R. A. Fisher's fiducial inference has been the subject of many discussions and controversies ever since he introduced the idea during the 1930's. The idea experienced a bumpy ride, to say the least, during its early years and one can safely say that it eventually fell into disfavor among mainstream statisticians. However, it appears to have made a resurgence recently under various names and modifications.

Model Validation is often a difficult task with many different objectives and their associated metrics. Many times these metrics can address a single issue such as model fit, predictive performance, uncertainty quantification, etc. One quantity of interest is the amount of uncertainty associated with model misspecification.  In this talk we present a bootstrap based method for uncertainty quantification under model misspecification.

Bayesian approach is developed for estimating the thermal conductivity of a homogeneous material from the temperature evolution acquired in few internal points. Temperature evolution is described by the classical one-dimensional heat equation, in which the thermal conductivity is one of the coefficients.

Stochastic partial differential equations (SPDEs) play fundamental and important roles in many problems and their numerical simulations become useful and important in practical applications. The mathematical treatment of SPDEs represents however a greater challenge in comparison to deterministic partial differential equations.

Gaussian beams are asymptotically valid high frequency solutions to hyperbolic partial differential equations, concentrated on a curve through the domain. Superpositions of Gaussian beams provide a tool to generate more general high frequency solutions. In contrast to the standard geometrical optics, the Gaussian beam approximation does not break down at caustics.

One of the most versatile parametrizations of stochastic processes or random fields for large-scale computations is provided by the Karhunen-Loève expansion. Its computational realization requires (at least the partial) solution of an eigenvalue problem for a covariance operator.

We consider the problem of approximating the marginal distribution of the solution of a stochastic differential equation (SDE) by probability measures with finite support, i.e., by quadrature formulas with positive weights summing up to one.  We study deterministic algorithms in a worst case analysis with respect to classes of SDEs, which are defined in terms of smoothness constraints for the coefficients of the equation.

The multi-level Monte Carlo approach is a powerful variance reduction technique, which is applied, in particular, in the context of SDEs. While the standard task is to compute the expectation of a real-valued functional, we discuss how to approximate a distribution function on a compact interval. In this talk, we establish upper bounds for the error of suitable multi-level algorithms.