A variation of functional equations related to sine type functions

In this paper, we consider a class of functional equation Q(λ)Y (λ) −P(λ)Z(λ) = η related to sine type functions, where the known P,Q are appropriate entire functions of exponential type. We are concerned with the existence and uniqueness of the solution (Y,Z) under certain circumstances. Furthermore, we modify the Lagrange interpolation to deal with the situation of the interpolation nodes being counted by multiplicities, which is significant to solve the above functional equation.

An Interpolation-Based Polynomial Method of Estimating the Objective Function Value in Scheduling Problems of Minimizing the Maximum Lateness

An approach to estimating the objective function value of minimization maximum lateness problem is proposed. It is shown how to use transformed instances to define a new continuous objective function. After that, using this new objective function, the approach itself is formulated. We calculate the objective function value for some polynomially solvable transformed instances and use them as interpolation nodes to estimate the objective function of the initial instance. What is more, two new polynomial cases, that are easy to use in the approach, are proposed. In the end of the paper numeric experiments are described and their results are provided.

Constructing Steklov-type cubature formulas for a finite element in the shape of a bipyramid

This paper reports the construction of cubature formulas for a finite element in the form of a bipyramid, which have a second algebraic order of accuracy. The proposed formulas explicitly take into consideration the parameter of bipyramid deformation, which is important when using irregular grids. The cubature formulas were constructed by applying two schemes for the location of interpolation nodes along the polyhedron axes: symmetrical and asymmetrical. The intervals of change in the elongation (compression) parameter of a bipyramid semi-axis have been determined, within which interpolation nodes of the constructed formulas belong to the integration region, while the weight coefficients are positive, which warrants the stability of calculations based on these cubature formulas. If the deformation parameter of the bipyramid is equal to unity, then both cubature formulas hold for the octahedron and have a third algebraic order of accuracy. The resulting formulas make it possible to find elements of the local stiffness matrix on a finite element in the form of a bipyramid. When calculating with a finite number of digits, a rounding error occurs, which has the same order for each of the two cubature formulas. The intervals of change in the elongation (compression) parameter of the bipyramid semi-axis have been determined, which meet the requirements, which are employed in the ANSYS software package, for deviations in the volume of the bipyramid from the volume of the octahedron. Among the constructed cubature formulas for a bipyramid, the optimal formula in terms of the accuracy of calculations has been chosen, derived from applying a symmetrical scheme of the arrangement of nodes relative to the center of the bipyramid. This formula is invariant in relation to any affinity transformations of the local bipyramid coordinate system. The constructed cubature formulas could be included in libraries of methods for approximate integration used by those software suites that implement the finite element method.

Rational approximation in initial boundary problems with the fronts

Предложен спектральный метод на основе дробно-рациональной аппроксимации. На примере решений уравнения Бюргерса с особенностями в виде фронтов показано, что дробно-рациональное приближение решений имеет существенные преимущества перед полиномиальным. Для эффективной реализации дробно- рациональной аппроксимации в работе использована барицентрическая интерполяционная формула Лагранжа, обеспечивающая быстроту вычислений и численную устойчивость. Для адаптации узлов интерполяции использован метод, основанный на аппроксимации положения особенности аналитического продолжения решения в комплексной плоскость. Предложено обобщение метода на случай нескольких особенностей. Описано построение спектрального метода и проведены расчеты на модельных задачах, в т. ч. с двумя фронтами. A spectral method with adaptive rational approximation is proposed. In traditional spectral polynomial interpolation, the interpolation points are fixed, usually at the roots or extrema of orthogonal polynomials. Free selection of interpolation points is impossible due to the effect described in the Runge example. The key feature of rational interpolation is the free distribution of interpolation nodes without the occurrence of the Runge phenomenon. Nevertheless, in practice it is very important to implement rational approximation effectively. Here rational approximation is implemented using the barycentric Lagrange form. This leads to fast computations and numerical stability comparable with the polynomial interpolation. It is shown that rational interpolation has significant advantages over polynomial on functions that have singularities in the form of fronts. The key idea is that rational interpolation allows adapting interpolation points according to function singularities. An effective method of grid adaptation that accounts for singularity location was used. Method was generalized to the case of several singularities, for example, for solutions with several fronts. For the solutions of the Burgers equation with singularities in the form of fronts, it is shown that rational interpolation has significant advantages over polynomial. The implementation of spectral method is described, and calculations results on model problems, including problems with two fronts, are presented.

Approximation and Uncertainty Quantification of Systems with Arbitrary Parameter Distributions Using Weighted Leja Interpolation

Approximation and uncertainty quantification methods based on Lagrange interpolation are typically abandoned in cases where the probability distributions of one or more system parameters are not normal, uniform, or closely related distributions, due to the computational issues that arise when one wishes to define interpolation nodes for general distributions. This paper examines the use of the recently introduced weighted Leja nodes for that purpose. Weighted Leja interpolation rules are presented, along with a dimension-adaptive sparse interpolation algorithm, to be employed in the case of high-dimensional input uncertainty. The performance and reliability of the suggested approach is verified by four numerical experiments, where the respective models feature extreme value and truncated normal parameter distributions. Furthermore, the suggested approach is compared with a well-established polynomial chaos method and found to be either comparable or superior in terms of approximation and statistics estimation accuracy.

Rational interpolation of the function |x|α by an extended system of Chebyshev – Markov nodes

In this paper, we study the approximations of a function |x|α, α > 0 by interpolation rational Lagrange functions on a segment [–1,1]. The zeros of the even Chebyshev – Markov rational functions and a point x = 0 are chosen as the interpolation nodes. An integral representation of an interpolation remainder and an upper bound for the considered uniform approximations are obtained. Based on them, a detailed study is made:a) the polynomial case. Here, the authors come to the famous asymptotic equality of M. N. Hanzburg;b) at a fixed number of geometrically different poles, the upper estimate is obtained for the corresponding uniform approximations, which improves the well-known result of K. N. Lungu;c) when approximating by general Lagrange rational interpolation functions, the estimate of uniform approximations is found and it is shown that at the ends of the segment [–1,1] it can be improved.The results can be applied in theoretical research and numerical methods.

A treecode based on barycentric Hermite interpolation for electrostatic particle interactions

A particle-cluster treecode based on barycentric Hermite interpolation is presented for fast summation of electrostatic particle interactions in 3D. The interpolation nodes are Chebyshev points of the 2nd kind in each cluster. It is noted that barycentric Hermite interpolation is scale-invariant in a certain sense that promotes the treecode’s efficiency. Numerical results for the Coulomb and screened Coulomb potentials show that the treecode run time scales like O(N log N), where N is the number of particles in the system. The advantage of the barycentric Hermite treecode is demonstrated in comparison with treecodes based on Taylor approximation and barycentric Lagrange interpolation.