A unified approach to evaluation algorithms for multivariate polynomials

Lodha, Suresh K.; Goldman, Ron

doi:10.1090/s0025-5718-97-00862-4

Cited by 16 publications

(12 citation statements)

References 42 publications

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“…What is the corresponding dual algorithm? We plan to investigate the dual evaluation algorithms for L-patches [LG95c], dual de Casteljau subdivision algorithm for Bernstein Be zier surfaces [LG95b], and duality between degree elevation and differentiation formulas [LG94a] in forthcoming papers. Although we have discussed point-line duality, we have observed that this duality is not self-dual.…”

Section: Discussionmentioning

confidence: 99%

“…Thus once we develop a formula or algorithm for one type of basis we can often obtain, almost for free, a dual formula or algorithm for the dual basis. Formulas and algorithms for change of bases [LG95a], evaluation [LG95c], differentiation [LG94a], degree elevation [LG94a], and subdivision [LG95b] each have dual analogues for B-bases and L-bases., This observation allows us to develop a formula or algorithm for whichever scheme is easier to analyze and then map this to a dual formula or algorithm for the dual scheme.…”

Section: Applications Of Dualitymentioning

confidence: 99%

“…Here we shall present for the first time the special case of the change of basis algorithm between the Lagrange and Bernstein Be zier L-bases. The inverse transformation from Bernstein Be zier form to Lagrange form yields an evaluation algorithm for degree n Bernstein Be zier surfaces with an amortized cost of O(n) computations per point [LG95c].…”

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confidence: 99%

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Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Lodha¹,

Goldman²

1998

Journal of Approximation Theory

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L-Bases and B-bases are two important classes of polynomial bases used for representing surfaces in approximation theory and computer aided geometric design. It is well known that the Bernstein and multinomial (or Taylor) bases are special cases of both L-bases and B-bases. We establish that certain proper subclasses of bivariate Lagrange and Newton bases are L-bases. Furthermore, we present a rich collection of lattices (or point-line configurations) that admit unique Lagrange or Hermite interpolation problems which can be solved quite naturally in terms of Lagrange and Newton L-bases. A new geometric point-line duality between L-bases and B-bases is described: lines in L-bases correspond to points or vectors in B-bases and concurrent lines map to collinear points and vice versa. This duality between L-bases and B-bases is then used to establish that certain proper subclasses of power bases are B-bases and are dual to Lagrange L-bases. This geometric duality is further used to describe the lattices that admit power B-bases. B-bases dual to Newton L-bases are also investigated. Duality can also be used to develop change of basis algorithms with computational complexity O(n 3 ) between any two L-bases andÂor B-bases. We describe, in particular, a new change of basis algorithm from a bivariate Lagrange L-basis to a bivariate Bernstein basis with computational complexity O(n 3 ).

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Section: Discussionmentioning

confidence: 99%

Section: Applications Of Dualitymentioning

confidence: 99%

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Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Lodha¹,

Goldman²

1998

Journal of Approximation Theory

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“…While the most efficient scheme for evaluating multivariate polynomials is not presently known [78], it is an active area of research. In general, Eq.…”

Section: Appendix F: Other Approaches For Waveform Predictionmentioning

confidence: 99%

Fast Prediction and Evaluation of Gravitational Waveforms Using Surrogate Models

et al. 2014

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We propose a solution to the problem of quickly and accurately predicting gravitational waveforms within any given physical model. The method is relevant for both real-time applications and more traditional scenarios where the generation of waveforms using standard methods can be prohibitively expensive. Our approach is based on three offline steps resulting in an accurate reduced order model in both parameter and physical dimensions that can be used as a surrogate for the true or fiducial waveform family. First, a set of m parameter values is determined using a greedy algorithm from which a reduced basis representation is constructed. Second, these m parameters induce the selection of m time values for interpolating a waveform time series using an empirical interpolant that is built for the fiducial waveform family. Third, a fit in the parameter dimension is performed for the waveform's value at each of these m times. The cost of predicting L waveform time samples for a generic parameter choice is of order OðmL þ mc fit Þ online operations, where c fit denotes the fitting function operation count and, typically, m ≪ L. The result is a compact, computationally efficient, and accurate surrogate model that retains the original physics of the fiducial waveform family while also being fast to evaluate. We generate accurate surrogate models for effective-one-body waveforms of nonspinning binary black hole coalescences with durations as long as 10 5 M, mass ratios from 1 to 10, and for multiple spherical harmonic modes. We find that these surrogates are more than 3 orders of magnitude faster to evaluate as compared to the cost of generating effective-one-body waveforms in standard ways. Surrogate model building for other waveform families and models follows the same steps and has the same low computational online scaling cost. For expensive numerical simulations of binary black hole coalescences, we thus anticipate extremely large speedups in generating new waveforms with a surrogate. As waveform generation is one of the dominant costs in parameter estimation algorithms and parameter space exploration, surrogate models offer a new and practical way to dramatically accelerate such studies without impacting accuracy.

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“…. , n. The number of arithmetic operations required for the evaluation of p(x) by using (47) is (κ + 1) n − 1 additions and the same amount for multiplications which is identical to the amount of operations of the Horner scheme [17].…”

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confidence: 99%

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

Titi

Garloff

2017

Applied Mathematics and Computation

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In this paper, multivariate polynomials in the Bernstein basis over a simplex (simplicial Bernstein representation) are considered. Two matrix methods for the computation of the polynomial coefficients with respect to the Bernstein basis, the so-called Bernstein coefficients, are presented. Also matrix methods for the calculation of the Bernstein coefficients over subsimplices generated by subdivision of the standard simplex are proposed and compared with the use of the de Casteljau algorithm. The evaluation of a multivariate polynomial in the power and in the Bernstein basis is considered as well. All the methods solely use matrix operations such as multiplication, transposition, and reshaping; some of them rely also on the bidiagonal factorization of the lower triangular Pascal matrix or the factorization of this matrix by a Toeplitz matrix. The latter one enables the use of the Fast Fourier Transform hereby reducing the amount of arithmetic operations.

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A unified approach to evaluation algorithms for multivariate polynomials

Cited by 16 publications

References 42 publications

Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Fast Prediction and Evaluation of Gravitational Waveforms Using Surrogate Models

Matrix methods for the simplicial Bernstein representation and for the evaluation of multivariate polynomials

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