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Left Inverses for B-spline Subdivision Matrices in Tensor-Product Spaces

Marcelo Actis, Silvano Figueroa, Eduardo M. Garau

TL;DR

This work addresses left inverses for dyadic subdivision matrices between coarse and fine univariate and tensor-product B-spline spaces on uniform grids. It constructs banded left inverses $B$ via a local least-squares approach, ensuring $BA=I$ and delivering coarse representations $oldsymbol{ ilde{c}}=Boldsymbol{c}$ that closely match global $L^2$-best approximations while remaining computationally efficient. The central design uses a locality width $r$ and selects the minimum-norm LS solution to optimize stability, with explicit expressions for the interior weight vector $oldsymbol{ u}_r$ and a clear ancestor structure for coarse-spline representations. The tensor-product extension leverages Kronecker structure, yielding $oldsymbol{eta}^x(x)$, $oldsymbol{eta}^y(y)$ transformations and $ ext{vec}( ilde{C})=(B_yoldsymbol{ackslash} ext{ } B_x) ext{vec}(C)$, enabling scalable multivariate coarsening. Experimental results demonstrate stable, local, and near-optimal $L^2$-projection performance across degrees and dimensions, supporting applications in multilevel and adaptive spline methods with reduced computational cost and localized error propagation.

Abstract

In this article, we study dyadic coarsening operators in univariate spline spaces and in tensor-product spline spaces over uniform grids. Our construction is strongly motivated by the work of Bartels, Golub, and Samavati (2006), Some observations on local least squares, BIT, 46(3):455--477. The proposed operators are local in nature and yield approximations to a given spline that are comparable to the global L2-best approximation, while being significantly faster to compute and computationally inexpensive.

Left Inverses for B-spline Subdivision Matrices in Tensor-Product Spaces

TL;DR

This work addresses left inverses for dyadic subdivision matrices between coarse and fine univariate and tensor-product B-spline spaces on uniform grids. It constructs banded left inverses via a local least-squares approach, ensuring and delivering coarse representations that closely match global -best approximations while remaining computationally efficient. The central design uses a locality width and selects the minimum-norm LS solution to optimize stability, with explicit expressions for the interior weight vector and a clear ancestor structure for coarse-spline representations. The tensor-product extension leverages Kronecker structure, yielding , transformations and , enabling scalable multivariate coarsening. Experimental results demonstrate stable, local, and near-optimal -projection performance across degrees and dimensions, supporting applications in multilevel and adaptive spline methods with reduced computational cost and localized error propagation.

Abstract

In this article, we study dyadic coarsening operators in univariate spline spaces and in tensor-product spline spaces over uniform grids. Our construction is strongly motivated by the work of Bartels, Golub, and Samavati (2006), Some observations on local least squares, BIT, 46(3):455--477. The proposed operators are local in nature and yield approximations to a given spline that are comparable to the global L2-best approximation, while being significantly faster to compute and computationally inexpensive.

Paper Structure

This paper contains 13 sections, 1 theorem, 53 equations, 18 figures, 11 tables.

Table of Contents

  1. Introduction
  2. B-splines and subdivision matrices
  3. Left inverses for subdivision matrix for the univariate case
  4. An efficient construction of left inverses through a local least-squares method
  5. On the importance of choosing the minimum-norm solution
  6. Some explicit expressions for $\boldsymbol{\omega}_r$.
  7. B-spline ancestors and local coarsening
  8. Coarsening operators for tensor-product splines
  9. Extension to $D$-directional tensor-product spline spaces.
  10. Experimental analysis
  11. Stability and approximation quality for coarsening operators
  12. Successive coarsening in tensor product spline spaces
  13. Single uniform coarsening for a tensor product spline

Key Result

Proposition 3.3

Let $A \in \mathbb{R}^{n \times \hat{n}}$ be a subdivision matrix defined by knot insertion 1D, and let $B \in \mathbb{R}^{\hat{n} \times n}$ be the matrix constructed as above. Then,

Figures (18)

  • Figure 1: Uniform partition $\hat{Z}$ with $\hat{N} = 6$ elements and its dyadic refinement $Z$. The associated open knot vectors $\hat{{\bm\xi}}$ and ${\bm\xi}$ correspond to polynomial degree $p = 2$, with multiplicity $p+1 = 3$ at the endpoints of the interval.
  • Figure 2: Coarse and fine quadratic B-spline bases are shown at the top left and top right, respectively. At the bottom left, a coarse B-spline basis function (in dashed line) can be expressed as a linear combination of the fine B-spline basis functions shown at the bottom right.
  • Figure 3: We present the subdivision matrices $A_p$ for $p = 1, 2, 3, 4$. Only the nonzero entries are displayed. Note that, except for the first and last $p$ columns, the columns of $A_p$ are generated by the vector $\boldsymbol{\eta}_p \in \mathbb{R}^{p+2}$, which slides downward with a stride of 2 from one column to the next.
  • Figure 4: Structure of a subdivision matrix $A$ with relevant blocks.
  • Figure 5: Some examples of submatrices $A_{\text{in}}$ of the subdivision matrix $A$, corresponding to polynomial degrees $p = 2$ and $p = 3$.
  • ...and 13 more figures

Theorems & Definitions (4)

  • Remark 3.2: On the parameters involved in the construction of the left inverse
  • Proposition 3.3
  • proof
  • Remark 3.4