Ali Siahkoohi
University of Central Florida
Title
Posterior sampling in dual space for constrained inverse problems
Abstract
Inverse problems constrained by partial differential equations are often ill-conditioned due to noisy, incomplete data and a nonlinear map from the model to the data. A Bayesian formulation describes their solution by a posterior distribution, whose samples quantify the uncertainty. Sampling in the reduced space, where the state equation is solved exactly at every step, however, is hampered by spurious local modes that trap the particles of gradient-based samplers. To address this challenge, we sample in the dual space, keeping the state as an unknown and enforcing the state equation through an augmented Lagrangian. We achieve this by integrating the alternating direction method of multipliers with Stein variational gradient descent (SVGD), where each particle has its own state and multipliers and every update is a gradient step. While the sampled density changes with the multipliers, we show for the Helmholtz equation with Gaussian noise that the fixed points of this sampler are exactly those of SVGD on the reduced posterior, for every penalty parameter. At a fixed particle the multipliers converge to the adjoint state at an explicit rate, and converging particles stop at such a fixed point. Because the state is minimized rather than integrated out, no Jacobian determinant appears in the sampled density. We validate the method on a Rosenbrock problem with a closed-form posterior and on full waveform inversion of the Marmousi II model, demonstrating calibrated uncertainty and posterior contraction with increasing data coverage. Applied to traveltime tomography with the constraint placed on the data, the same idea scales to three-dimensional problems.
Joint work with K. Aghazade, A. Vishwakarma, and A. Gholami.
