Model Order Reduction for (Stochastic-) Delay Equations With Error Bounds

We analyze a structure-preserving model order reduction technique for delay and stochastic delay equations based on the balanced truncation method and provide a system theoretic interpretation. Transferring error bounds based on Hankel operators to delay systems, we find error estimates for the difference between the dynamics of the full and reduced model. This analysis also yields new error bounds for bilinear systems and stochastic systems with multiplicative noise and non-zero initial states

An optimal control perspective on diffusion-based generative modeling

We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs) such as recently developed diffusion probabilistic models. In particular, we derive a Hamilton-Jacobi-Bellman equation that governs the evolution of the log-densities of the underlying SDE marginals. This perspective allows to transfer methods from optimal control theory to generative modeling. First, we show that the evidence lower bound is a direct consequence of the well-known verification theorem from control theory. Further, we develop a novel diffusion-based method for sampling from unnormalized densities -- a problem frequently occurring in statistics and computational sciences.Comment: Accepted for oral presentation at NeurIPS 2022 Workshop on Score-Based Method

From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs

The numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination of Monte Carlo methods and variational formulations, using neural networks for function approximation. Extending previous work (Richter et al., 2021), we argue that tensor trains provide an appealing framework for parabolic PDEs: The combination of reformulations in terms of backward stochastic differential equations and regression-type methods holds the promise of leveraging latent low-rank structures, enabling both compression and efficient computation. Emphasizing a continuous-time viewpoint, we develop iterative schemes, which differ in terms of computational efficiency and robustness. We demonstrate both theoretically and numerically that our methods can achieve a favorable trade-off between accuracy and computational efficiency. While previous methods have been either accurate or fast, we have identified a novel numerical strategy that can often combine both of these aspects

Does low soil base saturation affect fine root properties of European beech ( Fagus sylvatica L.)?

It is generally believed that high soil solution Al3+ in acidic soils with low base saturation (BS), negatively influences the properties of fine roots. Fine roots from European beech (Fagus sylvatica L.) trees growing in highly acidic soils with very low BS and potentially high Al3+ concentration in the soil solution were analysed and the dependency of fine root properties on soil BS was measured. The fine roots were sampled down to 1m depth at seven forest sites located on the Swiss Plateau. These sites varied in their BS from 1.4 to 11.4% in the mineral layers. We evaluated relationships between the BS of these mineral layers and fine root properties, such as ratio between bio- and necromass (live/dead ratio), specific root length (SRL), root tip abundance (RTA), root branching abundance (RBA), O2-consumption, and the Ca/Al molar ratio in the fine root tissue. The fine root properties were compared not only with the BS of the soil, but also with the Ca/Al molar ratio in the fine root tissues. Significant relations of fine root properties occurred when the soils of the seven sites were grouped into two BS groups (<5 and 5-10%). The live/dead ratio, the RTA, the RBA, the O2-consumption, and Ca/Al molar ratio were lower in the group of BS <5% than in the group 5-10%. Decreases in the morphological properties and in the O2-consumption were related to decrease in the Ca/Al molar ratio of the fine root tissues. There is evidence that the fine root properties are negatively influenced, nevertheless, fine root systems of mature European beech in their natural ecological environment seem to be able to compensate adverse effects of low B

Theory and Algorithms

The article surveys and extends variational formulations of the thermodynamic free energy and discusses their information-theoretic content from the perspective of mathematical statistics. We revisit the well-known Jarzynski equality for nonequilibrium free energy sampling within the framework of importance sampling and Girsanov change-of-measure transformations. The implications of the different variational formulations for designing efficient stochastic optimization and nonequilibrium simulation algorithms for computing free energies are discussed and illustrated. View Full-Tex

Fundamental absorption edges in heteroepitaxial YBiO3 thin films

The dielectric function of heteroepitaxial YBiO3 grown on a-Al2O3 single crystals via pulsed laser deposition is determined in the spectral range from 0.03 eV to 4.5 eV by a simultaneous modeling of the spectroscopic ellipsometry and optical transmission data of YBiO3 films of different thicknesses. The (111)-oriented YBiO3 films are nominally unstrained and crystallize in a defective fluorite-type structure with a Fm3⎯⎯m space group. From the calculated absorption spectrum, a direct electronic bandgap energy of 3.6(1) eV and the signature of an indirect electronic transition around 0.5 eV are obtained. These values provide necessary experimental feedback to previous conflicting electronic band structure calculations predicting either a topologically trivial or a non-trivial insulating ground state in YBiO3

Variational approach to rare event simulation using least-squares regression

We propose an adaptive importance sampling scheme for the simulation of rare events when the underlying dynamics is given by a diffusion. The scheme is based on a Gibbs variational principle that is used to determine the optimal (i.e. zero-variance) change of measure and exploits the fact that the latter can be rephrased as a stochastic optimal control problem. The control problem can be solved by a stochastic approximation algorithm, using the Feynman-Kac representation of the associated dynamic programming equations, and we discuss numerical aspects for high-dimensional problems along with simple toy examples.Comment: 28 pages, 7 figure

Application of the ESEM Technique in Wood Research: Part I. Optimization of Imaging Parameters and Working Conditions

A study using the ESEM (Environmental Scanning Electron Microscopy) technique was performed on wood objects in order to assess the particular advantages, possibilities, and limitations of this microscopic tool. In contrast to conventional high vacuum SEM, in ESEM specimens can be investigated in a gaseous atmosphere, usually of water vapor. This enables the observation of non-conductive, polymeric, composite, and porous materials (such as wood) in their natural state, without drying, evacuating, or sputtering them with a layer of carbon or metal. Further advantages include observations in a wide range of temperatures (-15° to 1000°C), conduction of dynamic processes such as condensation, freezing, and thawing of the specimen during observation, or mechanical testing.The imaging quality of ESEM for natural samples, however, is inferior to that of conventional SEM, and the specimens are liable to beam damage. The process of acquiring an image in ESEM is more complex than in SEM, demanding the optimization of a number of interacting parameters. These include the physical conditions of the specimen, conditions of the chamber environment, and electronic parameters of the formation and optimization of the image.The work on the ESEM can be performed through several operational modes that offer various sets of environmental and imaging conditions. This article presents guidance for assessment of influential operating parameters and their selection for the optimization of the ESEM work with wood

A New Ultra-dense Group of Obscured Emission-Line Galaxies

We present the discovery of an isolated compact group of galaxies that is extremely dense (median projected galaxy separation: 6.9 kpc), has a very low velocity dispersion ($\sigma_{\rm 2D}$ = 67 km s$^{-1}$), and where all observed members show emission lines and are morphologically disturbed. These properties, together with the lack of spirals and the presence of a prominent tidal tail make this group one of the most evolved compact groups.Comment: 15 pages,LaTeX, 2figures. A Postscript figure with spectra is available at ftp://astro.uibk.ac.at/pub/weinberger/ . Accepted for publication in ApJ Letter