In this work, we are interested in the non-linear inverse problem which consists in retrieving the basal drag factor, an important parameter for scientists who want to understand the dynamics of ice sheets in the Antarctic. This drag factor takes the form of a Robin boundary condition at the bottom of the ice sheet in our PDE problem, and varies spatially along the boundary. Due to the thickness of the ice, the drag cannot be measured directly, and the only data available to us is the velocity of the ice at the surface.

We present a computational routine to estimate posterior densities of parameters for this Robin boundary condition.

The talk will consider Bayesian nonparametric inference in multi-dimensional diffusion models from low-frequency data. Implementation of Bayesian procedures in such settings is a notoriously delicate task, due to the intractability of the likelihood, often requiring involved augmentation techniques. For the nonlinear inverse problem of inferring the diffusivity function in a stochastic differential equation, we rather propose to exploit the underlying PDE characterization of the transition densities, which allows the numerical evaluation of the likelihood via standard numerical methods for elliptic eigenvalue problems. A simple Metropolis-Hastings-type MCMC algorithm for inference on the diffusivity is then constructed, based on Gaussian process priors. The performance of the the algorithm will be illustrated via the results of numerical experiments. The talk will then discuss theoretical computational guarantees for MCMC methods in the considered inferential problem, based on derived local curvature properties for the log-likelihood, and connected to the `hot spots’ conjecture from spectral geometry.

Joint work with S. Wang (MIT).

Spatially inhomogeneous functions, which may be smooth in some regions and rough in other regions, are modelled naturally in a Bayesian manner using so-called Besov priors which are given by random wavelet expansions with Laplace-distributed coefficients. This talk considers frequentist theoretical guarantees for Bayes methods with Besov priors, in the setting of non-linear inverse problems with Gaussian white noise. Our results are first derived under a general local Lipschitz assumption on the forward map. We then verify the assumption for two non-linear inverse problems arising from elliptic partial differential equations. We also discuss novel convergence rate results for penalized least squares estimators with $\ell_{1}$ wavelet penalty, which have a natural interpretation as maximum a posteriori (MAP) estimators. The true parameter is assumed to belong to some spatially inhomogeneous Besov class. In a setting with direct observations, we complement these upper bounds with a lower bound on the rate of contraction for arbitrary Gaussian priors. Consequently, while Laplace priors can achieve minimax-optimal rates over spatially inhomogeneous classes, Gaussian priors are limited to a (by a polynomial factor) slower contraction rate. This gives information-theoretical justification for the intuition that Laplace priors are more compatible with $\ell_{1}$ regularity structure in the underlying parameter.

With large scale availability of precise real time data, their incorporation into physically based predictive models, became increasingly important. This procedure of combining the prediction and observation is called data assimilation. One especially popular algorithm of the class of Bayesian sequential data assimilation methods is the ensemble Kalman filter which successfully extends the ideas of the Kalman filter to the non-linear situation. However, in case of spatio-temporal models one regularly relies on some version of localization, to avoid spurious oscillations.

In this work we develop a-priori error estimates for a time continuous variant of the ensemble Kalman filter, known as localized ensemble Kalman--Bucy filter. More specifically we aim for the scenario of sparse observations applied to models from fluid dynamics and space weather.