Arnoldi-Enhanced Multivariate Hermite Interpolation of Manifold-Valued Data
This paper presents a robust enhancement of the Tangent space Hermite Interpolation (THI) method for manifold-valued data by integrating the multivariate Arnoldi process. To circumvent the inherent numerical instability of multivariate confluent Vandermonde matrices, we use a $G$-Arnoldi-based recurrence to construct a discrete orthogonal polynomial basis directly on the tangent space. The method generates better numerical conditioning for high-order approximations. We analyze the convergence rates for both $C^0$ and $C^1$ errors in the multivariate setting. When only function values are used, the $C^0$ approximation error decays as $\mathcal{O}\left(\sqrt{M} n^{-m}\right)$. For the $C^1$ error without derivative data, the rate becomes $\mathcal{O}\left(\sqrt{M} h^{-1} n^{-m}\right)$, where $h$ is the fill distance of the sampling set. When derivative data are additionally available, the $C^1$ error is $\mathcal{O}\left(\sqrt{M} n^{-(m-1)}\right)$. In all cases, $n$ is the polynomial degree, $m$ denotes the regularity of the target function, and $M$ is the number of sampling points. Importantly, as $n$ increases, the required number of points $M$ must also increase. This reveals the interplay among approximation order, sampling density ($M$), fill distance ($h$), dimension ($d$), and the regularity ($m$) of the target function. Extensive numerical experiments conducted on the special orthogonal group $SO(3)$ and the unit sphere $S^2$ show that the Arnoldi-enhanced THI method outperforms the Kriging-based approaches in terms of both computational efficiency and accuracy.