An AI-aided algorithm for multivariate polynomial reconstruction on Cartesian grids and the PLG finite difference method

TL;DR

The paper tackles the challenge of unisolvent multivariate polynomial interpolation on irregular domains by formulating and solving the poised lattice generation (PLG) problem in $\mathbb{Z}^D$ with a cap of $n\le 6$. It introduces the Triangular Lattice Generation (TLG) algorithm, grounded in a group-theoretic view of ${D}$-permutations acting on triangular lattices, and proves that a depth-first search with backtracking yields optimal efficiency for generating poised lattices. Building on this, the PLG-FD method discretizes PDEs on Cartesian grids by fitting multivariate polynomials on poised lattices, enabling fourth-order accuracy even near complex boundaries and for cross-derivative terms, while maintaining grid simplicity. The approach yields high-accuracy, dimension-agnostic PDE solvers with controllable conditioning via stencil enrichment, and numerical tests across elliptic, parabolic, and time-dependent problems demonstrate its effectiveness and potential for broad impact in irregular-domain simulations.

Abstract

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 $\mathbb{Z}^{\textsf{D}}$ a lattice $\mathcal{T}\subset K$ 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 (AI). 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 AI 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, efficiency, and fourth-order accuracy of the PLG finite difference method.

Figures (13)

• Figure 1: Two solutions of the PLG problem, where the feasible set $K$ is obvious.
• Figure 2: An example search problem of finding the shortest path from Hangzhou to Taizhou. The initial state and the goal state are Hangzhou and Taizhou, respectively. This search problem can be solved by search algorithms such as the depth-first search, the best-first search, and the $A^*$ search. See russell21:_artif_intel for more details.
• Figure 3: An illustration of the efficiency advantage of test sets of type II over test sets of type I on spanning subtrees in the solution space. The TLG problem $(K,\mathbf{q})$ is shown in subplot (a), with ${\scriptsize \textsf{D}}=2$, $n=2$, $\mathbf{q}$ being the hollow dot at $(1,0)$, and the set of feasible nodes $K$ consisting of the grid nodes within the dot-dashed box and to the left of the solid curve. In both (b) and (c), a solid edge ended with a partial ${\scriptsize \textsf{D}}$-permutation represents a subtree spanning by an acceptable leaf node, a solid edge ended with "$\cdots$" a subtree spanning with the subtree omitted, a thick solid edge part of the path from the root node to the solution in subplot (a), and an edge labeled with a pair of scissors a subtree pruning. The only criterion for either accepting or rejecting $A^{(\ell,m)}$ is whether $A^{(\ell,m)}W_{(\ell,m)}\subset K$ holds or not, respectively. For example, the last child of the root node in subplot (b) is rejected because $A^{(1,0)}W_{(1,0)}$ with $A^{(1,0)}(1,0)=2$ is not a subset of $K$. As another example, $W_{(1,1)}=\emptyset$ dictates that all immediate children of the first partial ${\scriptsize \textsf{D}}$-permutation in the third row be enumerated, despite the fact that the second partial ${\scriptsize \textsf{D}}$-permutation in the fourth row maps some points in the principal lattice out of $K$. Any partial ${\scriptsize \textsf{D}}$-permutation in the penultimate line already fully determines a ${\scriptsize \textsf{D}}$-permutation: in each row of its matrix the first $n$ entries settle the last one since each 1-permutation is a bijection. Hence we combine the last ${\scriptsize \textsf{D}}$ steps of subtree spanning into one. Compared to that in subplot (c), the subtree spanned in subplot (b) according to test sets of type II is much smaller and thus the resulting backtracking algorithm in Definition \ref{['def:testOrderingOneNorm']} is more efficient. Suppose the depth-first search scans the solution space from the right to the left, then for test sets of type II it would take 5 prunings and 6 spannings to obtain the solution in subplot (a) while the numbers of prunings and spannings for test sets of type I are 14 and 18, respectively.
• Figure 4: Classify cell centers in the PLG-FD method. The curve represents part of the domain boundary; a gray square represents an exterior node, a white square an interior node, a yellow square a boundary node, and a cross mark a boundary point.
• Figure 5: Our strategy (KQN) to determine the set $K$ of feasible nodes from an even $n$ and an irregular FD node $\mathbf{q}$ for PLG discretization in the PLG-FD method. A solid dot represents the irregular FD node, to which the starting point $\mathbf{q}$ is always assigned. Gray squares represent exterior nodes and the curve the domain boundary $\partial \Omega$. A box of dot-dashed line represents a shifted cube centered at the first pivot. Yellow squares represent pivots in the sequence of pivot sets.

Theorems & Definitions (82)

• Definition 1.1: Lagrange interpolation problem (LIP) carnicer06:_inter
• Definition 1.2: PLG in $\mathbb{Z}^{{\scriptsize \textsf{D}}}$
• Definition 2.1
• Example 2.2
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Definition 2.5
• Definition 2.6