Coarse Problem Research Articles

A family of Nonoverlapping Spectral Additive Schwarz methods (NOSAS) and their economic versions

NOSAS (Nonoverlapping Spectral Additive Schwarz) methods are nonoverlapping iterative domain decomposition methods for efficiently solving large sparse linear systems arising from elliptic problems with heterogeneous coefficients. NOSAS methods are of additive Schwarz types with local Dirichlet solver on each nonoverlapping subdomain Ωi, plus a coarse problem on the interface of the subdomains. The coarse problem has global and local interaction components which are associated respectively to low-frequency and high-frequency modes obtained locally from a generalized eigenvalue problem defined on each subdomain’s interface. In this paper, we propose several variants of the NOSAS methods by selecting different generalized eigenvalue problems and different coarse space extensions. Additionally, we propose an economic variant to reduce the complexity of these generalized eigenvalue problems. The analysis of all the variants is established, and numerical tests are provided.

Einstein material balance and modeling of the flow of compressible fluid near the boundary

We consider sewing machinery between nite difference and analytical solutions de ned at different scales: far away and near the source of the perturbation of the flow. One of the essences of the approach is that the coarse problem and the boundary-value problem in the proxy of the source model two different flows. In his remarkable paper, Peaceman proposes a framework for dealing with solutions de ned on different scales for linear time independent problems by introducing the famous Peaceman well block radius. In this article, we consider a novel problem: how to solve this issue for transient flow generated by the compressibility of the fluid. We are proposing a method to glue solution via total fluxes, which are prede ned on coarse grid, and changes in pressure, due to compressibility, in the block containing production (injection) well. It is important to mention that the coarse solution “does not see” the boundary. From an industrial point of view, our report provides a mathematical tool for the analytical interpretation of simulated data for compressible fluid flow around a well in a porous medium. It can be considered a mathematical “shirt” on the Peaceman well-block radius formula for linear (Darcy) transient flow but can be applied in much more general scenarios. In the article, we use the Einstein approach to derive the material balance equation, a key instrument to de ne R0. We will expand the Einstein approach for three regimes of Darcy and non-Darcy flows for compressible fluids (time-dependent): 1. stationary; 2. pseudostationary; 3. boundary dominated. Note that in all known authors literature, the rate of production on the well is time-independent.

Non-overlapping domain decomposition algorithms with only primal velocity unknowns for the discontinuous viscosity Stokes problem

In this paper, non-overlapping domain decomposition algorithms for the Stokes problem with a discontinuous viscosity coefficient are proposed and analyzed. A pair of inf–sup stable finite element spaces with discontinuous pressure finite element functions is employed to obtain a discrete Stokes system. By introducing a non-overlapping subdomain partition for the given finite element spaces, the discrete Stokes system is reformulated into an algebraic system on dual unknowns, where the continuity on the velocity unknowns at the subdomain vertices is enforced strongly and the continuity on the remaining interface velocity unknowns is enforced weakly using the dual unknowns. The resulting algebraic system on the dual unknowns is then solved by an iterative method. The coarse problem in the resulting algebraic system is only related to the strongly coupled primal velocity unknowns and it is thus obtained as a symmetric positive definite matrix equation. To accelerate the iteration convergence, practical versions of lumped and Dirichlet preconditioners are proposed and their condition numbers are also analyzed for the resulting dual algebraic system. Numerical results are also included to confirm our condition number estimates.

A domain decomposition strategy for mCRE-based model updating in dynamics

In structural dynamics, the modified constitutive relation error is a widely used functional for model updating. The method involves a two-step iterative process to minimize the functional: first, regions with high modeling errors are localized, then their associated parameters are updated. The localization step requires solving large linear systems at each iteration, which makes the method less suitable for industrial models. In this work, a domain decomposition formulation is proposed. It makes the method more flexible, as subdomains can use different models which are independently modified during the updating algorithm without any additional cost. Moreover, this approach is very efficient numerically as it benefits from the same advantages as domain decomposition methods. In its standard form, it lacks a coarse problem which seems problematic; this is however addressed by using a multi-preconditioned solver such as MPGMRES. The performance of the proposed approach is illustrated by means of academic and industrial examples, with practical implementation in a commercial software.

A comparative study of scalable multilevel preconditioners for cardiac mechanics

In this work, we provide a performance comparison between the Balancing Domain Decomposition by Constraints (BDDC) and the Algebraic Multigrid (AMG) preconditioners for cardiac mechanics on both structured and unstructured finite element meshes. The mechanical behavior of myocardium can be described by the equations of three-dimensional finite elasticity, which are discretized by finite elements in space and yield the solution of a large scale nonlinear algebraic system. This problem is solved by a Newton-Krylov method, where the solution of the Jacobian linear system is accelerated by BDDC/AMG preconditioners. We thoroughly explore the main parameters of the BDDC preconditioner in order to make the comparison fair. We focus on: the performance of different direct solvers for the local and coarse problems of the BDDC algorithm; the impact of the different choices of BDDC primal degrees of freedom; and the influence of the finite element degree. Scalability tests are performed on Linux clusters up to 1024 processors, and we conclude with a performance study on a realistic electromechanical simulation.

A Multilevel Extension of the GDSW Overlapping Schwarz Preconditioner in Two Dimensions

Abstract Multilevel extensions of overlapping Schwarz domain decomposition preconditioners of Generalized Dryja–Smith–Widlund (GDSW) type are considered in this paper. The original GDSW preconditioner is a two-level overlapping Schwarz domain decomposition preconditioner, which can be constructed algebraically from the fully assembled stiffness matrix. The FROSch software, which belongs to the ShyLU package of the Trilinos software library, provides parallel implementations of different variants of GDSW preconditioners. The coarse problem can limit the parallel scalability of two-level GDSW preconditioners. As a remedy, in the past, three-level GDSW approaches have been proposed, which can significantly extend the range of scalability. Here, a multilevel extension of the GDSW preconditioner is introduced and analyzed. Finally, parallel results for the implementation in FROSch for up to 40 000 cores of the SuperMUC-NG supercomputer at Leibniz Supercomputing Centre (LRZ) and to 48 000 cores of the JUWELS supercomputer at Jülich Supercomputing Centre (JSC) are presented.

Two-level additive Schwarz methods for three-dimensional unsteady Stokes flows in patient-specific arteries with parameterized one-dimensional central-line coarse preconditioner

We consider the numerical solution of unsteady Stokes equations in patient-specific arterial-like domains in 3D. A Stokes-like solver is a necessary component in a more sophisticated nonlinear Navier-Stokes method, for which several multilevel domain decomposition methods have been introduced recently. Because of the complex geometry, the construction and the solve of the coarse problem usually take a large percentage of the total compute time. In this paper, we introduce a parameterized one-dimensional Stokes solver defined along the centerline of the artery and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable 3D-to-1D restriction and 1D-to-3D extension operators on fully unstructured meshes, a two-level additive Schwarz preconditioner can be constructed. Some numerical experiments for flows in realistic arteries are presented to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than the existing three-dimensional coarse preconditioners.

Scalable mesh partitioning for multibody-3D finite element based rotary-wing structures

This paper presents a new and specialized mesh partitioner for large-scale three-dimensional multibody rotor dynamic models. This partitioner enables parallel solution with modern domain decomposition algorithms of complex, multibody, three-dimensional finite element problems. A parallel solver using a state-of-the-art iterative substructuring algorithm, FETI-DP, is developed in this paper to solve the partitioned data structures. The main feature of the partitioner is the ability to robustly partition any generic multibody structure, although it has several special features for rotary-wing structures. The NASA Tilt Rotor Aeroacoustic Model (TRAM), a 1/4 scale V-22 model, was specially released by NASA as a challenge test case. This model contains four flexible parts, six joints, 18 composite material decks, a blade fluid–structure interface, and control angle inputs. The parallel solver scalability is studied for progressively increasing complexity, from the isolated rotor blade to the blade and hub assembly. The use of a skyline solver for the coarse problem is shown to eliminate the coarse problem barrier. The special partitioner features are demonstrated to enable efficient parallel solution and significantly improve the performance and scalability. The principle barrier of computational time that prevented the use of high-fidelity three-dimensional structures in rotorcraft is thus resolved.

Parallel Scalability of Three-Level FROSch Preconditioners to 220000 Cores using the Theta Supercomputer

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 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 (Argonne Leadership Computing Facility, Argonne, IL) using up to 220 000 cores are discussed and compared to results obtained on the SuperMUC-NG supercomputer (Leibniz Supercomputing Centre, 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, which has a dragonfly interconnect, for 100 000 message passing interface (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, which uses a fat tree network. Using hybrid MPI/OpenMP parallelization, the onset of the MPI communication problem on Theta can be delayed. Further analysis of the performance of FROSch on large supercomputers with dragonfly interconnects will be necessary.

Inexact and primal multilevel FETI‐DP methods: a multilevel extension and interplay with BDDC

We study a framework that allows to solve the coarse problem in the FETI-DP method approximately. It is based on the saddle-point formulation of the FETI-DP system with a block-triangular preconditioner. One of the blocks approximates the coarse problem, for which we use the multilevel BDDC method as the main tool. This strategy then naturally leads to a version of multilevel FETI-DP method, and we show that the spectra of the multilevel FETI-DP and BDDC preconditioned operators are essentially the same. The theory is illustrated by a set of numerical experiments, and we also present a few experiments when the coarse solve is approximated by algebraic multigrid.

An aggregation-based nonlinear multigrid solver for two-phase flow and transport in porous media

A nonlinear multigrid solver for two-phase flow and transport in a mixed fractional-flow velocity-pressure-saturation formulation is proposed. The solver, which is under the framework of the full approximation scheme (FAS), extends our previous work on nonlinear multigrid for heterogeneous diffusion problems. The coarse spaces in the multigrid hierarchy are constructed by first aggregating degrees of freedom, and then solving some local flow problems. The mixed formulation and the choice of coarse spaces allow us to assemble the coarse problems without visiting finer levels during the solving phase, which is crucial for the scalability of multigrid methods. Specifically, a natural generalization of the upwind flux can be evaluated directly on coarse levels using the precomputed coarse flux basis vectors. The resulting solver is applicable to problems discretized on general unstructured grids. The performance of the proposed nonlinear multigrid solver in comparison with the standard single level Newton's method is demonstrated through challenging numerical examples. It is observed that the proposed solver is robust for highly nonlinear problems and clearly outperforms Newton's method in the case of high Courant-Friedrichs-Lewy (CFL) numbers.

Multilevel K-Means Density Based Flow Clustering Algorithm for Data Streams

The Data stream clustering is an active area of research that has recently emerged with the goal of discovering new knowledge from a large amount and variability of constantly generated data. In this context, different-different algorithm for unsupervised learning that clusters multiple data streams has been proposed by many researchers. There is a need for a more efficient and efficient data analysis method. This paper introduces a multi-level K-Means density-based flow clustering algorithm (MKDCSTREAM) for clustering problems. This approach proposes to view the problem of clustering is a optimization process hierarchy that follow different levels, from unrefined to subtle. In the clustering problem, for the solution first divide the problem in parts, by following different levels to make the first clustering a coarser problem than calculated. Coarse problem clustering is mapped level by level and improves the Clustering the original problem by improving intermediate clustering using the general K-means algorithm. Compare the performance of the hierarchical approach with its single-tier approach using tests with a set of datasets collected from different areas.

A parallel non-nested two-level domain decomposition method for simulating blood flows in cerebral artery of stroke patient.

Numerical simulation of blood flows in patient-specific arteries can be useful for the understanding of vascular diseases, as well as for surgery planning. In this paper, we simulate blood flows in the full cerebral artery of stroke patients. To accurately resolve the flow in this rather complex geometry with stenosis is challenging and it is also important to obtain the results in a short amount of computing time so that the simulation can be used in pre- and/or post-surgery planning. For this purpose, we introduce a highly scalable, parallel non-nested two-level domain decomposition method for the three-dimensional unsteady incompressible Navier-Stokes equations with an impedance outlet boundary condition. The problem is discretized with a stabilized finite element method on unstructured meshes in space and a fully implicit method in time, and the large nonlinear systems are solved by a preconditioned parallel Newton-Krylov method with a two-level Schwarz method. The key component of the method is a non-nested coarse problem solved using a subset of processor cores and its solution is interpolated to the fine space using radial basis functions. To validate and verify the proposed algorithm and its highly parallel implementation, we consider a case with available clinical data and show that the computed result matches with the measured data. Further numerical experiments indicate that the proposed method works well for realistic geometry and parameters of a full size cerebral artery of an adult stroke patient on a supercomputers with thousands of processor cores.

A central-line coarse preconditioner for Stokes flows in artery-like domains

We consider numerical simulation of blood flows in the artery using multilevel domain decomposition methods. Because of the complex geometry, the construction and the solve of the coarse problem take a large percentage of the total compute time in the multilevel method. In this paper, we introduce a one-dimensional central-line model of the blood flow and use its stabilized finite element discretization to construct a coarse preconditioner. With suitable restriction and extension operators, we obtain a two-level additive Schwarz preconditioner for two- and three-dimensional problems. We present some numerical experiments with different arteries to show the efficiency and robustness of the new coarse preconditioner whose computational cost is considerably lower than other coarse preconditioners constructed using the two- or three-dimensional geometry of the artery.

Adaptive-multilevel BDDC algorithm for three-dimensional plane wave Helmholtz systems

In this paper, we are concerned with the weighted plane wave least-squares (PWLS) method for three-dimensional Helmholtz equations, and develop the multi-level adaptive BDDC algorithms for solving the resulting discrete system. In order to form the adaptive coarse components, the local generalized eigenvalue problems for each common face and each common edge are carefully designed. The condition number of the two-level adaptive BDDC preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is dependent on the maximum number of faces and edges per subdomain and the number of subdomains sharing a common edge. The efficiency of these algorithms is illustrated on a benchmark problem. The numerical results show the robustness of our two-level adaptive BDDC algorithms with respect to the wave number, the number of subdomains and the mesh size, and illustrate that our multi-level adaptive BDDC algorithm can reduce the scale of the coarse problem and can be used to solve large wave number problems efficiently.

Comparison of selected FETI coarse space projector implementation strategies

This paper deals with scalability improvements of the FETI (Finite Element Tearing and Interconnecting) domain decomposition method solving elliptic PDEs. The main bottleneck of FETI is the solution of a coarse problem that is part of the projector onto the natural coarse space. This paper introduces and compares two strategies for the FETI coarse problem solution. The first one is a classical solution with either direct (factorization + forward/backward substitutions) or iterative solvers (conjugate gradient and deflated conjugate gradient methods). The second one is the assembly of an explicit inverse using a direct solver with the coarse problem solution realised by dense matrix-vector products. MPI subcommunicators are employed to increase arithmetic intensity and, crucially, to decrease the communication cost. PERMON library for quadratic programming implementing the Total FETI variant of FETI was used for the numerical experiments.

Analysis of BDDC algorithms for Stokes problems with hybridizable discontinuous Galerkin discretizations

The BDDC (balancing domain decomposition by constraints) methods have been applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization of the incompressible Stokes problem. In the BDDC algorithms, the coarse problem is composed by the edge/face constraints across the subdomain interface for each velocity component. As for the standard approaches of the BDDC algorithms for saddle point problems, these constraints ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a subspace where the preconditioned operator is positive definite. However, there are several popular choices of the local stabilization parameters used in the HDG discretizations. Different stabilization parameters change the properties of the resulting discretized operators, and some special observations and tools are needed in the analysis of the condition numbers of the BDDC preconditioned Stokes operators. In this paper, condition number estimates for different choices of stabilization parameters are provided. Numerical experiments confirm the theory.

Accelerating the convergence of AFETI partitioned analysis of heterogeneous structural dynamical systems

Variationally based algorithms for the partitioned solution of structural mechanics problems are presented. Two key features of the present algorithms are the judicious application of the d’Alembert-Lagrange principal equations and the use of dominant substructural deformation modes. The paper includes three developments:1. Variational derivation of AFETI parallel solution methods.2. One-level and two-level AFETI implicit transient analysis algorithms with coarse problem included in the projector and based on free floating rigid body modes.3. A new AFETI implicit transient solution algorithm derived by constraining the interface equilibrium equations with the floating and dominant deformational modes.In addition to variational derivations of solution algorithms, the present paper is strived to offer new physical and/or numerical insight as each of variational derivational steps is succinctly explained. Performance evaluations of the algorithms described herein are presented.

A two-level overlapping Schwarz method with energy-minimizing multiscale coarse basis functions

A two-level overlapping Schwarz method is developed for second order elliptic problems with highly oscillatory and high contrast coefficients, for which it is known that the standard coarse problem fails to give a robust preconditioner. In this paper, we develop energy minimizing multiscale finite element functions to form a more robust coarse problem. First, a local spectral problem is solved in each non-overlapping coarse subdomain, and dominant eigenfunctions are selected as auxiliary functions, which are crucial for the high contrast case. The required multiscale basis functions are then obtained by minimizing an energy subject to some orthogonality conditions with respect to the auxiliary functions. Due to an exponential decay property, the minimization problem is solved locally on oversampling subdomains, that are unions of a few coarse subdomains. The coarse basis functions are therefore local and can be computed efficiently. The resulting preconditioner is shown to be robust with respect to the contrast in the coefficients as well as the overlapping width in the subdomain partition. Numerical results are included to confirm the theory and show the performance.

Generalized multiscale multicontinuum model for fractured vuggy carbonate reservoirs

Simulating flow in a highly heterogeneous reservoir with multiscale characteristics is computationally demanding. To tackle this problem, we propose a numerical scheme based on the Generalized Multiscale Finite Element Method (GMsFEM) and a triple-continuum model. The aim is to obtain a more efficient numerical approach that can explicitly represent the interactions among different continua. To enhance the applicability of our proposed model, we combine the Discrete Fracture Model (DFM) to model the local effects of discrete fractures. In the proposed model, the GMsFEM, as an advanced model reduction technique, enables capturing the multiscale flow dynamics. This is accomplished by systematically generating an approximation space through solving a series of local snapshot and spectral problems. The resulting eigenfunctions can pass the local features to the global level when acting as basis functions in the global coarse problems. Our goal in this paper is to further improve the accuracy of flow simulation in complicated reservoirs especially for the case when multiple discrete fractures located in single coarse neighborhood. Several numerical experiments are conducted to confirm the success of our proposed method, and a rigorous convergence proof is also given.