Balancing Domain Decomposition By Constraints Research Articles

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.

BDDC Algorithms for Oseen problems with HDG Discretizations

Abstract The balancing domain decomposition by constraints (BDDC) methods are applied to the linear system arising from the hybridizable discontinuous Galerkin (HDG) discretization of the Oseen equation of incompressible flow. The generalized minimal residual method (GMRES) is used to accelerate the convergence. The original system is first reduced to a subdomain interface problem that is asymmetric indefinite, but can be positive definite in a special subspace. Edge/face average constraints can ensure all BDDC-preconditioned GMRES iterates stay in this special subspace. The convergence of the algorithm is analyzed, and additional edge/face constraints are used to improve the convergence. If the subdomain size is small enough, the number of iterations is independent of the number of subdomains, and depends only slightly on the subdomain problem size when the viscosity is large. When the viscosity is small, the convergence deteriorates. However, the numerical examples can give satisfactory results for the linear HDG discretization even with a small viscosity.

Parallel Newton--Krylov BDDC and FETI-DP Deluxe Solvers for Implicit Time discretizations of the Cardiac Bidomain Equations

Two novel parallel Newton-Krylov Balancing Domain Decomposition by Constraints (BDDC) and Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) solvers are here constructed, analyzed and tested numerically for implicit time discretizations of the three-dimensional Bidomain system of equations. This model represents the most advanced mathematical description of the cardiac bioelectrical activity and it consists of a degenerate system of two non-linear reaction-diffusion partial differential equations (PDEs), coupled with a stiff system of ordinary differential equations (ODEs). A finite element discretization in space and a segregated implicit discretization in time, based on decoupling the PDEs from the ODEs, yields at each time step the solution of a non-linear algebraic system. The Jacobian linear system at each Newton iteration is solved by a Krylov method, accelerated by BDDC or FETI-DP preconditioners, both augmented with the recently introduced {\em deluxe} scaling of the dual variables. A polylogarithmic convergence rate bound is proven for the resulting parallel Bidomain solvers. Extensive numerical experiments on linux clusters up to two thousands processors confirm the theoretical estimates, showing that the proposed parallel solvers are scalable and quasi-optimal.

Robust and scalable adaptive BDDC preconditioners for virtual element discretizations of elliptic partial differential equations in mixed form

The Virtual Element Method (VEM) is a recent numerical technology for the solution of partial differential equations on computational grids constituted by polygonal or polyhedral elements of very general shape. The aim of this work is to develop effective linear solvers for a general order VEM approximation designed to approximate three-dimensional scalar elliptic equations in mixed form. The proposed Balancing Domain Decomposition by Constraints (BDDC) preconditioner allows to use conjugate gradient iterations, albeit the algebraic linear systems arising from the discretization of the problem are indefinite, ill-conditioned, and of saddle point nature. The condition number of the resulting positive definite preconditioned system is adaptively controlled by means of deluxe scaling operators and suitable local generalized eigenvalue problems for the selection of optimal primal constraints. Numerical results confirm the theoretical estimates and the reliability of the adaptive procedure, with the experimental condition numbers always very close to the prescribed adaptive tolerance parameter. The scalability and quasi-optimality of the preconditioner are demonstrated, and the performances of the proposed solver are compared with state-of-the-art parallel direct solvers and block preconditioning techniques in a distributed memory setting.

Robust BDDC algorithms for finite volume element methods

The balancing domain decomposition by constraints (BDDC) method is applied to the linear system arising from the finite volume element method (FVEM) discretization of a scalar elliptic equation. The FVEMs share nice features of both finite element and finite volume methods and are flexible for complicated geometries with good conservation properties. However, the resulting linear system usually is asymmetric. The generalized minimal residual (GMRES) method is used to accelerate convergence. The proposed BDDC methods allow for jumps of the coefficient across subdomain interfaces. When jumps of the coefficient appear inside subdomains, the BDDC algorithms adaptively choose the primal variables deriving from the eigenvectors of some local generalized eigenvalue problems. The adaptive BDDC algorithms with advanced deluxe scaling can ensure good performance with highly discontinuous coefficients. A convergence analysis of the BDDC method with a preconditioned GMRES iteration is provided, and several numerical experiments confirm the theoretical estimate.

Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization

Abstract We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 {L_{2}} -norm regularization is replaced by the H - 1 {H^{-1}} -norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximation u ϱ ⁢ h {u_{\varrho h}} to the state u and the desired state u ¯ {\overline{u}} in terms of the mesh-size h and the regularization parameter ϱ. The choice ϱ = h 2 {\varrho=h^{2}} ensures optimal convergence the rate of which only depends on the regularity of the target function u ¯ {\overline{u}} . The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to h, ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

Fully-coupled micro\u2013macro finite element simulations of the Nakajima test using parallel computational homogenization

The Nakajima test is a well-known material test from the steel and metal industry to determine the forming limit of sheet metal. It is demonstrated how FE2TI, our highly parallel scalable implementation of the computational homogenization method FE^2, can be used for the simulation of the Nakajima test. In this test, a sample sheet geometry is clamped between a blank holder and a die. Then, a hemispherical punch is driven into the specimen until material failure occurs. For the simulation of the Nakajima test, our software package FE2TI has been enhanced with a frictionless contact formulation on the macroscopic level using the penalty method. The appropriate choice of suitable boundary conditions as well as the influence of symmetry assumptions regarding the symmetric test setup are discussed. In order to be able to solve larger macroscopic problems more efficiently, the balancing domain decomposition by constraints (BDDC) approach has been implemented on the macroscopic level as an alternative to a sparse direct solver. To improve the computational efficiency of FE2TI even further, additionally, an adaptive load step approach has been implemented and different extrapolation strategies are compared. Both strategies yield a significant reduction of the overall computing time. Furthermore, a strategy to dynamically increase the penalty parameter is presented which allows to resolve the contact conditions more accurately without increasing the overall computing time too much. Numerically computed forming limit diagrams based on virtual Nakajima tests are presented.

Learning Adaptive Coarse Spaces of BDDC Algorithms for Stochastic Elliptic Problems with Oscillatory and High Contrast Coefficients

In this paper, we consider the balancing domain decomposition by constraints (BDDC) algorithm with adaptive coarse spaces for a class of stochastic elliptic problems. The key ingredient in the construction of the coarse space is the solutions of local spectral problems, which depend on the coefficient of the PDE. This poses a significant challenge for stochastic coefficients as it is computationally expensive to solve the local spectral problems for every realization of the coefficient. To tackle this computational burden, we propose a machine learning approach. Our method is based on the use of a deep neural network (DNN) to approximate the relation between the stochastic coefficients and the coarse spaces. For the input of the DNN, we apply the Karhunen–Loève expansion and use the first few dominant terms in the expansion. The output of the DNN is the resulting coarse space, which is then applied with the standard adaptive BDDC algorithm. We will present some numerical results with oscillatory and high contrast coefficients to show the efficiency and robustness of the proposed scheme.

Block FETI–DP/BDDC preconditioners for mixed isogeometric discretizations of three-dimensional almost incompressible elasticity

A block FETI–DP/BDDC preconditioner for mixed formulations of almost incompressible elasticity is constructed and analyzed; FETI–DP (Finite Element Tearing and Interconnecting Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints) are two very successful domain decomposition algorithms for a variety of elliptic problems. The saddle point problems of the mixed problems are discretized with mixed isogeometric analysis with continuous pressure fields. As in previous work by Tu and Li (2015) for finite element discretizations of the incompressible Stokes system, the proposed preconditioner is applied to a reduced positive definite system involving only the pressure interface variable and the Lagrange multiplier of the FETI–DP algorithm. In this work, we extend the theory to a wider class of saddle point problems and we propose a novel block-preconditioning strategy, which consists in using BDDC with deluxe scaling for the interface pressure block as well as deluxe scaling for the FETI–DP preconditioner for the Lagrange multiplier block. A convergence rate analysis is presented with a condition number bound for the preconditioned operator which depends on the inf-sup parameter of the fully assembled problem and the condition number of a closely related BDDC algorithm for compressible elasticity. This bound is scalable in the number of subdomains, poly-logarithmic in the ratio of subdomain and element sizes, and robust with respect to material incompressibility. Parallel numerical experiments validate the theory, demonstrate robustness in the presence of discontinuities of the Lamé parameters, and indicate how the rate of convergence varies with respect to the spline polynomial degree and regularity and the deformation of the domain.

An adaptive BDDC method enhanced with prior selected primal constraints

An adaptive BDDC (Balancing Domain Decomposition by Constraints) method is considered for three dimensional elliptic problems with coefficients of random variation and high contrast. For such model problems, a certain generalized eigenvalue problem on each subdomain interface is formed to select primal constraints adaptively. In three dimensions, eigenvalue problems are formed on each face or edge nodal equivalence classes. For the case of edges, proposed eigenvalue problems in previous studies are not satisfactory while those for faces perform very effectively. The eigenvalue problems can be enhanced by utilizing prior selected primal constraints. In this paper, this new idea is adopted when forming edge eigenvalue problems and the resulting adaptive BDDC preconditioner is analyzed. In numerical experiments, the new edge eigenvalue problem is also shown to provide a more optimal set of adaptive primal constraints compared to those in the previous studies.

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.

A frugal FETI-DP and BDDC coarse space for heterogeneous problems

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 262144 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.

English

We study a method based on Balancing Domain Decomposition by Constraints (BDDC) for a numerical solution of a single-phase flow in heterogenous porous media. The method solves for both flux and pressure variables. The fluxes are resolved in three steps: the coarse solve is followed by subdomain solves and last we look for a divergence-free flux correction and pressures using conjugate gradients with the BDDC preconditioner. Our main contribution is an application of the adaptive algorithm for selection of flux constraints. Performance of the method is illustrated on the benchmark problem from the 10th SPE Comparative Solution Project (SPE 10). Numerical experiments in both 2D and 3D demonstrate that the first two steps of the method exhibit some numerical upscaling properties, and the adaptive preconditioner in the last step allows a significant decrease in number of iterations of conjugate gradients at a small additional cost.

Local Fourier Analysis of Balancing Domain Decomposition By Constraints Algorithms

Local Fourier analysis is a commonly used tool for the analysis of multigrid and other multilevel algorithms, providing both insight into observed convergence rates and predictive analysis of the performance of many algorithms. In this paper, for the first time, we adapt local Fourier analysis to examine variants of two- and three-level balancing domain decomposition by constraints (BDDC) algorithms, to better understand the eigenvalue distributions and condition number bounds on these preconditioned operators. This adaptation is based on a modified choice of basis for the space of Fourier harmonics that greatly simplifies the application of local Fourier analysis in this setting. The local Fourier analysis is validated by considering the two-dimensional Laplacian and predicting the condition numbers of the preconditioned operators with different sizes of subdomains. Several variants are analyzed, showing that the two- and three-level performance of the “lumped” variant can be greatly improved when used in multiplicative combination with a weighted diagonal scaling preconditioner, with weight optimized through the use of local Fourier analysis.

Preconditioning the coarse problem of BDDC methods ‐ three-level, algebraic multigrid, and vertex-based preconditioners

A fair comparison of three Balancing Domain Decomposition by Constraints (BDDC) methods with an approximate coarse space solver is attempted for the first time. The comparison is made for a BDDC method with an algebraic multigrid preconditioner for the coarse problem, a three-level BDDC method, and a BDDC method with a vertex-based coarse preconditioner which was recently introduced by Clark Dohrmann, Kendall Pierson, and Olof Widlund. For the first time, all methods are presented and discussed in a common framework. Condition number bounds are provided for all approaches. All methods are implemented in a common highly parallel scalable BDDC software package based on PETSc, to allow for a fair comparison. Numerical results showing the parallel scalability are presented for the equations of linear elasticity. For the first time, this includes parallel scalability tests for the vertex-based approximate BDDC method.

Isogeometric BDDC deluxe preconditioners for linear elasticity

Balancing Domain Decomposition by Constraints (BDDC) preconditioners have been shown to provide rapidly convergent preconditioned conjugate gradient methods for solving many of the very ill-conditioned systems of algebraic equations which often arise in finite element approximations of a large variety of problems in continuum mechanics. These algorithms have also been developed successfully for problems arising in isogeometric analysis. In particular, the BDDC deluxe version has proven very successful for problems approximated by Non-Uniform Rational B-Splines (NURBS), even those of high order and regularity. One main purpose of this paper is to extend the theory, previously fully developed only for scalar elliptic problems in the plane, to problems of linear elasticity in three dimensions. Numerical experiments supporting the theory are also reported. Some of these experiments highlight the fact that the development of the theory can help to decrease substantially the dimension of the primal space of the BDDC algorithm, which provides the necessary global component of these preconditioners, while maintaining scalability and good convergence rates.

A BDDC algorithm for the Stokes problem with weak Galerkin discretizations

The BDDC (balancing domain decomposition by constraints) methods have been applied successfully to solve the large sparse linear algebraic systems arising from conforming finite element discretizations of second order elliptic and Stokes problems. In this paper, the Stokes equations are discretized using the weak Galerkin method, a newly developed nonconforming finite element method. A BDDC algorithm is designed to solve the linear system such obtained. Edge/face velocity interface average and mean subdomain pressure are selected for the coarse problem. The condition number bounds of the BDDC preconditioned operator are analyzed, and the same rate of convergence is obtained as for conforming finite element methods. Numerical experiments are conducted to verify the theoretical results.

Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems

In FETI-DP (Finite Element Tearing and Interconnecting) and BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods, the transformation-of-basis approach is used to improve the convergence by combining the local assembly with a change of basis. Suitable basis vectors can be constructed by the recently introduced adaptive coarse space approaches. The resulting FETI-DP and BDDC methods fulfill a condition number bound independent of heterogeneities in the problem. The adaptive method with a transformation of basis presented here builds on a recently introduced adaptive FETI-DP approach for elliptic problems in three dimensions and uses a coarse space constructed from solving small, local eigenvalue problems on closed faces and on a small number of edges. In contrast to our earlier work on adaptive FETI-DP, the coarse space correction is not implemented by using balancing (or deflation), which requires the use of an exact coarse space solver, but by using local transformations. This will make it simpler to extend the method to a large number of subdomains and large supercomputers. The recently established theory of a generalized transformation-of-basis approach yields a condition number estimate for the preconditioned operator that is independent of jumps of the coefficients across and inside subdomains when using the local adaptive constraints. It is shown that all results are also valid for BDDC. Numerical results are presented in three dimensions for FETI-DP and BDDC. We also provide a comparison of different scalings, i.e., deluxe, rho, stiffness, and multiplicity for our adaptive coarse space in 3D.

Nonlinear BDDC Methods with approximate solvers

New nonlinear BDDC (Balancing Domain Decomposition by Constraints) domain decomposition methods using inexact solvers for the subdomains and the coarse problem are proposed. In nonlinear domain decomposition methods, the nonlinear problem is decomposed before linearization to improve concurrency and robustness. For linear problems, the new methods are equivalent to known inexact BDDC methods. The new approaches are therefore discussed in the context of other known inexact BDDC methods for linear problems. Relations are pointed out, and the advantages of the approaches chosen here are highlighted. For the new approaches, using an algebraic multigrid method as a building block, parallel scalability is shown for more than half a million (524288) MPI ranks on the JUQUEEN IBM BG/Q supercomputer (JSC Jülich, Germany) and on up to 193600 cores of the Theta Xeon Phi supercomputer (ALCF, Argonne National Laboratory, USA), which is based on the recent Intel Knights Landing (KNL) many-core architecture. One of our nonlinear inexact BDDC domain decomposition methods is also applied to three-dimensional plasticity problems. Comparisons to standard Newton-Krylov-BDDC methods are provided.

BDDC and FETI-DP algorithms with a change of basis formulation on adaptive primal constraints

BDDC (Balancing Domain Decomposition by Constraints) and FETI-DP (Dual-Primal Finite Element Tearing and Interconnecting) algorithms with adaptively enriched primal constraints are considered. The coarse component of the two algorithms is built on the set of primal unknowns consisting of those at subdomain vertices and those from the adaptive primal constraints after a change of basis. For the FETI-DP algorithm, a more general form of a preconditioner is proposed to extend the algorithm to the set of primal unknowns including those from the adaptive primal constraints. In addition, it can be shown that the two algorithms share the same spectra except those equal to one or zero when the same set of adaptive primal constraints are employed. Numerical results are included for both two and three dimensional model problems.