Moving-Least-Squares

Summary of Moving Least Squares (MLS) Method

Introduction

The Moving Least Squares (MLS) method is a meshless approximation technique used for function interpolation and numerical analysis. Unlike finite element or finite difference methods, MLS does not require a predefined mesh, making it highly flexible for irregular geometries and adaptive refinement.

Key Concepts

1. Approximation Function

MLS constructs an approximation of a function $$f(x)$$ in the form:

$$f(x) = \sum_{i=1}^{m} p_i(x) a_i(x)$$

where:

• $$p_i(x)$$ are basis functions (e.g., monomials from a Pascal triangle expansion).
• $$a_i(x)$$ are coefficients that minimize a weighted least squares error.

2. Pascal Triangle Basis Functions

The basis functions are selected from a 2D Pascal Triangle up to second-order terms:

$$ p(x, y) = \begin{bmatrix} 1, & x - x_0, & y - y_0, & (x - x_0)^2, & (x - x_0)(y - y_0), & (y - y_0)^2 \end{bmatrix}^T $$

This basis includes:

• Constant term: $1$
• Linear terms: $x - x_0$, $y - y_0$
• Quadratic terms: $(x - x_0)^2$, $(x - x_0)(y - y_0)$, $(y - y_0)^2$

These polynomials ensure local smoothness and accuracy in function approximation.

3. Weighting Function

A weight function $w(x)$ is introduced to ensure that nearby points influence the approximation more than distant points. A common choice is the Gaussian weight function:

$$ w(x) = e^{-\left(\frac{|x - x_i|}{d_i}\right)^2} $$

where $d_i$ is a characteristic distance defining the influence zone.

In the code, 30 nearest neighbors are used for each evaluation point. The radius of influence is determined dynamically using the distance to the 30th nearest neighbor.

4. Normal Equation System

The MLS method determines the unknown coefficients $a_i(x)$ by solving:

$$ A(x) a(x) = B(x) f $$

where:

• $A(x)$ is the moment matrix, formed from basis functions and weight functions.
• $B(x)$ is a matrix related to the weight function and sample points.
• $f$ is the vector of function values at known sample points.

5. Derivatives Computation

Once the shape functions are computed, derivatives (gradients, Hessians) can be found using:

$$ \frac{\partial f}{\partial x} = \sum_{i=1}^{m} \frac{\partial p_i(x)}{\partial x} a_i(x) $$

This allows MLS to be used for solving partial differential equations (PDEs).

Implementation Details

• Our Python implementation uses 30 nearest neighbors for each evaluation point.
• The moment matrix is regularized using the pseudo-inverse (pinv) to handle potential singularities.
• Shape functions and their derivatives are computed and stored as sparse matrices.
• The condition number of $A(x)$ is recorded to monitor numerical stability.

Advantages of MLS

✅ Meshless: No need for predefined connectivity between points.
✅ High Accuracy: Smooth and continuous function representation.
✅ Adaptability: Can handle complex geometries and irregular point distributions.
✅ Locality Control: Influence is restricted to a fixed number of nearest neighbors (30 in this case).

Applications

🔹 Computational Fluid Dynamics (CFD)
🔹 Structural Analysis (Meshfree methods like SPH)
🔹 Image Processing (Interpolation, Denoising)
🔹 Machine Learning (Kernel-Based Regression)

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References:

• Liu, G.R. (2003). Meshfree Methods: Moving Beyond the Finite Element Method. CRC Press.

  Code Availability

⚠️ This method has been used in the following research paper:
➡ Boutopoulos, Ioannis D., et al. "Two-phase biofluid flow model for magnetic drug targeting." Symmetry 12.7 (2020): 1083.

📌 If you want access to the implementation codes from the paper, please let me know.