arXiv subjects

Congpei An

Publications and source records attributed to Congpei An.

33 records · Page 2Linked to original sources

arXiv2024

Hybrid hyperinterpolation over general regions

We present an $\ell^2_2+\ell_1$-regularized discrete least squares approximation over general regions under assumptions of hyperinterpolation, named hybrid hyperinterpolation. Hybrid hyperinterpolation, using a soft thresholding operator and 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 parameter and the filter function are chosen well. We derive $L_2$ errors in theoretical analysis for hybrid hyperinterpolation to approximate continuous functions with noise data on sampling points. Numerical examples illustrate the theoretical results and show that well chosen regularization parameters can enhance the approximation quality over the unit-sphere and the union of disks.

math.NA↗

arXiv2023

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 $[-π,π],$ named Lasso trigonometric interpolation. This approximation is an $\ell_1$-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 $L_2$ 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 $[-π,π]$, with or without the presence of data errors.

math.NA↗

arXiv2022

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 $n$ 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.

math.NA↗

arXiv2022

The springback penalty for robust signal recovery

We propose a new penalty, the springback penalty, for constructing models to recover an unknown signal from incomplete and inaccurate measurements. Mathematically, the springback penalty is a weakly convex function. It bears various theoretical and computational advantages of both the benchmark convex $\ell_1$ penalty and many of its non-convex surrogates that have been well studied in the literature. We establish the exact and stable recovery theory for the recovery model using the springback penalty for both sparse and nearly sparse signals, respectively, and derive an easily implementable difference-of-convex algorithm. In particular, we show its theoretical superiority to some existing models with a sharper recovery bound for some scenarios where the level of measurement noise is large or the amount of measurements is limited. We also demonstrate its numerical robustness regardless of the varying coherence of the sensing matrix. The springback penalty is particularly favorable for the scenario where the incomplete and inaccurate measurements are collected by coherence-hidden or -static sensing hardware due to its theoretical guarantee of recovery with severe measurements, computational tractability, and numerical robustness for ill-conditioned sensing matrices.

math.NA↗

arXiv2022

Enhanced total variation minimization for stable image reconstruction

The total variation (TV) regularization has phenomenally boosted various variational models for image processing tasks. We propose to combine the backward diffusion process in the earlier literature of image enhancement with the TV regularization, and show that the resulting enhanced TV minimization model is particularly effective for reducing the loss of contrast. The main purpose of this paper is to establish stable reconstruction guarantees for the enhanced TV model from noisy subsampled measurements with two sampling strategies, non-adaptive sampling for general linear measurements and variable-density sampling for Fourier measurements. In particular, under some weaker restricted isometry property conditions, the enhanced TV minimization model is shown to have tighter reconstruction error bounds than various TV-based models for the scenario where the level of noise is significant and the amount of measurements is limited. Advantages of the enhanced TV model are also numerically validated by preliminary experiments on the reconstruction of some synthetic, natural, and medical images.

eess.IV↗

arXiv2022

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 $n$ requires a positive-weight quadrature rule with exactness degree $2n$. We examine the behavior of such approximation when the required exactness degree $2n$ is relaxed to $n+k$ with $0<k\leq n$. Aided by the Marcinkiewicz--Zygmund inequality, we affirm that the $L^2$ norm of the exactness-relaxing hyperinterpolation operator is bounded by a constant independent of $n$, and this approximation scheme is convergent as $n\rightarrow\infty$ 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. 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 $t$-designs.

math.NA↗

arXiv2022

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

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 $\mathcal{C}(Ω)$ 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.

math.NA↗

arXiv2021

Lasso hyperinterpolation over general regions

This paper develops a fully discrete soft thresholding polynomial approximation over a general region, named Lasso hyperinterpolation. This approximation is an $\ell_1$-regularized discrete least squares approximation under the same conditions of hyperinterpolation. Lasso hyperinterpolation also uses a high-order quadrature rule to approximate the Fourier coefficients of a given continuous function with respect to some orthonormal basis, and then it obtains its coefficients by acting a soft threshold operator on all approximated Fourier coefficients. Lasso hyperinterpolation is not a discrete orthogonal projection, but it is an efficient tool to deal with noisy data. We theoretically analyze Lasso hyperinterpolation for continuous and smooth functions. The principal results are twofold: the norm of the Lasso hyperinterpolation operator is bounded independently of the polynomial degree, which is inherited from hyperinterpolation; and the $L_2$ error bound of Lasso hyperinterpolation is less than that of hyperinterpolation when the level of noise becomes large, which improves the robustness of hyperinterpolation. Explicit constructions and corresponding numerical examples of Lasso hyperinterpolation over intervals, discs, spheres, and cubes are given.

math.NA↗

arXiv2021

Tikhonov regularization for polynomial approximation problems in Gauss quadrature points

This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing only a multiplicative correction factor into both classical barycentric formulae. An $L_2$ error bound and a uniform error bound are derived, providing similar information that Tikhonov regularization is able to reduce the operator norm (Lebesgue constant) and the error term related to the level of noise, both by multiplying a correction factor which is less than one. Numerical examples show the benefits of Tikhonov regularization when data is noisy or data size is relatively small.

math.NA↗

arXiv2020

The Numerical Stability of Regularized Barycentric Interpolation Formulae for Interpolation and Extrapolation

The $\ell_2-$ and $\ell_1-$regularized modified Lagrange interpolation formulae over $[-1,1]$ are deduced in this paper. This paper mainly analyzes the numerical characteristics of regularized barycentric interpolation formulae, which are presented in [C. An and H.-N. Wu, 2019], and regularized modified Lagrange interpolation formulae for both interpolation and extrapolation. Regularized barycentric interpolation formulae can be carried out in $\mathcal{O}(N)$ operations based on existed algorithms [H. Wang, D. Huybrechs and S. Vandewalle, Math. Comp., 2014], and regularized modified Lagrange interpolation formulae can be realized in an algorithm of $\mathcal{O}(N\log N)$ operations. For interpolation, the regularized modified Lagrange interpolation formulae are blessed with backward stability and forward stability, whereas the regularized barycentric interpolation formulae are only provided with forward stability. For extrapolation, the regularized barycentric interpolation formulae meet loss of accuracy outside $[-1,1]$, but the regularized modified Lagrange interpolation formulae still work. Consistent results for extrapolation are also verified outside the Chebfun ellipse (a special Bernstein ellipse) in the complex plain. Finally, we illustrate that regularized interpolation formulae perform better than classical interpolation formulae without regularization in noise reduction.

math.NA↗

arXiv2019

Regularized Weighted Discrete Least Squares Approximation Using Gauss Quadrature Points

We consider polynomial approximation over the interval $[-1,1]$ by regularized weighted discrete least squares methods with $\ell_2-$ or $\ell_1-$regularization, respectively. As the set of nodes we use Gauss quadrature points (which are zeros of orthogonal polynomials). The number of Gauss quadrature points is $N+1$. For $2L\leq2N+1$, with the aid of Gauss quadrature, we obtain approximation polynomials of degree $L$ in closed form without solving linear algebra or optimization problems. In fact, these approximation polynomials can be expressed in the form of the barycentric interpolation formula when an interpolation condition is satisfied. We then study the approximation quality of the $\ell_2-$regularized approximation polynomial in terms of Lebesgue constants, and the sparsity of the $\ell_1-$regularized approximation polynomial. Finally, we give numerical examples to illustrate these theoretical results and show that a well-chosen regularization parameter can lead to good performance, with or without contaminated data.

math.NA↗

arXiv2019

Numerical construction of spherical $t$-designs by Barzilai-Borwein method

A point set $\mathrm X_N$ on the unit sphere is a spherical $t$-design is equivalent to the nonnegative quantity $A_{N,t+1}$ vanished. We show that if $\mathrm X_N$ is a stationary point set of $A_{N,t+1}$ and the minimal singular value of basis matrix is positive, then $\mathrm X_N$ is a spherical $t$-design. Moreover, the numerical construction of spherical $t$-designs is valid by using Barzilai-Borwein method. We obtain numerical spherical $t$-designs with $t+1$ up to $127$ at $N=(t+2)^2$.

math.OC↗

arXiv2016

Numerical Integration over the Unit Sphere by using spherical t-design

This paper studies numerical integration over the unit sphere $ \mathbb{S}^2 \subset \mathbb{R}^{3} $ by using spherical $t$-design, which is an equal positive weights quadrature rule with polynomial precision $t$. We investigate two kinds of spherical $t$-designs with $t$ up to 160. One is well conditioned spherical $t$-design(WSTD), which was proposed by [1] with $ N=(t+1)^{2} $. The other is efficient spherical $t$-design(ESTD), given by Womersley [2], which is made of roughly of half cardinality of WSTD. Consequently, a series of persuasive numerical evidences indicates that WSTD is better than ESTD in the sense of worst-case error in Sobolev space $ \mathbb{H}^{s}(\mathbb{S}^2) $. Furthermore, WSTD is employed to approximate integrals of various of functions, especially including integrand has a point singularity over the unit sphere and a given ellipsoid. In particular, to deal with singularity of integrand, Atkinson's transformation [3] and Sidi's transformation [4] are implemented with the choices of `grading parameters' to obtain new integrand which is much smoother. Finally, the paper presents numerical results on uniform errors for approximating representive integrals over sphere with three quadrature rules: Bivariate trapezoidal rule, Equal area points and WSTD.

math.NA↗

arXiv2014

A Note On Characterizations of Spherical t-Designs

A set ${X}_{N}=\{x_1,\ldots,x_N\}$ of $N$ points on the unit sphere $\mathbb{S}^d,\,d\geq 2$ is a spherical $t$-design if the average of any polynomial of degree at most $t$ over the sphere is equal to the average value of the polynomial over ${X}_{N}$. This paper extends characterizations of spherical $t$-designs in previous paper from $\mathbb{S}^2$ to general $\mathbb{S}^d$. We show that for $N\geq\dim(\mathbb{P}_{t+1})$, $X_N$ is a stationary point set of a certain non-negative quantity $A_{N,\,t}$ and a fundamental system for polynomial space over $\mathbb{S}^d$ with degree at most $t$, then $X_N$ is a spherical $t$-design. In contrast, we present that with $N \geq \dim( \mathbb{P}_{t})$, a fundamental system $X_N$ is a spherical $t$-design if and only if non-negative quantity $D_{N,\,t}$ vanishes. In addition, the still unanswered questions about construction of spherical $t$-designs are discussed.

math.MG↗

arXiv2014

Error Bounds for Numerical Integration of Oscillatory Bessel Transforms with Algebraic or Logarithmic Singularities

In this paper, we present and analyze the Clenshaw-Curtis-Filon methods for computing two classes of oscillatory Bessel transforms with algebraic or logarithmic singularities. More importantly, for these quadrature rules we derive new computational sharp error bounds by rigorous proof. These new error bounds share the advantageous property that some error bounds are optimal on $ω$ for fixed $N$, while other error bounds are optimal on $N$ for fixed $ω$. Furthermore, we prove from the presented error bounds in inverse powers of $ω$ that the accuracy improves greatly, for fixed $N$, as $ω$ increases.

math.NA↗