Bayesian Methodologies for PDE Parameter Estimation: Model Problems, Algorithm Development and Applications

About This Grant

This project involves the development of methodologies to estimate unknown quantities from complex, sparse, and noisy data sets by incorporating first-principles physics into statistical modeling procedures. The approach pairs new inverse problem formulations with novel probabilistic high performance computing methods to resolve crucial statistical quantities (means, correlations, confidence bands). This framework has applications in medical imaging, weather modeling, robotics, geology, and geophysics, with mechanisms for improving methods to better formulate and quantify uncertainties for real world challenges in the above domains. Furthermore, the project will incorporate the comprehensive training and the promotion of excellence in applied mathematics and statistics for a new generation of scholars who will be involved in these research activities. This project leverages an emerging Bayesian inversion formalism to estimate non-parametric physical parameters from data modeled as sparse and noisy observations of solutions to partial differential equations (PDEs). The proposed strategy blends rigorous analysis, algorithm development, numerical case studies, and modeling to build a foundational understanding of these methodologies and to expand the scope of this statistical approach to PDE inference. The project is organized around three interconnected objectives. The first objective involves the development of fluid measurement problems featuring infinite-dimensional unknowns and nonlinear parameter dependence. This includes estimating a flow field from a passively advected solute subject to diffusive and reactive effects, determining domain geometry in a Stokes flow, appraising bottom boundary heating from bulk and top boundary observations in Rayleigh-Bénard convection, and specifying a flow underlying an observed mixture in a two-phase fluid. These concrete problems will have intrinsic interest for domain applications while serving as numerical and analytical test beds. The second objective involves the derivation and analysis of Markov Chain Monte Carlo (MCMC) algorithms. This project will formulate adjoint and approximate methods to resolve gradients in the newly developed PDE parameter-to-solution maps, study large proposal and high dimensional limits in multi-proposal algorithms, rigorously assess mixing rates in infinite dimensional methods, and develop diagnostics of bias in inexact MCMC procedures. This approach will be adapted to the fluid measurement problems but will also have wider significance for other areas of computational statistics. The third objective involves the study of concentration properties. This includes formulating general frameworks which isolate crucial properties of the PDE parameter-to-solution map, structure of observations, and the prior distribution to establish conditions for concentration in the large data limit, assessing large data behavior for posteriors when the non-invertibility of the PDE parameter to solution map prevents concentration in principle, and addressing experimental design for the situation when a fixed number of location for observations can be optimized. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.