A tomographic technique called full-waveform inversion has recently shown promise as a fast, affordable, and safe modality to image the brain using ultrasound. However, its high computational cost and memory footprint currently limit its clinical applicability. Here, we address these challenges through a frequency-adaptive discretisation of the imaging domain and lossy compression techniques. Because full-waveform inversion relies on the adjoint-state method, every iteration involves solving the wave equation over a discretised spatiotemporal grid and storing the numerical solution to calculate gradient updates. The computational cost depends on the grid size, which is controlled by the maximum frequency being modelled. Since the propagated frequency typically varies during the reconstruction, we reduce reconstruction time and memory use by allowing the grid size to change throughout the inversion. Moreover, we combine this approach with multiple lossy compression techniques that exploit the sparsity of the wavefield to further reduce its memory footprint. We explore applying these techniques in the spatial, wavelet, and wave atom domains. Numerical experiments using a human-head model show that our methods lead to a 30% reduction in reconstruction time and up to three orders of magnitude less memory, while negligibly affecting the accuracy of the reconstructions.