12 research outputs found
Lasso trigonometric polynomial approximation for periodic function recovery in equidistant points
In this paper, we propose a fully discrete soft thresholding trigonometric
polynomial approximation on named Lasso trigonometric
interpolation. This approximation is an -regularized discrete least
squares approximation under the same conditions of classical trigonometric
interpolation on an equidistant grid. Lasso trigonometric interpolation is
sparse and meanwhile it is an efficient tool to deal with noisy data. We
theoretically analyze Lasso trigonometric interpolation for continuous periodic
function. The principal results show that the error bound of Lasso
trigonometric interpolation is less than that of classical trigonometric
interpolation, which improved the robustness of trigonometric interpolation.
This paper also presents numerical results on Lasso trigonometric interpolation
on , with or without the presence of data errors.Comment: 18 pages, 5 figure
Hard thresholding hyperinterpolation over general regions
This paper proposes a novel variant of hyperinterpolation, called hard
thresholding hyperinterpolation. This approximation scheme of degree
leverages a hard thresholding operator to filter all hyperinterpolation
coefficients which approximate the Fourier coefficients of a continuous
function by a quadrature rule with algebraic exactness . We prove that hard
thresholding hyperinterpolation is the unique solution to an
-regularized weighted discrete least squares approximation problem.
Hard thresholding hyperinterpolation is not only idempotent and commutative
with hyperinterpolation, but also satisfies the Pythagorean theorem. By
estimating the reciprocal of the Christoffel function, we demonstrate that the
upper bound of the uniform norm of hard thresholding hyperinterpolation
operator is not greater than that of hyperinterpolation operator. Hard
thresholding hyperinterpolation possesses denoising and basis selection
abilities as Lasso hyperinterpolation. To judge the denoising effects of hard
thresholding and Lasso hyperinterpolations, this paper yields a criterion that
combines the regularization parameter and the product of noise coefficients and
signs of hyperinterpolation coefficients. Numerical examples on the spherical
triangle and the cube demonstrate the denoising performance of hard
thresholding hyperinterpolation.Comment: 19 pages, 7 figure
Bypassing the quadrature exactness assumption of hyperinterpolation on the sphere
This paper focuses on the approximation of continuous functions on the unit
sphere by spherical polynomials of degree via hyperinterpolation.
Hyperinterpolation of degree is a discrete approximation of the
-orthogonal projection of degree with its Fourier coefficients
evaluated by a positive-weight quadrature rule that exactly integrates all
spherical polynomials of degree at most . 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.Comment: 22 pages, 7 figure
On the quadrature exactness in hyperinterpolation
This paper investigates the role of quadrature exactness in the approximation
scheme of hyperinterpolation. Constructing a hyperinterpolant of degree
requires a positive-weight quadrature rule with exactness degree . We
examine the behavior of such approximation when the required exactness degree
is relaxed to with . Aided by the Marcinkiewicz--Zygmund
inequality, we affirm that the norm of the exactness-relaxing
hyperinterpolation operator is bounded by a constant independent of , and
this approximation scheme is convergent as if is
positively correlated to . 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. Our theoretical results are
asserted by numerical experiments on three of the best-known quadrature rules:
the Gauss quadrature, the Clenshaw--Curtis quadrature, and the spherical
-designs.Comment: 16 pages, 5 figures, 1 tabl
Hybrid hyperinterpolation over general regions
We present an -regularized discrete least squares
approximation over general regions under assumptions of hyperinterpolation,
named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft
thresholding operator as well a filter function to shrink the Fourier
coefficients approximated by a high-order quadrature rule of a given continuous
function with respect to some orthonormal basis, is a combination of lasso and
filtered hyperinterpolations. Hybrid hyperinterpolation inherits features of
them to deal with noisy data once the regularization parameters and filter
function are chosen well. We not only provide errors in theoretical
analysis for hybrid hyperinterpolation to approximate continuous functions with
noise and noise-free, but also decompose errors into three exact computed
terms with the aid of a prior regularization parameter choices rule. This rule,
making fully use of coefficients of hyperinterpolation to choose a
regularization parameter, reveals that errors for hybrid
hyperinterpolation sharply decline and then slowly increase when the sparsity
of coefficients ranges from one to large values. Numerical examples show the
enhanced performance of hybrid hyperinterpolation when regularization
parameters and noise vary. Theoretical errors bounds are verified in
numerical examples on the interval, the unit-disk, the unit-sphere and the
unit-cube, the union of disks.Comment: 25 pages, 7 figure