[Paper Review] Selection of Sparse Sets of Influence for Meshless Finite Difference Methods

Oleg Davydov|arXiv (Cornell University)|Aug 5, 2019

Advanced Numerical Methods in Computational Mathematics7 references4 citations

TL;DR

This paper proposes an efficient algorithm based on pivoted QR decomposition of weighted polynomial collocation matrices to select sparse sets of influence for meshless finite difference methods. The method reduces the number of nodes in the influence set while preserving the accuracy of numerical differentiation formulas, achieving consistency error comparable to that of the full set with significantly sparser system matrices.

ABSTRACT

We suggest an efficient algorithm for the selection of sparse subsets of a set of influence for the numerical discretization of differential operators on irregular nodes with polynomial consistency of a given order with the help of the QR decomposition of an appropriately weighted polynomial collocation matrix, and prove that the accuracy of the resulting numerical differentiation formulas is comparable with that of the formulas generated on the original set of influence.

Motivation & Objective

• To reduce the size of influence sets in meshless finite difference methods without sacrificing numerical accuracy.
• To develop a computationally efficient algorithm for selecting minimal yet accurate influence sets on irregular node distributions.
• To maintain polynomial consistency of a given order while minimizing the number of nodes used in the discretization of differential operators.
• To enable high-order accuracy with sparse linear systems, improving efficiency for irregular or poorly graded node sets.

Proposed method

• The method uses pivoted QR factorization on a weighted polynomial collocation matrix constructed from the influence set and a reference point.
• It selects a minimal subset of nodes by identifying the most significant columns through rank-revealing QR decomposition, ensuring polynomial consistency of order q.
• The algorithm ensures that the resulting numerical differentiation formula satisfies exactness conditions for all polynomials of degree ≤ q.
• A key component is the use of a weighted norm in the minimization, with weights proportional to the distance from the point of interest, to control consistency error.
• The method guarantees that the resulting sparse formula maintains an error bound proportional to the original full-set formula, with a factor depending on the condition number of the QR factorization.
• The algorithm is applied iteratively to identify a subset Y° with at most rank(A) nodes, where A is the collocation matrix, ensuring optimal sparsity.

Experimental results

Research questions

• RQ1 Can sparse influence sets be selected efficiently while preserving the accuracy of numerical differentiation formulas?
• RQ2 How can the number of nodes in influence sets be reduced without degrading the consistency error of meshless finite difference schemes?
• RQ3 What is the theoretical bound on the consistency error of the resulting sparse formula compared to the full influence set?
• RQ4 Can pivoted QR decomposition be used to identify minimal, accurate influence sets in a way that is robust to irregular node distributions?

Key findings

• The proposed method achieves a consistency error bound for the sparse formula that is within a factor of (1 + ||R₁⁻¹R₂||₂²)¹ᐟ² of the original full influence set, where R₁ and R₂ are blocks from the QR factorization.
• The size of the selected influence set is at most the rank of the polynomial collocation matrix, ensuring significant sparsity.
• The method maintains polynomial consistency of order q, ensuring high-order accuracy for smooth solutions.
• The algorithm is robust to irregular node distributions, making it suitable for use with simple, non-optimized node generation methods.
• The resulting linear systems are much sparser than those from nearest-neighbor selection, improving computational efficiency without sacrificing accuracy.