  PDE-constrained inverse problems are some of the most challenging and
computationally demanding problems in computational science today. Fine meshes
that are required to accurately compute the PDE solution introduce an enormous
number of parameters and require large scale computing resources such as more
processors and more memory to solve such systems in a reasonable time. For
inverse problems constrained by time dependent PDEs, the adjoint method that is
often employed to efficiently compute gradients and higher order derivatives
requires solving a time-reversed, so-called adjoint PDE that depends on the
forward PDE solution at each timestep. This necessitates the storage of a high
dimensional forward solution vector at every timestep. Such a procedure quickly
exhausts the available memory resources. Several approaches that trade
additional computation for reduced memory footprint have been proposed to
mitigate the memory bottleneck, including checkpointing and compression
strategies. In this work, we propose a close-to-ideal scalable compression
approach using autoencoders to eliminate the need for checkpointing and
substantial memory storage, thereby reducing both the time-to-solution and
memory requirements. We compare our approach with checkpointing and an
off-the-shelf compression approach on an earth-scale ill-posed seismic inverse
problem. The results verify the expected close-to-ideal speedup for both the
gradient and Hessian-vector product using the proposed autoencoder compression
approach. To highlight the usefulness of the proposed approach, we combine the
autoencoder compression with the data-informed active subspace (DIAS) prior to
show how the DIAS method can be affordably extended to large scale problems
without the need of checkpointing and large memory.
