The study examines the problem of approximate integration of multivariable functions. These functions are taken from a space with Gaussian measure. According to it, we calculated the average value of the integral standard deviation from theintegral sum. The paper gives the vanishing order for the standard deviation depending on the parameters that define the integral sum. we obtained probabilistic estimates of approximate integration errors.
An approach is described to implementation of the Method of Four Russians for reducing the dense matrices over GF(2) to row echelon form using the NVIDIA CUDA platform. Estimates of the algorithm running time and recommendations on choosing the algorithm parameters are given. It is shown that the developed implementation is most effective in comparison with the existing solutions for matrices of a size 2^17 x 2^17.
The main object of this paper consists in solving of the inverse problem of optical tomography through the development of a method of dynamical statistical spatial-temporal reconstruction of sizes and shape of moving three-dimensional objects with the usage of their projected images.
The newness of the given method consists in dynamical, few view, spatial-temporal reconstruction of sizes and shape not of a convex three-dimensional object itself, but its adequate approximation represented as a three-dimensional image of the ellipsoid of general form. Along with this, the geometrical sizes of a three-dimensional object are specified by numerical values of the axes of this three-dimensional image of ellipsoid and the average projected diameter of the image (D), and the shape factor (K) is specified by the ratio of the maximum and minimum overall dimensions (axes) of the image of the approximating ellipsoid.
The contours of three discrete, two-dimensional projected images of an object are specified as the optimal form of the primary geometrical information by a simulation modeling method. In this case, their spatial orientation is their mutual orthogonality. Their maximum and minimum overall dimensions are chosen as the optimal basic geometrical characteristics of projected images (the most informative characteristics according to the method of maximal entropy).
The mathematical model of object reconstruction is defined by the functional dependences of linear sizes of three mutually orthogonal axis of the approximating ellipsoid to the maximum and minimum dimensions of its three projected images.
In the result of statistical studies it is determined that the relative error of computing of the average projected diameter of an object is about 0.25% (at the reliability of PD = 0.7 and K = 1,3 relative units). The relative error of computing of form factor of an object is from 2.3% (PK = 0.7 and K = 1.3 relative units) to 0.6% (with PK = 0.96 and K = 1.05 relative units), and the total control time of object sizes and object shape does not exceed 10 Ms.
Thus, the proposed method of dynamic reconstruction has a new combination of characteristics of accuracy, reliability and performance. It has been successfully employed for a high performance, manufacturing, televisional size and form control for elements of the nuclear fuel and can be applied for remote control of various moving convex objects in real-time.
The article is devoted to the solution of stochastic recovery problem for a square-integrable (with respect to the Lebesgue measure) functions defined on the real number line from observations with additive white Gaussian noise for the case of discrete time. This is the problem of nonparametric (infinitedimensional) estimation. We prove optimality of the suggested recovery procedure (in the mean-square sense) with respect to the product of the Lebesgue measure and the Gaussian measure. Recovery algorithm for such square-integrable functions is provided. It is proved, that constructed nonparametric recovery procedure for a square-integrable function provides unbiased and consistent recovery for an unknown function. This new result has not been published yet. Furthermore, we suggest and prove almostoptimality of recovery procedure for a smooth recovered functions which gives non-improvable (in the sense of magnitude order) estimation of how number of orthogonal functions depends on observations' number. That is important, the error of constructed almost-optimal recovery procedure in comparison with optimal recovery procedure is no more than fifty per cent.