Parallel algebraic hybrid solvers for large 3D convection-diffusion problems

Giraud, Luc; Haidar, Azzam

doi:10.1007/s11075-008-9248-x

Cited by 8 publications

(13 citation statements)

References 25 publications

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“…See, e.g., [3,9,36]. In these cases, often a large problem is first solved by direct methods, and a much smaller Schur complement system corresponding to a small subdomain or interface is solved with iterative ones.…”

Section: Generalizationsmentioning

confidence: 99%

Effective matrix-free preconditioning for the augmented immersed interface method

Xia

2015

Journal of Computational Physics

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Please cite this article in press as: J. Xia et al., Effective matrix-free preconditioning for the augmented immersed interface method, J. Comput. Phys. (2015), http://dx. AbstractWe present effective and efficient matrix-free preconditioning techniques for the augmented immersed interface method (AIIM). AIIM has been developed recently and is shown to be very effective for interface problems and problems on irregular domains. GMRES is often used to solve for the augmented variable(s) associated with a Schur complement A in AIIM that is defined along the interface or the irregular boundary. The efficiency of AIIM relies on how quickly the system for A can be solved. For some applications, there are substantial difficulties involved, such as the slow convergence of GMRES (particularly for free boundary and moving interface problems), and the inconvenience in finding a preconditioner (due to the situation that only the products of A and vectors are available). Here, we propose matrix-free structured preconditioning techniques for AIIM via adaptive randomized sampling, using only the products of A and vectors to construct a hierarchically semiseparable matrix approximation to A. Several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability. The significance of the preconditioners includes: (1) they do not require the entries of A or the multiplication of A T with vectors; (2) constructing the preconditioners needs only O(log N) matrix-vector products and O(N) storage, where N is the size of A; (3) applying the preconditioners needs only O(N) flops;(4) they are very flexible and do not require any a priori knowledge of the structure of A. The preconditioners are observed to significantly accelerate the convergence of GMRES, with heuristical justifications of the effectiveness. Comprehensive tests on several important applications are provided, such as Navier-Stokes equations on irregular domains with traction boundary conditions, interface problems in incompressible flows, mixed boundary problems, and free boundary problems. The preconditioning techniques are also useful for several other problems and methods.

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Section: Generalizationsmentioning

confidence: 99%

Effective matrix-free preconditioning for the augmented immersed interface method

Xia

2015

Journal of Computational Physics

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“…Typically, they partially factor a matrix using direct methods and use iterative methods on the remaining Schur complement. Parallel codes of this type include HIPS [13], MaPhys [14], and PDSLin [15]. ShyLU is similar to these solvers in a conceptual way that all these solvers fall into the broad Schur complement framework described in section II.…”

Section: B Previous Workmentioning

confidence: 99%

ShyLU: A Hybrid-Hybrid Solver for Multicore Platforms

Rajamanickam

Boman

Heroux

2012

2012 IEEE 26th International Parallel and Distributed Processing Symposium

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Abstract-With the ubiquity of multicore processors, it is crucial that solvers adapt to the hierarchical structure of modern architectures. We present ShyLU, a "hybrid-hybrid" solver for general sparse linear systems that is hybrid in two ways: First, it combines direct and iterative methods. The iterative part is based on approximate Schur complements where we compute the approximate Schur complement using a value-based dropping strategy or structure-based probing strategy.Second, the solver uses two levels of parallelism via hybrid programming (MPI+threads). ShyLU is useful both in sharedmemory environments and on large parallel computers with distributed memory. In the latter case, it should be used as a subdomain solver. We argue that with the increasing complexity of compute nodes, it is important to exploit multiple levels of parallelism even within a single compute node.We show the robustness of ShyLU against other algebraic preconditioners. ShyLU scales well up to 384 cores for a given problem size. We also study the MPI-only performance of ShyLU against a hybrid implementation and conclude that on present multicore nodes MPI-only implementation is better. However, for future multicore machines (96 or more cores) hybrid/ hierarchical algorithms and implementations are important for sustained performance.

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“…To achieve a high scalability, algebraic domain decomposition methods are commonly employed to split a large size linear system into smaller size linear systems. To achieve this goal, the Schur complement method is often used to design sparse hybrid linear solvers [2], [3], [4], [5].…”

Section: Introductionmentioning

confidence: 99%

“…Section II-A presents the context in more details. Section II-B presents the basics of domain decomposition Schur complement methods, which are common to most sparse hybrid solvers [2], [3], [4], [5]. Sections II-C and II-D present the method used for preconditioning the reduced system in MAPHYS and the parallel implementation of the solver, respectively.…”

Section: Introductionmentioning

confidence: 99%

On the Resilience of Parallel Sparse Hybrid Solvers

Agullo¹,

Giraud²,

Zounon³

2015

2015 IEEE 22nd International Conference on High Performance Computing (HiPC)

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As the computational power of high performance computing (HPC) systems continues to increase by using a huge number of CPU cores or specialized processing units, extreme-scale applications are increasingly prone to faults. Consequently, the HPC community has proposed many contributions to design resilient HPC applications. These contributions may be system-oriented, theoretical or numerical. In this study we consider an actual fully-featured parallel sparse hybrid (direct/iterative) linear solver, MAPHYS, and we propose numerical remedies to design a resilient version of the solver. The solver being hybrid, we focus in this study on the iterative solution step, which is often the dominant step in practice. We furthermore assume that a separate mechanism ensures fault detection and that a system layer provides support for setting back the environment (processes, . . . ) in a running state. The present manuscript therefore focuses on (and only on) strategies for recovering lost data after the fault has been detected (a separate concern beyond the scope of this study), once the system is restored (another separate concern not studied here). The numerical remedies we propose are twofold. Whenever possible, we exploit the natural data redundancy between processes from the solver to perform exact recovery through clever copies over processes. Otherwise, data that has been lost and no longer available on any process is recovered through a so-called interpolation-restart mechanism. This mechanism is derived from [1] by carefully taking into account the properties of the target hybrid solver. These numerical remedies have been implemented in the MAPHYS parallel solver so that we can assess their efficiency on a large number of processing units (up to 12, 288 CPU cores) for solving large-scale real-life problems.

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Parallel algebraic hybrid solvers for large 3D convection-diffusion problems

Cited by 8 publications

References 25 publications

Effective matrix-free preconditioning for the augmented immersed interface method

Effective matrix-free preconditioning for the augmented immersed interface method

ShyLU: A Hybrid-Hybrid Solver for Multicore Platforms

On the Resilience of Parallel Sparse Hybrid Solvers

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