  3D image registration is one of the most fundamental and computationally
expensive operations in medical image analysis. Here, we present a
mixed-precision, Gauss--Newton--Krylov solver for diffeomorphic registration of
two images. Our work extends the publicly available CLAIRE library to GPU
architectures. Despite the importance of image registration, only a few
implementations of large deformation diffeomorphic registration packages
support GPUs. Our contributions are new algorithms to significantly reduce the
run time of the two main computational kernels in CLAIRE: calculation of
derivatives and scattered-data interpolation. We deploy (i) highly-optimized,
mixed-precision GPU-kernels for the evaluation of scattered-data interpolation,
(ii) replace Fast-Fourier-Transform (FFT)-based first-order derivatives with
optimized 8th-order finite differences, and (iii) compare with state-of-the-art
CPU and GPU implementations. As a highlight, we demonstrate that we can
register $256^3$ clinical images in less than 6 seconds on a single NVIDIA
Tesla V100. This amounts to over 20$\times$ speed-up over the current version
of CLAIRE and over 30$\times$ speed-up over existing GPU implementations.
