Regularly striped preconditioner for implicit clothing simulation

Abstract

This paper investigates the question of whether the Jacobi preconditioner can be made to run faster by including a few additional outlying diagonals, for the case of mesh-based implicit clothing simulation. For the given $N \times N$ block system matrix ${\mathbf {A}}$, we investigate two regularly striped (RS) preconditioners: (1) 3-RS which has two symmetric outlying diagonals with offset $\frac{N}{2}$ from the main diagonal and (2) 5-RS which has four symmetric outlying diagonals with offsets $\frac{N}{3}$ and $\frac{2N}{3}$ from the main diagonal. This paper finds that both 3- and 5-RS preconditioners are consistently superior to the Jacobi preconditioner in the performance. Based on the loop iteration count and time measurement, we finally recommend 5-RS rather than 3-RS.

A Calculating the inverse of the block k-RS preconditioner

In this section, we show how to calculate the inverse of the block k-RS preconditioner. A key point of the derivation is using the footprint. For a block k-RS preconditioner ${\mathbf {P}}$, when multiplying ${\mathbf {P}}$ and ${\mathbf {P}}^{-1}$, due to the sparsity pattern of ${\mathbf {P}}$ and ${\mathbf {P}}^{-1}$ (see [18]), it turns out only footprint(i, i) of ${\mathbf {P}}$ are multiplied with footprint(i, i) of ${\mathbf {P}}^{-1}$. Let us denote the footprint(i, i) of ${\mathbf {P}}$ and ${\mathbf {P}}^{-1}$ as $\hat{{\mathbf {P}}}_{(i,i)}$ and $\hat{{\mathbf {P}}}^{-1}_{(i,i)}$, respectively.

For the case of the block 3-RS preconditioner ${\mathbf {P}}$,

$$\begin{aligned} \hat{{\mathbf {P}}}_{(i,i)} = \begin{bmatrix} {\mathbf {P}}_{(i,i)} &{} {\mathbf {P}}_{(i,i+\frac{N}{2})} \\ {\mathbf {P}}_{(i+\frac{N}{2},i)} &{} {\mathbf {P}}_{(i+\frac{N}{2},i+\frac{N}{2})} \end{bmatrix} = \begin{bmatrix} {\mathbf {A}} &{} {\mathbf {B}} \\ {\mathbf {C}} &{} {\mathbf {D}} \end{bmatrix}, \end{aligned}$$

(19)

where ${\mathbf {P}}_{(a,b)}$ represents the block (a, b) of ${\mathbf {P}}$. We must have

$$\begin{aligned} \hat{{\mathbf {P}}}_{(i,i)} \hat{{\mathbf {P}}}^{-1}_{(i,i)} = \begin{bmatrix} {\mathbf {I}} &{} {\mathbf {0}} \\ {\mathbf {0}} &{} {\mathbf {I}} \end{bmatrix}, \end{aligned}$$

(20)

for $i = 0,\dots , \frac{N}{2} - 1$. Replacing $\hat{{\mathbf {P}}}_{(i,i)}$ of Eq. 20 with Eq. 19 and comparing the left- and right-hand sides of Eq. 20, it follows that

$$\begin{aligned} {\mathbf {P}}^{-1}_{(i,i)} = \begin{bmatrix} {\mathbf {A}}^{-1} + \mathbf {FCA}^{-1} &{} -{\mathbf {F}} \\ -\mathbf {ECA}^{-1} &{} {\mathbf {E}} \end{bmatrix}, \end{aligned}$$

(21)

where ${\mathbf {E}} = ({\mathbf {D}}-\mathbf {CA}^{-1}{\mathbf {B}})^{-1}$ and ${\mathbf {F}} = {\mathbf {A}}^{-1}\mathbf {BE}$. (You can see the detailed derivation in [8].)

For the case of the block k-RS preconditioner with $k>3$, the above procedure can be applied recursively. For example, if ${\mathbf {P}}$ is 5-RS preconditioner, as shown in Fig. 10, it can be decomposed into the 3-RS preconditioner ${\mathbf {A}}$ and three matrices $\mathbf {B, C, D}$. Then, ${\mathbf {P}}^{-1}$ can be calculated according to Eq. 21.

Fig. 10

a 3-RS preconditioner can be decomposed into four diagonal matrices. b 5-RS preconditioner can be decomposed into a 3-RS preconditioner and three matrices