Lasso Hyperinterpolation Over General Regions

An, Congpei; Wu, Hao-Ning

doi:10.1137/20m137793x

Cited by 15 publications

(16 citation statements)

References 57 publications

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“…We close this paper a toy illustration on the sphere, making use of the well-conditioned spherical t-designs [1] with m = (t+ 1) 2 . We are interested in a 25-degree hyperinterpolant L 25 f of a Wendland function f ; see the definition of Wendland functions in [14] and the precise definition of the testing function in [2,Equation (5.5)]. According to the original definition of hyperinterpolation (1.3), one shall use a spherical 50-design and its corresponding quadrature rule to construct L S 25 f , see the upper row in Figure 2.…”

Section: The Spherementioning

confidence: 99%

“…2) is a central assumption in constructing hyperinterpolants and their variants, such as filtered hyperinterpolants [12] (even more degrees are required) and Lasso hyperinterpolants [2]. This assumption has also potentially spurred the development of quadrature theory and orthogonal polynomials on some specific manifolds, such as the disk, the square, and the cube, if one considers hyperinterpolation on these manifolds.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 1.4 In the light of Theorem 1.2, we expect that the required exactness degree in constructing other variants of hyperinterpolants, such as filtered hyperinterpolants [12] and Lasso hyperinterpolants [2], can also be reduced, and corresponding theory can be developed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the quadrature exactness in hyperinterpolation

An¹,

Wu²

2022

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This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires an m-point positive-weight quadrature rule with exactness degree 2n. Aided by the Marcinkiewicz-Zygmund inequality, we affirm that when the required exactness degree 2n is relaxed to n + k with 0 < k ≤ n, the L 2 norm of the hyperinterpolation operator is bounded by a constant independent of n. The resulting scheme is convergent as n → ∞ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness.

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Section: The Spherementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the quadrature exactness in hyperinterpolation

An¹,

Wu²

2022

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“…We refer the reader to [17,19,21,29,32,36,37,42] for some follow-up works on the general analysis of hyperinterpolation and [2,22,28,38] for some variants of classical hyperinterpolation.…”

Section: The Approximation Basicsmentioning

confidence: 99%

Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?

An¹,

Wu²

2022

Preprint

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Singular and oscillatory functions feature in numerous applications. The high-accuracy approximation of such functions shall greatly help us develop high-order methods for solving applied mathematics problems. This paper demonstrates that hyperinterpolation, a discrete projection method with coefficients obtained by evaluating the L 2 orthogonal projection coefficients using some numerical integration methods, may be inefficient for approximating singular and oscillatory functions. A relatively large amount of numerical integration points are necessary for satisfactory accuracy. Moreover, in the spirit of product-integration, we propose an efficient modification of hyperinterpolation for such approximation. The proposed approximation scheme, called efficient hyperinterpolation, achieves satisfactory accuracy with fewer numerical integration points than the original scheme. The implementation of the new approximation scheme is relatively easy. Theorems are also given to explain the outperformance of efficient hyperinterpolation over the original scheme in such approximation, with the functions assumed to belong to L 1 (Ω), L 2 (Ω), and C(Ω) spaces, respectively. These theorems, as well as numerical experiments on the interval and the sphere, show that efficient hyperinterpolation has better accuracy in such approximation than the original one when the amount of numerical integration points is limited.

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“…A critical task of spherical modeling is to find an effective data fitting strategy to approximate the underlying mapping between input and output data. Hyperinterpolation, introduced by Sloan in [56], is a simple yet powerful method for fitting spherical data, and it has received a great deal of interest since its birth, see, e.g., [3,28,38,41,52,53,57,59,66]. Given sampled data {(x j , y j )} m j=1 ⊂ S d × R, the underlying mapping can be modeled as a spherical hyperinterpolant of degree n in the form of…”

Section: Introductionmentioning

confidence: 99%

Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere

An¹,

Wu²

2022

Preprint

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This paper focuses on the approximation of continuous functions on the unit sphere by spherical polynomials of degree n via hyperinterpolation. Hyperinterpolation of degree n is a discrete approximation of the L 2 -orthogonal projection of degree n with its Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all spherical polynomials of degree at most 2n. This paper aims to bypass this quadrature exactness assumption by replacing it with the Marcinkiewicz-Zygmund property proposed in a previous paper. Consequently, hyperinterpolation can be constructed by a positive-weight quadrature rule (not necessarily with quadrature exactness). This scheme is referred to as unfettered hyperinterpolation. This paper provides a reasonable error estimate for unfettered hyperinterpolation. The error estimate generally consists of two terms: a term representing the error estimate of the original hyperinterpolation of full quadrature exactness and another introduced as compensation for the loss of exactness degrees. A guide to controlling the newly introduced term in practice is provided. In particular, if the quadrature points form a quasi-Monte Carlo (QMC) design, then there is a refined error estimate. Numerical experiments verify the error estimates and the practical guide.

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Lasso Hyperinterpolation Over General Regions

Cited by 15 publications

References 57 publications

On the quadrature exactness in hyperinterpolation

On the quadrature exactness in hyperinterpolation

Is hyperinterpolation efficient in the approximation of singular and oscillatory functions?

Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere

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