Numerical conditioning

Higham, Nicholas J.

doi:10.1007/978-1-4614-7034-2_5

Cited by 3 publications

(3 citation statements)

References 21 publications

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“…The study of conditioning of Vandermonde matrices can be traced to Gautschi (1983), where orthogonal polynomials are used to improve the condition number. See also Gautschi (2011Gautschi ( , 2012, and, for a short review, Higham (2014). For different studies around Vandermonde-like matrices see Kuian et al (2019), Demmel and Koev (2006), Pan (2015), Reichel and Opfer (1991).…”

Section: Further Readingmentioning

confidence: 99%

Reduced order and surrogate models for gravitational waves

Tiglio

Villanueva

2022

Living Rev Relativ

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We present an introduction to some of the state of the art in reduced order and surrogate modeling in gravitational-wave (GW) science. Approaches that we cover include principal component analysis, proper orthogonal (singular value) decompositions, the reduced basis approach, the empirical interpolation method, reduced order quadratures, and compressed likelihood evaluations. We divide the review into three parts: representation/compression of known data, predictive models, and data analysis. The targeted audience is practitioners in GW science, a field in which building predictive models and data analysis tools that are both accurate and fast to evaluate, especially when dealing with large amounts of data and intensive computations, are necessary yet can be challenging. As such, practical presentations and, sometimes, heuristic approaches are here preferred over rigor when the latter is not available. This review aims to be self-contained, within reasonable page limits, with little previous knowledge (at the undergraduate level) requirements in mathematics, scientific computing, and related disciplines. Emphasis is placed on optimality, as well as the curse of dimensionality and approaches that might have the promise of beating it. We also review most of the state of the art of GW surrogates. Some numerical algorithms, conditioning details, scalability, parallelization and other practical points are discussed. The approaches presented are to a large extent non-intrusive (in the sense that no differential equations are invoked) and data-driven and can therefore be applicable to other disciplines. We close with open challenges in high dimension surrogates, which are not unique to GW science.

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Section: Further Readingmentioning

confidence: 99%

Reduced order and surrogate models for gravitational waves

Tiglio

Villanueva

2022

Living Rev Relativ

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“…The study of conditioning of Vandermonde matrices can be tracked to [59], where orthogonal polynomials are used to improve the condition number. See also [60,61], and, for a short review, [72]. For different studies around Vandermonde-like matrices see [78,46,104,115].…”

Section: Further Readingmentioning

confidence: 99%

Reduced Order and Surrogate Models for Gravitational Waves

Tiglio,

Villanueva

2021

Preprint

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We present an introduction to some of the state of the art in reduced order and surrogate modeling in gravitational wave (GW) science. Approaches that we cover include Principal Component Analysis, Proper Orthogonal Decomposition, the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order Quadratures, and Compressed Likelihood evaluations. We divide the review into three parts: representation/compression of known data, predictive models, and data analysis. The targeted audience is that one of practitioners in GW science, a field in which building predictive models and data analysis tools that are both accurate and fast to evaluate, especially when dealing with large amounts of data and intensive computations, are necessary yet can be challenging. As such, practical presentations and, sometimes, heuristic approaches are here preferred over rigor when the latter is not available. This review aims to be self-contained, within reasonable page limits, with little previous knowledge (at the undergraduate level) requirements in mathematics, scientific computing, and other disciplines. Emphasis is placed on optimality, as well as the curse of dimensionality and approaches that might have the promise of beating it. We also review most of the state of the art of GW surrogates. Some numerical algorithms, conditioning details, scalability, parallelization and other practical points are discussed. The approaches presented are to large extent non-intrusive and data-driven and can therefore be applicable to other disciplines. We close with open challenges in high dimension surrogates, which are not unique to GW science.

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“…The condition number of a matrix A quantifies how sensitive the linear system Ax = b is with respect to changes in the right-hand side vector b [14,15]. If the condition number is large, tiny changes in b can cause significant changes in the solution vector x.…”

Section: Introductionmentioning

confidence: 99%

Variable-Size Batched Condition Number Calculation on GPUs

Anzt

Dongarra

Flegar

et al. 2018

2018 30th International Symposium on Computer Architecture and High Performance Computing (SBAC-PAD)

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We present a kernel that is designed to quickly compute the condition number of a large collection of tiny matrices on a graphics processing unit (GPU). The matrices can differ in size and the process integrates the use of pivoting to ensure a numerically-stable matrix inversion. The performance assessment reveals that, in double precision arithmetic, the new GPU kernel achieves up to 550 GFLOPs (billions of floating-point operations per second) and 800 GFLOPs on NVIDIA's P100 and V100 GPUs, respectively. The results also demonstrate a considerable speed-up with respect to a workflow that computes the condition number via launching a set of four batched kernels. In addition, we present a variable-size batched kernel for the computation of the matrix infinity norm. We show that this memory-bound kernel achieves up to 90% of the sustainable peak bandwidth.

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Numerical conditioning

Cited by 3 publications

References 21 publications

Reduced order and surrogate models for gravitational waves

Reduced order and surrogate models for gravitational waves

Reduced Order and Surrogate Models for Gravitational Waves

Variable-Size Batched Condition Number Calculation on GPUs

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