Bayesian inversion for forward partial differential equations in mixed forms - Yang, Juntao (Nanyang Technological University, 2022)
In this thesis we investigate the Bayesian inverse problems with the forward partial differential equations in the mixed form. Partial differential equations in the mixed form are common governing equations for many physical laws, particularly in solid and fluid mechanics. We focus on linear elasticity and fluid mechanics problems in the mixed form. Bayesian inverse problems with governing equations in the mixed form occur frequently in real world applications, such as numerical weather prediction, climate modeling, oceanography modeling, subsurface oil detection, non-destructive defect detection. We are interested in finding some quantities of interest which depend on the coefficients, or on the forcing and the initial condition in the fluid mechanics case, which belong to infinite dimensional prior probability spaces, given noisy observation data which are functionals of the solution of the governing equation. We aim to find the posterior expectation of the quantity of interest under the Bayesian framework. We consider both the uniform prior when the functions we wish to infer are bounded, and the log Gaussian prior where the functions are positive but can be arbitrarily close to 0 and arbitrarily large. A general introduction is included in chapter 1, which covers the background and current status of Bayesian inversion techniques and the multilevel Markov Chain Monte Carlo method for partial differential equation governed problems. Chapter 2 considers the Bayesian inverse problems for linear elasticity problems in the Hellinger and Reissner mixed form with uniform prior. We consider Bayesian inverse problems to infer unknown quantities of interest from noisy observations on the stress. We set up the Bayesian inverse problem in infinite dimen- sion with the Lamé constants depend on a countable sequence of random coefficients which are uniformly distributed in a compact interval. We perform analysis on the approximation of the posterior by truncation of the Lamé constants and the finite element approximation of the truncated problem. The multilevel Markov Chain Monte Carlo method developed for elliptic equations by Hoang et al. (V. H. Hoang, Ch. Schwab, A. M. Stuart, Inverse problems, Vol. 29, Page. 37, 2013) is reviewed and implemented for the Bayesian inverse problem with linear elasticity equation in the Hellinger and Reissner mixed form. The multilevel Markov Chain Monte Carlo method is capable of achieving a prescribed level of accuracy for approximating the posterior expectation with an optimal number of total degrees of freedom and floating point operators, which are essentially equivalent to those for solving a single realization of the forward equation. We show the theoretical convergence rate of the multilevel Markov Chain Monte Carlo method for the Bayesian inverse problem for linear elasticity problems. We performed some numerical experiments in the last section of the chapter to validate the theory. In chapter 3, the Bayesian inverse problems for linear elasticity problems in the mixed form of Hellinger and Reissner with log Gaussian prior are discussed. The logarithm of the Lamé constants are linearly dependent on a sequence of countable number of random variables which are normally distributed. The multilevel Markov Chain Monte Carlo method developed by Hoang et al. for uniform prior relies on the uniform boundedness of the solution which is not the case with log Gaussian prior. Hence, we employ the multilevel Markov Chain Monte Carlo method developed for the log normal prior in (V. H. Hoang, J.H. Quek, Ch. Schwab, Inverse problems, Vol. 36, Page. 46, 2020). The theoretical convergence of the multilevel Markov Chain Monte Carlo method for Bayesian inverse problems with linear elasticity equations in the Hellinger and Reissner form is shown. We provide numerical experiments to demonstrate the theoretical result, where both independent sampler and preconditioned Crank-Nicolson (pCN) samplers are also used. We consider the Bayesian inverse problems for linear elasticity problems in the dual mixed form of Hellinger and Reissner with uniform prior in chapter 4. The Lamé constants depend on a countable sequence of random variables of uniform distribution in a compact interval. We consider the dual mixed problem with weak symmetry. Hence we could utilize PEERS element. We approximate the posterior expectation of the quantity of interest with multilevel Markov Chain Monte Carlo method developed for uniform prior by Hoang, Schwab, Stuart (2013). Numerical results confirm the theoretical convergence rate. Log Gaussian prior is considered in chapter 5 for the Bayesian inverse problems for linear elasticity problems in the dual mixed form of Hellinger and Reissner. The Lamé constants depend linearly on a countable sequence of random variables which are normally distributed. Log normal prior guarantees the positivity of the Lamé constants, but the Lamé constants can be arbitrarily close to zero or arbitrarily large. We consider the dual mixed form of Hellinger and Reissner with weak symmetry in the Bayesian inverse problem. We show the theoretical convergence rate of the multilevel Markov Chain Monte Carlo developed by Hoang, Quek, Schwab (2020). for the Bayesian inverse problem for linear elasticity equation in the dual mixed form of Hellinger and Reissner. Numerical experiments using both independent sampler and pCN sampler are performed to support the theory. In chapter 6, we consider Bayesian inverse problems for the displacement formulation of linear elasticity problem with nearly incompressible and incompressible materials with uniform prior. We write the governing equation of the nearly incompressible material as a saddle point problem with a penalty term where the Lamé constant λ being much larger than µ. Incompressible materials are modeled with saddle point problems. We present the finite element approximation of the posterior with stable finite element pairs, e.g. P1-iso-P2/P1. The optimal convergence rate of multilevel Markov Chain Monte Carlo for both problems are shown and numerical experiments are included to show case the theoretical results. In chapter 7, we continue the discussion on Bayesian inverse problems where the governing elasticity equation is in the mixed form with a penalty term to model a nearly incompressible materials as in chapter 6, and also the case of incompressible materials. However the Lamé constants are of the log normal form.We perform the finite element approximation of the truncated problem. We perform the multilevel Markov Chain Monte Carlo to verify the theoretical error estimate. In the last chapter we consider Bayesian inverse problems where the forward equation is the two dimen- sional Navier Stokes equations. Our interest is to recover the initial condition and the random forcing given noisy Eulerian observation of velocity field at some time moments. The uniform prior is considered for the Bayesian inverse problem. We approximate the posterior by taking a finite number of random variables (the truncated forward problem). Implicit/Explicit Euler time scheme coupled with finite element method is used to approximate the solution of the truncated forward problem. We show the theoretical convergence rate of the multilevel Markov Chain Monte Carlo method for uniform prior with proper balancing between truncation error, finite element approximation error and sampling error. Numerical experiments of approx- imating quantities of interest with multilevel Markov Chain Monte Carlo are shown. We do not include a rigorous analysis for Bayesian inversion problem with Gaussian prior due to the technical difficulty arising from finite element approximation of Navier-Stokes equations. The finite element approximation error grows as an exponential form of a polynomial function of the norms of the initial condition and the forcing due to the Gronwall’s inequality. This error estimate is not integrable with respect to Gaussian so a rigorous analysis is not possible. Nevertheless, we show by numerical experiments that the multilevel Markov Chain Monte Carlo method developed for the Gaussian prior works for the Bayesian inverse problem with the forward Navier Stokes equations. However we show numerically that the multilevel Markov Chain Monte Carlo method developed for uniform prior fail. This shows that it is necessary to take into account the unboundedness of the solution of partial differential equations where the prior is Gaussian in developing a multilevel sampling method.