On the Stability of Polynomial Interpolation Using Hierarchical Sampling

Abstract

Motivated by the development of non-intrusive methods for high dimensional parametric PDE's, we study the stability of a sparse high dimensional polynomial interpolation procedure introduced in [6]. A key aspect of this procedure is its hierarchical structure: the sampling set is progressively enriched together with the polynomial space. The evaluation points are selected from a grid obtained by tensorization of a univariate sequence. The Lebesgue constant that quantifies the stability of the resulting interpolation operator depends on the choice of this sequence. Here we study the -Leja sequence, obtained by the projection of Leja sequences on the complex unit circle, with initial value 1, onto [−1, 1]. For this sequence, we prove cubic growth in the number of points for the Lebesgue constant of the multivariate interpolation operator, independently of the number of variable and of the shape of the polynomial space.

Shape Holomorphy of the Stationary Navier--Stokes Equations

“…One example of a sequence z := (z j ) j≥1 such that the Lebesgue constants of the n-sections are bounded polynomially is the so-called R-Leja sequence, see [7]. The above results are a natural motivation for establishing holomorphy of the parametric mapping U y → (û(y),p(y)) = (û Ty ,p Ty ) of the Stokes and the Navier-Stokes pullback solutions.…”

Multivariate Interpolation on Unisolvent Nodes -- Lifting the Curse of Dimensionality

“…Alternative approaches [3,15,17,18,19,72] are available to realize mD Weierstrass-type approximations. However, the linear convergence rate of the Bernstein approximation [7] is reflected in the circumstance that these approaches are prevented from approximating a generic class of functions, but are limited to wellbehaving a-priori bounded analytical or holomorphic functions occurring, for instance, as solutions of elliptic PDEs.…”

“…Approximation power of multivariate splines. A prominent and well established alternative to the already discussed Weier-strass-type approximation schemes [3,15,17,18,19,72] is the by Carl de Boor et al developed multivariate spline interpolation [23,24,25,26]. The approximation result [27] for bivariate splines due to de Boor is given as: Theorem 1.1 (Carl de Boor).…”