For inpainting with linear partial differential equations (PDEs) such as homogeneous or biharmonic diffusion, sophisticated data optimisation strategies have been found recently. These allow high-quality reconstructions from sparse known data. While they have been explicitly developed with compression in mind, they have not entered actual codecs so far: Storing these optimised data efficiently is a nontrivial task. Since this step is essential for any competetive codec, we propose two new compression frameworks for linear PDEs: Efficient storage of pixel locations obtained from an optimal control approach, and a stochastic strategy for a locally adaptive, tree-based grid. Suprisingly, our experiments show that homogeneous diffusion inpainting can surpass its often favoured biharmonic counterpart in compression. Last but not least, we demonstrate that our linear approach is able to beat both JPEG2000 and the nonlinear state-of-the-art in PDE-based image compression.