FFTW 3.3.5: Transposed distributions

Internally, FFTW’s MPI transform algorithms work by first computing Chris@42: transforms of the data local to each process, then by globally Chris@42: transposing the data in some fashion to redistribute the data Chris@42: among the processes, transforming the new data local to each process, Chris@42: and transposing back. For example, a two-dimensional n0 by Chris@42: n1 array, distributed across the n0 dimension, is Chris@42: transformd by: (i) transforming the n1 dimension, which are Chris@42: local to each process; (ii) transposing to an n1 by n0 Chris@42: array, distributed across the n1 dimension; (iii) transforming Chris@42: the n0 dimension, which is now local to each process; (iv) Chris@42: transposing back. Chris@42: Chris@42:

However, in many applications it is acceptable to compute a Chris@42: multidimensional DFT whose results are produced in transposed order Chris@42: (e.g., n1 by n0 in two dimensions). This provides a Chris@42: significant performance advantage, because it means that the final Chris@42: transposition step can be omitted. FFTW supports this optimization, Chris@42: which you specify by passing the flag FFTW_MPI_TRANSPOSED_OUT Chris@42: to the planner routines. To compute the inverse transform of Chris@42: transposed output, you specify FFTW_MPI_TRANSPOSED_IN to tell Chris@42: it that the input is transposed. In this section, we explain how to Chris@42: interpret the output format of such a transform. Chris@42: Chris@42: Chris@42:

Suppose you have are transforming multi-dimensional data with (at Chris@42: least two) dimensions n₀ × n₁ × n₂ × … × n_d-1. As always, it is distributed along Chris@42: the first dimension n₀. Now, if we compute its DFT with the Chris@42: FFTW_MPI_TRANSPOSED_OUT flag, the resulting output data are stored Chris@42: with the first two dimensions transposed: n₁ × n₀ × n₂ ×…× n_d-1, Chris@42: distributed along the n₁ dimension. Conversely, if we take the Chris@42: n₁ × n₀ × n₂ ×…× n_d-1 data and transform it with the Chris@42: FFTW_MPI_TRANSPOSED_IN flag, then the format goes back to the Chris@42: original n₀ × n₁ × n₂ × … × n_d-1 array. Chris@42:

There are two ways to find the portion of the transposed array that Chris@42: resides on the current process. First, you can simply call the Chris@42: appropriate ‘local_size’ function, passing n₁ × n₀ × n₂ ×…× n_d-1 (the Chris@42: transposed dimensions). This would mean calling the ‘local_size’ Chris@42: function twice, once for the transposed and once for the Chris@42: non-transposed dimensions. Alternatively, you can call one of the Chris@42: ‘local_size_transposed’ functions, which returns both the Chris@42: non-transposed and transposed data distribution from a single call. Chris@42: For example, for a 3d transform with transposed output (or input), you Chris@42: might call: Chris@42:

Chris@42:

ptrdiff_t fftw_mpi_local_size_3d_transposed(
Chris@42:                 ptrdiff_t n0, ptrdiff_t n1, ptrdiff_t n2, MPI_Comm comm,
Chris@42:                 ptrdiff_t *local_n0, ptrdiff_t *local_0_start,
Chris@42:                 ptrdiff_t *local_n1, ptrdiff_t *local_1_start);
Chris@42:

Here, local_n0 and local_0_start give the size and Chris@42: starting index of the n0 dimension for the Chris@42: non-transposed data, as in the previous sections. For Chris@42: transposed data (e.g. the output for Chris@42: FFTW_MPI_TRANSPOSED_OUT), local_n1 and Chris@42: local_1_start give the size and starting index of the n1 Chris@42: dimension, which is the first dimension of the transposed data Chris@42: (n1 by n0 by n2). Chris@42:

(Note that FFTW_MPI_TRANSPOSED_IN is completely equivalent to Chris@42: performing FFTW_MPI_TRANSPOSED_OUT and passing the first two Chris@42: dimensions to the planner in reverse order, or vice versa. If you Chris@42: pass both the FFTW_MPI_TRANSPOSED_IN and Chris@42: FFTW_MPI_TRANSPOSED_OUT flags, it is equivalent to swapping the Chris@42: first two dimensions passed to the planner and passing neither Chris@42: flag.) Chris@42:

6.4.3 Transposed distributions