# LMS Workshop: Bayesian Inverse Problems

# Talks

### Speaker: Tim Sullivan

**Title:** Bayesian Probabilistic Numerical Methods.

**Abstract:** Numerical computation --- such as numerical solution of a PDE --- can modelled as a statistical inverse problem in its own right. The popular Bayesian approach to inversion is considered, wherein a posterior distribution is induced over the object of interest by conditioning a prior distribution on the same finite information that would be used in a classical numerical method, thereby restricting attention to a meaningful subclass of probabilistic numerical methods distinct from classical average-case analysis and information-based complexity. The main technical consideration here is that the data are non-random and thus the standard Bayes' theorem does not hold. General conditions will be presented under which numerical methods based upon such Bayesian probabilistic foundations are well-posed, and a sequential Monte-Carlo method will be shown to provide consistent estimation of the posterior. The paradigm is extended to computational ``pipelines'', through which a distributional quantification of numerical error can be propagated. A sufficient condition is presented for when such propagation can be endowed with a globally coherent Bayesian interpretation, based on a novel class of probabilistic graphical models designed to represent a computational work-flow. The concepts are illustrated through explicit numerical experiments involving both linear and non-linear PDE models.

### Speaker: Kody Law

**Title:** Strategies for Multilevel Monte Carlo

**Abstract:** This talk will concern the problem of inference when the posterior measure involves continuous models which require approximation before inference can be performed. Typically one cannot sample from the posterior distribution directly, but can at best only evaluate it, up to a normalizing constant. Therefore one must resort to computationally-intensive inference algorithms in order to construct estimators. These algorithms are typically of Monte Carlo type, and include for example Markov chain Monte Carlo, importance samplers, and sequential Monte Carlo samplers. The multilevel Monte Carlo method provides a way of optimally balancing discretization and sampling error on a hierarchy of approximation levels, such that cost is optimized. Recently this method has been applied to computationally intensive inference. This non-trivial task can be achieved in a variety of ways. This talk will review 3 primary strategies which have been successfully employed to achieve optimal (or canonical) convergence rates – in other words faster convergence than i.i.d. sampling at the finest discretization level. Some of the specific resulting algorithms, and applications, will also be presented.

### Speaker: Katherine Tant

**Title:** A Transdimensional Bayesian Approach to Material Mapping and Imaging in Industrial Applications

**Abstract:** Ultrasonic non-destructive testing is the practice of transmitting mechanical waves into industrial components and using the resulting scattered wave data to construct images of internal defects. Detecting defects in complex media presents a significant challenge, as strong scattering and refraction of the waves by the underlying material microstructure can cause defocussing in the resulting images. However, if the spatially varying material properties of the medium are known a priori, this can be accounted and corrected for. Hence we are presented with an inverse problem: can we map the material properties of a heterogeneous medium from scattered wave data? In this talk we present an efficient solver for the Eikonal equation, to model wave front propagation in locally anisotropic, polycrystalline materials. This is implemented within a Bayesian inversion framework to estimate the material map which gave rise to the recorded scattered wave data. The forward model and material map are then used as the basis for an advanced imaging algorithm and the improved reconstructions of defects embedded within polycrystalline materials are used to measure the success of the approach.

### Speaker: Michela Ottobre

**Title:** Sampling with Irreversible Dynamics.

**Abstract:** In recent years the observation that "irreversible processes converge to equilibrium faster than their reversible counterparts" has sparked a significant amount of research to exploit irreversibility within sampling schemes, thereby accelerating convergence of the resulting Markov Chains and MCMC methods. It is now understood how to design irreversible continuous time dynamics with prescribed invariant measure. However, for sampling/simulation purposes, such dynamics still need to undergo discretization (or some other form of fiddling) and, as it is well known, naive discretizations can completely destroy all the good properties of the continuous-time process. In this talk we will i) give some background on irreversibility and review the progress made so far on the study of non-reversible processes; ii) present a non-reversible version of the well-known Hamiltonian monte Carlo algorithm iii) make further considerations on how to use (or not to use) irreversibility for algorithmic purposes.

### Speaker: Yoann Altman

**Title:** Bayesian 3D Scene Reconstruction from Sparse Multispectral Lidar Waveforms

**Abstract:** In this talk, we will compare sampling strategies and associated unsupervised Bayesian algorithms to reconstruct scenes sensed via sparse multispectral Lidar measurements. In the presence of a target, Lidar waveforms usually consist of a peak, whose position and amplitude depend on the target distance and reflectivity, respectively. Using multiple wavelengths (e.g., multiple laser sources), it becomes possible to discriminate spectrally the main objects in the scene, in addition to extracting range profiles. We compare different sampling strategies illustrated via experiments conducted with real multispectral Lidar data and the results demonstrate the possibility to infer scene content from extremely sparse photon counts using different acquisition scenarios.

# Posters

Presenter | Title |

Ying, Yik Keung | Bayesian Inference of Ocean Diffusivity from Lagrangian Trajectories |

Sam Power | Challenges for MCMC in Infinite-Dimensional Bayesian Inverse Problems |

Jenovah Rodrigues | Contraction Rates for Bayesian Inverse Problems with Heterogeneous Variance |

Steven Kleinegesse | Efficient Bayesian Experimental Design for Implicit Models |

Ana Fernadez Vidal | Maximum Likelihood Estimation of Regularisation Parameters |

Prateek Kumar Dongre | Application of Bayesian Framework for Information content analysis of a novel TES-based Hyperspectral Microwave Atmospheric Sounding Instrument |