A Robust Parallel Preconditioner for Indefinite Systems Using Hierarchical Matrices and Randomized Sampling

Ghysels, Pieter; Li, Xiaoye Sherry; Gorman, Christopher; Rouet, François-Henry

doi:10.1109/ipdps.2017.21

Cited by 38 publications

(36 citation statements)

References 32 publications

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“…We thank the anonymous referees for their suggestions and remarks. The shift-invert [13] and time-evolution computations were performed using the PETSc [96], SLEPc [97] and STRUMPACK [98,99] libraries. Data analysis and plotting was performed using the NumPy, Pandas [100] and Matplotlib [101] ANR-16-CE30-0023-02 of the French National Research Agency (ANR) and by the French Programme Investissements d'Avenir under the program ANR-11-IDEX-0002-02, reference ANR-10-LABX-0037-NEXT.…”

Section: Acknowledgementsmentioning

confidence: 99%

Many-body localization in a quasiperiodic Fibonacci chain

2019

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We study the many-body localization (MBL) properties of a chain of interacting fermions subject to a quasiperiodic potential such that the non-interacting chain is always delocalized and displays multifractality. Contrary to naive expectations, adding interactions in this systems does not enhance delocalization, and a MBL transition is observed. Due to the local properties of the quasiperiodic potential, the MBL phase presents specific features, such as additional peaks in the density distribution. We furthermore investigate the fate of multifractality in the ergodic phase for low potential values. Our analysis is based on exact numerical studies of eigenstates and dynamical properties after a quench. 12 5.3 One-particle density matrix 13 6 Dynamical probes of many-body localization 15 6.1 Imbalance 15 6.2 Entanglement growth 16 1 arXiv:1811.01912v3 [cond-mat.dis-nn]

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Section: Acknowledgementsmentioning

confidence: 99%

Many-body localization in a quasiperiodic Fibonacci chain

2019

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“…They now appear in major reference works and textbooks Van Loan 2013, Strang 2019). Key methods are being incorporated into standard software libraries (The Numerical Algorithms Group (NAG) 2019, Langou 2017, Ghysels, Li, Gorman andRouet 2017).…”

Section: Randomized Algorithms Emergementioning

confidence: 99%

“…For details, see Martinsson and Tropp (2020, Section 20) or Martinsson (2019, Section 17.4). Techniques that partially avoid the constraint that individual matrix elements must be accessible are described in Xia (2013) and Ghysels et al (2017).…”

Section: Linear Complexity Algorithms For Rank-structured Matricesmentioning

confidence: 99%

Randomized numerical linear algebra: Foundations and algorithms

2020

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This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues.Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nyström approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing.

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“…A similar relation holds for the V i basis matrices, and is referred to as the nested basis property. The HSS data structure is implemented in the STRUMPACK (STRUctured Matrix PACKage) library [32,14,18]. STRUMPACK is a sparse direct solver and preconditioner.…”

Section: Hierarchically Semi-separable Matrix Representationmentioning

confidence: 99%

A Study of Clustering Techniques and Hierarchical Matrix Formats for Kernel Ridge Regression

Rebrova

Chávez

Liu

et al. 2018

2018 IEEE International Parallel and Distributed Processing Symposium Workshops (IPDPSW)

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We present memory-efficient and scalable algorithms for kernel methods used in machine learning. Using hierarchical matrix approximations for the kernel matrix the memory requirements, the number of floating point operations, and the execution time are drastically reduced compared to standard dense linear algebra routines. We consider both the general H matrix hierarchical format as well as Hierarchically Semi-Separable (HSS) matrices. Furthermore, we investigate the impact of several preprocessing and clustering techniques on the hierarchical matrix compression. Effective clustering of the input leads to a ten-fold increase in efficiency of the compression. The algorithms are implemented using the STRUMPACK solver library. These results confirm that -with correct tuning of the hyperparameters -classification using kernel ridge regression with the compressed matrix does not lose prediction accuracy compared to the exact -not compressed -kernel matrix and that our approach can be extended to O(1M ) datasets, for which computation with the full kernel matrix becomes prohibitively expensive. We present numerical experiments in a distributed memory environment up to 1,024 processors of the NERSC's Cori supercomputer using wellknown datasets to the machine learning community that range from dimension 8 up to 784.

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A Robust Parallel Preconditioner for Indefinite Systems Using Hierarchical Matrices and Randomized Sampling

Cited by 38 publications

References 32 publications

Many-body localization in a quasiperiodic Fibonacci chain

Many-body localization in a quasiperiodic Fibonacci chain

Randomized numerical linear algebra: Foundations and algorithms

A Study of Clustering Techniques and Hierarchical Matrix Formats for Kernel Ridge Regression

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