Alexander Heinlein

recent publications (full list)

1. SISC

   Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods

   Heinlein, Alexander, Klawonn, Axel, Knepper, Jascha, Rheinbach, Oliver, and Widlund, Olof B.

   SIAM Journal on Scientific Computing. 2022

   Abstract Bib doi HTML PDF Preprint

   A new reduced-dimension adaptive generalized Dryja–Smith–Widlund (GDSW) 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 independent of the coefficient contrast and only dependent on a user-prescribed tolerance. The interface of the nonoverlapping domain decomposition is partitioned into nonoverlapping patches. The new coarse space is obtained by selecting a few eigenvectors of certain local eigenproblems which are defined on these patches. These eigenmodes are energy-minimally extended to the interior of the nonoverlapping subdomains and added to the coarse space. By using a new interface decomposition, the reduced-dimension adaptive GDSW overlapping Schwarz method usually has a smaller coarse space than existing GDSW and adaptive GDSW domain decomposition methods. A robust condition number estimate is proven for the new reduced-dimension adaptive GDSW method which is also valid for existing adaptive GDSW methods. Numerical results for the equations of isotropic linear elasticity in three dimensions confirming the theoretical findings are presented.

   @article{Heinlein:2022:AGD,
     author = {Heinlein, Alexander and Klawonn, Axel and Knepper, Jascha and Rheinbach, Oliver and Widlund, Olof B.},
     title = {Adaptive GDSW Coarse Spaces of Reduced Dimension for Overlapping Schwarz Methods},
     journal = {SIAM Journal on Scientific Computing},
     volume = {44},
     number = {3},
     pages = {A1176-A1204},
     year = {2022},
     doi = {10.1137/20M1364540},
     abbr = {SISC},
     url = {https://doi.org/10.1137/20M1364540},
     preprint = {https://kups.ub.uni-koeln.de/id/eprint/12113},
     pdf = {publications/2022/heinlein-2022-agd.pdf},
     keywords = {published, reviewed, selected, recent},
     bibtex_show = {true}
   }

2. SISC

   FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica

   Heinlein, Alexander, Perego, Mauro, and Rajamanickam, Sivasankaran

   SIAM Journal on Scientific Computing. 2022

   Abstract Bib doi HTML Preprint

   Numerical simulations of Greenland and Antarctic ice sheets involve the solution of large-scale highly nonlinear systems of equations on complex shallow geometries. This work is concerned with the construction of Schwarz preconditioners for the solution of the associated tangent problems, which are challenging for solvers mainly because of the strong anisotropy of the meshes and wildly changing boundary conditions that can lead to poorly constrained problems on large portions of the domain. Here, two-level generalized Dryja–Smith–Widlund (GDSW)–type Schwarz preconditioners are applied to different land ice problems, i.e., a velocity problem, a temperature problem, as well as the coupling of the former two problems. We employ the message passing interface (MPI)–parallel implementation of multilevel Schwarz preconditioners provided by the package FROSch (fast and robust Schwarz) from the Trilinos library. The strength of the proposed preconditioner is that it yields out-of-the-box scalable and robust preconditioners for the single physics problems. To the best of our knowledge, this is the first time two-level Schwarz preconditioners have been applied to the ice sheet problem and a scalable preconditioner has been used for the coupled problem. The preconditioner for the coupled problem differs from previous monolithic GDSW preconditioners in the sense that decoupled extension operators are used to compute the values in the interior of the subdomains. Several approaches for improving the performance, such as reuse strategies and shared memory OpenMP parallelization, are explored as well. In our numerical study we target both uniform meshes of varying resolution for the Antarctic ice sheet as well as nonuniform meshes for the Greenland ice sheet. We present several weak and strong scaling studies confirming the robustness of the approach and the parallel scalability of the FROSch implementation. Among the highlights of the numerical results are a weak scaling study for up to 32K processor cores (8K MPI ranks and 4 OpenMP threads) and 566M degrees of freedom for the velocity problem as well as a strong scaling study for up to 4K processor cores (and MPI ranks) and 68M degrees of freedom for the coupled problem.

   @article{Heinlein:2022:FRO,
     author = {Heinlein, Alexander and Perego, Mauro and Rajamanickam, Sivasankaran},
     title = {FROSch Preconditioners for Land Ice Simulations of Greenland and Antarctica},
     journal = {SIAM Journal on Scientific Computing},
     volume = {44},
     number = {2},
     pages = {B339-B367},
     year = {2022},
     doi = {10.1137/21M1395260},
     abbr = {SISC},
     url = {https://doi.org/10.1137/21M1395260},
     preprint = {https://kups.ub.uni-koeln.de/id/eprint/30668},
     keywords = {published, reviewed, selected, recent},
     bibtex_show = {true}
   }

3. ETNA

   Surrogate convolutional neural network models for steady computational fluid dynamics simulations

   Eichinger, Matthias, Heinlein, Alexander, and Klawonn, Axel

   Electronic Transactions on Numerical Analysis (ETNA). 2022

   Abstract Bib doi HTML Preprint

   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 SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD’16, New York, USA, 2016, ACM, pp. 481–490]. In particular, the neural networks are trained in order to predict images of the flow field in a channel with varying obstacle based on an image of the geometry of the channel. A classical CNN with bottleneck structure and a U-Net are compared while varying the input format, the number of decoder paths, as well as the loss function used to train the networks. This approach yields very low prediction errors, in particular, when using the U-Net architecture. Furthermore, the models are also able to generalize to unseen geometries of the same type. A transfer learning approach enables the model to be trained to a new type of geometries with very low training cost. Finally, based on this transfer learning approach, a sequential learning strategy is introduced, which significantly reduces the amount of necessary training data.

   @article{Eichinger:2022:SCN,
     author = {Eichinger, Matthias and Heinlein, Alexander and Klawonn, Axel},
     title = {Surrogate convolutional neural network models for steady computational fluid dynamics simulations},
     journal = {Electronic Transactions on Numerical Analysis (ETNA)},
     volume = {56},
     pages = {235--255},
     year = {2022},
     doi = {10.1553/etna_vol56s235},
     abbr = {ETNA},
     url = {https://epub.oeaw.ac.at/?arp=0x003d4c21},
     preprint = {https://kups.ub.uni-koeln.de/29760/},
     keywords = {published, reviewed, selected, recent},
     bibtex_show = {true}
   }

4. SISC

   Parallel Scalability of Three-Level FROSch Preconditioners to 220 000 Cores using the Theta Supercomputer

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

   Accepted for publication in SIAM Journal on Scientific Computing. April 2022

   Abstract Bib Preprint

   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 (Generalized Dryja–Smith–Widlund) type. The three-level extension is constructed by a recursive application of the FROSch preconditioner to the coarse problem. In this paper, the additional steps in the implementation in order to apply the FROSch preconditioner recursively are described in detail. Furthermore, it is shown that no explicit geometric information is needed in the recursive application of the preconditioner. In particular, the rigid body modes, including the rotations, can be interpolated on the coarse level without additional geometric information. Parallel results for a three-dimensional linear elasticity problem obtained on the Theta supercomputer (ALCF, Argonne, USA) using up to 220 000 cores are discussed and compared to results obtained on the SuperMUC-NG supercomputer (LRZ, Garching, Germany). Notably, it can be observed that a hierarchical communication operation in FROSch related to the coarse operator starts to dominate the computing time on Theta for 100 000 MPI ranks or more. The same operation, however, scales well and stays within the order of a second in all experiments performed on SuperMUC-NG. Using hybrid MPI/OpenMP parallelization, better performance is then achieved on Theta.

   @techreport{Heinlein:2021:PST,
     author = {Heinlein, Alexander and Rheinbach, Oliver and R{\"o}ver, Friederike},
     title = {Parallel Scalability of Three-Level FROSch Preconditioners to 220 000 Cores using the Theta Supercomputer},
     note = {Accepted for publication in SIAM Journal on Scientific Computing. April 2022},
     abbr = {SISC},
     preprint = {https://tu-freiberg.de/sites/default/files/media/fakultaet-fuer-mathematik-und-informatik-fakultaet-1-9277/prep/2021-03.pdf},
     keywords = {accepted, reviewed, selected, recent},
     bibtex_show = {true}
   }

5. SISC

   Adaptive nonlinear domain decomposition methods

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

   Accepted for publication in SIAM Journal on Scientific Computing. March 2022

   Abstract Bib Preprint

   In this article, different nonlinear domain decomposition methods are applied to nonlinear problems with highly-heterogeneous coefficient functions with jumps. In order to obtain a robust solver with respect to nonlinear as well as linear convergence, adaptive coarse spaces are employed. First, as an example for a nonlinearly left-preconditioned domain decomposition method, the two-level restricted nonlinear Schwarz method H1-RASPEN (Hybrid Restricted Additive Schwarz Preconditioned Exact Newton) is combined with an adaptive generalized Dryja–Smith–Widlund (GDSW) coarse space. Second, as an example for a nonlinearly right-preconditioned domain decomposition method, a nonlinear FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) method is equipped with an edge-based adaptive coarse space. Both approaches are compared with the respective nonlinear domain decomposition methods with classical coarse spaces as well as with the respective Newton-Krylov methods with adaptive coarse spaces. For some two-dimensional pLaplace model problems with different spatial coefficient distributions, it can be observed that the best linear and nonlinear convergence can only be obtained when combining the nonlinear domain decomposition methods with adaptive coarse spaces.

   @techreport{Heinlein:2021:AND,
     author = {Heinlein, Alexander and Klawonn, Axel and Lanser, Martin},
     title = {Adaptive nonlinear domain decomposition methods},
     note = {Accepted for publication in SIAM Journal on Scientific Computing. March 2022},
     abbr = {SISC},
     preprint = {https://kups.ub.uni-koeln.de/id/eprint/52537},
     keywords = {accepted, reviewed, selected, recent},
     bibtex_show = {true}
   }

6. Towards parallel time-stepping for the numerical simulation of atherosclerotic plaque growth

   Frei, Stefan, and Heinlein, Alexander

   Submitted March 2022

   Abstract Bib 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. In this work we combine a temporal homogenization approach, which separates the problem in computationally expensive FSI problems on a micro scale and a reaction-diffusion problem on the macro scale, with parallel time-stepping algorithms. It has been found in the literature that parallel time-stepping algorithms do not perform well when applied directly to the FSI problem. To circumvent this problem, a parareal algorithm is applied on the macro scale reaction-diffusion problem instead of the micro scale FSI problem. We investigate modifications in the coarse propagator of the parareal algorithm, in order to further reduce the number of costly micro problems to be solved. The approaches are tested in detailed numerical investigations based on serial simulations.

   @techreport{Frei:2022:TPT,
     author = {Frei, Stefan and Heinlein, Alexander},
     title = {Towards parallel time-stepping for the numerical simulation of atherosclerotic plaque growth},
     note = {Submitted March 2022},
     preprint = {https://arxiv.org/abs/2203.06526},
     keywords = {submitted, reviewed, recent},
     bibtex_show = {true}
   }

7. Comparison of Arterial Wall Models in Fluid-Structure Interaction Simulations

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

   Submitted March 2022

   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 = {Submitted March 2022},
     preprint = {http://kups.ub.uni-koeln.de/id/eprint/55586},
     keywords = {submitted, reviewed, recent},
     bibtex_show = {true}
   }