Related papers: Arnoldi-Enhanced Multivariate Hermite Interpolatio…
High-order interpolation on the Grassmann manifold $\Gr(n, p)$ is often hindered by the computational overhead and derivative instability of SVD-based geometric mappings. To solve the challenges, we propose a stabilized framework that…
In this paper, we propose two methods for multivariate Hermite interpolation of manifold-valued functions. On the one hand, we approach the problem via computing suitable weighted Riemannian barycenters. To satisfy the conditions for…
The Arnoldi process provides an efficient framework for approximating functions of a matrix applied to a vector, i.e., of the form $f(M)\bm{b}$, by repeated matrix-vector multiplications. In this paper, we derive error estimates for…
In the least-squares fitting framework, the Vandermonde with Arnoldi (V+A) method presented in [Brubeck, Nakatsukasa, and Trefethen, {SIAM Review}, 63 (2021), pp. 405-415] is an effective approach to compute a polynomial that approximates…
In this note, we extend the Vandermonde with Arnoldi method recently advocated by P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen to dealing with the confluent Vandermonde matrix. To apply the Arnoldi process, it is critical to find a…
Vandermonde matrices are exponentially ill-conditioned, rendering the familiar "polyval(polyfit)" algorithm for polynomial interpolation and least-squares fitting ineffective at higher degrees. We show that Arnoldi orthogonalization fixes…
Interpolation of data on non-Euclidean spaces is an active research area fostered by its numerous applications. This work considers the Hermite interpolation problem: finding a sufficiently smooth manifold curve that interpolates a…
The main contribution of this paper is twofold: On the one hand, a general framework for performing Hermite interpolation on Riemannian manifolds is presented. The method is applicable, if algorithms for the associated Riemannian…
We propose a method for interpolating divergence-free continuous magnetic fields via vector potential reconstruction using Hermite interpolation, which ensures high-order continuity for applications requiring adaptive, high-order ordinary…
In recent years, a great deal of attention has been focused on numerically solving exponential integrators. The important ingredient to the implementation of exponential integrators is the efficient and accurate evaluation of the so called…
This paper presents a novel method for polynomial approximation (Hermite approximation) using the fusion of value and derivative information. Therefore, the least-squares error in both domains is simultaneously minimized. A covariance…
The Hermite-Taylor method evolves all the variables and their derivatives through order $m$ in time to achieve a $2m+1$ order rate of convergence. The data required at each node of the staggered Cartesian meshes used by this method makes…
We present a sampling strategy suitable for optimization problems characterized by high-dimensional design spaces and noisy outputs. Such outputs can arise, for example, in time-averaged objectives that depend on chaotic states. The…
We describe a general approach for constructing a broad class of operators approximating high-dimensional curves based on geometric Hermite data. The geometric Hermite data consists of point samples and their associated tangent vectors of…
The randomized Arnoldi process has been used in large-scale scientific computing because it produces a well-conditioned basis for the Krylov subspace more quickly than the standard Arnoldi process. However, the resulting Hessenberg matrix…
Numerical integration methods are central to the study of self-gravitating systems, particularly those comprised of many bodies or otherwise beyond the reach of analytical methods. Predictor-corrector schemes, both multi-step methods and…
We propose an efficient and easy-to-implement gradient-enhanced least squares Monte Carlo method for computing price and Greeks (i.e., derivatives of the price function) of high-dimensional American options. It employs the sparse Hermite…
Polynomial preconditioning can improve the convergence of the Arnoldi method for computing eigenvalues. Such preconditioning significantly reduces the cost of orthogonalization; for difficult problems, it can also reduce the number of…
In this work, we present a trust-region optimization framework that employs Hermite kernel surrogate models. The method targets optimization problems with computationally demanding objective functions, for which direct optimization is often…
Multivariate quantum signal processing (M-QSP) has recently been shown to be applicable for non-Hermitian Hamiltonian simulation, opening several problems regarding the optimization landscape, angle-finding, and constant-factor analysis. We…