Multivariate confluent Vandermonde with G-Arnoldi and applications (2404.09266v1)
Abstract: 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 its certain 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.
- A. P. Austin, M. Krishnamoorthy, S. Leyffer, S. Mrenna, J. Müller and H. Schulz, Practical algorithms for multivariate rational approximation, Comput. Phys. Commun., 261 (2021), 107663.
- T. Bagby, L. Bos and N. Levenberg, Multivariate simultaneous approximation, Constr. Approx., 18 (2002), 569–577.
- M. V. Barel and A. Chesnokov, A method to compute recurrence relation coefficients for bivariate orthogonal polynomials by unitary matrix transformations, Numer. Algorithms, 55 (2009), 383–402.
- B. Beckermann, The condition number of real Vandermonde, Krylov and positive definite Hankel matrices, Numer. Math., 85 (2000), 553–577.
- P. D. Brubeck, Y. Nakatsukasa and L. N. Trefethen, Vandermonde with Arnoldi, SIAM Rev., 63 (2021), 405–415.
- P. D. Brubeck and L. N. Trefethen, Lightning Stokes solver, SIAM J. Sci. Comput., 44 (2022), A1205–A1226.
- M. Caliari, S. D. Marchi and M. Vianello, Bivariate polynomial interpolation on the square at new nodal sets, J. Comput. Appl. Math., 165 (2005), 261–274.
- A. M. Delgado, L. Fernández, T. E. Pérez, M. A. Piñar and Y. Xu, Orthogonal polynomials in several variables for measures with mass points, Numer. Algorithms, 55 (2010), 245–264.
- W. Gautschi, On the inverses of Vandermonde and confluent Vandermonde matrices, Numer. Math., 4 (1962), 117–123.
- W. Gautschi, Orthogonal polynomials: computation and approximation, OUP Oxford, 2004.
- I. Gohberg, P. Lancaster and L. Rodman, Matrices and indefinite scalar products, Birkhäuser Verlag, Basel, Boston, Stuttgart, 1983.
- N. J. Higham, Stability analysis of algorithms for solving confluent Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990), 23–41.
- J. M. Hokanson, Multivariate rational approximation using a stabilized Sanathanan-Koerner iteration, 2020, URL arXiv:2009.10803v1.
- I.-P. Kim and A. R. Kräuter, VDR decomposition of Chebyshev-Vandermonde matrices with the Arnoldi process, Lin. Multilin. Alg., URL https://doi.org/10.1080/03081087.2024.2335487.
- R.-C. Li, Asymptotically optimal lower bounds for the condition number of a real Vandermonde matrix, SIAM J. Matrix Anal. Appl., 28 (2006), 829–844.
- R.-C. Li, Lower bounds for the condition number of a real confluent Vandermonde matrix, Math. Comp., 75 (2006), 1987–1995.
- R.-C. Li, Vandermonde matrices with Chebyshev nodes, Linear Algebra Appl., 428 (2008), 1803–1832.
- R.-C. Li, Structural Preserving Model Reductions, Technical Report 04-02, University of Kentucky, Lexington, 2004.
- Y. Nakatsukasa and L. N. Trefethen, Reciprocal-log approximation and planar PDE solvers, SIAM J. Numer. Anal., 59 (2021), 2801–2822.
- Q. Niu, H. Zhang and Y. Zhou, Confluent Vandermonde with Arnoldi, Appl. Math. Lett., 135 (2023), 108420.
- V. Y. Pan, How bad are Vandermonde matrices?, SIAM J. Matrix Anal. Appl., 37 (2016), 676–694.
- P.-O. Persson and G. Strang, A simple mesh generator in MATLAB, SIAM Rev., 46 (2004), 329–345.
- L. Reichel, Construction of polynomials that are orthogonal with respect to a discrete bilinear form, Adv. in Comput. Math., 1 (1993), 241–258.
- J. H. Wilkinson, The Algebraic Eigenvalue Problem, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1988.
- Y. Xu, On discrete orthogonal polynomials of several variables, Adv. Appl. Math., 33 (2004), 615–632.
- L. Yang, L.-H. Zhang and Y. Zhang, The Lq-weighted dual programming of the linear Chebyshev approximation and an interior-point method, 2023, URL https://arxiv.org/abs/2308.07636.
- L.-H. Zhang, Y. Su and R.-C. Li, Accurate polynomial fitting and evaluation via Arnoldi, Numerical Algebra, Control and Optimization, to appear, Doi:10.3934/naco.2023002.
- L.-H. Zhang, L. Yang, W. H. Yang and Y.-N. Zhang, A convex dual programming for the rational minimax approximation and Lawson’s iteration, 2023, URL https://arxiv.org/pdf/2308.06991v1.
- W. Zhu and Y. Nakatsukasa, Convergence and near-optimal sampling for multivariate function approximations in irregular domains via Vandermonde with Arnoldi, 2023, URL https://arxiv.org/abs/2301.12241.