arXiv subjects

Mou Cai

Publications and source records attributed to Mou Cai.

8 recordsLinked to original sources

arXiv2026

Multivariate integration and approximation in weighted Sobolev spaces of low fractional smoothness

The weighted half-period cosine space has often been employed in the theory of quasi-Monte Carlo methods for multivariate integration and approximation of non-periodic functions. For integer-order smoothness, its norm equivalence to certain weighted unanchored Sobolev spaces has been established in the literature. In this work, we extend this equivalence to fractional-order smoothness up to $2$. By introducing an explicit representation via Slobodeckij-type seminorms, we prove a norm equivalence between the half-period cosine spaces and the corresponding weighted unanchored Sobolev spaces. Our Sobolev norm representation clarifies how the fractional regularity dictates the presence or absence of boundary constraints and (non-)periodic structures. Furthermore, we investigate the limiting behavior of these fractional spaces as the smoothness parameter approaches integer boundaries, establishing a continuous bridge to the classical integer-order Sobolev spaces. These equivalence results enable us to transfer the near-optimal error bounds and tractability results for multivariate integration and function approximation from the half-period cosine settings to our newly introduced fractional Sobolev spaces.

math.NA↗

arXiv2026

Worst-case $L_p$-approximation of periodic functions using median lattice algorithms

We study the worst-case approximation of multivariate periodic functions from the weighted Korobov space $H_{d,\alpha,\gamma}$ with smoothness $\alpha>1/2$ in the Lebesgue norm $L_p([0,1]^d)$ for $1\le p\le\infty$. We analyze a \emph{median lattice algorithm} that reconstructs a truncated Fourier series by approximating the coefficients on a hyperbolic-cross-type index set using $R$ rank-1 lattice sampling rules with independent randomly chosen generating vectors, and then aggregating the resulting coefficient estimators via the componentwise median. For an odd number of repetitions $R>1$ and an odd prime lattice size $N$, we prove high-probability error bounds in both $L_\infty$ and $L_2$. Interpolation then yields the result for all $1 \le p\le\infty$. In particular, with a high probability, the algorithm satisfies \[ \mathrm{err}(H_{d,\alpha,\gamma},L_p,A)\ \le\ C_{d,\alpha,\beta,\boldsymbol{\gamma},p}\, N^{- \alpha + (\frac12 - \frac1p)_+ + \beta }, \qquad 1 \le p\le\infty,\ \beta>0, \] where $(x)_+ = \max\{x, 0\}$, $N$ is the number of function evaluations, and the weights $\boldsymbol{\gamma}$ and the constant $C_{d,\alpha,\beta,\boldsymbol{\gamma},p}$ are independent of $N$. For $p=\infty$, $C_{d,\alpha,\beta,\boldsymbol{\gamma},\infty}$ is dimension-independent under the summability condition $\sum_{j=1}^\infty \gamma_j^{1/(2\alpha)}<\infty$. These results extend recent analyses of median-based lattice approximation in $L_2$ and complement related multiple-shift lattice approaches, showing that median aggregation yields nearly optimal $L_p$-approximation rates (up to logarithmic factors and an arbitrarily small loss) in weighted Korobov spaces.

math.NA↗

arXiv2026

A note on approximation in weighted Korobov spaces via multiple rank-1 lattices

This paper studies the multivariate approximation of functions in weighted Korobov spaces using multiple rank-1 lattice rules. It has been shown by K\"{a}mmerer and Volkmer (2019) that algorithms based on multiple rank-1 lattices achieve the optimal convergence rate for the $L_{\infty}$ error in Wiener-type spaces, up to logarithmic factors. While this result was translated to weighted Korobov spaces in the recent monograph by Dick, Kritzer, and Pillichshammer (2022), the analysis requires the smoothness parameter $\alpha$ to be greater than $1$ and is restricted to product weights. In this paper, we extend this result for multiple rank-1 lattice-based algorithms to the case where $1/2<\alpha\le 1$ and for general weights, covering a broader range of periodic functions with low smoothness and general relative importance of variables. We also provide a summability condition on the weights to ensure strong polynomial tractability for any $\alpha>1/2$. Furthermore, by incorporating random shifts into multiple rank-1 lattice-based algorithms, we prove that the resulting randomized algorithm achieves a nearly optimal convergence rate in terms of the worst-case root mean squared $L_2$ error, while retaining the same tractability property.

math.NA↗

arXiv2025

A lattice algorithm with multiple shifts for function approximation in Korobov spaces

In this paper, we propose a novel algorithm for function approximation in a weighted Korobov space based on shifted rank-1 lattice rules. To mitigate aliasing errors inherent in lattice-based Fourier coefficient estimation, we employ $\mathcal{O}((\log N)^{2d-1})$ shifted copies of a single rank-1 lattice and recover each Fourier coefficient via a least-squares procedure. Writing $p$ for the total number of function evaluations, we show that the resulting approximation achieves the optimal convergence rate for the $L_{\infty}$-approximation error in the worst-case setting, namely $\mathcal{O}(p^{-\alpha+1/2+\varepsilon})$ for arbitrarily small $\varepsilon>0$. Moreover, by incorporating random shifts, the algorithm attains the optimal rate for the $L_{2}$-approximation error in the randomized setting, which is $\mathcal{O}(p^{-\alpha+\varepsilon})$. Numerical experiments illustrate the practical performance of the algorithms and the qualitative behavior predicted by the theoretical analysis.

math.NA↗

arXiv2025

Quasi-Monte Carlo hyperinterpolation

This paper studies a generalization of hyperinterpolation over the high-dimensional unit cube. Hyperinterpolation of degree $ m $ serves as a discrete approximation of the $ L_2 $-orthogonal projection of the same degree, using Fourier coefficients evaluated by a positive-weight quadrature rule that exactly integrates all polynomials of degree up to $ 2m $. Traditional hyperinterpolation methods often depend on exact quadrature assumptions, which can be impractical in high-dimensional contexts. We address the challenges and advancements in hyperinterpolation, bypassing the assumption of exactness for quadrature rules by replacing it with quasi-Monte Carlo (QMC) rules and propose a novel approximation scheme with an index set $ I $, which is referred to as QMC hyperinterpolation of range $ I $. In particular, we provide concrete construction algorithms for QMC hyperinterpolation with certain lattice rules. Consequently, we show that QMC hyperinterpolation achieves accuracy comparable to traditional hyperinterpolation while avoiding its higher computational costs. Furthermore, we introduce a Lasso-based approach to improve the robustness of QMC hyperinterpolation against noise from sampling processes. Numerical experiments validate the efficacy of our proposed methods.

math.NA↗

arXiv2024

$L_2$-approximation using randomized lattice algorithms

We propose a randomized lattice algorithm for approximating multivariate periodic functions over the $d$-dimensional unit cube from the weighted Korobov space with mixed smoothness $\alpha > 1/2$ and product weights $\gamma_1,\gamma_2,\ldots\in [0,1]$. Building upon the deterministic lattice algorithm by Kuo, Sloan, and Wo\'{z}niakowski (2006), we incorporate a randomized quadrature rule by Dick, Goda, and Suzuki (2022) to accelerate the convergence rate. This randomization involves drawing the number of points for function evaluations randomly, and selecting a good generating vector for rank-1 lattice points using the randomized component-by-component algorithm. We prove that our randomized algorithm achieves a worst-case root mean squared $L_2$-approximation error of order $M^{-\alpha(2\alpha+1)/(4\alpha+1)+\varepsilon}$ for an arbitrarily small $\varepsilon > 0$, where $M$ denotes the maximum number of function evaluations, and that the error bound is independent of the dimension $d$ if the weights satisfy $\sum_{j=1}^\infty \gamma_j^{1/\alpha} < \infty$. Our upper bound converges faster than a lower bound on the worst-case $L_2$-approximation error for deterministic rank-1 lattice-based approximation proved by Byrenheid, K\"{a}mmerer, Ullrich, and Volkmer (2017). We also show a lower error bound of order $M^{-\alpha/2-1/2}$ for our randomized algorithm, leaving a slight gap between the upper and lower bounds open for future research.

math.NA↗

arXiv2024

Parameter choice strategies for regularized least squares approximation of noisy continuous functions on the unit circle

This paper explores the incorporation of Tikhonov regularization into the least squares approximation scheme using trigonometric polynomials on the unit circle. This approach encompasses interpolation and hyperinterpolation as specific cases. With the aid of the de la Vall\'ee-Poussin approximation, we derive a uniform error bound and a concrete $L_2$ error bound. These error estimates demonstrate the effectiveness of Tikhonov regularization in the denoising process. A new regularity condition for the selection of regularization parameters is proposed. We investigate three strategies for choosing regularization parameters: Morozov's discrepancy principle, the L-curve, and generalized cross-validation, by explicitly combining these error bounds of the approximating trigonometric polynomial. We show that Morozov's discrepancy principle satisfies the proposed regularity condition, while the other two methods do not. Finally, numerical examples are provided to illustrate how the aforementioned methodologies, when applied with well-chosen parameters, can significantly improve the quality of approximation.

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↗