On sparse interpolation of rational functions and gcds

In this note, we present a variant of a probabilistic algorithm by Cuyt and Lee for the sparse interpolation of multivariate rational functions. We also present an analogous method for the computation of sparse gcds.

Minimum norm interpolation in the ℓ1(ℕ) space

We consider the minimum norm interpolation problem in the [Formula: see text] space, aiming at constructing a sparse interpolation solution. The original problem is reformulated in the pre-dual space, thereby inducing a norm in a related finite-dimensional Euclidean space. The dual problem is then transformed into a linear programming problem, which can be solved by existing methods. With that done, the original interpolation problem is reduced by solving an elementary finite-dimensional linear algebra equation. A specific example is presented to illustrate the proposed method, in which a sparse solution in the [Formula: see text] space is compared to the dense solution in the [Formula: see text] space. This example shows that a solution of the minimum norm interpolation problem in the [Formula: see text] space is indeed sparse, while that of the minimum norm interpolation problem in the [Formula: see text] space is not.

Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

Approximation and uncertainty quantification methods based on Lagrange interpolation are typically abandoned in cases where the probability distributions of one or more system parameters are not normal, uniform, or closely related distributions, due to the computational issues that arise when one wishes to define interpolation nodes for general distributions. This paper examines the use of the recently introduced weighted Leja nodes for that purpose. Weighted Leja interpolation rules are presented, along with a dimension-adaptive sparse interpolation algorithm, to be employed in the case of high-dimensional input uncertainty. The performance and reliability of the suggested approach is verified by four numerical experiments, where the respective models feature extreme value and truncated normal parameter distributions. Furthermore, the suggested approach is compared with a well-established polynomial chaos method and found to be either comparable or superior in terms of approximation and statistics estimation accuracy.