Determination of thermophysical characteristics of solid materials by electrical modelling of the solutions to the inverse problems in nonsteady heat conduction

The solution of the inverse problem of nonsteady heat conduction is discussed, based on finding the coefficient of the heat conduction and the coefficient of specific volumetric heat capacity. These findings are included in the equation used for the electrical model of this phenomenon

On-Line Learning of Linear Dynamical Systems: Exponential Forgetting in Kalman Filters

Kalman filter is a key tool for time-series forecasting and analysis. We show that the dependence of a prediction of Kalman filter on the past is decaying exponentially, whenever the process noise is non-degenerate. Therefore, Kalman filter may be approximated by regression on a few recent observations. Surprisingly, we also show that having some process noise is essential for the exponential decay. With no process noise, it may happen that the forecast depends on all of the past uniformly, which makes forecasting more difficult. Based on this insight, we devise an on-line algorithm for improper learning of a linear dynamical system (LDS), which considers only a few most recent observations. We use our decay results to provide the first regret bounds w.r.t. to Kalman filters within learning an LDS. That is, we compare the results of our algorithm to the best, in hindsight, Kalman filter for a given signal. Also, the algorithm is practical: its per-update run-time is linear in the regression depth

Oscillations and stability of numerical solutions of the heat conduction equation

The mathematical model and results of numerical solutions are given for the one dimensional problem when the linear equations are written in a rectangular coordinate system. All the computations are easily realizable for two and three dimensional problems when the equations are written in any coordinate system. Explicit and implicit schemes are shown in tabular form for stability and oscillations criteria; the initial temperature distribution is considered uniform

Variance Estimation For Dynamic Regression via Spectrum Thresholding

We consider the dynamic linear regression problem, where the predictor vector may vary with time. This problem can be modeled as a linear dynamical system, where the parameters that need to be learned are the variance of both the process noise and the observation noise. While variance estimation for dynamic regression is a natural problem, with a variety of applications, existing approaches to this problem either lack guarantees or only have asymptotic guarantees without explicit rates. In addition, all existing approaches rely strongly on Guassianity of the noises. In this paper we study the global system operator: the operator that maps the noise vectors to the output. In particular, we obtain estimates on its spectrum, and as a result derive the first known variance estimators with finite sample complexity guarantees. Moreover, our results hold for arbitrary sub Gaussian distributions of noise terms. We evaluate the approach on synthetic and real-world benchmarks