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The two-dimensional five-point Poisson matrix under nested dissection

Example

For $k \geq 1$ , let $G_{k}$ be the grid on coordinate vertices ${0, \dots, k - 1}^{2}$ , relabeled by the lexicographic zero-based bijection in Recursive coordinate-separator ordering for two- and three-dimensional grids, and let $A_{k} = 4 I - Adj (G_{k})$ , the Dirichlet five-point Poisson matrix. The coordinate nested-dissection ordering recursively places its median row and column after the child rectangles. This example checks positive definiteness and instantiates the two-dimensional factorization bounds.

Facts & Assumptions

Given: An integer $k \geq 1$ , the $k \times k$ nearest-neighbor grid $G_{k}$ , and $A_{k} = 4 I - Adj (G_{k})$ .

[F1]

Recursive coordinate-separator ordering for two- and three-dimensional grids: In dimension two the procedure orders child rectangles first and then appends the union of the selected median row and median column.

[F2]

Nested dissection gives the stated work and factor-storage bounds on regular grids: For a symmetric positive-definite matrix with this grid pattern and $k \geq 2$ , exact scalar multifrontal Cholesky under the stated ordering uses $O (n^{3 / 2})$ arithmetic work and $O (n \log n)$ symbolic factor storage for $n = k^{2}$ . Storage counts allocated factor entries, including structural zeros, and excludes temporary contribution arrays. For $k = 1$ , both costs are $O (1)$ .

Verification

1.1givenalgebra

For every grid vertex $v$ , let $b (v) = 4 - \deg_{G_{k}} (v)$ be its number of missing boundary neighbors. Expanding squares gives $x^{T} A_{k} x = \sum_{{v, w} \in E (G_{k})} (x_{v} - x_{w})^{2} + \sum_{v \in G_{k}} b (v) x_{v}^{2} .$ The right side is nonnegative. If it is zero, every adjacent pair has equal coordinates and every boundary vertex with $b (v) > 0$ has value zero. The grid is connected and has a boundary vertex, so all coordinates are equal to zero. Hence $A_{k}$ is positive definite.

2.1step 1.1F1F2algebra

By [F1], the order is the coordinate nested-dissection order on the five-point grid. The matrix is symmetric positive definite by step 1.1 and its off-diagonal pattern is exactly $G_{k}$ . For $k \geq 2$ , applying [F2] with $n = k^{2}$ to exact scalar multifrontal Cholesky gives arithmetic work $O ((k^{2})^{3 / 2}) = O (k^{3})$ and symbolic factor storage $O (k^{2} \log (k^{2})) = O (k^{2} \log k)$ , with the allocation convention of [F2]. For $k = 1$ , the matrix and factor each have one entry.

3.1step 1.1step 2.1∎

Step 1.1 verifies the matrix hypothesis and step 2.1 gives the nested-dissection cost specialization.

Depends on

Used by

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Sources

Jennifer Scott and Miroslav Tůma, Algorithms for Sparse Linear Systems, Chapter 8, §8.4 (standard reference, not scraped)
David Bindel, CS 5220, Lecture 16: Sparse Direct Solvers, slides 13–14 (standard reference, not scraped)