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The total variation (TV) penalty, as many other analysis-sparsity problems,does not lead to separable factors or a proximal operatorwith a closed-formexpression, such as soft thresholding for the $\ell\_1$ penalty. As a result,in a variational formulation of an inverse problem or statisticallearningestimation, it leads to challenging non-smooth optimization problemsthat areoften solved with elaborate single-step first-order methods. When thedata-fitterm arises from empirical measurements, as in brain imaging, it isoften veryill-conditioned and without simple structure. In this situation, in proximalsplitting methods, the computation cost of thegradient step can easily dominateeach iteration. Thus it is beneficialto minimize the number of gradientsteps.We present fAASTA, a variant of FISTA, that relies on an internal solverforthe TV proximal operator, and refines its tolerance to balancecomputationalcost of the gradient and the proximal steps. We give benchmarksandillustrations on "brain decoding": recovering brain maps fromnoisymeasurements to predict observed behavior. The algorithm as well astheempirical study of convergence speed are valuable for any non-exactproximaloperator, in particular analysis-sparsity problems.

FAASTA: A fast solver for total-variation regularization of ill-conditioned problems with application to brain imaging