Approximation properties of the Generalized Finite Element Method

In this paper, we have obtained an approximation result in the Generalized Finite Element Method (GFEM) that reflects the global approximation property of the Partition of Unity (PU) as well as the approximability of the local approximation spaces. We have considered a GFEM, where the underlying PU functions reproduce polynomials of degree l. With the space of polynomials of degree k serving as the local approximation spaces of the GFEM, we have shown, in particular, that the energy norm of the GFEM approximation error of a smooth function is O(h ^l + k ). This result cannot be obtained from the classical approximation result of GFEM, which does not reflect the global approximation property of the PU.     

Korteweg-de Vries hierarchy as an asymptotic limit of the Boussinesq system

For the model of surface waves, we perform an asymptotic analysis with respect to a small parameter ε for large times where corrections to the approximation described by the Korteweg-de Vries equation must be taken into account. We reveal the appearance of the Korteweg-de Vries hierarchy, which ensures the construction of an asymptotic representation up to the times t ≈ ε⁻², where the Korteweg-de Vries approximation becomes inapplicable. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 294–304, February, 2008.     

Dispersion analysis of triangle-based spectral element methods for elastic wave propagation

We study the numerical dispersion/dissipation of Triangle-based Spectral Element Methods (TSEM) of order N????1 when coupled with the Leap-Frog (LF) finite difference scheme to simulate the elastic wave propagation over a structured triangulation of the 2D physical domain. The analysis relies on the discrete eigenvalue problem resulting from the approximation of the dispersion relation. First, we present semi-discrete dispersion graphs by varying the approximation polynomial degree and the number of discrete points per wavelength. The fully-discrete ones are then obtained by varying also the time step. Numerical results for the TSEM, resp. TSEM-LF, are compared with those of the classical Quadrangle-based Spectral Element Method (QSEM), resp. QSEM-LF.     

Retardation effects in the form factor of the two-particle bound state

In the Logunov-Tavkhelidze quasipotential approach, convergence of the iteration procedure for taking the dynamic retardation effect in the electromagnetic form factor into account is investigated for a bound system of two scalar particles interacting via a separable Bethe-Salpeter kernel. The status of the relativistic impulse approximation within the framework of this approach is discussed. In the static approximation, the electromagnetic vertex operator obeying the current conservation and normalization conditions is constructed, taking the interaction current into account. A comparative analysis of the form factor in different approximations is carried out. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 272–290, February, 1997.     

Two-loop calculations of the matrix σ-model effective action in the background field formalism

We consider the matrix σ-model in the background field formalism. In the two-loop approximation, we demonstrate the equality of “running coupling constants” in the momentum cutoff regularization and in the dimensional regularization by direct calculation. We verify that the β-function coincides with the previously obtained data. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 354–362, February, 2008.     

Averaging the resolvent with a space-correlated random potential

The problem of the quasi-particle spectrum in a binary disordered alloy with a space-correlated random potential is considered. The extended space formalism is used to represent the average resolvent. To calculate the mass operator, some self-consistent approximation procedures are suggested that coincide with the well-known self-consistent approximations for α=0 (where α is the short-range order parameter). The elaborated theory ensures the correct passage to the Green's function of a perfect crystal in the limits α→1 and α→−1 for any concentration and 50% concentration, respectively. The approximations possess the correct analytic properties for all values of the parameter α. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 114, No. 2, pp. 296–313, February, 1998.     

A Note on the Strong Approximation of the Smoothed Empirical Process of α-mixing Sequences

A strong approximation of the smoothed empirical process of strictly stationary α-mixing random variables by a sequence of iid Gaussian processes will be studied. Here, the smoothing is done via kernel density estimators. No assumptions are made on the support of the kernel; in fact, our main results are stated for kernels with possibly an infinite support. Received June 2003; Accepted February 2004.     

Superconvergence in the generalized finite element method

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct. I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724. U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899. J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.     

Construction of asymptotically optimal controls for control and game problems

Recently the connection between control and game problems and Backward Stochastic Differential Equations has been established. This allows us to use an approximation scheme for such equations in order to construct an ɛ-optimal control. Received: 13 November 1995 / Revised version: 11 February 1998     

On finite-dimensional approximation of solutions of ill-posed problems

We show that the modified method for finite-dimensional approximation of solutions of Fredholm integral equations of the first kind presented in this paper is more economical than traditional methods for finite-dimensional approximation. Institute of Mathematies, Ukrainian Academy of Science, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 2, pp. 289–295, February, 1997.     

Numerical analysis of convergent perturbation expansions in quantum theory

We present the results of a numerical analysis of the convergence of the new perturbation expansion recently proposed by Belokurov, Solovyev, and Shavgulidze. Two particular examples are considered: the anharmonic oscillator in quantum mechanics and the renormalization group β-function in field theory. It is shown that in the first case, the series converges to an exact value in a wide range of expansion parameters. This range can be enlarged with the help of the Padé approximation. In field theory, the results have a stronger dependence on the regularization parameter. We discuss an algorithm for choosing this parameter that produces stable results. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 291–297, February, 1997.     

On the best linear approximation methods and the widths of certain classes of analytic functions

We discuss the best linear approximation methods in the Hardy spaceH _q q≥1, for classes of analytic functions studied by N. Ainulloev; these are generalizations (in a certain sense) of function sets introduced by L. V. Taikov. The exact values of their linear and Gelfandn-widths are obtained. The exact values of the Kolmogorov and Bernsteinn-widths of classes of analytic (in |z|<1) functions whose boundaryK-functionals are majorized by a prescribed functions are also obtained. Translated fromMatermaticheskie Zametki, Vol. 65, No. 2, pp. 186–193, February, 1999.     

A new rounding procedure for the assignment problem with applications to dense graph arrangement problems

We present a randomized procedure for rounding fractional perfect matchings to (integral) matchings. If the original fractional matching satisfies any linear inequality, then with high probability, the new matching satisfies that linear inequality in an approximate sense. This extends the well-known LP rounding procedure of Raghavan and Thompson, which is usually used to round fractional solutions of linear programs.?We use our rounding procedure to design an additive approximation algorithm to the Quadratic Assignment Problem. The approximation error of the algorithm is εn ² and it runs in n ^{O (log n /ε2)} time.?We also describe Polynomial Time Approximation Schemes (PTASs) for dense subcases of many well-known NP-hard arrangement problems, including MINIMUM LINEAR ARRANGEMENT, MINIMUM CUT LINEAR ARRANGEMENT, MAXIMUM ACYCLIC SUBGRAPH, and BETWEENNESS. Received: December 12, 1999 / Accepted: October 25, 2001?Published online February 14, 2002     

On the construction of bodies of constant width containing a given set

We discuss the problem of sparse representation of domains in ℝ ^d . We demonstrate how the recently developed general theory of greedy approximation in Banach spaces can be used in this problem. The use of greedy approximation has two important advantages: (1) it works for an arbitrary dictionary of sets used for sparse representation and (2) the method of approximation does not depend on smoothness properties of the domains and automatically provides a near optimal rate of approximation for domains with different smoothness properties. We also give some lower estimates of the approximation error and discuss a specific greedy algorithm for approximation of convex domains in ℝ².     

A remark on the behaviour ofL p-multipliers and the range of operators acting onL p-spaces

In this paper, new results are obtained concerning the uniform approximation property (UAP) inL ^p-spaces (p≠2,1,∞). First, it is shown that the “uniform approximation function” does not allow a polynomial estimate. This fact is rather surprising since it disproves the analogy between UAP-features and the presence of “large” euclidian subspaces in the space and its dual. The examples are translation invariant spaces on the Cantor group and this extra structure permits one to replace the problem with statements about the nonexistence of certain multipliers in harmonic analysis. Secondly, it is proved that the UAP-function has an exponential upper estimate (this was known forp=1, ∞). The argument uses Schauder’s fix point theorem. Its precise behaviour is left unclarified here. It appears as a difficult question, even in the translation invariant context.     

Perturbation theory for the periodic Anderson model

We investigate the system of conductivity electrons and f-localized electrons described by the periodic Anderson model. Single-site hybridization of the state of two constituent subsystems of electrons is treated as a perturbation. We develop a new diagram technique based on the use of multiparticle one-site irreducible Green’s functions for the f-electrons and the standard Wick theorem for the subsystem of conductivity electrons. We derive the Dyson equations for the one-particle Green’s functions and find the relation between these functions. These results are exact and can be used as a starting point for self-consistent approximations. In the Hubbard-I approximation, we analyze the spectrum of one-particle perturbations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 110, No. 2, pp. 308–322, February, 1997.     

Quadrature formulas for the generalized Riemann-Stieltjes integral

In this paper we propose a technique of approximation for the generalized Riemann-Stieltjes integral and we found an analogue for Newton-Cotes formulas in the case n = 2 and n = 3. *Beneficiary of a Socrates fellowship at the Department of Mathematics, University of Study of Cagliari, Via Ospedale, n. 72, Cagliari, 09124, Italy, in the period February – July 2002.     

Smoluchowski–Kramers approximation in the case of variable friction

We consider the small mass asymptotics (Smoluchowski–Kramers approximation) for the Langevin equation with a variable friction coefficient. The limit of the solution in the classical sense does not exist in this case. We study a modification of the Smoluchowski–Kramers approximation. Some applications of the Smoluchowski–Kramers approximation to problems with fast oscillating or discontinuous coefficients are considered. Bibliography: 15 titles.     

New asymptotic expansion for the Gamma function

Using a series transformation, the Stirling-De Moivre asymptotic series approximation to the Gamma function is converted into a new one with better convergence properties. The new formula is being compared with those of Stirling, Laplace, and Ramanujan for real arguments greater than 0.5 and turns out to be, for equal number of “correction” terms, numerically superior to all of them. As a side benefit, a closed-form approximation has turned up during the analysis which is about as good as 3rd order Stirling’s (maximum relative error smaller than 1e − 10 for real arguments greater or equal to 24).     

Analysis of the free-edge effect in composite laminates by the boundary finite element method

Because of the risk of delamination due to high interlaminar stresses in the vicinity of free edges of composite laminates, there is a strong interest in efficient methods for the analysis of this free-edge effect. By the example of a symmetric [0°/90°]_s cross-ply laminate, the Boundary Finite Element Method is presented as a very efficient numerical method, which combines the advantages of the finite element method and the boundary element method. Analogously to the boundary element method, only the boundary is discretized, while the element formulation is finite element based. The resultant stress field is shown to be in very good agreement qualitatively and quantitatively with the comparative finite element analysis. Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000). Published in Mekhanika Kompozitnykh Materialov, Vol. 36, No. 3, pp. 355–366, March–April, 2000.