FFTW 3.3.5: Transposed distributions

Internally, FFTW’s MPI transform algorithms work by first computing cannam@127: transforms of the data local to each process, then by globally cannam@127: transposing the data in some fashion to redistribute the data cannam@127: among the processes, transforming the new data local to each process, cannam@127: and transposing back. For example, a two-dimensional n0 by cannam@127: n1 array, distributed across the n0 dimension, is cannam@127: transformd by: (i) transforming the n1 dimension, which are cannam@127: local to each process; (ii) transposing to an n1 by n0 cannam@127: array, distributed across the n1 dimension; (iii) transforming cannam@127: the n0 dimension, which is now local to each process; (iv) cannam@127: transposing back. cannam@127: cannam@127:

However, in many applications it is acceptable to compute a cannam@127: multidimensional DFT whose results are produced in transposed order cannam@127: (e.g., n1 by n0 in two dimensions). This provides a cannam@127: significant performance advantage, because it means that the final cannam@127: transposition step can be omitted. FFTW supports this optimization, cannam@127: which you specify by passing the flag FFTW_MPI_TRANSPOSED_OUT cannam@127: to the planner routines. To compute the inverse transform of cannam@127: transposed output, you specify FFTW_MPI_TRANSPOSED_IN to tell cannam@127: it that the input is transposed. In this section, we explain how to cannam@127: interpret the output format of such a transform. cannam@127: cannam@127: cannam@127:

Suppose you have are transforming multi-dimensional data with (at cannam@127: least two) dimensions n₀ × n₁ × n₂ × … × n_d-1. As always, it is distributed along cannam@127: the first dimension n₀. Now, if we compute its DFT with the cannam@127: FFTW_MPI_TRANSPOSED_OUT flag, the resulting output data are stored cannam@127: with the first two dimensions transposed: n₁ × n₀ × n₂ ×…× n_d-1, cannam@127: distributed along the n₁ dimension. Conversely, if we take the cannam@127: n₁ × n₀ × n₂ ×…× n_d-1 data and transform it with the cannam@127: FFTW_MPI_TRANSPOSED_IN flag, then the format goes back to the cannam@127: original n₀ × n₁ × n₂ × … × n_d-1 array. cannam@127:

There are two ways to find the portion of the transposed array that cannam@127: resides on the current process. First, you can simply call the cannam@127: appropriate ‘local_size’ function, passing n₁ × n₀ × n₂ ×…× n_d-1 (the cannam@127: transposed dimensions). This would mean calling the ‘local_size’ cannam@127: function twice, once for the transposed and once for the cannam@127: non-transposed dimensions. Alternatively, you can call one of the cannam@127: ‘local_size_transposed’ functions, which returns both the cannam@127: non-transposed and transposed data distribution from a single call. cannam@127: For example, for a 3d transform with transposed output (or input), you cannam@127: might call: cannam@127:

cannam@127:

ptrdiff_t fftw_mpi_local_size_3d_transposed(
cannam@127:                 ptrdiff_t n0, ptrdiff_t n1, ptrdiff_t n2, MPI_Comm comm,
cannam@127:                 ptrdiff_t *local_n0, ptrdiff_t *local_0_start,
cannam@127:                 ptrdiff_t *local_n1, ptrdiff_t *local_1_start);
cannam@127:

Here, local_n0 and local_0_start give the size and cannam@127: starting index of the n0 dimension for the cannam@127: non-transposed data, as in the previous sections. For cannam@127: transposed data (e.g. the output for cannam@127: FFTW_MPI_TRANSPOSED_OUT), local_n1 and cannam@127: local_1_start give the size and starting index of the n1 cannam@127: dimension, which is the first dimension of the transposed data cannam@127: (n1 by n0 by n2). cannam@127:

(Note that FFTW_MPI_TRANSPOSED_IN is completely equivalent to cannam@127: performing FFTW_MPI_TRANSPOSED_OUT and passing the first two cannam@127: dimensions to the planner in reverse order, or vice versa. If you cannam@127: pass both the FFTW_MPI_TRANSPOSED_IN and cannam@127: FFTW_MPI_TRANSPOSED_OUT flags, it is equivalent to swapping the cannam@127: first two dimensions passed to the planner and passing neither cannam@127: flag.) cannam@127:

6.4.3 Transposed distributions