Sobolev Seminorm of Quadratic Functions with Applications to Derivative-Free Optimization

Notes on the Sobolev (Semi)Norms of Quadratic Functions

This paper studies the $H^0$ norm and $H^1$ seminorm of quadratic functions. The (semi)norms are expressed explicitly in terms of the coefficients of the quadratic function under consideration when the underlying domain is an $l_p$-ball (1 \leq p \leq \infty) in $R^n$ .

关于各向异性能量情况下基本频率与扭转刚度之间的关系

On the relations between fundamental frequency and torsional rigidity in the case of anisotropic energies

本文研究了各向异性能量情况下基本频率与扭转刚度之间的关系，分析了半范数诱导的特征值和扭转刚度的优化问题。

A Derivative-Free Approach to Total Variation Regularization

The goal of this paper is to present a novel approach for total variation regularization and Sobolev minimization, which are prominent tools for variational imaging. Thereby we use derivative free characterizations of the total variation semi-norm and Sobolev semi-norms of functions recently derived by Bourgain, Brézis, Mironescu and Dávila. Their analysis is to approximate the semi-norms of a function by singular integral operators. With this characterization we derive a series of novel regularization methods for total variation minimization which have as a novel feature a non-local double in…

Fractional Sobolev Spaces and Functions of Bounded Variation

We investigate the 1D Riemann-Liouville fractional derivative focusing on the connections with fractional Sobolev spaces, the space $BV$ of functions of bounded variation, whose derivatives are not functions but measures and the space $SBV$, say the space of bounded variation functions whose derivative has no Cantor part. We prove that $SBV$ is included in $W^{s,1} $ for every $s \in (0,1)$ while the result remains open for $BV$. We study examples and address open questions.

Global Convergence of Sobolev Training for Overparameterized Neural Networks

Sobolev loss is used when training a network to approximate the values and derivatives of a target function at a prescribed set of input points. Recent works have demonstrated its successful applications in various tasks such as distillation or synthetic gradient prediction. In this work we prove that an overparameterized two-layer relu neural network trained on the Sobolev loss with gradient flow from random initialization can fit any given function values and any given directional derivatives, under a separation condition on the input data.

非线性算子及其导数的通用逼近

Universal Approximation of Nonlinear Operators and Their Derivatives

本文提出通过运算学习架构证明非线性算子及其导数的通用逼近定理，扩展了经典结果到无限维空间，并探讨了其在高阶精度、约束优化和无限维PDE数值方法中的应用。