Posterior Error Control in Bayesian Uncertainty Quantification
Abstract
This research proposal is concerned with the calibration and Uncertainty Quantification of complex, nonlinear, Partial Differential Equations (PDEs) computer models in the presence of data. These computers models are numerical approximations of mathematically well de ned PDE models of the phenomena in question. Since we only use approximate numerical solutions for the PDE models we will only obtain approximate answers when performing statistical inference from the data at hand. However, recent research suggest that by correctly controlling the error in our computer model we may in fact obtain exact answers (for all practical considerations), having explicit error estimates in the solvers or not. Making these novel ideas truly applicable in both cases, and applying them in real data situations, is the core of this research proposal. Uncertainty Quantification (UQ) is an emerging research field in the intersection of applied mathematics, computer science and statistics. In broad terms UQ is concerned with the study of complex models in the presence of data. In particular the bayesian approach to UQ has attracted substantial attention in recent years, covering a wide range of applications both in well established fields as well as in emerging areas of science and technology. Some recent examples of UQ applications include x-ray tomography, oxygen muscle consumption, image recovery, image recognition, fluid mechanics, geothermal reservoir modeling, emission tomography, network analysis, heat transfer analysis, pollution source analysis, anti-personal mine detection, electrical impedance tomography, among many others. In our group we have worked on epidemic outbreak analysis, acoustic wave scattering and elastography, water reservoir pro ling, among other applications. Improving the Uncertainty Quantification in the presence of data, with respect to precision and computational speed, will have an impact in the above mentioned application areas and many others of relevance for society, in science, technology, decision making and culture in general. The main question of this research proposal may be summarized as follows: How do we calibrate the tolerance or error in our numerical approximation of the PDEs involved, that is, in our computer model, in order to have a satisfactory UQ answer (that is, a reasonable numerical approximation to the posterior probability distribution of all unknowns given current a priori knowledge and data)? All this, while keeping CPU computer cost as low as possible. At least theoretically [1] have provided and answer in the case of ODEs: For higher order numerical solvers there exists a tolerance h* such that, for all h < h*, the numerical posterior probability distribution at tolerance h is basically error free, for all practical considerations. That is, we could have it all: use an approximate numerical model with relatively low CPU cost but nevertheless get error free UQ answers. However, the approach proposed in [1] still have a series of shortcomings to be of practical use in real situations and the core of this research proposal is to make the these ideas truly applicable in a wide range of practical scenarios and areas of applications. [1] Capistran, M.A., J.A. Christen, and S. Donnet (2016), Bayesian Analysis of ODEs: solver optimal accuracy and Bayes factors, Journal of Uncertainty Quantification, 4(1), 829-849.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Oct 16, 2018
- Source ID
- W911NF1710414
Entities
People
- J.a. Christen
Organizations
- Army Contracting Command
- CIMAT Center for Mathematical Research
- United States Army