On the Generalization of the Random Projection Method for Problems of the Recovery of Object Signal Described by Models of Convolution Type

Introduction. In technical systems, there is a common situation when transformation input-output is described by the integral equation of convolution type. This situation accurses if the object signal is recovered by the results of remote measurements. For example, in spectrometric tasks, for an image deblurring, etc. Matrices of the discrete representation for the output signal and the kernel of convolution are known. We need to find a matrix of the discrete representation of a signal of the object. The well known approach for solving this problem includes the next steps. First, the kernel matrix has to be represented as the Kroneker product. Second, the input-output transformation has to be presented with the usage of Kroneker product matrices. Third, the matrix of the discrete representation of the object has to be found. The object signal matrix estimation obtained with the help of pseudo inverting of Kroneker decomposition matrices is unstable. The instability of the object signal estimation in the case of usage of Kroneker decomposition matrices is caused by their discrete ill posed matrix properties (condition number is big and the series of the singular numbers smoothly decrease to zero). To find solutions of discrete ill-posed problems we developed methods based on the random projection and the random projection with an averaging by the random matrices. These methods provide a stable solutions with a small computational complexity. We consider the problem of object signals recovering in the systems where an input-output transformation is described by the integral equation of a convolution. To find a solution for these problems we need to build a generalization for two-dimensional signals case of the random projection method. Purpose. To develop a stable method of the recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. Results and conclusions. We developed the method of a stable recovery of object signal for the case in which an input-output transformation is described by the integral equation of a convolution. The stable estimation of the object signal is provided by Kroneker decomposition of the kernel matrix of convolution, computation of random projections for Kroneker factorization matrices, and a selection of the optimal dimension of a projector matrix. The method is illustrated by its application in technical problems.

On the Finiteness of Hulls

Let β≥φΛ. Is it possible to characterize normal, almost surely semi-degenerate categories? We show that K ̄is comparable to k. The ground breaking work of P. Laplace on vectors was a major advance. Recent developments in advanced discrete representation theory[1] have raised the question of whether the re exists a sub-continuous factor.

Detecting the Number of States in Raw Trajectories

In this paper, we present a simple method to detect the number of states in a stochastic trajectory. The method quantifies the degree of correlations in stochastic trajectories, computes the correlation function with two variables (the three-point correlation function), then finds the rank of the computed matrix (the method identifies the signal singular values, those that are beyond the noise). The computed rank is the number of states in the discrete trajectory, yet meaningful also in continuous trajectories; in such cases, the rank is compiled with the number of terms in the correlation function to determine the number of fluctuating independent potential profiles in the approximated discrete representation of the process.

On-the-Fly Treatment of Discrete Representation Thermal Neutron Scattering Data in RMC Code

A proper treatment of thermal neutron scattering data is required for the high-fidelity neutronics calculation of thermal reactors. Monte Carlo codes typically use an S(α, β) treatment to describe scattering events in the thermal region if the S(α, β) data is available for the material. The S(α,β) model stores a large majority of scattering physics and can handle thermal scattering process accurately. In neutronic-thermohydraulic coupling calculations, the temperature effect on nuclear data must be treated properly. The on-the-fly sampling method or the on-the-fly interpolation method are typically used in thermal region. In this paper, the on-the-fly interpolation method for the discrete representation S(α,β) data was introduced. The two-dimensional linear-linear interpolation was used to calculate the scattering cross sections and the secondary information for inelastic scattering, coherent elastic scattering and incoherent elastic scattering. The implemented on-the-fly capability was tested by a series of benchmarks that contain various thermal materials, including light water, beryllium and beryllium oxide. The integral kinf eigenvalues, the efficiency and the fine energy spectra of the on-the-fly treatment capacity were compared with those of the references. Results show that the on-the-fly treatment capability has high accuracy, and the computational time increases up to 20%.