Guest lectures and seminars - Page 20
Stalagmites grow on the floor of caves by precipitation of calcium ions found in the residual water film covering the top of the stalagmite, which is progressively drained away. Drops dripping from stalactites ensure the renewal of these ions.
Previous models of stalagmite growth assumed that drops fall on a straight vertical line from stalactites. Through high-speed imaging during field experiments in caves, we however observe that the impact point position of the drops is scattered. Using a Langevin-like equation to describe the fall of drops in response to gravity and aerodynamic forces, we then propose a prediction of the impact point dispersal. We show that measured stalagmite widths are correlated to the impact point dispersal of the drops.
In a second time, we focus on the mixing of calcium ions between the drop and the film during impact. The drop produces a crown when impacting the film, accompanied by a large amount of secondary droplet ejections. This is at the very heart of the film thickness variability post-impact. We record high-speed imaging of drop impacts on films of controlled thickness in a lab environment and assess the mixing between the drop and the film. We deduce how much liquid coming from the initial drop remains in the film.
A flexible predictive density combination is introduced for large financial data sets which allows for model set incompleteness. Dimension reduction procedures that include learning allocate the large sets of predictive densities and combination weights to relatively small subsets. Given the representation of the probability model in extended nonlinear state-space form, efficient simulation-based Bayesian inference is proposed using parallel dynamic clustering as well as nonlinear filtering, implemented on graphics processing units. The approach is applied to combine predictive densities based on a large number of individual US stock returns of daily observations over a period that includes the Covid-19 crisis period. Evidence on dynamic cluster composition, weight patterns and model set incompleteness gives valuable signals for improved modelling. This enables higher predictive accuracy and better assessment of uncertainty and risk for investment fund management.
Online changepoint detection algorithms based on likelihood-ratio tests have excellent statistical properties. However, a simple exact online implementation is computationally infeasible as, at time T, it involves considering O(T) possible locations for the change. To improve on this, we use functional pruning ideas to reduce the set of changepoint locations that need to be stored at time T to approximately log T. Furthermore, we show how we need only maximise the likelihood-ratio test statistic over a small subset of these possible locations. Empirical results show that the resulting exact online algorithm, which can detect changes under a wide range of models, has a constant-per-iteration cost on average. We consider applications of this algorithm in the context of detecting increases in radiation count that represent astronomical or nuclear events of interest.
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The Kolmogorov N-width describes the best possible error one can achieve by elements of an N-dimensional linear space. Its decay has extensively been studied in Approximation Theory and for the solution of Partial Differential Equations (PDEs). Particular interest has occurred within Model Order Reduction (MOR) of parameterized PDEs e.g. by the Reduced Basis Method (RBM). While it is known that the N-width decays exponentially fast (and thus admits efficient MOR) for certain problems, there are examples of the linear transport and the wave equation, where the decay rate deteriorates to N-1/2. On the other hand, it is widely accepted that a smooth parameter dependence admits a fast decay of the N-width. However, a detailed analysis of the influence of properties of the data (such as regularity or slope) on the rate of the N-width seems to lack. In this work, we use techniques from Fourier Analysis to derive exact representations of the N-width in terms of initial and boundary conditions of the linear transport equation modeled by some function g for half-wave symmetric data. For arbitrary functions g, we derive bounds and prove that these bounds are sharp. In particular, we prove that the N-width decays as cr N(-r-1/2) for functions in the Sobolev space, g ∈ Hr. Our theoretical investigations are complemented by numerical experiments which confirm the sharpness of our bounds and give additional quantitative insigh.