On spline approximation with a reproducing kernel method

Spline approximation with a reproducing kernel of a semi-Hilbert space is studied. Conditions are formulated that uniquely identify the natural Hilbert space by a reproducing kernel, a trend of the spline, and the approximation domain. The construction of a spline with external drift is proposed. It allows one to approximate functions having areas of large gradients or first-kind discontinuities. The conditional positive definiteness of some known radial basis functions is proved.     

On a certain method for the approximation of attractors

Dissipative semidynamical systems in the metric space X of all possible bounded closed subsets of a metric space X are considered. The obtained results are applied for the construction of a method of approximation of the attractor of an arbitrary semidynamical system in X in the case when the action of this system is only approximately known.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 188, pp. 87–104, 1991.     

On the dynamics of a third order Newton's approximation method

We provide an answer to a question raised by S. Amat, S. Busquier, S. Plaza on the qualitative analysis of the dynamics of a certain third order Newton type approximation function , by proving that for functions f twice continuously differentiable and such that both f and its derivative do not have multiple roots, with at least four roots and infinite limits of opposite signs at , has periodic points of any prime period and that the set of points a at which the approximation sequence does not converge is uncountable. In addition, we observe that in their Scaling Theorem analyticity can be replaced with differentiability.     

On a method of approximation of solutions to delay differential equations

Connections between solutions to a class of systems of ordinary differential equations of a large dimension and delay equations are studied. A new method is justified for approximation of solutions to delay equations.     

On an approximation method of a discontinuous solution of an ill-posed problem

Some new algorithms of solving approximately linear and semilinear ill-posed problems with discontinuous exact solutions are proposed.     

On approximation in the fictitious domain method for a bidimensional transport equation

In this article a numerical method for solving a two‐dimensional transport equation in the stationary case is presented. Using the techniques of the variational calculus, we find the approximate solution for a homogeneous boundary‐value problem that corresponds to a square domain D₂. Then, using the method of the fictitious domain, we extend our algorithm to a boundary value problem for a set D that has an arbitrary shape. In this approach, the initial computation domain D (called physical domain) is immersed in a square domain D₂. We prove that the solution obtained by this method is a good approximation of the exact solution. The theoretical results are verified with the help of a numerical example. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

On finite difference approximation of a matrix-vector product in the Jacobian-free Newton-Krylov method

The Jacobian-free Newton-Krylov (JFNK) method is a special kind of Newton-Krylov algorithm, in which the matrix-vector product is approximated by a finite difference scheme. Consequently, it is not necessary to form and store the Jacobian matrix. This can greatly improve the efficiency and enlarge the application area of the Newton-Krylov method. The finite difference scheme has a strong influence on the accuracy and robustness of the JFNK method. In this paper, several methods for approximating the Jacobian-vector product, including the finite difference scheme and the finite difference step size, are analyzed and compared. Numerical results are given to verify the effectiveness of different finite difference methods.     

On a variational approximation method for a class of elliptic eigenvalue problems in composite structures

We consider a second-order elliptic eigenvalue problem on a convex polygonal domain, divided in nonoverlapping subdomains. The conormal derivative of the unknown function is continuous on the interfaces, while the function itself is discontinuous. We present a general finite element method to obtain a numerical solution of the eigenvalue problem, starting from a nonstandard formally equivalent variational formulation in an abstract setting in product Hilbert spaces. We use standard Lagrange finite element spaces on the subdomains. Moreover, the bilinear forms are approximated by suitable numerical quadrature formulas. We obtain error estimates for both the eigenfunctions and the eigenvalues, allowing for the case of multiple exact eigenvalues, by a pure variational method.

     

On the range of use of the tight-binding approximation method for a complex-valued potential

We consider a one-dimensional Schrödinger operator with a periodic potential constructed as the sum of the shifts of a given complex-valued potential q that decreases at infinity. Mathematical justification of the tight-binding approximation method is presented. Let λ₀ be an isolated eigenvalue of the Schrödinger operator with a potential q. Then, for the corresponding operator with a periodic potential, a continuous spectrum exists in the neighborhood of λ₀. The asymptotic behavior of this part of the spectrum as the period increases infinitely is studied for the cases of one- and two-dimensional invariant subspaces corresponding to λ₀.     

On saturation for a spline approximation

We obtain a saturation theorem for a wavelet operator which is constructed by B-spline function.     

An approximation method for a fokker-planck-vlasov equation

A simplified Fokker-Planck equation of statistical plasma physics is mathematically investigated. For the Cauchy problem a constructive approximation method is introduced. This procedure yields a sequence (fn) of approximate densities, uniformly converging to the problem's global classical solution with linear convergence velocity.     

On approximation of a certain differential operator

This paper consider the approximation problem of the image of an analytic function under the action of a higher-order differential operator with polynomial coefficients. The obtained formula generalizes the corresponding formulas of numerical differentiation. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Spline approximation method

A system of relations is presented for determining optimal coefficients for a set consisting of an arbitrary number of smoothly matched cubic polynomial dependences. As an example, the method is used to construct optimal approximating formulas for the viscosity of air in thermochemical equilibrium as a function of enthalpy.     

On a limit class of approximation spaces

We describe the behaviour under interpolation of a limit class of approximation spaces. We characterize them as extrapolation spaces. Moreover, we study the boundedness of certain operators on these spaces. As an application, we derive several results on Macaev operator ideals.