Consensus ADMM for inverse problems governed by multiple PDE models

The alternating direction method of multipliers (ADMM) provides a natural way of solving inverse problems with multiple forward models and nonsmooth regularization. ADMM allows splitting these large-scale inverse problems into smaller, simpler sub-problems, for which computationally efficient solvers are available. In this work, we are interested in the case in which the forward models stem from partial differential equations and the inversion parameter is a scalar or vector field belonging to an infinite-dimensional Hilbert space. Then, the ADMM methods allow us to split the original inverse problems into several (one for each forward model) single-PDE inverse problems with a smooth Tikhonov-like regularization and, an unconstrained denoising-like problem with non-smooth regularization to update the consensus variable. We discuss several adaptations of the ADMM needed to maintain consistency with the underlining infinite-dimensional problem and ensure scalability. Specifically, we show how using the correct norm in the consensus equations (ie, the one of the underlining Hilbert space) improves both the accuracy and computational efficiency of ADMM. Moreover, we use the Lagrangian formalism to derive expressions of first and second-order optimality conditions for the continuous form of each subproblem, which are then discretized with the finite element method using FEniCS. In particular, for solving the decoupled Tikhonov regularized inverse problems we utilized an inexact Newton conjugate gradient solver in hIPPYlib, an extensible software framework for large-scale inverse problems governed by PDEs. To handle the denoising problem stemming from the consensus variable update we apply the PETScTAOSolver built into FEniCS. We present two imaging applications inspired by electrical impedance tomography and quantitative photoacoustic tomography to demonstrate the proposed method's effectiveness.