  Inverse problems occur in a variety of parameter identification tasks in
engineering. Such problems are challenging in practice, as they require
repeated evaluation of computationally expensive forward models. We introduce a
unifying framework of multilevel optimization that can be applied to a wide
range of optimization-based solvers. Our framework provably reduces the
computational cost associated with evaluating the expensive forward maps
stemming from various physical models. To demonstrate the versatility of our
analysis, we discuss its implications for various methodologies including
multilevel (accelerated, stochastic) gradient descent, a multilevel ensemble
Kalman inversion and a multilevel Langevin sampler. We also provide numerical
experiments to verify our theoretical findings.
