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Fast 3D butterfly algorithm

Equation 10 is the discretized form of a 3D oscillatory integral of the type

$\displaystyle u(\mathbf{x})=\int_K e^{ 2\pi i \Phi(\mathbf{x},\mathbf{k})}v(\mathbf{k})  d\mathbf{k},\quad \mathbf{x}\in X,$ (11)

whose fast evaluation can be realized by a butterfly algorithm (Candès et al., 2009).

The overall structure of the 3D butterfly algorithm basically follows its 2D analogue. The idea is to partition the computational domains $ X$ and $ K$ recursively into a pair of octrees, $ T_X$ and $ T_K$ , ending at level $ l=\log N$ (see Figure 1 for an illustration). Here $ N$ is chosen as an integer power of two, which is on the order of the maximum of $ \vert\Phi(\mathbf{x},\mathbf{k})\vert$ for all possible $ \mathbf{x}$ and $ \mathbf{k}$ (so it is mainly determined by the range of variables $ (f,x,y)$ and $ (\tau,p,q)$ ). A crucial property of this structure is that at arbitrary level $ l$ , the side lengths $ w(A)$ of a box $ A$ in $ T_X$ and $ w(B)$ of a box $ B$ in $ T_K$ always satisfy $ w(A)w(B)=1/N$ . Then when $ \mathbf{x}$ , $ \mathbf{k}$ restricted in $ A$ and $ B$ respectively, one can construct a low-rank, separated expansion for the kernel function $ e^{2\pi i\Phi(\mathbf{x},\mathbf{k})}$ (via a Chebyshev interpolation):

$\displaystyle \left \vert e^{2\pi i \Phi(\mathbf{x},\mathbf{k})} -\sum_{r=1}^{r_{\epsilon}}\alpha_r^{AB}(\mathbf{x})\beta_r^{AB}(\mathbf{k})\right\vert<\epsilon.$ (12)

By embedding this approximation in the above tree structure and traversing $ T_X$ from top to bottom, $ T_K$ from bottom to top, we arrive at a fast algorithm running in complexity $ O(N^3\log N)$ (there are $ N^3$ pairs of boxes $ (A,B)$ on every level, and there are $ \log N$ levels in total). Detailed description of the algorithm can be found in Hu et al. (2013), where the difference between 2D and 3D formulations should be clear from the context.

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Figure 1.
Butterfly tree structure for the special case of $ N=4$ .
[pdf] [png]

Considering the initial Fourier transform for preparing data in $ (f,x,y)$ domain, the overall complexity of our algorithm is roughly $ O(N_xN_yN_t\log N_t)+ O(C_1(r_{\epsilon})(N_fN_xN_y+N_{\tau}N_pN_q))+O(C_2(r_{\epsilon})N^3\log N)$ ( $ r_{\epsilon}$ terms are due to low-rank approximations, and $ C_2(r_{\epsilon})$ is bigger than $ C_1(r_{\epsilon})$ ).

By comparison, the conventional straightforward velocity scan requires at least $ O(N_{\tau}N_pN_qN_xN_y)$ computations, which may quickly become a bottleneck as the problem size increases. Yet the efficiency of our algorithm is controlled mainly by $ O(N^3\log N)$ with an $ \epsilon$ -dependent constant, where $ N$ is determined by the degree of oscillations in the kernel $ e^{2\pi i\Phi(\mathbf{x},\mathbf{k})}$ . Generally speaking, $ N$ depends on the maximum frequency and offset in the dataset, and the range of parameters in the model space. In practice, $ N$ can often be chosen smaller than the grid size.

The significance of above analysis for the fast algorithm lies in the fact that the input and output data sizes $ N_tN_xN_y$ and $ N_{\tau}N_p N_q$ have little impact on the final computational cost; a dense sampling therefore becomes affordable.


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Next: Numerical examples Up: Theory Previous: Basic formulation

2015-03-27