Quasi-interpolation for high-dimensional function approximation

TL;DR

This paper introduces a quasi-interpolation method for high-dimensional functions that balances convolution and discretization errors, effectively mitigating the curse of dimensionality through a two-step approximation process.

Contribution

It presents a novel quasi-interpolation scheme viewed as a regularization technique, with a concrete sparse grid example for high-dimensional approximation.

Findings

The scheme effectively balances errors to improve approximation accuracy.

Numerical results demonstrate robustness against the curse of dimensionality.

Theoretical analysis confirms the method's convergence and stability.

Abstract

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.