Multispace and multilevel BDDC

Mandel, Jan; Sousedík, Bedřich; Dohrmann, Clark R.

doi:10.1007/s00607-008-0014-7

Cited by 66 publications

(111 citation statements)

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“…The subassembled nature of the coarse problem (24) naturally leads to a multilevel extension of the algorithm, where a subdomain at a finer level is considered an element of a coarser mesh; we note that these ideas were first presented in [55,56] for a three-level algorithm, and then generalized to an arbitrary number of levels in [36]. Here we extend the three-level BDDC algorithm for saddle point problems introduced in [57] to an arbitrary number of levels.…”

Section: Multilevel Extensionsmentioning

confidence: 99%

Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media

Zampini¹,

Tu²

2017

SIAM J. Sci. Comput.

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Abstract. Multilevel balancing domain decomposition by constraints (BDDC) deluxe algorithms are developed for the saddle point problems arising from mixed formulations of Darcy flow in porous media. In addition to the standard no-net-flux constraints on each face, adaptive primal constraints obtained from the solutions of local generalized eigenvalue problems are included to control the condition number. Special deluxe scaling and local generalized eigenvalue problems are designed in order to make sure that these additional primal variables lie in a benign subspace in which the preconditioned operator is positive definite. The current multilevel theory for BDDC methods for porous media flow is complemented with an efficient algorithm for the computation of the so-called malign part of the solution, which is needed to make sure the rest of the solution can be obtained using the conjugate gradient iterates lying in the benign subspace. We also propose a new technique, based on the Sherman-Morrison formula, that lets us preserve the complexity of the subdomain local solvers. Condition number estimates are provided under certain standard assumptions. Extensive numerical experiments confirm the theoretical estimates; additional numerical results prove the effectiveness of the method with higher order elements and high-contrast problems from real-world applications.Key words. adaptive coarse space, BDDC, Darcy flow, domain decomposition, PETSc AMS subject classifications. 65F08, 65N55, 65Y05, 68W10 DOI. 10.1137/16M10806531. Introduction. The purpose of this work is to construct and analyze a balancing domain decomposition by constraints (BDDC) method [11] for the finite element (FE) discretization of saddle point problems arising from the mixed formulation of Darcy flows in porous media. Most previous work on domain decomposition methods for saddle point problems is based on the benign subspace approach [15,38,39,40,42], where the original saddle point is reduced to a positive definite problem in the case of discontinuous pressure discretization spaces. Multilevel extensions of these ideas have been presented in [48,47,57]. For domain decomposition methods with continuous pressures, see [29,58,59] and the references therein. We note that our two-level results are equally valid for the FE tearing and interconnecting dual-primal (FETI-DP) method [17], due to the well-known duality between BDDC and FETI-DP [34].After the pioneering work [35], recent research on BDDC methods has focused on controlling the condition number of the preconditioned operators through the adaptive generation of coarse spaces [6,8,23,24,25,37,44,48,64]; these techniques lead to robust preconditioning techniques with tunable rates of convergence; see, in particular, [45]. The adaptive enrichment of the primal space is accomplished by means of solving

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Section: Multilevel Extensionsmentioning

confidence: 99%

Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media

Zampini¹,

Tu²

2017

SIAM J. Sci. Comput.

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“…(P2) Coarse and fine components can be computed in parallel, since the basis for the coarse space is constructed in such a way that it is orthogonal to the fine component space with respect to the inner product endowed by the system matrix [5]. (P3) Due to the fact that the coarse matrix has a similar structure as the original system matrix, a multilevel extension of the algorithm is possible [25,32]. Property (P1) is readily exploited in any BDDC implementation.…”

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“…With regard to (P3), a multilevel BDDC (MLBDDC) algorithm has been proposed in [25], where the coarse problem at the next BDDC level is approximated by its BDDC approximation. An implementation of the MLBDDC method that does not exploit (P2) can be found in [29].…”

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confidence: 99%

Multilevel Balancing Domain Decomposition at Extreme Scales

Badia¹,

Martı́n²,

Principe³

2016

SIAM J. Sci. Comput.

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Abstract. In this paper we present a fully-distributed, communicator-aware, recursive, and interlevel-overlapped message-passing implementation of the multilevel balancing domain decomposition by constraints (MLBDDC) preconditioner. The implementation highly relies on subcommunicators in order to achieve the desired effect of coarse-grain overlapping of computation and communication, and communication and communication among levels in the hierarchy (namely inter-level overlapping). Essentially, the main communicator is split into as many non-overlapping subsets of MPI tasks (i.e., MPI subcommunicators) as levels in the hierarchy. Provided that specialized resources (cores and memory) are devoted to each level, a careful re-scheduling and mapping of all the computations and communications in the algorithm lets a high degree of overlapping to be exploited among levels. All subroutines and associated data structures are expressed recursively, and therefore MLBDDC preconditioners with an arbitrary number of levels can be built while re-using significant and recurrent parts of the codes. This approach leads to excellent weak scalability results as soon as level-1 tasks can mask coarser-levels duties. We provide a model to indicate how to choose the number of levels and coarsening ratios between consecutive levels and determine qualitatively the scalability limits for a given choice. We have carried out a comprehensive weak scalability analysis of the proposed implementation for the 3D Laplacian and linear elasticity problems. Excellent weak scalability results have been obtained up to 458,752 IBM BG/Q cores and 1.8 million MPI tasks, being the first time that exact domain decomposition preconditioners (only based on sparse direct solvers) reach these scales.1. Introduction. The simulation of scientific and engineering problems governed by partial differential equations (PDEs) involves the solution of sparse linear systems. The time spent in an implicit simulation at the linear solver relative to the overall execution time grows with the size of the problem and the number of cores [22]. In order to satisfy the ever increasing demand of reality and complexity in the simulations, scientific computing must advance in the development of numerical algorithms and implementations that will efficiently exploit the largest amounts of computational resources, and a massively parallel linear solver is a key component in this process.The growth in computational power passes now through increasing the number of cores in a chip, instead of making cores faster. The next generation of supercomputers, able to reach 1 exaflop/s, is expected to reach billions of cores. Thus, the future of scientific computing will be strongly related to the ability to efficiently exploit these extreme core counts [1].Only numerical algorithms with all their components scalable will efficiently run on extreme scale supercomputers. On extreme core counts, it will be a must to reduce communication and synchronization among cores, and overlap communication ...

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“…In fact, the eigenvalues of the preconditioned systems in the two approaches are almost identical [44,41,15]. The BDDC method is particularly well suited for extreme scale simulations, since it allows for a very aggressive coarsening, the computations at different levels can be computed in parallel, the subdomain problems can be solved inexactly [19,42] by, e.g., one AMG cycle, and it can straightforwardly be extended to multiple levels [50,45]. All of these properties have been carefully exploited in the series of papers [6,7,8,9], where an extremely scalable implementation of these algorithms has been proposed, leading to excellent weak scalability on nearly half a million cores in its multilevel version (see also [29,30] for weak scalability at extreme scales of the FETI-DP method).…”

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Balancing Domain Decomposition by Constraints and Perturbation

Badia¹,

Nguyen²

2016

SIAM J. Numer. Anal.

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Abstract. In this paper, we formulate and analyze a perturbed formulation of the balancing domain decomposition by constraints (BDDC) method. We prove that the perturbed BDDC has the same polylogarithmic bound for the condition number as the standard formulation. Two types of properly scaled zero-order perturbations are considered: one uses a mass matrix, and the other uses a Robin-type boundary condition, i.e, a mass matrix on the interface. With perturbation, the wellposedness of the local Neumann problems and the global coarse problem is automatically guaranteed, and coarse degrees of freedom can be defined only for convergence purposes but not well-posedness. This allows a much simpler implementation as no complicated corner selection algorithm is needed. Minimal coarse spaces using only face or edge constraints can also be considered. They are very useful in extreme scale calculations where the coarse problem is usually the bottleneck that can jeopardize scalability. The perturbation also adds extra robustness as the perturbed formulation works even when the constraints fail to eliminate a small number of subdomain rigid body modes from the standard BDDC space. This is extremely important when solving problems on unstructured meshes partitioned by automatic graph partitioners since arbitrary disconnected subdomains are possible. Numerical results are provided to support the theoretical findings.Key words. BDDC, preconditioner, coarse space, parallel solver, scalability AMS subject classifications. 65N55, 65N22, 65F08 DOI. 10.1137/15M10456481. Introduction. The development of highly scalable linear solvers for the solution of large scale linear systems arising from the finite element (FE) discretization of second-order elliptic problems on distributed-memory machines is of great importance in many applications. In this work, we consider nonoverlapping domain decomposition (DD) methods [49], which take advantage of the partition of the FE mesh into submeshes to define effective preconditioners that can exploit large levels of concurrency. In particular, we focus on a variant of the balancing DD by constraints (BDDC) method. The BDDC method was first introduced in 2003 by Dohrmann [18]. It can be regarded as an improved version of the balancing domain decomposition (BDD) method by Mandel [43]. It also has a very close connection with the dual primal finite element tearing and interconnecting (FETI-DP) method [25,24]. In fact, the eigenvalues of the preconditioned systems in the two approaches are almost identical [44,41,15]. The BDDC method is particularly well suited for extreme scale simulations, since it allows for a very aggressive coarsening, the computations at different levels can be computed in parallel, the subdomain problems can be solved inexactly

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Multispace and multilevel BDDC

Cited by 66 publications

References 19 publications

Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media

Multilevel Balancing Domain Decomposition by Constraints Deluxe Algorithms with Adaptive Coarse Spaces for Flow in Porous Media

Multilevel Balancing Domain Decomposition at Extreme Scales

Balancing Domain Decomposition by Constraints and Perturbation

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