Showing papers on "Trigonometric interpolation published in 2021"

Measuring Smoothness of Trigonometric Interpolation Through Incomplete Sample Points

Abstract: In this paper we present a metric to assess the smoothness of a trigonometric interpolation through an in-complete set of sample points. We measure smoothness as the power of a particular derivative of a 2π-periodic Dirichlet interpolant through some sample points. We show that we do not need to explicitly complete the sample set or perform the interpolation, but can simply work with the available sample points, under the assumption that any missing points are chosen to minimise the metric, and present a simple and robust approach to the computation of this metric. We assess the accuracy and computational complexity of this approach, and compare it to benchmarks.

Interpolation Methods for Molecular Potential Energy Surface Construction.

Abstract: The concept of a potential energy surface (PES) is one of the most important concepts in modern chemistry. A PES represents the relationship between the chemical system's energy and its geometry (i.e., atom positions) and can provide useful information about the system's chemical properties and reactivity. Construction of accurate PESs with high-level theoretical methodologies, such as density functional theory, is still challenging due to a steep increase in the computational cost with the increase of the system size. Thus, over the past few decades, many different mathematical approaches have been applied to the problem of the cost-efficient PES construction. This article serves as a short overview of interpolative methods for the PES construction, including global polynomial interpolation, trigonometric interpolation, modified Shepard interpolation, interpolative moving least-squares, and the automated PES construction derived from these.

Efficient Approximation of Potential Energy Surfaces with Mixed-Basis Interpolation.

Abstract: The potential energy surface (PES) describes the energy of a chemical system as a function of its geometry and is a fundamental concept in modern chemistry. A PES provides much useful information about the system, including the structures and energies of various stationary points, such as stable conformers (local minima) and transition states (first-order saddle points) connected by a minimum-energy path. Our group has previously produced surrogate reduced-dimensional PESs using sparse interpolation along chemically significant reaction coordinates, such as bond lengths, bond angles, and torsion angles. These surrogates used a single interpolation basis, either polynomials or trigonometric functions, in every dimension. However, relevant molecular dynamics (MD) simulations often involve some combination of both periodic and nonperiodic coordinates. Using a trigonometric basis on nonperiodic coordinates, such as bond lengths, leads to inaccuracies near the domain boundary. Conversely, polynomial interpolation on the periodic coordinates does not enforce the periodicity of the surrogate PES gradient, leading to nonconservation of total energy even in a microcanonical ensemble. In this work, we present an interpolation method that uses trigonometric interpolation on the periodic reaction coordinates and polynomial interpolation on the nonperiodic coordinates. We apply this method to MD simulations of possible isomerization pathways of azomethane between cis and trans conformers. This method is the only known interpolative method that appropriately conserves total energy in systems with both periodic and nonperiodic reaction coordinates. In addition, compared to all-polynomial interpolation, the mixed basis requires fewer electronic structure calculations to obtain a given level of accuracy, is an order of magnitude faster, and is freely available on GitHub.

Symmetries of discrete curves and point clouds via trigonometric interpolation.

Abstract: We formulate a simple algorithm for computing global exact symmetries of closed discrete curves in plane. The method is based on a suitable trigonometric interpolation of vertices of the given polyline and consequent computation of the symmetry group of the obtained trigonometric curve. The algorithm exploits the fact that the introduced unique assigning of the trigonometric curve to each closed discrete curve commutes with isometries. For understandable reasons, an essential part of the paper is devoted to determining rotational and axial symmetries of trigonometric curves. We also show that the formulated approach can be easily applied on nonorganized clouds of points. A functionality of the designed detection method is presented on several examples.

Low-Computational-Cost Hybrid FEM-Analytical Induction Machine Model for the Diagnosis of Rotor Eccentricity, Based on Sparse Identification Techniques and Trigonometric Interpolation.

Abstract: Since it is not efficient to physically study many machine failures, models of faulty induction machines (IMs) have attracted a rising interest. These models must be accurate enough to include fault effects and must be computed with relatively low resources to reproduce different fault scenarios. Moreover, they should run in real time to develop online condition-monitoring (CM) systems. Hybrid finite element method (FEM)-analytical models have been recently proposed for fault diagnosis purposes since they keep good accuracy, which is widely accepted, and they can run in real-time simulators. However, these models still require the full simulation of the FEM model to compute the parameters of the analytical model for each faulty scenario with its corresponding computing needs. To address these drawbacks (large computing power and memory resources requirements) this paper proposes sparse identification techniques in combination with the trigonometric interpolation polynomial for the computation of IM model parameters. The proposed model keeps accuracy similar to a FEM model at a much lower computational effort, which could contribute to the development and to the testing of condition-monitoring systems. This approach has been applied to develop an IM model under static eccentricity conditions, but this may extend to other fault types.

Mimicking relative continuum measurements by electrode data in two-dimensional electrical impedance tomography

Abstract: This paper introduces a constructive method for approximating relative continuum measurements in two-dimensional electrical impedance tomography based on data originating from either the point electrode model or the complete electrode model. The upper bounds for the corresponding approximation errors explicitly depend on the number (and size) of the employed electrodes as well as on the regularity of the continuum current that is mimicked. In particular, if the input current and the object boundary are infinitely smooth, the discrepancy associated with the point electrode model converges to zero faster than any negative power of the number of electrodes. The results are first proven for the unit disk via trigonometric interpolation and quadrature rules, and they are subsequently extended to more general domains with the help of conformal mappings.

Approximation of the Lebesgue Constant of a Lagrange Polynomial by a Logarithmic Function with Shifted Argument

Abstract: Well-known two-sided estimates for the Lebesgue constants of two classical trigonometric interpolation Lagrange polynomials are improved. Approximations of these Lebesgue constants are based on logarithmic functions with shifted arguments.

Perbandingan Metode Interpolasi Newton dan Lagrange dengan Bahasa Pemrograman C

Abstract: Interpolation is defined as an estimate of a known value. Extensive interpolation is an attempt to determine the approximate value of an analytic function that is unknown or alternatively a complex function whose analytic equation cannot be obtained. You can combine the use of math and math to analyze the price of an item. This study describes Newton's method and the polynomial method. Therefore, interpolation with Newton's method has an error value that is smaller than the error value of the Lagrange interpolation. The results in this research case study when x = 2.5 using 10 data performed using Newton's polynomial interpolation method have a result of 1.70956 where this value is lower than the value of the analysis using the Lagrange method which is 3.2163.

About some properties of simple trigonometric splines.

Abstract: The class $Ts(r,f)$ the trigonometric interpolation splines depending on the parameter vectors, selected convergence factors and interpolation factors is considered. The main properties of simple interpolation trigonometric splines are given, which are also transferred to periodic simple interpolation polynomial splines. These results lead to the possibility of combining the theory of simple polynomial interpolation splines and the basics of the theory of simple trigonometric splines into a single theory - the theory of interpolation splines

Approximation, regularization and smoothing of trigonometric splines.

Abstract: The methods of approximation, regularization and smoothing of trigonometric interpolation splines are considered in the paper. It is shown that trigonometric splines can be considered from two points of view - as a trigonometric Fourier series and as discrete trigonometric Fourier series according to certain systems of functions that are smoothness carriers. It is argued that with approximation and smoothing of trigonometric splines it is expedient to consider as discrete rows, since their differential properties are stored.