Kalman-like Inversion with ODE/SDE Formulations and Adaptive Algorithms - Yang, Yuchen (2020)
This PhD thesis conducts survey in numerical algorithms for inverse problems. The inverse problems are usually ill-posed in practice. They require additional regularization. The standard setting of inverse problems are variational approach and Bayesian approach. This thesis rewrites the standard setting into the tempering setting with an auxiliary parameter called the tempering parameter. The tempering setting has the similar mathematical structure as canonical ensemble in statistical mechanics. This mathematical skill has been widely applied in annealed importance sampling, simulated annealing, sequential Monte Carlo simulation, et al. In this thesis, we consider infinite-dimensional inverse problems, and uses continuous tempering parameter. Inverse problems with the tempering setting can be approximately simplified as continuous extend Kalman filter with a PDE formula, or continuous mean-field limit ensemble Kalman filter as a SDE formula. We propose the adaptive strategy called data-misfit controller to discretize the PDE and SDE, and the resulting algorithms keep both efficiency and accuracy. Additionally, we prove monotone properties of the tempering setting. Based on these properties, we propose the early stop criterion monitoring quality of estimates in filtering. This improves the robustness of the Kalman-like methods for highly nonlinear problems.