Multivariate confluent Vandermonde with G-Arnoldi and applications

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 an underlying univariate function $f$. Extensions of V+A include its multivariate version and the univariate confluent V+A; the latter enables us to use the information of the derivative of $f$ in obtaining the approximation polynomial. In this paper, we shall extend V+A further to the multivariate confluent V+A. Besides the technical generalization of the univariate confluent V+A, we also introduce a general and application-dependent $G$-orthogonalization in the Arnoldi process. We shall demonstrate with several applications that, by specifying an application-related $G$-inner product, the desired approximate multivariate polynomial, as well as certain of its partial derivatives can be computed accurately from a well-conditioned least-squares problem whose coefficient matrix is orthonormal. The desired multivariate polynomial is represented in a discrete $G$-orthogonal polynomials basis which admits an explicit recurrence, and therefore, facilitates evaluating function values and certain partial derivatives at new nodes efficiently. We demonstrate its flexibility by applying it to solve the multivariate Hermite least-squares problem and PDEs with various boundary conditions in irregular domains.