We 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 \(\ell ^1\) 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 \(L^2\)-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.
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This is not to be confused with expansions in Hermite polynomials, which we do not address in this paper. See  for some work in this direction.
As we discuss in Sect. 6, we use a slightly different method of proof to remove the factor \(\lambda \) in the error bound, at the expense of a slightly increased log factor.
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This work is supported in part by the NSERC Grant 611675 and an Alfred P. Sloan Research Fellowship. Yi Sui also acknowledges support from an NSERC PGSD scholarship.
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Communicated by Karlheinz Groechenig.
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Adcock, B., Sui, Y. Compressive Hermite Interpolation: Sparse, High-Dimensional Approximation from Gradient-Augmented Measurements. Constr Approx 50, 167–207 (2019). https://doi.org/10.1007/s00365-019-09467-0
- Multivariate approximation
- Orthogonal polynomials
- Nonlinear approximation
- Compressed sensing
Mathematics Subject Classification