Showing papers in "Ussr Computational Mathematics and Mathematical Physics in 1970"

Iterative methods using lagrange multipliers for solving extremal problems with constraints of the equation type

Abstract: Some methods of minimization under constraints of the equation type will be discussed, in which the iterative process is based both on the initial variables and on dual variables (Lagrange multipliers). In the classification given in [1], these methods are of the duality type. Alternatively, they may be interpreted as iterative methods for finding the stationary (in particular, saddle) points of the Lagrange function. We assume that both the functional to be minimized, and the constraints, are reasonably smooth. In Section 1 we describe the methods, prove their local convergence, and estimate the rate of convergence. In Section 2 we consider the merits and drawbacks of the methods from the computational point of view, and outline a means for selecting the initial approximation.

Completely conservative difference schemes for magnetohydrodynamic equations

Abstract: A class of schemes called completely conservative, which are free from this defect, exist. These schemes satisfy not only the difference analogs of the fundamental laws of the conservation of mass, momentum and total energy (as for ordinary conservative schemes), but the detailed energy balance is also valid, that is, the balance with respect to the individual forms of energy: internal and kinetic.

Completely conservative difference schemes for the equations of gas dynamics in Euler's variables

Abstract: 1. THE set of equations of gas dynamics can be written in a variety of ways, each capable of direct physical interpretation. For instance, the energy equation may be written in the divergent form, expressing the variation of the total energy. or in the non-divergent form, expressing the variation of the internal energy only, or in the entropic form etc. These forms are equivalent in the differential form, in the sense that one can be obtained from the other by means of the remaining equations of the set.

Minimization methods based on approximation of the initial functional by a convex functional

Abstract: The results obtained in [1] will be developed below. At the present time, minimization methods based on linearization of the initial functional (gradient methods) are well known [2–5] for problems on an unconstrained extremum and for minimization problems in a closed convex region. There are also well-known second-order methods, convergent from any initial approximation (they are in essence a generalization of Newton's method); such niethods were discussed in [6] for the unconstrained extremum problem, and in [1] for the constrained problem. Second-order methods are based on quadratic approximation of the functional. Theorem 1 of the present paper indicates a general approach to the construction of methods of this type; it shows that, under certain conditions, minimization algorithms can be based on approximation of the initial functional by a segment of a Taylor series with an arbitrary number of terms. All the methods that can be obtained by means of Theorem 1 are convergent from any initial approximation; this is achieved by regulating the length of step in the direction of motion. The algorithm for selecting the length of step proposed in the theorem is unified for all methods, and differs from previous algorithms. Its advantages and usefulness are discussed in the paper. We observe, for instance, that our algorithm for step length selection in secondorder methods enables a higher order estimate to be obtained for the rate of convergence than that obtained in [6], while at the same time reducing the labour of each iteration. This result follows from Theorem 2. We show in Theorem 3 that the convergence rate estimate obtained in [6] for a second-order method for solving unconstrained extremum problems, still holds for solving problems of minimization in a closed convex set (with the same method of step length selection). The approach used in Theorem 1 for constructing algorithms for solving problems of minimization in a closed region, may also be used for proving the convergence of algorithms employed for solving problems with supplementary restrictions in the form of equations (Theorem 4). Algorithms for solving such problems were considered in [1], but with a different method of step length selection from that used in Theorem 4. By using the method of Theorem 4, the number of computations in such problems may be considerably reduced.

Properties of optimal methods for the solution of problems of mathematical physics

Abstract: IN this paper we formulate hypotheses for the characteristics of optimal methods for solving problems of mathematical physics. The truth of the statement of one of the hypotheses is demonstrated by an example of the method of alternating directions for solving the heat-conduction equation.

Methods of the Runge-Kutta type for gas dynamics

Abstract: THE difference methods of direct calculation of the first and second order of accuracy used at the present time make it possible to calculate the pressure, velocity and temperature fields with an accuracy amounting to a few percent in typical conditions. The greater storage capacity and higher speed of modern computers make it possible to consider the problem of developing and incorporating complex methods for an increase of accuracy approaching that attained for ordinary differential equations.

Optimal search for the extrema of unimodal functions

Abstract: IN this paper search algorithms (optimal with respect to two different criteria) are found for the extrema of unimodal functions of one variable which satisfy a Lipschitz condition. The algorithms obtained are a generalization of the well-known Kiefer-Johnson method [1, 2].

The solution of minimaximin problems

Abstract: AT the present time considerable attention is being devoted to the solution of the problem of finding the minimum of a function of the form ϕ(x)= max y∈Ω f(x,y) Similar problems arise in various theoretical and practical fields (e.g. in the theory of optimal control, differential games, best approximation, etc. see [1, 2]). In the study of these questions the problem of minimising (or maximising) The differentiability with respect to direction of the function ϕ(x)= max y∈Ω1 min z∈Ω2 f(x,y,z) and the form of its directional derivative (with the satisfaction of certain conditions for the function f) was established in [3]. In the present paper a necessary condition for a minimum of the above function is given and a method of successive approximation is described. We here confine ourselves to the case where the sets Ω 1 and Ω 2 consist of a finite number of points, and the function ϑ(x) can then be written in the form (1.1), x ϵ En.

Solution of problems using computational models

Abstract: When solving the problem: “find the values of a set of variables V, given the values U of a set of variables W’, in essence we use information on the structure of the model which is used for solving the problem. The process of solution involves the formation of an algorithm and computation on the basis of this algorithm. Both stages are performed by using the computational model of some realising system.

Cubature formulae and orthogonal polynomials

Abstract: IN [1] the proposition (Theorem 2' ) was stated that k1 common zeros of two orthogonal polynomials of degree k of a plane domain and with a positive weight function, can be taken as the nodes of a cubature formula exact for polynomials of degree 2 k — 1. Here this statement is proved in a somewhat stronger form: the common zeros of the orthogonal polynomials are not assumed to be different.

Blunt body flow of a viscous radiating gas

Abstract: THE results of a number of papers D-41 show that in the calculation of the blunt body flow of a hypersonic air stream taking radiation into account, it is necessary to take into consideration the radiation cooling of the shock layer, the absorption of energy and the dependence of the absorption coefficient not only on the temperature and density, but also on the frequency. The radiation cooling may lead to the appearance of a significant temperature gradient in the main part of the shock wave, which throws some doubt on possibility of dividing the shock layer into a region of non-viscous flow and a boundary layer. When there is absorption the radiative transfer of energy leads to interaction between the main and boundary flow regions. It is extremely difficult to take account of consideration of these regions of the shock layer, especially if we take into account the complex nature of the dependence of the absorption coefficient of air on the frequency. Therefore, in studying the blunt flow of radiating air taking into account radiation cooling and absorption, it is necessary at the same time to consider the entire flow between the shock wave and the frontal surface of the body.

A method of minimal iterations for evaluating the eigenvalues of an elliptic operator

Abstract: The present paper describes experiments on evaluating the spectrum of the difference approximation of an elliptic operator by the familiar Lanczos algorithm of minimal iterations (see [1, 2]). We use this algorithm to reduce the matrix of the difference operator to the Jacobi form, and then evaluate the spectrum by employing the programme described in [3]. Knowing the eigenvalues, we can easily find the eigenvectors, and this is in fact done. It turns out that, for the difference problem in question, the reduction of the matrix to the tridiagonal form does not necessarily need to be carried to completion. The truncated Jacobi matrices give spectra which approximate more and more closely to the spectrum of the initial difference problem as their order increases. The rate of this convergence can be estimated by the usual methods (see [2, 4]). It follows from these estimates that, as the mesh of the difference approximation is refined, the convergence deteriorates. The present paper was inspired by the striking fact that, given 15 × 15 or 30 × 30 points in the difference mesh, the convergence of the method for evaluating the spectrum of the Laplace operator is still entirely satisfactory. We employed our process to evaluate several eigenfrequencies and eigenfunctions of piezo-electric resonators, and compared our results with those obtained in [5]; we also applied the process to a model problem on the spectrum of the Laplace operator. Our experience reveals that standard algebraic procedures can be applied to elliptic differences problems where the number of points is limited but adequate for a wide range of engineering purposes.