Grid methods for Bayes-optimal continuous-discrete filtering and utilizing a functional tensor train representation

Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.</p

Grid methods for Bayes-optimal continuous-discrete filtering and utilizing a functional tensor train representation

Optimal continuous-discrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.</p

A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the self-similarity parameter

By using chaos expansion into multiple stochastic integrals, we make a wavelet analysis of two self-similar stochastic processes: the fractional Brownian motion and the Rosenblatt process. We study the asymptotic behavior of the statistic based on the wavelet coefficients of these processes. Basically, when applied to a non-Gaussian process (such as the Rosenblatt process) this statistic satisfies a non-central limit theorem even when we increase the number of vanishing moments of the wavelet function. We apply our limit theorems to construct estimators for the self-similarity index and we illustrate our results by simulations

Core Mass Estimates in Strong Lensing Galaxy Clusters: A Comparison between Masses Obtained from Detailed Lens Models, Single-halo Lens Models, and Einstein Radii

The core mass of galaxy clusters is both an important anchor of the radial mass distribution profile and a probe of structure formation. With thousands of strong lensing galaxy clusters being discovered by current and upcoming surveys, timely, efficient, and accurate core mass estimates are needed. We assess the results of two efficient methods to estimate the core mass of strong lensing clusters: the mass enclosed by the Einstein radius (M(<θE), where θE is approximated from arc positions, and a single-halo lens model (MSHM), compared with measurements from publicly available detailed lens models (MDLM) of the same clusters. We use data from the Sloan Giant Arc Survey, the Reionization Lensing Cluster Survey, the Hubble Frontier Fields, and the Cluster Lensing and Supernova Survey with Hubble. We find a scatter of 18.1% (8.2%) with a bias of −7.1% (1.0%) between ${M}_{\mathrm{corr}}\left(\lt {\theta }_{\mathrm{arcs}}\right)$ (MSHM) and MDLM. Last, we compare the statistical uncertainties measured in this work to those from simulations. This work demonstrates the successful application of these methods to observational data. As the effort to efficiently model the mass distribution of strong lensing galaxy clusters continues, we need fast, reliable methods to advance the field

Fabrication of high quality plan-view TEM specimens using the focused ion beam

We describe a technique using a focused ion beam instrument to fabricate high quality plan-view specimens for transmission electron microscopy studies. The technique is simple, site-specific and is capable of fabricating multiple large, &gt;100 μm2 electron transparent windows within epitaxially-grown thin films. A film of La0.67Sr0.33MnO3 is used to demonstrate the technique and its structural and functional properties are surveyed by high resolution imaging, electron spectroscopy, atomic force microscopy and Lorentz electron microscopy. The window is demonstrated to have good thickness uniformity and a low defect density that does not impair the film’s Curie temperature. The technique will enable the study of in–plane structural and functional properties of a variety of epitaxial thin film systems

Dynamical approach to spectator fragmentation in Au+Au reactions at 35 MeV/A

The characteristics of fragment emission in peripheral $^{197}$Au+$^{197}$Au collisions 35 MeV/A are studied using the two clusterization approaches within framework of \emph{quantum molecular dynamics} model. Our model calculations using \emph{minimum spanning tree} (MST) algorithm and advanced clusterization method namely \emph{simulated annealing clusterization algorithm} (SACA) showed that fragment structure can be realized at an earlier time when spectators contribute significantly toward the fragment production even at such a low incident energy. Comparison of model predictions with experimental data reveals that SACA method can nicely reproduce the fragment charge yields and mean charge of the heaviest fragment. This reflects suitability of SACA method over conventional clusterization techniques to investigate spectator matter fragmentation in low energy domain.Comment: 6 pages, 5 figures, accepte