AbstractWe consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted $$\ell ^1$$ℓ1 minimization has recently been shown to be an effective procedure for computing such approximations. We extend this work to the gradient-augmented case. Our main results show that for the same asymptotic sample complexity, gradient-augmented measurements achieve an approximation error bound in a stronger Sobolev norm, as opposed to the $$L^2$$L2-norm in the unaugmented case. For Chebyshev and Legendre polynomial approximations, this sample complexity estimate is algebraic in the sparsity s and at most logarithmic in the dimension d, thus mitigating the curse of dimensionality to a substantial extent. We also present several experiments numerically illustrating the benefits of gradient information over an equivalent number of function samples only.

中文翻译：

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压缩 Hermite 插值：梯度增强测量的稀疏、高维近似

摘要我们从函数及其梯度的样本中考虑在张量积域上的多元函数的稀疏多项式逼近。当仅规定函数样本时，加权 $$\ell ^1$$ℓ1 最小化最近已被证明是计算此类近似值的有效程序。我们将这项工作扩展到梯度增强的情况。我们的主要结果表明，对于相同的渐近样本复杂度，梯度增强测量在更强的 Sobolev 范数中实现了近似误差界限，而不是在未增强情况下的 $$L^2$$L2 范数。对于切比雪夫和勒让德多项式近似，该样本复杂度估计在稀疏度 s 中是代数的，在维度 d 中至多是对数的，因此在很大程度上减轻了维数灾难。