Organizer: Andrés Christen, CIMAT, México
Dealing with nuisance parameters in Bayesian model calibration
We consider the problem of physical parameter estimation by combining experimental data with computer model simulations. The Bayesian Model Calibration (BMC) framework accommodates a wide variety of uncertainties, including model misspecification or model discrepancy, and is thus a useful tool for solving inverse problems. In the presence of a high-dimensional vector of nuisance parameters, the problem of inferring physical parameters is often poorly identified. We propose various statistical approaches for calibrating physical parameters in such situation. First, we consider regularization, the specification of hierarchical priors on the nuisance parameters, with the goal to preserve measurement uncertainty and to avoid over-fitting. Secondly, we consider modularization, an alternative to the full Bayesian approach which mimics the forward propagation of uncertainty by sampling the nuisance parameter via an alternative scheme, rather than updating them via full conditional distributions. The efficiency of these methods is illustrated with a comprehensive simulation study that accommodates for different scenarios of dimensionality for the nuisance parameters, along with different levels of complexity to infer the parameters of interest. Also, these methods are applied to a dynamic material setting to produce statistical inferences on the material properties of tantalum.
Bayesian mesh refinement technique for Magnetic Resonance Elastography (MRE)
Magnetic Resonance Elastography (MRE) is an imaging technique that allows us to infer quantitative results about the stiffness of the tissue from displacements generated by an external device. The confiability of MRE can be compromised due to the quality of the image. We propose to increase the resolution of the image based on information from the data, that is, a posterior mesh refinement. The map from parameters to observables is given implicitly by a partial differential equation. The linearization of the forward map around the MAP (maximum a posteriori) allow us to approximate a normalization constant and to use Bayes Factors to decide the region to increase the resolution level of the image.
Numerical Error Control in the Posterior Distribution for the Initial Condition Inverse Problem in a 2d Heat Equation
We present a 2D heat equation problem to determine the initial conditions from observations of transient temperature measurements taken within the domain at a time t = t_1. We define the inverse problem with a Bayesian approach. The Forward Mapping (FM) is analytically available. A numerical solution is also computed using the Finite Element Method (FEM) using FEniCS. The numeric error in the solution is then directly calculated.
With the numerical error calculated and the result obtained by Christen et. al (2018), on using Bayesian Factors (BF) to compare the theoretical and the numeric posteriors, we found a mesh size h for the numeric posterior. Any further refinements on the mesh size lead to a posterior distribution practically indistinguishable from the latter and with a BF provable nearly equal to 1 with the theoretical posterior. The additional CPU effort in refining the mesh is thereafter useless, once the numeric solver error is below the theoretical bound found in Christen et. al (2018).
Christen, J., Capistrán, M. A., Daza-Torres, M. L., Flores-Arguedas, H., & Cricelio Montesinos- López, J. (2017). Posterior distribution existence and error control in Banach spaces in the Bayesian approach to UQ in inverse problems. arXiv e-prints, arXiv:1712.03299.