Randomized QR with Column Pivoting

Abstract This paper revisits the Bézout, Sylvester, and power-basis matrix representations of the greatest common divisor (GCD) of sets of several polynomials. Furthermore, the present work introduces the application of the QR decomposition with column pivoting to a Bézout matrix achieving the computation of the degree and the coeffcients of the GCD through the range of the Bézout matrix. A comparison in terms of computational complexity and numerical effciency of the Bézout-QR, Sylvester-QR, and subspace-SVD methods for the computation of theGCDof sets of several polynomials with real coeffcients is provided.Useful remarks about the performance of the methods based on computational simulations of sets of several polynomials are also presented.

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Affordable Uncertainty Quantification for Industrial Problems: Application to Aero-Engine Fans

Journal of Turbomachinery ◽

10.1115/1.4038982 ◽

2018 ◽

Vol 140 (6) ◽

Cited By ~ 5

Author(s):

Tiziano Ghisu ◽

Shahrokh Shahpar

Keyword(s):

Uncertainty Quantification ◽

Uncertainty Distribution ◽

Blade Shape ◽

Shape Uncertainty ◽

Column Pivoting ◽

The Cost ◽

Sparse Grid Quadrature ◽

The Impact ◽

The Given ◽

Bypass Ratio

Uncertainty quantification (UQ) is an increasingly important area of research. As components and systems become more efficient and optimized, the impact of uncertain parameters is likely to become critical. It is fundamental to consider the impact of these uncertainties as early as possible during the design process, with the aim of producing more robust designs (less sensitive to the presence of uncertainties). The cost of UQ with high-fidelity simulations becomes therefore of fundamental importance. This work makes use of least-squares approximations in the context of appropriately selected polynomial chaos (PC) bases. An efficient technique based on QR column pivoting has been employed to reduce the number of evaluations required to construct the approximation, demonstrating the superiority of the method with respect to full-tensor quadrature (FTQ) and sparse-grid quadrature (SGQ). Orthonormal polynomials used for the PC expansion are calculated numerically based on the given uncertainty distribution, making the approach optimal for any type of input uncertainty. The approach is used to quantify the variability in the performance of two large bypass-ratio jet engine fans in the presence of shape uncertainty due to possible manufacturing processes. The impacts of shape uncertainty on the two geometries are compared, and sensitivities to the location of the blade shape variability are extracted. The mechanisms at the origin of the change in performance are analyzed in detail, as well as the differences between the two configurations.

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PRECONDITIONED PARALLEL BLOCK–JACOBI SVD ALGORITHM

Parallel Processing Letters ◽

10.1142/s0129626406002708 ◽

2006 ◽

Vol 16 (03) ◽

pp. 371-379 ◽

Cited By ~ 1

Author(s):

GABRIEL OKŠA ◽

MARIÁN VAJTERŠIC

Keyword(s):

Execution Time ◽

Parallel Architecture ◽

Singular Values ◽

Singular Value ◽

Parallel Execution ◽

Qr Factorization ◽

R Factor ◽

Order Of Magnitude ◽

Column Pivoting ◽

Parallel Iteration

We show experimentally, that the QR factorization with the complete column pivoting, optionally followed by the LQ factorization of the R-factor, can lead to a substantial decrease of the number of outer parallel iteration steps in the parallel block-Jacobi SVD algorithm, whereby the details depend on the condition number and on the shape of spectrum, including the multiplicity of singular values. Best results were achieved for well-conditioned matrices with a multiple minimal singular value, where the number of parallel iteration steps has been reduced by two orders of magnitude. However, the gain in speed, as measured by the total parallel execution time, depends decisively on how efficient is the implementation of the distributed QR and LQ factorizations on a given parallel architecture. In general, the reduction of the total parallel execution time up to one order of magnitude has been achieved.

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