Quasi-Monte Carlo Methods and Kernel-Based Approximation

The field of quasi-Monte Carlo methods and kernel-based approximation is rapidly advancing, with a focus on improving the accuracy and efficiency of these methods. Recent developments have led to the discovery of new techniques for achieving faster convergence rates and more accurate approximations. One of the key areas of research is the use of digital nets and quasi-Monte Carlo rules for approximating nonperiodic functions, where new error bounds and convergence rates have been established. Additionally, kernel-based approximation methods have been improved through the characterization of superconvergence for projections in general Hilbert spaces. Noteworthy papers include: the work on tent transformed order 2 nets, which achieves a quadratic decay rate for quasi-Monte Carlo quadrature of nonperiodic functions. The paper on general superconvergence for kernel-based approximation, which extends existing results and provides a broader framework for understanding and exploiting superconvergence.

Sources

Tent transformed order $2$ nets and quasi-Monte Carlo rules with quadratic error decay

General superconvergence for kernel-based approximation

Dimension-independent convergence rates of randomized nets using median-of-means

Variably Scaled Kernels for the regularized solution of the parametric Fourier imaging problem

Hyperbolic trigonometric functions as approximation kernels and their properties I: generalised Fourier transforms

Regularized least squares learning with heavy-tailed noise is minimax optimal

Fast Newton Transform: Interpolation in Downward Closed Polynomial Spaces

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