Asymptotically better algorithm for M2M
On master, M2M is $`O(p^2d)`$ for order p and dimension d, but CSE reduces this down to $`O(p^{2d-1})`$ sometimes. For eg: in Helmholtz 2D, full taylor is $`O(p^{2d-1})`$ and HelmholtzConformingTaylor is $`O(p^{2d-1.5})`$. This commit produces expressions in $`O(p^{2d-1})`$ consistently regardless of how good CSE is and $`O(p^{2d-2})`$ for compressed. The new algorithm uses the observation that M2M coefficients have the form in 2D, $`B_{m, n} = \sum_{i\le m, j\le n} A_{i, j} d_x^i d_y^j \binom{m}{i} \binom{n}{j}`$ and can be rewritten as follows, Let $`T_{m, n} = \sum_{i\le m} A_{i, n} d_x^i \binom{m}{i}`$. Then, $`B_{m, n} = \sum_{j\le n} T_{m, j} d_y^j \binom{n}{j}`$ and $`T_{m, n}`$ are $`p^2`$ number of temporary variables that are reused for different M2M coefficients and costs $`p`$ per variable. Total cost for calculating $`T_{m, n}`$ is $`p^3`$ and similar for $`B_{m, n}`$
parent
cfa3683a
Loading
Loading
Pipeline
#20327
failed
with stage
in
39 minutes and 20 seconds
Loading