Interpolatory Method Research Articles

A non-separable progressive multivariate WENO-2r point value

The weighted essentially non-oscillatory technique using a stencil of 2r points (WENO-2r) is an interpolatory method that consists in obtaining a higher approximation order from the non-linear combination of interpolants of r+1 nodes. The result is an interpolant of order 2r at the smooth parts and order r+1 when an isolated discontinuity falls at any grid interval of the large stencil except at the central one. Recently, a new WENO method based on Aitken-Neville's algorithm has been designed for interpolation of equally spaced data at the mid-points and presents progressive order of accuracy close to discontinuities. This paper is devoted to constructing a general progressive WENO method for non-necessarily uniformly spaced data and several variables interpolating in any point of the central interval. Also, we provide explicit formulas for linear and non-linear weights and prove the order obtained. Finally, some numerical experiments are presented to check the theoretical results.

A framework for fitting quadratic-bilinear systems with applications to models of electrical circuits

We propose a method for fitting quadratic-bilinear models from data. Although the dynamics characterizing the original model consist of general analytic nonlinearities, we rely on lifting techniques for equivalently embedding the original model into the quadratic-bilinear structure. Here, data are given by generalized transfer function values that can be sampled from the time-domain steady-state response. This method is an extension of methods that perform bilinear, or quadratic inference, separately. It is based on first identifying the underline minimal linear model with the interpolatory method known as the Loewner framework, and then on inferring the best supplementing nonlinear (quadratic and bilinear) operators, by solving an optimization problem. The proposed data-driven method finds applications in engineering serving the scopes of robust simulation, design, and control. Model examples of electrical circuits with nonlinear components (diodes) were used to test the method's performance.

Tangential interpolatory projections for a class of second-order index-1 descriptor systems and application to Mechatronics

This paper studies the model order reduction of second-order index-1 descriptor systems using a tangential interpolation projection method based on the Iterative Rational Krylov Algorithm (IRKA). Our primary focus is to reduce the system into a second-order form so that the structure of the original system can be preserved. For this purpose, the IRKA based tangential interpolatory method is modified to deal with the second-order structure of the underlying descriptor system efficiently in an implicit way. The paper also shows that by exploiting the symmetric properties of the system the implementing computational costs can be reduced significantly. Theoretical results are verified for the model reduction of the piezo actuator based adaptive spindle support which is second-order index-1 differential-algebraic form. The efficiency and accuracy of the method are demonstrated by analyzing the numerical results.

A systematic interpolatory method for an impurity in a one-dimensional fermionic background

We explore a new numerical method for studying one-dimensional quantum systems in a trapping potential. We focus on the setup of an impurity in a fermionic background, where a single distinguishable particle interacts through a contact potential with a number of identical fermions. We can accurately describe this system, for various particle numbers, different trapping potentials and arbitrary finite repulsion, by constructing a truncated basis containing states at both zero and infinite repulsion. The results are compared with matrix product states methods and with the analytical result for two particles in a harmonic well.

Planar interpolation with extreme deformation, topology change and dynamics

We present a mesh-based, interpolatory method for interactively creating artist-directed inbetweens from arbitrary sets of 2D drawing shapes without rigging. To enable artistic freedom of expression we remove prior restrictions on the range of possible changes between shapes; we support interpolation with extreme deformation and unrestricted topology change. To do this, we extend discrete variational interpolation by introducing a consistent multimesh structure over drawings, a Comesh Optimization algorithm that optimizes our multimesh for both intra- and inter-mesh quality, and a new shape-space energy that efficiently supports arbitrary changes and can prevent artwork overlap when desired. Our multimesh encodes specified correspondences that guide interpolation paths between shapes. With these correspondences, an efficient local-global minimization of our energy interpolates n-way between drawing shapes to create inbetweens. Our Comesh Optimization enables artifact-free minimization by building consistent meshes across drawings that improve both the quality of per-mesh energy discretization and inter-mesh mapping distortions, while guaranteeing a single, compatible triangulation. We implement our method in a test-bed interpolation system that allows interactive creation and editing of animations from sparse key drawings with arbitrary topology and shape change.

Weighted Hermite-Fejér interpolation on the real line: L ∞ case

We give a weighted Hermite-Fejer-type interpolatory method on the real line, which is a positive operator on “good” matrices. We give an example on “good” interpolatory matrix by weighted Fekete points. To prove the convergence theorem we need the generalization of “Rodrigues’ property”.

On a Steffensen-Hermite type method for approximating the solutions of nonlinear equations

It is well known that the Steffensen and Aitken-Steffensen type methods are obtained from the chord method, using controlled nodes. The chord method is an interpolatory method, with two distinct nodes. Using this remark, the Steffensen and Aitken-Steffensen methods have been generalized using interpolatory methods obtained from the inverse interpolation polynomial of Lagrange or Hermite type. In this paper we study the convergence and efficiency of some Steffensen type methods which are obtained from the inverse interpolatory polynomial of Hermite type with two controlled nodes.

On an interpolatory method for high dimensional integration

We discuss numerical integration of smooth functions that are defined on a bounded or unbounded hyperrectangular region. We present numerical results that demonstrate the wide range of applications for our recently invented method. In addition, we survey theoretical results. In the fully symmetric case our method is a special fully symmetric rule. However, we also have error bounds (not available for general fully symmetric rules) and the method also can be used if the weight function is an arbitrary tensor product with different and/or nonsymmetric factors.

Generalized Information and Maximal Order of Iteration for Operator Equations

We consider stationary iterative methods for the solution of operator equations which use “generalized information”. We seek methods with maximal order provided the points of iterations are in “good position”. The new concepts of order of information and “generalized” interpolatory methods are introduced. The main result states that the maximal order is equal to the order of information which depends on generalized information and on the position of iteration points. We show that the maximal order is achievable by the generalized interpolatory method.

Maximal Stationary Iterative Methods for the Solution of Operator Equations

We study stationary iterative methods of maximal order for calculating zeros of operator equations. These methods use the values of the operator and its first s Frechet derivatives at n previous iteration points. We introduce a sufficient condition for an iterative method to have maximal order in a certain class of admissible methods. We prove the maximality of the interpolatory method $I_{n,s} $ in the scalar case (see Traub [11, p. 60 and ff.]). For the m-dimensional case, $2 \leqq m \leqq + \infty $, we prove that interpolatory iteration is maximal for $n = 0$ in the class of iterations using values of the first s derivatives at n previous points.

An Interpolatory Method of Approximating Distribution Functions with Applications to the Gamma and Related Variates

An Interpolatory Method of Approximating Distribution Functions with Applications to the Gamma and Related Variates

An Interpolatory Method of Approximating Distribution Functions with Applications to the Gamma and Related Variates

Abstract A method of approximating distribution functions is presented which is based on a weighted sum of exponential functions. The method is quite general but attention is confined here to approximating the distribution function of gamma and related variates. As a limiting case of X 2, the normal distribution is also considered. Tables of constants are provided for applying the approximation procedure. It can be carried out on a computer or on a desk-type calculator. Among the attractive features of the technique is the fact that the constants involved in the approximating function can be obtained by simple interpolation.