**Organisers:** Marie-Colette van Lieshout (CWI), Kathrin Smetana (University of Twente)

#### Robert Scheichl (Heidelberg University, Germany)

*Uncertainty Quantification: when numerics meets statistics*

Numerical methods for partial differential equations (PDEs) and their implementations on high performance computers have reached a level of sophistication that allows the numerical simulation of ever more complex, heterogeneous, nonlinear processes. Many openly available PDE software packages exist. However, model parameters and initial/boundary conditions are typically only partially available or measurable and those measurements are often indirect and/or noisy. On the other hand, statistical methods for inverse problems that allow to infer distributions of unknown or uncertain parameters given (noisy) measurements of the system have also reached a high level of sophistication, in particular Bayesian techniques such as the gold-standard Metropolis-Hastings MCMC or filtering techniques. Nevertheless, the application of Bayesian inference to complex PDE constrained inverse problems is still a hugely challenging task. This is partly due to their high computational complexity, but also to the typically very high dimensional parameter domain. To overcome these challenges requires a concerted effort of statisticians, numerical analysts and application scientists, which has spawned the new scientific research area of Uncertainty Quantification (UQ). In this talk, I will present a number of examples from my research where classical ideas from numerical analysis and scientific computing are used to increase the efficiency of Bayesian statistical methods for PDE problems. In particular, I will highlight hierarchical approaches such as multilevel Monte Carlo as well as low-rank matrix and tensor approximations to accelerate sampling methods.

#### Finn Lindgren (The University of Edinburgh, Scotland)

*Modelling and computation for multiscale spatio-temporal temperature reconstruction*

Combining multiple and large data sources of historical temperatures into unified spatio-temporal analyses is challenging from both modelling and computational points of view. As part of the H2020 EUSTACE project, new approaches needed to be developed not only due to the size of the problem, but also due to the highly heterogeneous data coverage and the latent heterogeneous physical processes. A specific aim was to design a system that could obtain realistic uncertainty estimates, due to both observation uncertainty and lack of spatio-temporal coverage.

To this end, a spatio-temporal multiscale statistical Gaussian random field model is constructed, with a hierarchy of spatially non-stationary spatio-temporal dependence structures, ranging from weather on a daily timescale to climate on a multidecadal timescale. Connections between SPDEs and Markov random fields are used to obtain sparse matrices for the practical computations. The extreme size of the problem necessitates the use of iterative solvers, which requires using the multiscale structure of the model to design an effective preconditioner.

#### Svetlana Dubinkina (CWI)

*Bayesian approach to elliptic inverse problems*

Predicting the amount of gas or oil extracted from a subsurface reservoir depends on the soil properties such as porosity and permeability. These properties, however, are highly uncertain due to the lack of measurements. Therefore decreasing these uncertainties is of a great importance.

Mathematically speaking, permeability can be represented by a random process, which in turn leads to a random partial differential equation. The solution of such a partial differential equation, for example pressure, is only partially observed and, moreover, contaminated with measurement errors. Therefore, instead of a well-posed forward problem of finding pressure from certain permeability, we are faced with an ill-posed inverse problem of finding uncertain random process from a few pressure measurements. We develop a Bayesian method for inverse problems, that is both general and computationally affordable.