Scalable Linear Solvers Based on Enlarged Krylov Subspaces with Dynamic Reduction of Search Directions

Grigori, Laura; Tissot, Olivier

doi:10.1137/18m1196285

Cited by 7 publications

(9 citation statements)

References 32 publications

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“…Then the unknown vector x is split into t vectors, X 1≤i≤t , such that we can still retrieve the original vector by summing them: x = T i=1 X (i) . The parameter t is called the enlarging factor, and the method is in practice not sensitive to the choice of the splitting scheme [16]. Following these considerations, the method can be derived in a similar way to the standard CG solver, replacing the residuals and search directions by N × t matrices, and the optimal step which is a scalar in CG becomes a t × t matrix.…”

Section: Enlarged Conjugate Gradient Solvermentioning

confidence: 99%

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MAPPRAISER: A massively parallel map-making framework for multi-kilo pixel CMB experiments

Bouhargani,

Jamal,

Beck

et al. 2021

Preprint

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Forthcoming cosmic microwave background (CMB) polarized anisotropy experiments have the potential to revolutionize our understanding of the Universe and fundamental physics. The sought-after, tale-telling signatures will be however distributed over voluminous data sets which these experiments will collect. These data sets will need to be efficiently processed and unwanted contributions due to astrophysical, environmental, and instrumental effects characterized and efficiently mitigated in order to uncover the signatures. This poses a significant challenge to data analysis methods, techniques, and software tools which will not only have to be able to cope with huge volumes of data but to do so with unprecedented precision driven by the demanding science goals posed for the new experiments.A keystone of efficient CMB data analysis are solvers of very large linear systems of equations. Such systems appear in very diverse contexts throughout CMB data analysis pipelines, however they typically display similar algebraic structures and can therefore be solved using similar numerical techniques. Linear systems arising in the so-called map-making problem are one of the most prominent and common ones.In this work we present a massively parallel, flexible and extensible framework, comprised of a numerical library, MIDAPACK, and a high level code, MAPPRAISER, which provide tools for solving efficiently such systems. The framework implements iterative solvers based on conjugate gradient techniques: enlarged and preconditioned using different preconditioners. We demonstrate the framework on simulated examples reflecting basic characteristics of the forthcoming data sets issued by ground-based and satelliteborne instruments, executing it on as many as 16,384 compute cores. The software is developed as an open source project freely available to the community at: https://github.com/B3Dcmb/midapack.

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Section: Enlarged Conjugate Gradient Solvermentioning

confidence: 99%

“…The rate of convergence of the ECG is given by the following result demonstrated in [16]: if x k is the approximate solution given by the Enlarged Conjugate Gradient with an enlarging factor t at step k, and x * the true solution then we have…”

Section: Enlarged Conjugate Gradient Solvermentioning

confidence: 99%

MAPPRAISER: A massively parallel map-making framework for multi-kilo pixel CMB experiments

Bouhargani,

Jamal,

Beck

et al. 2021

Preprint

Self Cite

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“…Reducing communication in iterative methods is a challenging topic, with an additional difficulty to consider, the fact that the convergence of iterative methods depends on the spectral properties of the matrices involved. Approaches such as s-step methods (see, e.g., [60], [61] and the references therein) or enlarged Krylov methods [62], [63] are actively investigated.…”

Section: Algorithms Minimizing Data Transfermentioning

confidence: 99%

“…Approaches such as s-step methods (e.g. [60,61] and the references therein) or enlarged Krylov methods [62,63] are actively investigated.…”

Section: Algorithms Minimizing Data Transfermentioning

confidence: 99%

Numerical algorithms for high-performance computational science

Dongarra

Grigori

Higham

2020

Phil. Trans. R. Soc. A.

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A number of features of today’s high-performance computers make it challenging to exploit these machines fully for computational science. These include increasing core counts but stagnant clock frequencies; the high cost of data movement; use of accelerators (GPUs, FPGAs, coprocessors), making architectures increasingly heterogeneous; and multi- ple precisions of floating-point arithmetic, including half-precision. Moreover, as well as maximizing speed and accuracy, minimizing energy consumption is an important criterion. New generations of algorithms are needed to tackle these challenges. We discuss some approaches that we can take to develop numerical algorithms for high-performance computational science, with a view to exploiting the next generation of supercomputers. This article is part of a discussion meeting issue ‘Numerical algorithms for high-performance computational science’.

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“…The iterative solution to large, sparse linear systems of the form 𝐴𝑥 = 𝑏 often requires many sparse matrix-vector multiplications and costly collective communication in the form of inner products; this is the case with the conjugate gradient (CG) method, and with Krylov methods in general. In this paper, we consider so-called enlarged Krylov methods [15], wherein block vectors are introduced to improve convergence, thereby reducing the amount of collective communication in exchange for denser point-to-point communication in the sparse matrix-block vector multiplication.…”

Section: Introductionmentioning

confidence: 99%

Performance Analysis and Optimal Node-Aware Communication for Enlarged Conjugate Gradient Methods

Lockhart¹,

Bienz²,

Gropp³

et al. 2022

Preprint

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Krylov methods are a key way of solving large sparse linear systems of equations, but suffer from poor strong scalabilty on distributed memory machines. This is due to high synchronization costs from large numbers of collective communication calls alongside a low computational workload. Enlarged Krylov methods address this issue by decreasing the total iterations to convergence, an artifact of splitting the initial residual and resulting in operations on block vectors. In this paper, we present a performance study of an Enlarged Krylov Method, Enlarged Conjugate Gradients (ECG), noting the impact of block vectors on parallel performance at scale. Most notably, we observe the increased overhead of point-to-point communication as a result of denser messages in the sparse matrix-block vector multiplication kernel. Additionally, we present models to analyze expected performance of ECG, as well as, motivate design decisions. Most importantly, we introduce a new point-to-point communication approach based on node-aware communication techniques that increases efficiency of the method at scale.

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Scalable Linear Solvers Based on Enlarged Krylov Subspaces with Dynamic Reduction of Search Directions

Cited by 7 publications

References 32 publications

MAPPRAISER: A massively parallel map-making framework for multi-kilo pixel CMB experiments

MAPPRAISER: A massively parallel map-making framework for multi-kilo pixel CMB experiments

Numerical algorithms for high-performance computational science

Performance Analysis and Optimal Node-Aware Communication for Enlarged Conjugate Gradient Methods

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