Trivariate polynomial approximation on Lissajous curves

Bos, Len; Marchi, Stefano De; Vianello, Marco

doi:10.1093/imanum/drw013

Cited by 10 publications

(9 citation statements)

References 29 publications

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“…Since 2|LD int n,p | + |LD out n,p | = n(n + p) + 1, we even have equality in the last formula. This together with the definitions of LD int n,p and LD out n,p in (4) and (5) implies equation (6) as well as equation (11). Finally, (7) and (8) follow from (6).…”

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confidence: 84%

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Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

Erb

2016

Applied Mathematics and Computation

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In this article, we study bivariate polynomial interpolation on the node points of degenerate Lissajous figures. These node points form Chebyshev lattices of rank 1 and are generalizations of the well-known Padua points. We show that these node points allow unique interpolation in appropriately defined spaces of polynomials and give explicit formulas for the Lagrange basis polynomials. Further, we prove mean and uniform convergence of the interpolating schemes. For the uniform convergence the growth of the Lebesgue constant has to be taken into consideration. It turns out that this growth is of logarithmic nature.

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confidence: 84%

“…This approximate interpolation is known as hyperinterpolation [28] and is frequently used in applications. For trivariate Lissajous curves, it is studied in the recent work [5].…”

Section: Introductionmentioning

confidence: 99%

Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

Erb

2016

Applied Mathematics and Computation

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“…Interpolation by polynomials of prescribed total degree, however, is not straightforward because of the challenge of selecting interpolation points that guarantee a unique interpolant, the condition known as unisolvency. (In two dimensions, the Padua points are suitable [7], and there are generalizations to higher dimensions [8,19].) Nevertheless, much progress has been made in investigating cubature formulas that integrate all multivariate polynomials of degree m exactly, an idea going back to Maxwell [18,35,46,49,59].…”

Section: Polynomial-based Cubature and Padua Pointsmentioning

confidence: 99%

Cubature, Approximation, and Isotropy in the Hypercube

Trefethen¹

2017

SIAM Rev.

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Algorithms that combat the curse of dimensionality take advantage of nonuniformity properties of the underlying functions, which may be rotational (e.g., grid alignment) or translational (e.g., near-singularities localized at certain points of the domain). The significance of such effects is explored for four different classes of algorithms: low-rank compression, quasi-Monte Carlo integration, sparse grids, and cubature. The exponentially pronounced computational consequences of the anisotropy of the hypercube are described, notably its mismatch with the isotropy of the set of multivariate polynomials of a fixed degree on which some cubature formulas are based, and it is observed that the tensor product Gauss quadrature rule in [−1, 1] d requires up to (π/2) d times fewer points if it is transformed to a nonpolynomial basis.

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“…Alternative choices of discretization nodes constructed directly on the multidimensional domain may improve the computational cost required for cubature formulas and differentiation matrices. The Padua points in two dimensions [24,25] and their generalizations to higher dimension [26,27] are well-suited to this aim.…”

Section: Discussionmentioning

confidence: 99%

Bivariate collocation for computing $R_{0}$ in epidemic models with two structures

Breda¹,

Reggi²,

Scarabel³

et al. 2021

Preprint

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Structured epidemic models can be formulated as first-order hyperbolic PDEs, where the "spatial" variables represent individual traits, called structures. For models with two structures, we propose a numerical technique to approximate R 0 , which measures the transmissibility of an infectious disease and, rigorously, is defined as the dominant eigenvalue of a next-generation operator. Via bivariate collocation and cubature on tensor grids, the latter is approximated with a finite-dimensional matrix, so that its dominant eigenvalue can easily be computed with standard techniques. We use test examples to investigate experimentally the behavior of the approximation: the convergence order appears to be infinite when the corresponding eigenfunction is smooth, and finite for less regular eigenfunctions. To demonstrate the effectiveness of the technique for more realistic applications, we present a new epidemic model structured by demographic age and immunity, and study the approximation of R 0 in some particular cases of interest.

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Trivariate polynomial approximation on Lissajous curves

Cited by 10 publications

References 29 publications

Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

Bivariate Lagrange interpolation at the node points of Lissajous curves – the degenerate case

Cubature, Approximation, and Isotropy in the Hypercube

Bivariate collocation for computing $R_{0}$ in epidemic models with two structures

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