Analysis of a tensor POD-ROM for parameter dependent parabolic problems

A space-time-parameters structure of the parametric parabolic PDEs motivates the application of tensor methods to define reduced order models (ROMs). Within a tensor-based ROM framework, the matrix SVD -- a traditional dimension reduction technique -- yields to a low-rank tensor decomposition (LRTD). Such tensor extension of the Galerkin proper orthogonal decomposition ROMs (POD-ROMs) benefits both the practical efficiency of the ROM and its amenability for the rigorous error analysis when applied to parametric PDEs. The paper addresses the error analysis of the Galerkin LRTD-ROM for an abstract linear parabolic problem that depends on multiple physical parameters. An error estimate for the LRTD-ROM solution is proved, which is uniform with respect to problem parameters and extends to parameter values not in a sampling/training set. The estimate is given in terms of discretization and sampling mesh properties, and LRTD accuracy. The estimate depends on the smoothness rather than on the Kolmogorov n-widths of the parameterized manifold of solutions. Theoretical results are illustrated with several numerical experiments

Interpolatory tensorial reduced order models for parametric dynamical systems

The paper introduces a reduced order model (ROM) for numerical integration of a dynamical system which depends on multiple parameters. The ROM is a projection of the dynamical system on a low dimensional space that is both problem-dependent and parameter-specific. The ROM exploits compressed tensor formats to find a low rank representation for a sample of high-fidelity snapshots of the system state. This tensorial representation provides ROM with an orthogonal basis in a universal space of all snapshots and encodes information about the state variation in parameter domain. During the online phase and for any incoming parameter, this information is used to find a reduced basis that spans a parameter-specific subspace in the universal space. The computational cost of the online phase then depends only on tensor compression ranks, but not on space or time resolution of high-fidelity computations. Moreover, certain compressed tensor formats enable to avoid the adverse effect of parameter space dimension on the online costs (known as the curse of dimension). The analysis of the approach includes an estimate for the representation power of the acquired ROM basis. We illustrate the performance and prediction properties of the ROM with several numerical experiments, where tensorial ROM's complexity and accuracy is compared to those of conventional POD-ROM

Tensorial parametric model order reduction of nonlinear dynamical systems

For a nonlinear dynamical system that depends on parameters, the paper introduces a novel tensorial reduced-order model (TROM). The reduced model is projection-based, and for systems with no parameters involved, it resembles proper orthogonal decomposition (POD) combined with the discrete empirical interpolation method (DEIM). For parametric systems, TROM employs low-rank tensor approximations in place of truncated SVD, a key dimension-reduction technique in POD with DEIM. Three popular low-rank tensor compression formats are considered for this purpose: canonical polyadic, Tucker, and tensor train. The use of multilinear algebra tools allows the incorporation of information about the parameter dependence of the system into the reduced model and leads to a POD-DEIM type ROM that (i) is parameter-specific (localized) and predicts the system dynamics for out-of-training set (unseen) parameter values, (ii) mitigates the adverse effects of high parameter space dimension, (iii) has online computational costs that depend only on tensor compression ranks but not on the full-order model size, and (iv) achieves lower reduced space dimensions compared to the conventional POD-DEIM ROM. The paper explains the method, analyzes its prediction power, and assesses its performance for two specific parameter-dependent nonlinear dynamical systems

Neutrino wave function and oscillation suppression

We consider a thought experiment, in which a neutrino is produced by an electron on a nucleus in a crystal. The wave function of the oscillating neutrino is calculated assuming that the electron is described by a wave packet. If the electron is relativistic and the spatial size of its wave packet is much larger than the size of the crystal cell, then the wave packet of the produced neutrino has essentially the same size as the wave packet of the electron. We investigate the suppression of neutrino oscillations at large distances caused by two mechanisms: 1) spatial separation of wave packets corresponding to different neutrino masses; 2) neutrino energy dispersion for given neutrino mass eigenstates. We resolve contributions of these two mechanisms.Comment: 7 page

Experience in Developing Early Warning System for Financial Crises and the Forecast of Russian Banking Sector Dynamic in 2012

The article summarizes the key results of researches in the field of early warning systems for financial crises, conducted by the Center for Macroeconomic Analysis and Short-Term Forecasting (CMASF) since 2005. The proposed early warning system consists of three major blocks: the leading indicators of certain types of risks and the composite leading indicator of a systemic banking crisis; the medium-term scenario forecasting of key macroeconomic and financial indicators; stress testing of credit and liquidity risks of banks. On the basis of this early warning system we estimate the risks of financial crisis and some kinds of systemic risks in the different scenarios for the Russian economy in 2012. The analysis, in particular, revealed a sensitivity threshold of the domestic financial sector to changes in the world oil prices. Furthermore, it was found that the greatest destabilizing effect on the Russian financial sector may be caused by systemic credit risk.systemic financial crises, credit risks, liquidity risks, currency risks, early warning system, leading indicators, stress testing

Self-energy limited ion transport in sub-nanometer channels

The current-voltage characteristics of the alpha-Hemolysin protein pore during the passage of single-stranded DNA under varying ionic strength, C, are studied experimentally. We observe strong blockage of the current, weak super-linear growth of the current as a function of voltage, and a minimum of the current as a function of C. These observations are interpreted as the result of the ion electrostatic self-energy barrier originating from the large difference in the dielectric constants of water and the lipid bilayer. The dependence of DNA capture rate on C also agrees with our model.Comment: more experimental material is added. 4 pages, 7 figure