Backpropagation (adjoint) timing on the same 4-GPU configuration. Open symbols denote per-call backward evaluation time after compilation; filled symbols show the one-time compilation cost of the backward/adjoint kernels. The MG adjoint compiles and evaluates robustly up to the largest tested grids, with evaluation times that scale smoothly with resolution. FFT compilation and evaluation are reliable only up to 512³ cells in this setup; for larger grids, we were unable to obtain a working distributed-FFT backward pass, likely due to memory pressure and/or communication-plan constraints associated with large multi-GPU transposes. Parameters for MG were chosen at each resolution to require <0.1% accuracy over all pixels for net force calculation. Absolute timings vary with system configuration and interconnect.