Unstructured MLS for Manifolds & MPM

• The paper presents a unified MLS approach achieving smooth, high-order approximations on unstructured manifolds with error bounds of O(h^(m+1)).
• It leverages local chart reconstruction and high-order polynomial fitting to ensure robust function regression and continuous shape functions in MPM applications.
• Empirical results demonstrate lower RMSE and enhanced computational efficiency while eliminating cell-crossing artifacts in complex geometries.

Unstructured Moving Least Squares for Manifolds and Material Point Methods (UMLS-MPM) encompasses a set of algorithms applying moving least squares (MLS) techniques to function approximation on sampled manifolds and Lagrangian–Eulerian discretizations in computational mechanics. The unifying characteristic is their ability to yield smooth, high-accuracy approximations and stable field reconstructions on data defined over non-uniform and unstructured domains—without reliance on global embeddings or structured grids. UMLS-MPM approaches are of critical importance in geometric data analysis, machine learning on manifolds, and large-deformation computational mechanics, particularly in settings where the underlying domain is known only through noisy samples or unstructured tessellation.

1. Manifold MLS: Chart Reconstruction and Atlas Generation

Let $M\subset\mathbb R^n$ be an unknown $d$-dimensional $C^2$ manifold. Given a set of noisy samples $R=\{r_i = \tilde r_i + n_i\}_1^I$ near $M$ ($\|\mathbf n_i\|\leq\sigma_M$), the first stage constructs a local coordinate chart around any query $r$ by solving

$$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$

where $H = q + \text{Span}\{e_1, ..., e_d\}$ is an affine $d$-subspace, $d$0, $d$1 is a compactly supported, smooth weight, and $d$2 is the fill distance of clean samples. The minimization is subject to

• $d$3,
• $d$4 within a ball $d$5, $d$6,
• $d$7 contains at least one sample.

Under mild hypotheses, there exists a unique, $d$8-smooth mapping $d$9. The projection property $C^2$0, $C^2$1 for $C^2$2 provides chart consistency over neighborhoods. The collection $C^2$3 with $C^2$4 defines a $C^2$5 atlas for a manifold $C^2$6 close to $C^2$7 in Hausdorff distance (Sober et al., 2017).

2. High-Order Polynomial MLS on Local Charts

With the local chart $C^2$8, each sample $C^2$9 is paired as $R=\{r_i = \tilde r_i + n_i\}_1^I$0 and $R=\{r_i = \tilde r_i + n_i\}_1^I$1. The moving least squares approximant solves for $R=\{r_i = \tilde r_i + n_i\}_1^I$2, the space of polynomials of degree $R=\{r_i = \tilde r_i + n_i\}_1^I$3 in $R=\{r_i = \tilde r_i + n_i\}_1^I$4 variables: $R=\{r_i = \tilde r_i + n_i\}_1^I$5 where $R=\{r_i = \tilde r_i + n_i\}_1^I$6 is a $R=\{r_i = \tilde r_i + n_i\}_1^I$7 weight of compact support. The approximation at $R=\{r_i = \tilde r_i + n_i\}_1^I$8 is $R=\{r_i = \tilde r_i + n_i\}_1^I$9. The Backus–Gilbert representation gives

$M$0

where $M$1 are smooth in $M$2 (Sober et al., 2017).

3. Approximation Order, Smoothness, and Error Guarantees

Let $M$3 denote the fill distance for $M$4. Under $M$5 smoothness of the target and sufficient density, projected samples in each chart are $M$6-dense. The resulting MLS fit achieves

$M$7

in the noiseless case, recovering the classical $M$8 order (Sober et al., 2017). The full map $M$9 is $\|\mathbf n_i\|\leq\sigma_M$0-smooth, guaranteed by the $\|\mathbf n_i\|\leq\sigma_M$1 chart construction and smooth, compactly supported weights.

4. UMLS-MPM Algorithms for Function and Tensor Field Approximation

Two algorithmic forms are canonical:

Manifold function regression (Sober et al., 2017):

• $\|\mathbf n_i\|\leq\sigma_M$2 Iterative least-squares fits return an affine subspace around $\|\mathbf n_i\|\leq\sigma_M$3.
• $\|\mathbf n_i\|\leq\sigma_M$4 Projects samples, builds the polynomial MLS normal matrix, and evaluates at the origin to yield $\|\mathbf n_i\|\leq\sigma_M$5.

Material Point Method on Tessellations (Cao et al., 2023):

• For a point $\|\mathbf n_i\|\leq\sigma_M$6 in an unstructured mesh, neighbors $\|\mathbf n_i\|\leq\sigma_M$7 are assigned raw MLS weights $\|\mathbf n_i\|\leq\sigma_M$8, then scaled by a “diminishing” factor $\|\mathbf n_i\|\leq\sigma_M$9 based on mesh adjacency and barycentric coordinates.
• The sample weight is $r$0, ensuring $r$1 continuity of the resulting MLS shape functions $r$2 across cell faces.
• Shape functions are constructed as

$r$3

where $r$4 aggregates the weighted second moments. The field and its gradient are then

$r$5

Steps for explicit MPM include particle–grid transfers, mass/momentum assembly, and velocity updates using MLS kernels (Cao et al., 2023).

5. Stability, Mesh Independence, and $r$6 Kernel Construction

The central innovation of UMLS-MPM for material point simulation is the construction of weights via a product of a smooth, compactly supported kernel $r$7 and a diminishing factor $r$8. For each mesh cell, $r$9 is given by

$$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$0

where $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$1 is the simplex containing $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$2, $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$3 are its barycentric coordinates, and $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$4 encodes mesh adjacency. This guarantees:

• $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$5 for all vertices in the current cell;
• $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$6 as $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$7 approaches a boundary losing/gaining the node.

When these conditions hold, both MLS moment matrices and hence shape functions $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$8 are at least $$J(r; q, H) = \sum_{i=1}^I d(r_i, H)^2\, \theta\left(\|r_i - q\|/h\right)$$9, which removes the cell-crossing artifacts characteristic of piecewise-linear interpolation on unstructured meshes (Cao et al., 2023).

6. Computational Complexity and Implementation Properties

For manifold regression, the dominant cost is in local chart construction: linear least-squares steps cost $H = q + \text{Span}\{e_1, ..., e_d\}$0 in ambient dimension $H = q + \text{Span}\{e_1, ..., e_d\}$1, with constant iterations; chart-based local MLS in $H = q + \text{Span}\{e_1, ..., e_d\}$2 requires $H = q + \text{Span}\{e_1, ..., e_d\}$3 per fit and $H = q + \text{Span}\{e_1, ..., e_d\}$4 for evaluation. The overall complexity is $H = q + \text{Span}\{e_1, ..., e_d\}$5, yielding linear scaling in $H = q + \text{Span}\{e_1, ..., e_d\}$6. No global eigendecomposition, kernel matrix, or dimension reduction is performed (Sober et al., 2017).

In MPM, MLS assembly over a first-ring vertex neighborhood is local per timestep. Explicit UMLS-MPM remains subject to a standard CFL stability constraint, $H = q + \text{Span}\{e_1, ..., e_d\}$7 (Cao et al., 2023). Mesh quality (simplex conditioning, support radius $H = q + \text{Span}\{e_1, ..., e_d\}$8) crucially affects invertibility and convergence; extremely skewed elements can lead to poor conditioning.

7. Numerical Experiments and Performance

Empirical evaluations demonstrate:

• Function regression: On smooth manifolds (helix in $H = q + \text{Span}\{e_1, ..., e_d\}$9, $d$0), the error decays as $d$1. High-dimensional tests (image regression in $d$2) achieve RMSE $d$3 0.0066 with $d$4 samples and $d$5. Against regression baselines (MALLER, NEDE), UMLS-MPM attains lower RMSE and 2–3$d$6 faster CPU time for Klein-bottle tasks in $d$7 (Sober et al., 2017).
• MPM on unstructured meshes: In all cases, UMLS-MPM eliminates cell-crossing errors, matches or exceeds accuracy of structured-grid B-spline MPM, and delivers $d$8nd-order spatial convergence (e.g., in the 1D vibrating bar, 2D cantilever, and 3D slope failure tests). Complex geometries and contact boundaries (3D sphere expansion) are resolved without producing stress discontinuities or artifacts (Cao et al., 2023).

Table: Algorithmic Core Components

Domain	Charting/Neighborhood	MLS Core	Continuity
Manifold regression	Affine tangent chart	Weighted poly MLS	$d$9
MPM on meshes	Vertex/element adjacency	$d$00 diminished MLS	$d$01

The provided empirical and theoretical guarantees establish UMLS-MPM as an intrinsic, local, mesh-agnostic, and smooth high-order regression/interpolation framework for both geometric learning and continuum mechanics without global embeddings or kernel matrices (Sober et al., 2017, Cao et al., 2023).