Approximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equation

This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.

Closed Analytic Formulas for the Approximation of the Legendre Complete Elliptic Integrals of the First and Second Kinds

The author proposes two sets of closedanalytic functions for the approximate calculus of thecomplete elliptic integrals of the first and secondkinds in the normal form due to Legendre, therespective expressions having a remarkablesimplicity and accuracy. The special usefulness of theproposed formulas consists in that they allowperforming the analytic study of variation of thefunctions in which they appear, by using thederivatives. Comparative tables including theapproximate values obtained by applying the two setsof formulas and the exact values, reproduced fromspecial functions tables are given (all versus therespective elliptic integrals modulus, k = sin ). It is tobe noticed that both sets of approximate formulas aregiven neither by spline nor by regression functions,but by asymptotic expansions, the identity with theexact functions being accomplished for the left end k= 0 ( = 0) of the domain. As one can see, the secondset of functions, although something more intricate,gives more accurate values than the first one andextends itself more closely to the right end k = 1 ( =90) of the domain. For reasons of accuracy, it isrecommended to use the first set until  = 70.5 only,and if it is necessary a better accuracy or a greaterupper limit of the validity domain, to use the secondset, but on no account beyond  = 88.2.

AN APPROXIMATE METHOD FOR CALCULATING GRAIN FLOW IN A VERTICAL CYLINDRICAL VIBROSIEVE

A modified hydrodynamic model of a stable grain flow of an inhomogeneous mixture over the surface of a vertical cylindrical vibrating sieve is proposed under the assumption that the porosity of the mixture in the moving annular layer depends on the velocity of movement. A linear dependence of the porosity of the mixture on the velocity of movement are accepted, where higher speed corresponds to higher porosity.. The calculation of the velocity is reduced to solving an inhomogeneous differential equation of the Bessel type. Further, by "freezing" the variable coefficient of this equation, the problem has been simplified. This simplification is permissible due to the fact that the thickness of the moving layer of the mixture is much less than the radius of the vibrating sieve. As a result, the dependence of the velocity on the radial coordinate is expressed through the elementary functions. A compact formula for determining the maximum grain flow velocity is obtained. By integrating, in elementary functions, the formula for the average velocity in the layer is obtained. An approximate formula for the performance of the vibrating sieve by the mass of the exit fraction is derived. For this, it is proposed to calculate the corresponding integral approximately by the Simpson formula, so as not to calculate the values of special functions of large argument using the asymptotic formulas. It is shown that the named productivity significantly depends on the porosity of the grain mixture. In order to obtain information about the actual errors of the approximate formulas, we additionally carried out the numerical integration of the original non-simplified Bessel-type equation on a computer. A comparative analysis of the calculation results confirmed the small errors of the simplifications introduced into the equation of motion, as well as the adequacy of the theoretical results obtained. By passing to a simplified differential equation, approximate formulas were derived and tested for calculating the main characteristics of the grain flow along a vertical cylindrical vibrating sieve, taking into account the change in porosity in the grain mixture layer from the velocity of movement. The work summarizes the known theoretical results obtained using hydrodynamic models of the motion of grain mixtures fluidized by vibrations. The generalization carried out slightly complicated the theory, because the final calculation formulas are quite compact and convenient in practical implementation.

Determination of the effective stiffness of a unidirectional layer by the finite element method and approximate formulas

Layered composites formed by unidirectional layers are widely used in aviation in the most loaded areas of the aircraft. Data on the elastic properties of the layers are required for the strength and stiffness calculation of structural elements made of such materials. There are two possible approaches to address the problem. The first approach is based on solving the problem of micromechanics using methods of the theory of elasticity. The second approach consists in developing a simplified model of a unidirectional layer. Analysis of the model can provide for fairly simple formulas for determination of the effective stiffness of a unidirectional layer. A comparative analysis of the results obtained in both approaches revealed the limits of applicability of approximate formulas derived for evaluating the effective characteristics of the different types of composites depending on the volume content of fibers. The effective elastic characteristics of unidirectional composites are determined by the finite element method in the framework of the linear theory of elasticity. The boundary value problem is solved for a characteristic representative element selected in accordance with the physical and geometric parameters of the medium of an ordered structure. A set of algorithm-programs has been developed under ANSYS environment which automates calculations of the elastic characteristics of materials depending on the volume content of fibers at different ratios of the elastic properties of fibers and binder, and on the parameters of the curvature of the fiber cross-sectional profile. The results obtained by the numerical method are compared with the data obtained experimentally and by approximate formulas.

An approximate formula for calculating the expectations of functionals from random processes based on using the Wiener chaos expansion

In this work, we propose a new method for calculating the mathematical expectation of nonlinear functionals from random processes. The method is based on using Wiener chaos expansion and approximate formulas, exact for functional polynomials of given degree. Examples illustrating approximation accuracy are considered.

Fast and accurate approximation of the joint site frequency spectrum of multiple populations

The site frequency spectrum (SFS) is a statistic that summarizes the distribution of derived allele frequencies in a sample of DNA sequences. The SFS provides useful information about genetic variation within and among populations and it can used to make population genetic inferences. Methods for computing the SFS based on the diffusion approximation are computationally efficient when computing all terms of the SFS simultaneously and they can handle complicated demographic scenarios. However, in practice it is sometimes only necessary to compute a subset of terms of the SFS, in which case coalescent-based methods can achieve greater computational efficiency. Here, we present simple and accurate approximate formulas for the expected joint SFS for multiple populations connected by migration. Compared with existing exact approaches, our approximate formulas greatly reduce the complexity of computing each entry of the SFS and have simple forms. The computational complexity of our method depends on the index of the entry to be computed, rather than on the sample size, and the accuracy of our approximation improves as the sample size increases.

Approximate Formulas for Zeta Functions of Selberg’s Type in Quotients of SL4

The goal of the paper is to derive some approximate formulas for the logarithmic derivative of several zata functions of Selberg’s type for compact symmetric spaces formed as quotients of the Lie group SL4 (R). Such formulas, known in literature as Tutchmarsh-Landau style approximate formulas, are usually applied in order to obtain prime geodesic theorems in various settings of underlying locally symmetric spaces.

Bayesian Time-lapse Difference Inversion Based on the exact Zoeppritz Equations with Blockiness Constraint

Accurately inverting changes in the reservoir elastic parameters that are caused by oil and gas exploitation is of great importance in accurately describing reservoir dynamics and enhancing recovery. Previously numerous time-lapse seismic inversion methods based on the approximate formulas of exact Zoeppritz equations or wave equations have been used to estimate these changes. However the low accuracy of calculations using approximate formulas and the significant calculation effort for the wave equations seriously limits the field application of these methods. However, these limitations can be overcome by using exact Zoeppritz equations. Therefore, we study the time-lapse seismic difference inversion method using the exact Zoeppritz equations. Firstly, the forward equation of time-lapse seismic difference data is derived based on the exact Zoeppritz equations. Secondly, the objective function based on Bayesian inversion theory is constructed using this equation, with the changes in elastic parameters assumed to obey a Gaussian distribution. In order to capture the sharp time-lapse changes of elastic parameters and further enhance the resolution of the inversion results, the blockiness constraint, which follows the differentiable Laplace distribution, is added to the prior Gaussian background model. All examples of its application show that the proposed method can obtain stable and reasonable P- and S-wave velocities and density changes from the difference data. The accuracy of estimation is higher than for existing methods, which verifies the effectiveness and feasibility of the new method. It can provide high-quality seismic inversion results for dynamic detailed reservoir description and well location during development.