On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

Badia, Santiago; Martı́n, Alberto F.; Principe, Javier

doi:10.1016/j.parco.2015.09.004

Cited by 21 publications

(34 citation statements)

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“…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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Balancing Domain Decomposition by Constraints and Perturbation

Badia¹,

Nguyen²

2016

SIAM J. Numer. Anal.

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“…By definition, the cardinality of this space is n r = n sbd i=1 n i Γ , which is equivalent to R nr . 4 We denote by S sub : V × V → R the subassembled Cartesian product Schur complement matrix composed by S (i) , i.e., S sub = diag(S (1) , S (2) , . .…”

Section: General Frameworkmentioning

confidence: 99%

“…This work has been performed for exact (direct) solvers for both local and coarse problems and implemented as a MPMD model of execution. The extension of this approach to inexact (AMG) solvers can be found in [4].…”

Section: Bddc For 3d Poisson With a Complex Domain And Unstructured Mmentioning

confidence: 99%

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A Highly Scalable Parallel Implementation of Balancing Domain Decomposition by Constraints

Badia

Martı́n

Principe

2014

SIAM J. Sci. Comput.

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In this work we propose a novel parallelization approach of two-level balancing domain decomposition by constraints preconditioning based on overlapping of fine-grid and coarsegrid duties in time. The global set of MPI tasks is split into those that have fine-grid duties and those that have coarse-grid duties, and the different computations and communications in the algorithm are then rescheduled and mapped in such a way that the maximum degree of overlapping is achieved while preserving data dependencies among them. In many ranges of interest, the extra cost associated to the coarse-grid problem can be fully masked by fine-grid related computations (which are embarrassingly parallel). Apart from discussing code implementation details, the paper also presents a comprehensive set of numerical experiments that includes weak scalability analyses with structured and unstructured meshes for the three-dimensional Poisson and linear elasticity problems on a pair of state-of-the-art multicore-based distributed-memory machines. This experimental study reveals remarkable weak scalability in the solution of problems with thousands of millions of unknowns on several tens of thousands of computational cores. Introduction.Scientific phenomena governed by partial differential equations (PDEs) can be approximated by finite element (FE) methods, which results in a sparse linear system of equations to be solved via numerical linear algebra. The everincreasing demand of reality in the simulation of the complex scientific and engineering three-dimensional (3D) problems faced nowadays involves the solution of very large sparse linear systems with several hundreds and even thousands of millions of equations/unknowns. The solution of these systems in a moderate time requires the vast amount of computational resources provided by current multicore-based distributedmemory machines. It is therefore essential to design parallel algorithms able to efficiently exploit their underlying architecture.Domain decomposition (DD) preconditioning of Krylov iterative solvers provides a natural framework for the development of fast parallel solvers tailored for distributedmemory machines, as it has by construction the desirable design principle of maximizing local computations while minimizing interprocessor communication. One-level DD preconditioners, such as the Neumann-Neumann preconditioner [29], are highly

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“…FEMPAR includes a highly scalable built-in numerical linear algebra module based on stateof-the-art domain decomposition solvers; the multilevel Balancing Domain Decomposition by Constraints (BDDC) solver in FEMPAR has scaled up to 1.75 million MPI tasks in the JUQUEEN Supercomputer [23,24]. This linear algebra framework has been designed to efficiently tackle the linear systems that arise from FE discretizations, exploiting the underlying mathematical structure of the PDEs.…”

Section: Introductionmentioning

confidence: 99%

A tutorial-driven introduction to the parallel finite element library FEMPAR v1.0.0

Badia

Martı́n

2020

Computer Physics Communications

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This work is a user guide to the FEMPAR scientific software library. FEMPAR is an open-source object-oriented framework for the simulation of partial differential equations (PDEs) using finite element methods on distributed-memory platforms. It provides a rich set of tools for numerical discretization and built-in scalable solvers for the resulting linear systems of equations. An application expert that wants to simulate a PDE-governed problem has to extend the framework with a description of the weak form of the PDE at hand (and additional perturbation terms for non-conforming approximations). We show how to use the library by going through three different tutorials. The first tutorial simulates a linear PDE (Poisson equation) in a serial environment for a structured mesh using both continuous and discontinuous Galerkin finite element methods. The second tutorial extends it with adaptive mesh refinement on octree meshes. The third tutorial is a distributed-memory version of the previous one that combines a scalable octree handler and a scalable domain decomposition solver. The exposition is restricted to linear PDEs and simple geometries to keep it concise. The interested user can dive into more tutorials available in the FEMPAR public repository to learn about further capabilities of the library, e.g., nonlinear PDEs and nonlinear solvers, time integration, multi-field PDEs, block preconditioning, or unstructured mesh handling.

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On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

Cited by 21 publications

References 38 publications

Balancing Domain Decomposition by Constraints and Perturbation

Balancing Domain Decomposition by Constraints and Perturbation

A Highly Scalable Parallel Implementation of Balancing Domain Decomposition by Constraints

A tutorial-driven introduction to the parallel finite element library FEMPAR v1.0.0

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