We consider the problem of interpolating a sparse multivariate polynomial over finite field, represented with black box. Building on algorithm Ben-Or and Tiwari (1988) for polynomials fields characteristic zero, we develop new Monte Carlo by doing additional probes. To interpolate $$f \in {\mathbb{F}_q}[{x_1}, \ldots ,{x_n}]$$ partial degree bound D term T, our costs $${O^ \sim }(nT{\log ^2}q ...

We consider the noisy polynomial interpolation problem of recovering an unknown s-sparse f(X) over ring Zpk residues modulo pk, where p is a small prime and k large integer parameter, from approximate values f(t)∈Zpk. Similar results are known for p, however case power modulus with k, new requires different techniques. give deterministic time algorithm, which almost given more than half bits f(...

We consider the problem of Lagrange polynomial interpolation in high or countably infinite dimension, motivated by the fast computation of solution to parametric/stochastic PDE’s. In such applications there is a substantial advantage in considering polynomial spaces that are sparse and anisotropic with respect to the different parametric variables. In an adaptive context, the polynomial space i...

Abstract: Multivariate polynomial metamodels are widely used for uncertainty quantification due to the development of polynomial chaos methods and stochastic collocation. However, these metamodels only provide point predictions. There is no known method that can quantify interpolation error probabilistically and design interpolation points using available data to reduce the error. We shall intr...

We investigate convergence of the rational-trigonometric-polynomial interpolation that performs convergence acceleration of the classical trigonometric interpolation by sequential application of polynomial and rational correction functions. Unknown parameters of the rational corrections are determined along the ideas of the Fourier-Pade approximations. The resultant interpolation we call as Fou...

Birkhoff (or lacunary) interpolation is an extension of polynomial interpolation that appears when observation gives irregular information about function and its derivatives. A Birkhoff interpolation problem is not always solvable even in the appropriate polynomial or rational space. In this paper we split up the initial problem in subproblems having a unique polynomial solution and use multino...