Bayesian Inverse Problems and Model Uncertainties

Case Western Reserve University, Cleveland OH

Investigators

Abstract

The traditional and very natural paradigm in science is to build predictive mathematical models that move from causes to consequences. However, it often happens that observations of consequences are available, and one needs to identify the causes that made the observations possible. The latter type of problems are known as inverse problems. Inverse problems are characterized by their high sensitivity to errors in the measurements and the models used, the existence of not just one but several possible solutions, and their computational complexity. This project focuses on one particular but central aspect in inverse problems: Assume that a very detailed and complex predictive model exists, known to be able to produce predictions that match well with observations. Furthermore, assume that the model is computationally very demanding, and it contains numerous parameters whose values are unknown or poorly known. To solve the inverse problem in the required time frame, it may be that a simplified, or reduced model, needs to be used. Given that inverse problems are sensitive to errors in the model, it typically happens that the model reduction introduces an uncontrolled error, or discrepancy between the model and reality, that may render the solution of the inverse problem completely useless. The investigator and his colleagues have proposed a general methodology to handle the modeling error problem in the statistical framework, and in this project, the aim is to develop the methodology further so that it allows a reliable way to find a useful solution with limited computational resources, and to quantify the reliability of such solution. The main applications in this project are in the field of medicine, including mapping of the brain activity, identification and localization of stroke using a portable equipment, and development of fast and portable computing tools to model blood flow, but the results also have applications beyond medical applications. The technical difficulty in handling the modeling error in an inverse problem is that it depends on the unknown cause that the inverse problem is seeking. However, the Bayesian statistical paradigm provides a very natural solution to this problem. In the Bayesian context, the unknown of primary interest is described as a random variable that has an a priori probability distribution, and therefore, it is possible to estimate a probability distribution of the modeling error and include it as part of the likelihood model. This basic observation has been shown to lead to algorithms that dramatically improve the estimates compared to results that ignore the modeling error. In this project, the methodology will be developed further, by carefully following how the inclusion of the modeling error distribution affects the Bayesian posterior distribution of the unknown, and conversely, how the modeling error distribution can be updated after the data is used to update the prior density of the unknown. Such tracking will hopefully lead to a computationally efficient way of quantifying uncertainties in the inverse solutions in the presence of modeling errors. One family of problems the project addresses is multi-scale inverse problems, in which the unknowns of primary interest are describing fine-scale behavior of the system, while the observation represents a macroscopic, coarse scale quantity. These types of problems often appear in biological applications, where the high-fidelity microscopic models are often stochastic in nature, and cannot be handled directly in the standard Bayesian framework.