Error bounds for kernel-based approximations of the Koopman operator

We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results.Comment: 28 page

Drilling into a deep buried valley (ICDP DOVE): a 252 m long sediment succession from a glacial overdeepening in northwestern Switzerland

The modern Alpine landscape and its foreland were strongly impacted by the numerous glacier ad- vance and retreat cycles during the Middle-to-Late Pleistocene. Due to the overall erosive character of each glaciation cycle, however, direct traces of older glaciations tend to be poorly preserved within the formerly glaciated domains of the pan-Alpine area. Nevertheless, sediments of older glaciations may occur hidden un- der the modern surface in buried glacially overdeepened troughs that reach below the normal level of fluvial erosion (fluvial base level). These sedimentary archives, partly dating back to the Middle Pleistocene period, are of great scientific value for reconstructing the timing and extent of extensive Alpine glaciation, paleocli- mate, and paleoenvironmental changes in the past and help to better understand ongoing and future changes in the pan-Alpine area. Therefore, the International Continental Scientific Drilling Program (ICDP) project DOVE (Drilling Overdeepened Alpine Valleys) targets several of these glacial overdeepened sedimentary basins to re- cover their sedimentary infills. In the frame of the DOVE project, a 252 m long drill core of unconsolidated Quaternary sediments was recovered in northern Switzerland from an over 300 m deep glacially overdeepened structure (“Basadingen Trough”) formed by the former Rhine Glacier lobe system. The recovered sedimentary succession was divided into three stratigraphic units on the basis of lithological and petrophysical characteristics. The lowest unit, deposited below the fluvial base level, consists of an over 200 m thick succession of glacial to (glacio)lacustrine sediments and contains remains of possibly two glaciation cycles. Overlying this lowermost succession, an ∼ 37 m thick fluvial-to-glaciofluvial gravel deposit occurs, which correlates to a locally outcrop- ping Middle Pleistocene formation (“Buechberg Gravel Complex”). The sediment succession is capped by an ∼ 11 m thick diamictic succession interpreted as the subglacial till from the later extensive glaciation, including the regional glaciation during the Last Glacial Maximum. The recovered sediment succession thus supports the proposed multi-phase origin of trough formation and its infill

Partial observations, coarse graining and equivariance in Koopman operator theory for large-scale dynamical systems

The Koopman operator has become an essential tool for data-driven analysis, prediction and control of complex systems, the main reason being the enormous potential of identifying linear function space representations of nonlinear dynamics from measurements. Until now, the situation where for large-scale systems, we (i) only have access to partial observations (i.e., measurements, as is very common for experimental data) or (ii) deliberately perform coarse graining (for efficiency reasons) has not been treated to its full extent. In this paper, we address the pitfall associated with this situation, that the classical EDMD algorithm does not automatically provide a Koopman operator approximation for the underlying system if we do not carefully select the number of observables. Moreover, we show that symmetries in the system dynamics can be carried over to the Koopman operator, which allows us to massively increase the model efficiency. We also briefly draw a connection to domain decomposition techniques for partial differential equations and present numerical evidence using the Kuramoto--Sivashinsky equation

All-Body-Cavity (ABC)-scopy: an approach for a feasible method of minimally invasive autopsy to allow for postmortem tissue sampling in cases where a conventional autopsy is denied

Objectives The decreasing autopsy numbers in many western countries have been partially attributed to the invasiveness of the autopsy, which causes relatives to decline postmortem examination. This issue has been addressed by developing methods of minimally or non-invasive autopsy, which could be shown to increase acceptance for autopsies. The aim of this study is to compare the All-Body-Cavity-scopy (ABC-scopy) to conventional autopsies for diagnostic accuracy. Methods The ABC-scopy is an endoscopic approach for minimally invasive autopsy involving laparoscopic and thoracoscopic evaluation of the accessible organs, followed by excision biopsies of relevant organs and conspicuous findings. The method was performed in 10 cases on deceased patients scheduled for autopsy, each followed by a conventional autopsy. Results The results gathered from ABC-scopy through observation and histopathological evaluation provided an acceptable diagnostic accuracy in 9 out of 10 autopsies when compared to those of the conventional autopsy for diagnostic findings. Conclusions The ABC-scopy is a feasible approach for minimally invasive autopsy that provides acceptable diagnostic value. Despite its minimally invasive nature, the procedure enables representative histology through providing large size excision biopsies from intraabdominal and thoracic organs, which is especially useful for examining disseminated diseases such as metastasized tumors

Garcinol from Garcinia indica inhibits HIV-1 reverse transcriptase-associated ribonuclease H

The bioactive components of Garcinia indica, garcinol (camboginol), and isogarcinol (cambogin), are suitable drug candidates for the treatment of various human diseases. HIV-1-RNase H assay was used to study the RNase H inhibition by garcinol and isogarcinol. Docking of garcinol into the active site of the enzyme was carried out to rationalize the difference in activities between the two compounds. Garcinol showed higher HIV-1-RNase H inhibition than the known inhibitor RDS1759 and retained full potency against the RNase H of a drug-resistant HIV-1 reverse transcriptase form. Isogarcinol was distinctly less active than garcinol, indicating the importance of the enolizable β-diketone moiety of garcinol for anti-RNase H activity. Docking calculations confirmed these findings and suggested this moiety to be involved in the chelation of metal ions of the active site. On the basis of its HIV-1 reverse transcriptase-associated RNase H inhibitory activity, garcinol is worth being further explored concerning its potential as a cost-effective treatment for HIV patients

Non-universal Voronoi cell shapes in amorphous ellipsoid packings

In particulate systems with short-range interactions, such as granular matter or simple fluids, local structure plays a pivotal role in determining the macroscopic physical properties. Here, we analyse local structure metrics derived from the Voronoi diagram of configurations of oblate ellipsoids, for various aspect ratios $\alpha$ and global volume fractions $\phi_g$. We focus on jammed static configurations of frictional ellipsoids, obtained by tomographic imaging and by discrete element method simulations. In particular, we consider the local packing fraction $\phi_l$, defined as the particle's volume divided by its Voronoi cell volume. We find that the probability $P(\phi_l)$ for a Voronoi cell to have a given local packing fraction shows the same scaling behaviour as function of $\phi_g$ as observed for random sphere packs. Surprisingly, this scaling behaviour is further found to be independent of the particle aspect ratio. By contrast, the typical Voronoi cell shape, quantified by the Minkowski tensor anisotropy index $\beta=\beta_0^{2,0}$, points towards a significant difference between random packings of spheres and those of oblate ellipsoids. While the average cell shape $\beta$ of all cells with a given value of $\phi_l$ is very similar in dense and loose jammed sphere packings, the structure of dense and loose ellipsoid packings differs substantially such that this does not hold true. This non-universality has implications for our understanding of jamming of aspherical particles.Comment: 6 pages, 5 figure

From quantum heat engines to laser cooling: Floquet theory beyond the Born–Markov approximation

We combine the formalisms of Floquet theory and full counting statistics with a Markovian embedding strategy to access the dynamics and thermodynamics of a periodically driven thermal machine beyond the conventional Born–Markov approximation. The working medium is a two-level system and we drive the tunneling as well as the coupling to one bath with the same period. We identify four different operating regimes of our machine which include a heat engine and a refrigerator. As the coupling strength with one bath is increased, the refrigerator regime disappears, the heat engine regime narrows and their efficiency and coefficient of performance decrease. Furthermore, our model can reproduce the setup of laser cooling of trapped ions in a specific parameter limit.DFG, 163436311, SFB 910: Kontrolle selbstorganisierender nichtlinearer Systeme: Theoretische Methoden und AnwendungskonzepteDFG, 87159868, GRK 1558: Kollektive Dynamik im Nichtgleichgewicht: in kondensierter Materie und biologischen SystemenEC/H2020/681456/EU/Energy Conversion and Information Processing at Small Scales/NanoTherm