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Hermitian Interpolation Subject to Uncertainties

Abstract : This contribution is a sequel of the report [1]. In PDE-constrained global optimization (e.g., [3]), iterative algorithms are commonly efficiently accelerated by techniques relying on approximate evaluations of the functional to be minimized by an economical but lower-fidelity model ("meta-model"), in a so-called "Design of Experiment" (DoE) [6]. Various types of meta-models exist (interpolation polynomials, neural networks, Kriging models, etc.). Such meta-models are constructed by pre-calculation of a database of functional values by the costly high-fidelity model. In adjoint-based numerical methods, derivatives of the functional are also available at the same cost, although usually with poorer accuracy. Thus, a question arises: should the derivative information, available but known to be less accurate, be used to construct the meta-model or be ignored? As the first step to investigate this issue, we consider the case of the Hermitian interpolation of a function of a single variable, when the function values are known exactly, and the derivatives only approximately, assuming a uniform upper bound ε on this approximation is known. The classical notion of best approximation is revisited in this context, and a criterion is introduced to define the best set of interpolation points. This set is identified by either analytical or numerical means. If n+1 is the number of interpolation points, it is advantageous to account for the derivative information when ε -> ε_0, where ε_0 decreases with n, and this is in favor of piecewise, low-degree Hermitian interpolants. In all our numerical tests, we have found that the distribution of Chebyshev points is always close to optimal, and provides bounded approximants with close-to-least sensitivity to the uncertainties.
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Jean-Antoine Désidéri, Manuel Bompard, Jacques Peter. Hermitian Interpolation Subject to Uncertainties. Repin, S. and Tiihonen, T. and Tuovinen, T. Numerical Methods for Differential Equations, Optimization, and Technological Problems, 27, Springer Dordrecht, pp.193-218, 2013, Computational Methods in Applied Sciences, 978-94-007-5288-7. ⟨10.1007/978-94-007-5288-7⟩. ⟨hal-00765276⟩

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