Speaker
Description
The parametric adaptive Antoulas–Anderson (p-AAA) algorithm is an effective method for multivariate rational approximation [Carracedo Rodriguez et al., 2023], inspired by the AAA framework for univariate rational approximation [Nakatsukasa et al., 2018].
In its original formulation, p-AAA aims to approximate a function $\mathbf{f} : \mathbb{C}^d \rightarrow \mathbb{C}$ via a multivariate rational function $\mathbf{r} \approx \mathbf{f}$ based on the samples
$$
\begin{equation*}
\mathbf{D} = \{ \mathbf{f}(z^{(1)},\ldots,z^{(d)}) \; | \; (z^{(1)},\ldots,z^{(d)}) \in \mathbf{S} \},
\end{equation*}
$$
with the sampling points given by the Cartesian grid
$$
\begin{equation*}
\mathbf{S} = \{ z_1^{(1)}, \ldots, z_{n_1}^{(1)} \} \times \cdots \times \{ z_1^{(d)}, \ldots, z_{n_d}^{(d)} \} \subset \mathbb{C}^d.
\end{equation*}
$$
The main idea of p-AAA is to iteratively construct $\mathbf{r}$ such that it interpolates $\mathbf{f}$ on a subgrid of $\mathbf{S}$ and minimizes the least-squares error over the remaining data. Despite its effectiveness, the original p-AAA algorithm can not always be applied in practice due to its grid data requirement, which may not always be available or feasible to obtain in practice. In this talk, we discuss how the p-AAA algorithm can be formulated with sampling points given by scattered data sets, i.e., arbitrary subsets of $\mathbb{C}^d$ that do not necessarily follow a grid structure. Towards this goal, we introduce a novel approach for enforcing interpolation conditions on scattered data sets via multivariate rational functions in barycentric form. To incorporate this approach into the p-AAA framework, a constrained linear least-squares problem is integrated into the algorithm. We show that this novel formulation of p-AAA is a strict generalization of the original algorithm, and demonstrate its effectiveness on several challenging examples.