Hermitian Interpolation Subject to Uncertainties

Abstract

This contribution is a sequel of the report (Bompard et al. in http://hal.inria.fr/inria-00526558/en/, 2010). In PDE-constrained global optimization (e.g., Jones (in J. Global Optim. 21(4):345–383, 2001)), 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) (Sacks et al. in Stat. Sci. 4(4):409–435, 1989). 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.