In this chapter the state-of-the-art in data assimilation for high-dimensional highly nonlinear
systems is reviewed, and recent developments are highlighted. This knowledge is available …

We present a review of methods for optimal experimental design (OED) for Bayesian inverse
problems governed by partial differential equations with infinite-dimensional parameters …

Regularization methods are a key tool in the solution of inverse problems. They are used to
introduce prior knowledge and allow a robust approximation of ill-posed (pseudo-) inverses …

We introduce a simple, rigorous, and unified framework for solving nonlinear partial
differential equations (PDEs), and for solving inverse problems (IPs) involving the …

A central research challenge for the mathematical sciences in the twenty-first century is the
development of principled methodologies for the seamless integration of (often vast) data …

These lecture notes highlight the mathematical and computational structure relating to the
formulation of, and development of algorithms for, the Bayesian approach to inverse …

Weak Convergence Page 1 1.3 Weak Convergence In this section IDl and IE are metric spaces
with metrics d and e, respectively. The set of all continuous, bounded functions f: IDl 1--+ IR is …

Questions of 'how best to acquire data'are essential to modelling and prediction in the
natural and social sciences, engineering applications, and beyond. Optimal experimental …

Many Bayesian inference problems require exploring the posterior distribution of high-
dimensional parameters that represent the discretization of an underlying function. This work …

This paper studies the learning of linear operators between infinite-dimensional Hilbert
spaces. The training data comprises pairs of random input vectors in a Hilbert space and …