Efficient Adaptive Elimination Strategies in Nonlinear FETI-DP Methods in Combination with Adaptive Spectral Coarse Spaces

Nonlinear FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) is a nonlinear nonoverlapping domain decomposition method (DDM) which has a superior nonlinear convergence behavior compared with classical Newton-Krylov-DDMs - at least for many problems. Its fast and robust nonlinear convergence is strongly influenced by the choice of the second level or, in other words, the choice of the coarse constraints. Additionally, the convergence is also affected by the choice of an elimination set, that is, a set of degrees of freedom which are eliminated nonlinearly before linearization. In this article, an adaptive coarse space is combined with a problem-dependent and residual-based choice of the elimination set. An efficient implementation exploiting sparse local saddle point problems instead of an explicit transformation of basis is used. Unfortunately, this approach makes a further adaption of the elimination sets necessary, that is, edges and faces with coarse constraints have to be either included in the elimination set completely or not at all. Different strategies to fulfill this additional constraint are discussed and compared with a solely residual-based approach. The latter approach has to be implemented with an explicit transformation of basis. In general, the residual which is used to choose the elimination set has to be transformed to a space which basis functions explicitly contain the coarse constraints. This is computationally expensive. Here, for the first time, it is suggested to use an approximation of the transformed residual instead to compute the elimination set

Monolithic Overlapping Schwarz Domain Decomposition Methods with GDSW Coarse Spaces for Saddle Point Problems

Monolithic overlapping Schwarz preconditioners for saddle point problems of Stokes, Navier-Stokes, and mixed linear elasticity ty e are presented. For the first time, coarse spaces obtained from the GDSW (Generalized Dryja-Smith-Widlund) approach are used in such a setting. Numerical results of our parallel implementation are presented for several model problems. In particular, cases are considered where the problem cannot or should not b e reduced using local static condensation, e.g., Stokes, Navier-Stokes or mixed elasticity 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 FROSch (Fast and Robust Overlapping Schwarz) library, which is part of the Trilinos package ShyLU. The implementation is algebraic in the sense that the preconditioners can be constructed from the fully assembled stiffness matrix and information about the block structure of the problem. Parallel scalability results for several thousand cores for Stokes, Navier-Stokes, and mixed linear elasticity model problems are reported. Each of the local problems is solved using a direct solver in serial mo de, whereas the coarse problem is solved using a direct solver in serial or MPI-parallel mode or using an MPI-parallel iterative Krylov solve

Surrogate Convolutional Neural Network Models for Steady Computational Fluid Dynamics Simulations

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

Adaptive Nonlinear Elimination in Nonlinear FETI-DP Methods

Highly scalable and robust Newton-Krylov domain decomposition approaches are widely used for the solution of nonlinear implicit problems. In these methods, the nonlinear problem is first linearized and then decomposed into subdomains. By changing this order and first decomposing the nonlinear problem, many nonlinear domain decomposition methods have been designed in the last two decades. These methods often show a higher robustness compared to classical Newton-Krylov variants due to a better resolution of nonlinear effects. Additionally, the balance between local work, communication, and synchronization is usually more favorable for modern computer architectures. In all our nonlinear FETI-DP methods, we introduce a nonlinear right-preconditioner that can be interpreted as a (partial) nonlinear elimination of variables. The choice of the elimination set has a huge impact on the nonlinear convergence behavior. In order to design a nonlinear FETI-DP method that is tailored to arbitrary problems, we introduce a strategy, based on the residual of the nonlinear saddle point system, to adaptively choose sets of variables for the nonlinear elimination. The new strategy is applied to challenging distributions of nonlinearity in problems based on the p-Laplace operator. Promising numerical results are presented

Three-level BDDC for Virtual Elements

The Virtual Element Method (VEM) is a discretization procedure for the solution of partial differential equations that allows for the use of nearly arbitrary polygonal/polyhedral grids. For the parallel scalable and iterative solution of large scale VE problems, the FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods have recently been applied. As for the case of finite element discretizations, a large global coarse problem that usually arises in large scale applications is a parallel scaling bottleneck of FETI-DP and BDDC. Nonetheless, the coarse problem/second level is usually necessary for the numerical robustness of the method. To alleviate this difficulty and to retain the scalability, the three-level BDDC method is applied to virtual element discretizations in this article. In this approach, to allow for a parallel solution of the coarse problem, the solution of it is only approximated by applying BDDC recursively, which automatically introduces a third level. Numerical results using several different configurations of the three-level approach and different polygonal meshes are presented and additionally compared with the classical two-level BDDC approach

Adaptive Three-level BDDC Using Frugal Constraints

The highly parallel scalable three-level BDDC (Balancing Domain Decomposition by Constraints) method is a very successful approach to overcome the scaling bottleneck of directly solving a large coarse problem in classical two-level BDDC. As long as the problem is homogeneous on each subregion, three-level BDDC is also provably robust in many cases. For problems with complex microstructures, as, e.g., stationary diffusion problems with jumps in the diffusion coefficient function, in two-level BDDC methods, advanced adaptive or frugal coarse spaces have to be considered to obtain a robust preconditioner. Unfortunately, these approaches result in even larger coarse problems on the second level and, additionally, computing adaptive coarse constraints is computationally expensive. Therefore, in this article, the three-level approach is combined with a provably robust adaptive coarse space and the computationally cheaper frugal coarse space. Both coarse spaces are used on the second as well as the third level. All different possible combinations are investigated numerically for stationary diffusion problems with highly varying coefficient functions

A short note on solving partial differential equations using convolutional neural networks

The approach of using physics-based machine learning to solve PDEs has recently become very popular. A recent approach to solve PDEs based on CNNs uses finite difference stencils to include the residual of the partial differential equation into the loss function. In this work, the relation between the network training and the solution of a respective finite difference linear system of equations using classical numerical solvers is discussed. It turns out that many beneficial properties of the linear equation system are neglected in the network training. Finally, numerical results which underline the benefits of classical numerical solvers are presented

Learning Adaptive FETI-DP Constraints for Irregular Domain Decompositions

Adaptive coarse spaces yield a robust convergence behavior for FETI-DP (Finite Element Tearing and Interconnecting - Dual Primal) and BDDC (Balancing Domain Decomposition by Constraints) methods for highly heterogeneous problems. However, the usage of such adaptive coarse spaces can be computationally expensive since, in general, it requires the setup and the solution of a relatively high amount of local eigenvalue problems on parts of the domain decomposition interface. In earlier works, see, e.g., [2], it has been shown that it is possible to train a neural network to make an automatic decision which of the eigenvalue problems in an adaptive FETI-DP method are actually necessary for robustness with a satisfactory accuracy. Moreover, these results have been extended in [6] by directly learning an approximation of the adaptive edge constraints themselves for regular, two-dimensional domain decompositions. In particular, this does not require the setup or the solution of any eigenvalue problems at all since the FETI-DP coarse space is, in this case, exclusively enhanced by the learned constraints obtained from the regression neural networks trained in an offline phase. Here, in contrast to [6], a regression neural network is trained with both, training data resulting from straight and irregular edges. Thus, it is possible to use the trained networks also for the approximation of adaptive constraints for irregular domain decompositions. Numerical results for a heterogeneous two-dimensional stationary diffusion problem are presented using both, a decomposition into regular and irregular subdomains

Parallel adaptive FETI-DP using lightweight asynchronous dynamic load balancing

A parallel FETI-DP domain decomposition method using an adaptive coarse space is presented. The implementation builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses small, local eigenvalue problems for faces and, additionally, for a small number of edges. The condition number of the preconditioned operator then satisfies a bound which is independent of coefficient heterogeneities in the problem. The computational cost of the local eigenvalue problems is not negligible, and also a significant load imbalance can be introduced. As a remedy, certain eigenvalue problems are discarded by a theory-guided heuristic strategy, based on the diagonal entries of the stiffness matrices. Additionally, a lightweight pairwise dynamic load balancing strategy is implemented for the eigenvalue problems. The load balancing is supervised by an orchestrating rank using asynchronous point-to-point communication. The resulting method shows good weak and strong scalability up to thousands of cores while fast convergence is obtained even for heterogeneous problems