arXiv subjects

Wenwu Gao

Publications and source records attributed to Wenwu Gao.

8 recordsLinked to original sources

arXiv2026

Structure-preserving quasi-interpolation for vector-valued function with multiple physical constraints

We develop a unified theory of vectorial quasi-interpolation that exactly preserves intrinsic physical structures characterized by systems of linear constant-coefficient differential operators, including divergence-free and curl-free constraints as special cases. The key ingredient is a family of matrix-valued kernels constructed from an orthogonal projector in the frequency domain, whose columns analytically satisfy the prescribed constraints. The resulting quasi-interpolant is a weighted combination of kernel translates with sampled function values as coefficients, and therefore requires no constrained optimization. Simultaneous error estimates for the approximand and its derivatives are established within a bias-variance framework. For numerical vector-field decomposition, we introduce a generalized Helmholtz--Hodge decomposition that splits a smooth vector field into two components with distinct physical structures, and develop corresponding structure-preserving quasi-interpolation schemes and error estimates. Explicit kernels are derived for several representative constraints, including steady-state acoustic constraints, combined divergence-curl constraints, coupled divergence-free constraints, and second-order Saint-Venant compatibility conditions. Numerical experiments confirm the theoretical error estimates and exact structure preservation, demonstrate accurate identification of sources and sinks, and show three-dimensional reconstruction of generalized Helmholtz--Hodge components for general smooth vector fields without prior constraint assumptions.

math.NA↗

arXiv2026

Quasi-interpolation using generalized Gaussian kernels

This paper focuses on developing a framework for constructing quasi-interpolation with the highest achievable approximation order from generalized Gaussian kernels with the help of kernel restriction trick and periodization technique. We first demonstrate that when we restrict generalized Gaussian kernels satisfying generalized Strang-Fix conditions of order s over a torus, the corresponding restricted kernels in tensor-product forms fulfill periodic Strang-Fix conditions of the same order s. Then, based on these restricted kernels, we construct a periodic quasi-interpolant in Schoenberg's form and derive its error estimates for periodic function approximation over a torus, which reveals that our quasiinterpolant attains the highest approximation order s. Finally, using the periodization technique, we extend the periodic quasi-interpolant to its nonperiodic counterpart with the highest approximation order s for approximating a general function defined over a cube via a torus-to-cube transformation. This result stands in stark contrast to classical quasi-interpolation counterparts, which often yield much lower approximation orders than those dictated by the generalized Strang-Fix conditions of generalized Gaussian kernels. Furthermore, we propose a sparse grid counterpart for high-dimensional function approximation to alleviate the curse of dimensionality. Numerical simulations confirm that our quasi-interpolation scheme is simple and computationally efficient.

math.NA↗

arXiv2026

Expectile Periodograms

This paper introduces a novel periodogram-like function, called the expectile periodogram, for modeling spectral features of time series and detecting hidden periodicities. The expectile periodogram is constructed from trigonometric expectile regression, in which a specially designed check function is used to substitute the squared $l_2$ norm that leads to the ordinary periodogram. The expectile periodogram retains the key properties of the ordinary periodogram as a frequency-domain representation of serial dependence in time series, while offering a more comprehensive understanding by examining the data across the entire range of expectile levels. We establish the asymptotic theory and investigate the relationship between the expectile periodogram and the so called expectile spectrum. Simulations demonstrate the efficiency of the expectile periodogram in the presence of hidden periodicities. Finally, by leveraging the inherent two-dimensional nature of the expectile periodogram, we train a deep learning (DL) model to classify earthquake waveform data. Remarkably, our approach outperforms alternative periodogram-based methods in terms of classification accuracy.

stat.ME↗

arXiv2025

Quasi-interpolation with random sampling centers

We propose and study a general quasi-interpolation framework for stochastic function approximation, which stems and draws motivation from convolution-type solutions for certain practical weighted variational problems. We obtain our quasi-interpolants using Monte Carlo discretization of the pertinent integrals and establish a family of $L^p$-McDiarmid-type concentration inequalities for $1\leq p\leq \infty$, which resulted in verifiable expected error estimates for the stochastic quasi-interpolants. The $L^1$-version of these concentration inequalities is dynamically-independent of dimensions, which offers a partial stochastic mitigation of the so called ``curse of dimensionality". The $L^\infty$-version of these concentration inequalities strengthens the existing expected $L^\infty$-error estimates in the literature. Numerical simulation results are provided at the end of the paper to validate the underlying theoretical analysis.

math.NA↗

arXiv2025

Quantile regression with generalized multiquadric loss function

Quantile regression (QR) is now widely used to analyze the effect of covariates on the conditional distribution of a response variable. It provides a more comprehensive picture of the relationship between a response and covariates compared with classical least squares regression. However, the non-differentiability of the check loss function precludes the use of gradient-based methods to solve the optimization problem in quantile regression estimation. To this end, This paper constructs a smoothed loss function based on multiquadric (MQ) function. The proposed loss function leads to a globally convex optimization problem that can be efficiently solved via (stochastic) gradient descent methods. As an example, we apply the Barzilai-Borwein gradient descent method to obtain the estimation of quantile regression. We establish the theoretical results of the proposed estimator under some regularity conditions, and compare it with other estimation methods using Monte Carlo simulations.

stat.ME↗

arXiv2025

Spherical quasi-interpolation using scaled zonal kernels

We propose and study a new quasi-interpolation method on spheres featuring the following two-phase construction and analysis. In Phase I, we analyze and characterize a large family of zonal kernels (e.g., the spherical version of Poisson kernel, Gaussian, compactly-supported radial kernels), so that the underlying spherical convolution operators (upon the introduction of a scaling parameter) attains a high-order of approximation to target functions. In Phase II, we discretize the spherical integrals utilizing quadrature rules of optimal order to produce the final quasi-interpolants. Numerical experiments demonstrate that the new quasi-interpolation algorithm is robust and amenable to integrated as well as distributed ways of implementation. Moreover, the underlying error-analysis shows that by fine-tuning the scaling parameter in the radial kernels employed, the resulting quasi-interpolants achieve a well-balanced trade-off between approximation and sampling errors.

math.NA↗

arXiv2024

Quasi-interpolation for the Helmholtz-Hodge decomposition

The paper aims at proposing an efficient and stable quasi-interpolation based method for numerically computing the Helmholtz-Hodge decomposition of a vector field. To this end, we first explicitly construct a matrix kernel in a general form from polyharmonic splines such that it includes divergence-free/curl-free/harmonic matrix kernels as special cases. Then we apply the matrix kernel to vector decomposition via the convolution technique together with the Helmholtz-Hodge decomposition. More precisely, we show that if we convolve a vector field with a scaled divergence-free (curl-free) matrix kernel, then the resulting divergence-free (curl-free) convolution sequence converges to the corresponding divergence-free (curl-free) part of the Helmholtz-Hodge decomposition of the field. Finally, by discretizing the convolution sequence via certain quadrature rule, we construct a family of (divergence-free/curl-free) quasi-interpolants for the Helmholtz-Hodge decomposition (defined both in the whole space and over a bounded domain). Corresponding error estimates derived in the paper show that our quasi-interpolation based method yields convergent approximants to both the vector field and its Helmholtz-Hodge decomposition

math.NA↗

arXiv2024

Quasi-interpolation for high-dimensional function approximation

The paper proposes a general quasi-interpolation scheme for high-dimensional function approximation. To facilitate error analysis, we view our quasi-interpolation as a two-step procedure. In the first step, we approximate a target function by a purpose-built convolution operator (with an error term referred to as convolution error). In the second step, we discretize the underlying convolution operator using certain quadrature rules at the given sampling data sites (with an error term called discretization error). The final approximation error is obtained as an optimally balanced sum of these two errors, which in turn views our quasi-interpolation as a regularization technique that balances convolution error and discretization error. As a concrete example, we construct a sparse grid quasi-interpolation scheme for high-dimensional function approximation. Both theoretical analysis and numerical implementations provide evidence that our quasi-interpolation scheme is robust and capable of mitigating the curse of dimensionality for approximating high-dimensional functions.

math.NA↗