Bayesian mesh adaptation for estimating distributed parameters

The problem of estimating numerically a distributed parameter from indirect measurements arises in many applications, and in that context the choice of the discretization plays an important role. In fact, guaranteeing a certain level of accuracy of the forward model that maps the unknown to the observations may require a fine discretization, adding to the complexity of the problem and to the computational cost. On the other hand, reducing the complexity of the problem by adopting a coarser discretization may increase the modeling error and can be very detrimental for ill-posed inverse problems. To balance accuracy and complexity, we propose an adaptive algorithm for adjusting the discretization level automatically and dynamically while estimating the unknown distributed parameter by an iterative scheme. In the Bayesian paradigm, all unknowns, including the metric that defines the discretization, are modeled as random variables. Our approach couples the discretization with a Bayesian hierarchical hyperparameter that is estimated simultaneously with the unknown parameter of primary interest. The viability of the proposed algorithm, the Bayesian mesh adaptation (BMA) is assessed on two test cases: a fan-beam X-ray tomography problem and an inverse source problem for a Darcy flow model.