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Optimal recovery of functions determined by second-order differential operators

Bo Ling, Yi Gu

TL;DR

This work addresses the optimal recovery of isotropic function classes $W^{P(D)}_ ty(\Omega)$ generated by a second-order differential operator $P(D)=D^2+pD+q$, using samples of both $f$ and $\nabla f$. The authors derive an explicit, operator-independent upper bound for the $n$-th recovery error and show asymptotic exactness in the self-adjoint case ($p=0$); they achieve this by reducing the multivariate problem to univariate extremal problems via Green's functions and linking the results to optimal sphere coverings through the covering density. A key contribution is an exact asymptotic formula for $P(D)=D^2+q$, with explicit special-case results for $D^2$, $D^2-\beta^2$, and $D^2+\beta^2$, thus connecting isotropic function recovery with geometric covering problems to yield sharp performance benchmarks. These results provide precise benchmarks for isotropic recovery and illuminate the interplay between operator structure, geometry, and sample placement in high-dimensional recovery tasks.

Abstract

We study the optimal recovery problem for isotropic functions defined by second-order differential operators using both function and gradient values. We derive the upper bound for n-th optimal error with an explicit constant, which is independent of the specific form of the differential operators. Furthermore, for self-adjoint operators, we obtain asymptotic exact results for the n-th optimal error.

Optimal recovery of functions determined by second-order differential operators

TL;DR

This work addresses the optimal recovery of isotropic function classes generated by a second-order differential operator , using samples of both and . The authors derive an explicit, operator-independent upper bound for the -th recovery error and show asymptotic exactness in the self-adjoint case (); they achieve this by reducing the multivariate problem to univariate extremal problems via Green's functions and linking the results to optimal sphere coverings through the covering density. A key contribution is an exact asymptotic formula for , with explicit special-case results for , , and , thus connecting isotropic function recovery with geometric covering problems to yield sharp performance benchmarks. These results provide precise benchmarks for isotropic recovery and illuminate the interplay between operator structure, geometry, and sample placement in high-dimensional recovery tasks.

Abstract

We study the optimal recovery problem for isotropic functions defined by second-order differential operators using both function and gradient values. We derive the upper bound for n-th optimal error with an explicit constant, which is independent of the specific form of the differential operators. Furthermore, for self-adjoint operators, we obtain asymptotic exact results for the n-th optimal error.
Paper Structure (8 sections, 8 theorems, 60 equations)

This paper contains 8 sections, 8 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Optimal Sphere Covering
  5. Green's Functions of Second-order Differential Equations
  6. Some Univariate Extremal Results
  7. Proof of the Explicit Upper Bound
  8. Proof of Exact Result for Optimal Recovery

Key Result

Theorem 1

Let $\Omega\subset \mathbb{R}^d$ be a bounded and convex body, and $P(D)=D^2+p D+q$ be a second-order differential operator with constant coefficients $p,q\in \mathbb{R}$. Then where $\mu_d(\Omega)$ is the volume of $\Omega$, $\nu_d$ is the volume of unit ball in $\mathbb{R}^d,$ and the constant $\mathop{\mathrm{dens}}\nolimits(d)$ is the least density of sphere covering of $\mathbb{R}^d$ as defi

Theorems & Definitions (15)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 3
  • proof
  • proof
  • ...and 5 more