximate local solver and using lower precision for computing the whole FROSch preconditioner. Overall, the solve time was reduced by factors of about 2x using GPUs, while the effects on the numerical setup time depend on the solver options and the local matrix sizes.

   @techreport{Yamazaki:2022:EST,
     author = {Yamazaki, Ichitaro and Heinlein, Alexander and Rajamanickam, Sivasankaran},
     title = {An Experimental Study of Two-level Schwarz Domain-Decomposition Preconditioners on GPUs},
     note = {Accepted for publication in the IPDPS’23 proceedings. December 2022},
     abbr = {IPDPS},
     keywords = {accepted, reviewed, recent},
     bibtex_show = {true}
   }

7. An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation

   Giammatteo, Elena, Heinlein, Alexander, and Schlottbom, Matthias

   Submitted March 2023

   Abstract Bib Preprint

   An extension of the approximate component mode synthesis (ACMS) method to the heterogeneous Helmholtz equation is proposed. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. This complements related results for elliptic problems where the focus is on the refinement of the domain decomposition instead. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the interface problems. In certain, practically relevant cases, even exponential convergence for the local Helmholtz problems can be achieved without oversampling.

   @techreport{Giammatteo:2023:EAC,
     author = {Giammatteo, Elena and Heinlein, Alexander and Schlottbom, Matthias},
     title = {An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation},
     note = {Submitted March 2023},
     preprint = {https://arxiv.org/abs/2303.06671},
     keywords = {submitted, reviewed, recent},
     bibtex_show = {true}
   }