Applied Math Colloquium: Jonathan Poterjoy (UMCP)

Title: 
Kernel and Generative Strategies for Handling Complex Observation Processes in Geophysical Data Assimilation
Abstract: 
Data assimilation in high-dimensional systems, such as numerical weather prediction, presents a formidable computational challenge. Operational centers routinely infer the probabilistic evolution of state vectors comprising more than a billion variables using physics-based models, noisy observations, and classic Bayesian filtering techniques. While many of these approaches rely on heavy approximations, recent advances make it feasible to move beyond rigid Gaussian assumptions for the prior. These non-Gaussian approaches are becoming increasingly attractive, as inexpensive surrogate models prove more effective at rapidly generating large Monte Carlo estimates of this density. Nevertheless, traditional likelihood estimation still relies on a well-defined measurement operator, or forward model, to link model states to observations, and considers uncertainty only in the form of an observation error covariance. In reality, this measurement process can be highly nonlinear, rely on incomplete physics, or remain fundamentally unknown.

To address this challenge, we present a suite of operator-free strategies that directly estimate likelihood functions from training data. These methods range from leveraging kernel mean embeddings to dynamically learn conditional distributions within a Reproducing Kernel Hilbert Space (RKHS) to employing probabilistic generative models such as conditional variational autoencoders (cVAEs). To explore the scalability of these techniques, we integrate them with contemporary filtering algorithms and assess their performance in a low-dimensional application that serves as an analog for weather forecasting and climate reconstruction. By weighing the trade-offs in accuracy and computational cost, this work describes a path toward implementation in next-generation Earth System models.