Quasi-interpolation with random sampling centers

2025/12/23 by Wenwu Gao, Gao, Wenwu, Le Hu +5
Mathematics · Computer Science · Decision Sciences · #Mathematical Approximation and Integration #Stochastic Gradient Optimization Techniques #Probabilistic and Robust Engineering Design

paper · doi:10.48550/arxiv.2512.19988

Abstract

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≤ p≤ ∞, which resulted in verifiable expected error estimates for the stochastic quasi-interpolants. The L¹-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^∞-version of these concentration inequalities strengthens the existing expected L^∞-error estimates in the literature. Numerical simulation results are provided at the end of the paper to validate the underlying theoretical analysis.

Citations

• Quasi-interpolation for high-dimensional function approximation
• Random sampling and polynomial-free interpolation by Generalized MultiQuadrics
• New methods for quasi-interpolation approximations: resolution of odd-degree singularities
• High Accuracy Quasi-Interpolation using a new class of generalized Multiquadrics
• Convergence of sparse grid Gaussian convolution approximation for multi-dimensional periodic function
• Distributed Learning via Filtered Hyperinterpolation on Manifolds
• Distributed filtered hyperinterpolation for noisy data on the sphere
• Optimal order Jackson type inequality for scaled Shepard approximation
• An extension of McDiarmid's inequality
• Sampling and cubature on sparse grids based on a B-spline\n quasi-interpolation

Related

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