Multivariate Confluent Vandermonde with G-Arnoldi: A Flexible Approach for Multivariate Polynomial Approximation and Partial Derivative Computation

Główne pojęcia

The paper presents a multivariate confluent Vandermonde with G-Arnoldi (MV+G-A) method that can efficiently compute a multivariate polynomial approximation and its partial derivatives using both function values and higher-order partial derivative information at a set of nodes.

Streszczenie

The paper introduces the multivariate confluent Vandermonde with G-Arnoldi (MV+G-A) method, which is an extension of the univariate confluent Vandermonde with Arnoldi (V+A) method to the multivariate case.
Key highlights:

The method uses the multivariate monomial basis in the grevlex ordering and derives recurrence relations for the first-order and second-order partial derivatives of the basis functions.
The Arnoldi process is used to construct a new discrete G-orthogonal polynomials basis, where the G-inner product can be specified based on the target application.
This allows the desired multivariate polynomial approximation and its partial derivatives to be computed accurately from a well-conditioned least-squares problem with an orthonormal coefficient matrix.
The new basis functions have an explicit recurrence, enabling efficient evaluation of the approximant and its partial derivatives at new nodes.
The method is demonstrated on applications such as the multivariate Hermite least-squares problem and solving PDEs with various boundary conditions in irregular domains.

The paper provides a flexible and efficient approach for multivariate polynomial approximation that can leverage both function values and partial derivative information.

Statystyki

The multivariate monomial basis {ϕi(x
x
x)}g
i=1 for the total degree polynomial space Pd,tol
n
satisfies the relation ϕi(x
x
x) = xuiϕsi(x
x
x), where si is the smallest index such that ∃ui ∈[d].
The first-order and second-order partial derivatives of the basis functions {ϕi(x
x
x)}g
i=1 can be computed using the recurrence relations in (2.4a) and (2.4b).

Cytaty

"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."
"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 including the multivariate Hermite least-squares problem and solving PDEs with various boundary conditions in irregular domains that, by specifying an application-related G-inner product in the Gram-Schmidt process within the Arnoldi process, the desired approximate multivariate polynomial as well as its certain partial derivatives can be computed accurately from a well-conditioned least-squares problem whose coefficient matrix is orthonormal."

Kluczowe wnioski z

Multivariate confluent Vandermonde with G-Arnoldi and applications

by Lei-Hong Zha... o arxiv.org 04-16-2024

https://arxiv.org/pdf/2404.09266.pdf

Głębsze pytania

How can the MV+G-A method be extended to handle higher-order partial derivatives beyond the second-order

To extend the MV+G-A method to handle higher-order partial derivatives beyond the second-order, we can modify the algorithm to incorporate the computation of third-order, fourth-order, and higher-order partial derivatives of the basis functions. This extension would involve updating the recurrence relations for the higher-order derivatives, similar to how the second-order derivatives were handled in the original algorithm. Additionally, the G-Arnoldi process would need to be adjusted to accommodate the increased complexity of the basis functions and their derivatives. By iteratively computing and orthogonalizing the higher-order derivatives, the MV+G-A method can be extended to handle arbitrary orders of partial derivatives for more accurate polynomial approximations.

What are the theoretical guarantees on the approximation accuracy and convergence rates of the MV+G-A method compared to other multivariate polynomial approximation techniques

Theoretical guarantees on the approximation accuracy and convergence rates of the MV+G-A method can be established through analysis of the properties of the G-orthogonalization process and the Arnoldi iteration. By considering the conditioning of the Vandermonde matrix, the orthogonality of the basis functions, and the convergence behavior of the Arnoldi process, one can derive bounds on the approximation error and convergence rates of the method. Comparisons with other multivariate polynomial approximation techniques can be made based on the stability, efficiency, and accuracy of the MV+G-A method in various applications. Theoretical results can provide insights into the performance of the method under different conditions and help assess its suitability for specific problems.

Can the MV+G-A method be adapted to handle non-polynomial basis functions or non-Cartesian domains

Adapting the MV+G-A method to handle non-polynomial basis functions or non-Cartesian domains would require modifications to accommodate the different types of functions and domains. For non-polynomial basis functions, the algorithm would need to be adjusted to work with the new basis functions, ensuring that the orthogonality and recurrence properties are maintained. In the case of non-Cartesian domains, the node set X and the inner product matrix G would need to be defined appropriately for the specific domain geometry. By customizing the algorithm to handle non-polynomial functions and non-Cartesian domains, the MV+G-A method can be applied to a wider range of problems with diverse function spaces and geometries.