6.5 Multi-dimensional MPI DFTs of Real Data

FFTW's MPI interface also supports multi-dimensional DFTs of real data, similar to the serial r2c and c2r interfaces. (Parallel one-dimensional real-data DFTs are not currently supported; you must use a complex transform and set the imaginary parts of the inputs to zero.)

The key points to understand for r2c and c2r MPI transforms (compared to the MPI complex DFTs or the serial r2c/c2r transforms), are:

• Just as for serial transforms, r2c/c2r DFTs transform n₀ × n₁ × n₂ × … × n_d-1 real data to/from n₀ × n₁ × n₂ × … × (n_d-1/2 + 1) complex data: the last dimension of the complex data is cut in half (rounded down), plus one. As for the serial transforms, the sizes you pass to the ‘plan_dft_r2c’ and ‘plan_dft_c2r’ are the n₀ × n₁ × n₂ × … × n_d-1 dimensions of the real data.
• Although the real data is conceptually n₀ × n₁ × n₂ × … × n_d-1, it is physically stored as an n₀ × n₁ × n₂ × … × [2 (n_d-1/2 + 1)] array, where the last dimension has been padded to make it the same size as the complex output. This is much like the in-place serial r2c/c2r interface (see Multi-Dimensional DFTs of Real Data), except that in MPI the padding is required even for out-of-place data. The extra padding numbers are ignored by FFTW (they are not like zero-padding the transform to a larger size); they are only used to determine the data layout.
• The data distribution in MPI for both the real and complex data is determined by the shape of the complex data. That is, you call the appropriate ‘local size’ function for the n₀ × n₁ × n₂ × … × (n_d-1/2 + 1)

  complex data, and then use the same distribution for the real data except that the last complex dimension is replaced by a (padded) real dimension of twice the length.

For example suppose we are performing an out-of-place r2c transform of L × M × N real data [padded to L × M × 2(N/2+1)], resulting in L × M × N/2+1 complex data. Similar to the example in 2d MPI example, we might do something like:

     #include <fftw3-mpi.h>
     
     int main(int argc, char **argv)
     {
         const ptrdiff_t L = ..., M = ..., N = ...;
         fftw_plan plan;
         double *rin;
         fftw_complex *cout;
         ptrdiff_t alloc_local, local_n0, local_0_start, i, j, k;
     
         MPI_Init(&argc, &argv);
         fftw_mpi_init();
     
         /* get local data size and allocate */
         alloc_local = fftw_mpi_local_size_3d(L, M, N/2+1, MPI_COMM_WORLD,
                                              &local_n0, &local_0_start);
         rin = fftw_alloc_real(2 * alloc_local);
         cout = fftw_alloc_complex(alloc_local);
     
         /* create plan for out-of-place r2c DFT */
         plan = fftw_mpi_plan_dft_r2c_3d(L, M, N, rin, cout, MPI_COMM_WORLD,
                                         FFTW_MEASURE);
     
         /* initialize rin to some function my_func(x,y,z) */
         for (i = 0; i < local_n0; ++i)
            for (j = 0; j < M; ++j)
              for (k = 0; k < N; ++k)
            rin[(i*M + j) * (2*(N/2+1)) + k] = my_func(local_0_start+i, j, k);
     
         /* compute transforms as many times as desired */
         fftw_execute(plan);
     
         fftw_destroy_plan(plan);
     
         MPI_Finalize();
     }

Note that we allocated rin using fftw_alloc_real with an argument of 2 * alloc_local: since alloc_local is the number of complex values to allocate, the number of real values is twice as many. The rin array is then local_n0 × M × 2(N/2+1) in row-major order, so its (i,j,k) element is at the index (i*M + j) * (2*(N/2+1)) + k (see Multi-dimensional Array Format).

As for the complex transforms, improved performance can be obtained by specifying that the output is the transpose of the input or vice versa (see Transposed distributions). In our L × M × N r2c example, including FFTW_TRANSPOSED_OUT in the flags means that the input would be a padded L × M × 2(N/2+1) real array distributed over the L dimension, while the output would be a M × L × N/2+1 complex array distributed over the M dimension. To perform the inverse c2r transform with the same data distributions, you would use the FFTW_TRANSPOSED_IN flag.