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We address the use of diffuse light to characterize the space-varying absorption coefficient in tissue, posed as an inverse problem, ill-posed due to the physics and limitations on source-detector location. Accurate reliable solutions require a priori constraints. We extend our previously utilized admissible solution approach, with convex constraint functions defining admissibility conditions, by using the deep-cut ellipsoid algorithm, iteratively choosing the most important constraint value, and introducing a multi-resolution grid method to decrease the computational burden. Simulations in representative 2-D scenarios indicate that we successfully reconstruct relatively deep anomalies while reducing the computational time more than 95%.