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. However, their construction requires the setup and solution of local generalized eigenvalue problems, which is typically computationally expensive. The idea of ML-FETI-DP is to interpret the coefficient distribution as image data and predict whether an eigenvalue problem has to be solved or can be neglected while still maintaining robustness of the adaptive FETI-DP method. For this purpose, neural networks are used as image classifiers. In the present work, the ML-FETI-DP algorithm is extended to three dimensions, which requires both a complex data preprocessing procedure to construct consistent input data for the neural network as well as a representative training and validation data set to ensure generalization properties of the machine learning model. Numerical experiments for stationary diffusion and linear elasticity problems with realistic coefficient distributions show that a large number of eigenvalue problems can be saved; in the best case of the numerical results presented here, 97% of the eigenvalue problems can be avoided being set up and solved.@en

Abstract 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 for representative examples. Additionally, a parallel domain decomposition preconditioner (FETI-DP) is applied to solve the ACMS finite element system. Weak parallel scalability results for ACMS are presented for up to 1024 cores. Our numerical results also suggest a quadratic-logarithmic condition number bound for the preconditioned FETI-DP method applied to ACMS discretizations.@en

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 mechanical response and thus, a predictive material model capturing all characteristics of the material behavior is required. Here, polyconvex strain energy functions are considered for the hyperelastic behavior and a simplified viscoelastic model is proposed which does not take into account an isochoric strain energy for the fiber response. Based thereon, a comparative study is presented, investigating the influence of viscoelasticity on the mechanical behavior of a simplified arterial wall and a rather small impact is found. Realistic predictions of transmural stress distributions require a simulation of the interaction between the blood flow and the vessel wall. We recall the equations that model fluid-structure interaction and the monolithic Convective Explicit algorithm for their numerical approximation, addressing both the cases when the fluid-structure meshes are conforming and noncon-forming at the interface. We also present numerical experiments, using the monolithic approach, for the fluid structure interaction problem in a curved tube using a hyperelastic material model for the structure and an absorbing boundary condition. The fluid structure interaction using a highly nonlinear anisotropic structural model for the solid in this context is one of the main contributions of this paper.@en

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 obtained by solving local generalized eigenvalue problems on subdomain edges. In order to reduce the number of eigenvalue problems and thus the computational cost, we use a neural network to predict the geometric location of critical edges, i.e., edges where the eigenvalue problem is indispensable. As input data for the neural network, we use function evaluations of the coefficient function within the two subdomains adjacent to an edge. In the present article, we examine the effect of computing the input data only in a neighborhood of the edge, i.e., on slabs next to the edge. We show numerical results for both the training data as well as for a concrete test problem in form of a microsection subsection for linear elasticity problems. We observe that computing the sampling points only in one half or one quarter of each subdomain still provides robust algorithms.@en

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 improved performance.@en

Solving partial differential equations (PDEs) is a common task in numerical mathematics and scientific computing. Typical discretization schemes, for example, finite element (FE), finite volume (FV), or finite difference (FD) methods, have the disadvantage that the computations have to be repeated once the boundary conditions (BCs) or the geometry change slightly; typical examples requiring the solution of many similar problems are time-dependent and inverse problems or uncertainty quantification.@en

A computational framework is presented to numerically simulate the effects of antihypertensive drugs, in particular calcium channel blockers, on the mechanical response of arterial walls. A stretch-dependent smooth muscle model by Uhlmann and Balzani is modified to describe the interaction of pharmacological drugs and the inhibition of smooth muscle activation. The coupled deformation-diffusion problem is then solved using the finite element software FEDDLib and overlapping Schwarz preconditioners from the Trilinos package FROSch. These preconditioners include highly scalable parallel GDSW (generalized Dryja–Smith–Widlund) and RGDSW (reduced GDSW) preconditioners. Simulation results show the expected increase in the lumen diameter of an idealized artery due to the drug-induced reduction of smooth muscle contraction, as well as a decrease in the rate of arterial contraction in the presence of calcium channel blockers. Strong and weak parallel scalability of the resulting computational implementation are also analyzed.@en

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.@en

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.@en

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 GDSW (generalized Dryja-Smith-Widlund) 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 p-Laplace 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.@en

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 and the SEIR models. The transition rates between the different compartments are defined by certain parameters that are specific for the respective virus. Often, these parameters are known from the literature or can be determined using statistics. However, the contact rate or the related effective reproduction number are in general not constant in time and thus cannot easily be determined. Here, a new machine learning approach based on physics-informed neural networks is presented that can learn the contact rate from given data for the dynamical systems given by the SIR and SEIR models. The new method generalizes an already known approach for the identification of constant parameters to the variable or time-dependent case. After introducing the new method, it is tested for synthetic data generated by the numerical solution of SIR and SEIR models. The case of exact and perturbed data is considered. In all cases, the contact rate can be learned very satisfactorily. Finally, the SEIR model in combination with physics-informed neural networks is used to learn the contact rate for COVID-19 data given by the course of the epidemic in Germany. The simulation of the number of infected individuals over the course of the epidemic, using the learned contact rate, shows a very promising accordance with the data. @en

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 then shown that the convergence of the resulting preconditioned Krylov method can be controlled by a user-specified tolerance and thus can be made independent of heterogeneities in the coefficient of the partial differential equation. The relations of this new approach to other known adaptive coarse space approaches for overlapping Schwarz methods are also discussed. Compared to one of the competing adaptive approaches, the new coarse space can be significantly smaller. Compared to other competing approaches, the eigenvalue problems are significantly cheaper to solve, i.e., the dimension of the eigenvalue problems is minimal among the competing adaptive approaches under consideration. Our local eigenvalue problems can be solved using one iteration of LobPCG for essentially the same cost as a Cholesky-decomposition of a Schur complement on a subdomain edge.@en

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 techniques for the solution of partial differential equations. The aim of the present work is to make an attempt of providing a review of existing and also new approaches within this field as well as to present some known results in a unified framework; no claim of completeness is made. As a concrete example of machine learning enhanced DDMs, an approach is presented which uses neural networks to reduce the computational effort in adaptive DDMs while retaining their robustness. More precisely, deep neural networks are used to predict the geometric location of constraints which are needed to define a robust coarse space. Additionally, two recently published deep domain decomposition approaches are presented in a unified framework. Both approaches use physics-constrained neural networks to replace the discretization and solution of the subdomain problems of a given decomposition of the computational domain. Finally, a brief overview is given of several further approaches which combine machine learning with ideas from DDMs to either increase the performance of already existing algorithms or to create completely new methods.@en

The convergence rate of domain decomposition methods is generally determined by the eigenvalues of the preconditioned system. For second-order elliptic partial differential equations, coefficient discontinuities with a large contrast can lead to a deterioration of the convergence rate. Only by implementing an appropriate coarse space, or second level, a robust domain decomposition method can be obtained. In this article, a new frugal coarse space for FETI-DP (Finite Element Tearing and Interconnecting-Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) methods is presented, which has a lower set-up cost than competing adaptive coarse spaces. In particular, in contrast to adaptive coarse spaces, it does not require the solution of any local generalized eigenvalue problems. The approach considered here aims at a low-dimensional approximation of the adaptive coarse space by using appropriate weighted averages, and it is robust for a broad range of coefficient distributions for diffusion and elasticity problems. However, in general, for completely arbitrary coefficient distributions with high contrast, some additional, adaptively chosen constraints are necessary in order to guarantee robustness. In this article, the robustness is heuristically justified as well as numerically shown for several coefficient distributions. The new coarse space is compared to adaptive coarse spaces, and parallel scalability up to 262 144 parallel cores for a parallel BDDC implementation with the new coarse space is shown. The superiority of the new coarse space over classic coarse spaces with respect to parallel weak scalability and time-to-solution is confirmed by numerical experiments. Since the new frugal coarse space is computationally inexpensive, it could serve as a new default coarse space, which, for very challenging coefficient distributions, could then still be enhanced by adaptively chosen constraints.@en

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. For second-order elliptic partial differential equations, coefficient discontinuities with a large contrast can lead to a deterioration of the convergence rate. A remedy can be obtained by enhancing the coarse space with elements, which are often called constraints, that are computed by solving small eigenvalue problems on portions of the interface of the domain decomposition, i.e., edges in two dimensions or faces and edges in three dimensions. In the present work, without restriction of generality, the focus is on two dimensions. In general, it is difficult to predict where these constraints have to be added, and therefore the corresponding local eigenvalue problems have to be computed, i.e., on which edges. Here, a machine learning based strategy using neural networks is suggested to predict the geometric location of these edges in a preprocessing step. This reduces the number of eigenvalue problems that have to be solved before the iteration. Numerical experiments for model problems and realistic microsections using regular decompositions as well as decompositions from graph partitioners are provided, showing very promising results.@en

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 and on the other hand the use of a material model for the vessel wall that is able to capture the relevant features of the material behavior. One of the main contributions of this paper is the application of a highly nonlinear, polyconvex anisotropic structural model for the solid in the context of fluid–structure interaction, together with a suitable discretization. Additionally, the influence of viscoelasticity is investigated. The fluid–structure interaction problem is solved using a monolithic approach; that is, the nonlinear system is solved (after time and space discretizations) as a whole without splitting among its components. The linearized block systems are solved iteratively using parallel domain decomposition preconditioners. A simple – but nonsymmetric – curved geometry is proposed that is demonstrated to be suitable as a benchmark testbed for fluid–structure interaction simulations in biomechanics where nonlinear structural models are used. Based on the curved benchmark geometry, the influence of different material models, spatial discretizations, and meshes of varying refinement is investigated. It turns out that often-used standard displacement elements with linear shape functions are not sufficient to provide good approximations of the arterial wall stresses, whereas for standard displacement elements or F-bar formulations with quadratic shape functions, suitable results are obtained. For the time discretization, a second-order backward differentiation formula scheme is used. It is shown that the curved geometry enables the analysis of non-rotationally symmetric distributions of the mechanical fields. For instance, the maximal shear stresses in the fluid–structure interface are found to be higher in the inner curve that corresponds to clinical observations indicating a high plaque nucleation probability at such locations.@en

Monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes and Navier-Stokes type are presented. In order to obtain numerically scalable algorithms, coarse spaces obtained from the generalized Dryja-Smith-Widlund (GDSW) approach are used. Numerical results of our parallel implementation are presented for various incompressible fluid flow problems. In particular, cases are considered where the problem cannot or should not be reduced using local static condensation, e.g., Stokes or Navier-Stokes problems with continuous pressure spaces. In the new monolithic preconditioners, the local overlapping problems and the coarse problem are saddle point problems with the same structure as the original problem. Our parallel implementation of these preconditioners is based on the fast and robust overlapping Schwarz (FROSch) library, which is part of the Trilinos package ShyLU. The implementation is essentially algebraic in the sense that, for the class of problems presented here, the preconditioners can be constructed from the fully assembled stiffness matrix and information about the block structure of the problem. Further information about the geometry or the null space of the underlying problem can improve the performance compared to the default settings. Parallel scalability results for several thousand cores for Stokes and Navier-Stokes model problems are reported. Each of the local problems is solved using a direct solver in serial mode, whereas the coarse problem is solved using a direct solver in serial or message passing interface (MPI)-parallel mode or using an MPI-parallel iterative Krylov solver.@en

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, the condition number only depends on a user-prescribed tolerance. The coarse space is based on discrete harmonic extensions of vertex, edge, and face interface functions, which are computed from the solutions of corresponding local generalized edge and face eigenvalue problems. The local eigenvalue problems are of the size of the edges and faces of the decomposition, and the eigenvalue problems can be constructed solely from the local subdomain stiffness matrices and the fully assembled global stiffness matrix. The new AGDSW (adaptive generalized Dryja-Smith-Widlund) coarse space always contains the classical GDSW coarse space by construction of the generalized eigenvalue problems. Numerical results supporting the theory are presented for several model problems in three dimensions using structured as well as unstructured meshes and unstructured decompositions.@en

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 parallel two-level Schwarz methods. We show parallel scalability for two- and three-dimensional scalar second-order elliptic and linear elasticity problems for several thousands of cores. We also discuss techniques for the parallel construction of coarse spaces which are also of interest for other parallel preconditioners and discretization methods using energy minimizing coarse functions. We finally show an application in monolithic fluid-structure interaction, where significant improvements are achieved compared to a standard algebraic, one-level overlapping Schwarz method.@en

Monolithic preconditioners for incompressible fluid flow problems can significantly improve the convergence speed compared with preconditioners based on incomplete block factorizations. However, the computational costs for the setup and the application of monolithic preconditioners are typically higher. In this article, several techniques are applied to monolithic two-level generalized Dryja-Smith-Widlund (GDSW) preconditioners to further improve the convergence speed and the computing time. In particular, reduced dimension GDSW coarse spaces, restricted and scaled versions of the first level, hybrid, and parallel coupling of the levels, and recycling strategies are investigated. Using a combination of all these improvements, for a small time-dependent Navier-Stokes problem on 240 message passing interface (MPI) ranks, a reduction of 86% of the time-to-solution can be obtained. Even without applying recycling strategies, the time-to-solution can be reduced by more than 50% for a larger steady Stokes problem on 4608 MPI ranks. For the largest problems with 11 979 MPI ranks, the scalability deteriorates drastically for the monolithic GDSW coarse space. On the other hand, using the reduced dimension coarse spaces, good scalability up to 11 979 MPI ranks, which corresponds to the largest problem configuration fitting on the employed supercomputer, could be achieved.@en