An AI-Aided Algorithm for Multivariate Polynomial Reconstruction on Cartesian Grids and the PLG Finite Difference Method

Polynomial reconstruction on Cartesian grids is fundamental in many scientific and engineering applications, yet it is still an open problem how to construct for a finite subset K of Z D a lattice T ⊂ K so that multivariate polynomial interpolation on this lattice is unisolvent. In this work, we sol...

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Bibliographic Details
Published in:Journal of scientific computing Vol. 101; no. 3; p. 66
Main Authors: Zhang, Qinghai, Zhu, Yuke, Li, Zhixuan
Format: Journal Article
Language:English
Published: New York Springer US 01.12.2024
Springer Nature B.V
Subjects:
Algorithms
Approximation
Artificial intelligence
Cartesian coordinates
Computational Mathematics and Numerical Analysis
Finite difference method
Group theory
Interdisciplinary studies
Isomorphism
Lattices (mathematics)
Mathematical analysis
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Methods
Multivariate analysis
Ordinary differential equations
Partial differential equations
Permutations
Polynomials
Reconstruction
Set theory
Stencils
Theoretical
Depth-first search with backtracking
Multivariate polynomial interpolation
Artificial intelligence (AI)
65N06
Poised lattice generation
65D05
Data reconstruction
Finite difference methods
ISSN:0885-7474, 1573-7691
Online Access:Get full text
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Summary:Polynomial reconstruction on Cartesian grids is fundamental in many scientific and engineering applications, yet it is still an open problem how to construct for a finite subset K of Z D a lattice T ⊂ K so that multivariate polynomial interpolation on this lattice is unisolvent. In this work, we solve this open problem of poised lattice generation (PLG) via an interdisciplinary research of approximation theory, abstract algebra, and artificial intelligence. Specifically, we focus on the triangular lattices in approximation theory, study group actions of permutations upon triangular lattices, prove an isomorphism between the group of permutations and that of triangular lattices, and dynamically organize the state space of permutations so that a depth-first search of poised lattices has optimal efficiency. Based on this algorithm, we further develop the PLG finite difference method that retains the simplicity of Cartesian grids yet overcomes the disadvantage of legacy finite difference methods in handling irregular geometries. Results of various numerical tests demonstrate the effectiveness of our algorithm and the simplicity, flexibility, efficiency, and fourth-order accuracy of the PLG finite difference method.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02706-y