On Optimal Quadratic Lagrange Interpolation: Extremal Node Systems with Minimal Lebesgue Constant via Symbolic Computation

ACM Computing Classification System (1998): G.1.1, G.1.2.

Utilização do método numérico interpolador de Lagrange para correção de bases de dados de radiometria solar

This work explores the suitability of the Lagrange interpolating polynomial as a tool to estimate and correct solar databases. From the knowledge of the irradiance distribution over a day, a portion of it was removed for applying Lagrange interpolation polynomial. After generation of the estimates by interpolation, the assessment was made by MBE and RMSE statistical indicators. The application of Lagrange interpolating generated the following results: underestimation of 0.27% (MBE = -1.83 W/m2) and scattering of 0.51% (RMSE = 3.48 W/m2).

Equidistribution of Fekete Points on the Sphere

Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of Fekete points in the sphere. The way we proceed is by showing their connection to other arrays of points, the so-called Marcinkiewicz-Zygmund arrays and interpolating arrays, that have been studied recently.

Equidistribution of Fekete Points on the Sphere

Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of Fekete points in the sphere. The way we proceed is by showing their connection to other arrays of points, the so-called Marcinkiewicz-Zygmund arrays and interpolating arrays, that have been studied recently.

Chebyshev-Laurent polynomials and weighted approximation

Let (a, b) subset of (0, infinity) and for any positive integer n, let S-n be the Chebyshev space in [a, b] defined by S-n:= span{x(-n/2+k),k= 0,...,n}. The unique (up to a constant factor) function tau(n) is an element of S-n, which satisfies the orthogonality relation S(a)(b)tau(n)(x)q(x) (x(b - x)(x - a))(-1/2) dx = 0 for any q is an element of Sn-1, is said to be the orthogonal Chebyshev S-n-polynomials. This paper is an attempt to exibit some interesting properties of the orthogonal Chebyshev S-n-polynomials and to demonstrate their importance to the problem of approximation by S-n-polynomials. A simple proof of a Jackson-type theorem is given and the Lagrange interpolation problem by functions from S-n is discussed. It is shown also that tau(n) obeys an extremal property in L-q, 1 less than or equal to q less than or equal to infinity. Natural analogues of some inequalities for algebraic polynomials, which we expect to hold for the S-n-pelynomials, are conjectured.

Avaliação de Técnicas de Interpolação de Imagens Digitais

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Optimization of chemical plant simulation using double collocation

A method has been constructed for the solution of a wide range of chemical plant simulation models including differential equations and optimization. Double orthogonal collocation on finite elements is applied to convert the model into an NLP problem that is solved either by the VF 13AD package based on successive quadratic programming, or by the GRG2 package, based on the generalized reduced gradient method. This approach is termed simultaneous optimization and solution strategy. The objective functional can contain integral terms. The state and control variables can have time delays. Equalities and inequalities containing state and control variables can be included into the model as well as algebraic equations and inequalities. The maximum number of independent variables is 2. Problems containing 3 independent variables can be transformed into problems having 2 independent variables using finite differencing. The maximum number of NLP variables and constraints is 1500. The method is also suitable for solving ordinary and partial differential equations. The state functions are approximated by a linear combination of Lagrange interpolation polynomials. The control function can either be approximated by a linear combination of Lagrange interpolation polynomials or by a piecewise constant function over finite elements. The number of internal collocation points can vary by finite elements. The residual error is evaluated at arbitrarily chosen equidistant grid-points, thus enabling the user to check the accuracy of the solution between collocation points, where the solution is exact. The solution functions can be tabulated. There is an option to use control vector parameterization to solve optimization problems containing initial value ordinary differential equations. When there are many differential equations or the upper integration limit should be selected optimally then this approach should be used. The portability of the package has been addressed converting the package from V AX FORTRAN 77 into IBM PC FORTRAN 77 and into SUN SPARC 2000 FORTRAN 77. Computer runs have shown that the method can reproduce optimization problems published in the literature. The GRG2 and the VF I 3AD packages, integrated into the optimization package, proved to be robust and reliable. The package contains an executive module, a module performing control vector parameterization and 2 nonlinear problem solver modules, GRG2 and VF I 3AD. There is a stand-alone module that converts the differential-algebraic optimization problem into a nonlinear programming problem.

The Zero-Removing Property and Lagrange-Type Interpolation Series

The classical Kramer sampling theorem, which provides a method for obtaining orthogonal sampling formulas, can be formulated in a more general nonorthogonal setting. In this setting, a challenging problem is to characterize the situations when the obtained nonorthogonal sampling formulas can be expressed as Lagrange-type interpolation series. In this article a necessary and sufficient condition is given in terms of the zero removing property. Roughly speaking, this property concerns the stability of the sampled functions on removing a finite number of their zeros.

Analytic Kramer kernels, Lagrange-type interpolation series and de Branges spaces

The classical Kramer sampling theorem provides a method for obtaining orthogonal sampling formulas. In particular, when the involved kernel is analytic in the sampling parameter it can be stated in an abstract setting of reproducing kernel Hilbert spaces of entire functions which includes as a particular case the classical Shannon sampling theory. This abstract setting allows us to obtain a sort of converse result and to characterize when the sampling formula associated with an analytic Kramer kernel can be expressed as a Lagrange-type interpolation series. On the other hand, the de Branges spaces of entire functions satisfy orthogonal sampling formulas which can be written as Lagrange-type interpolation series. In this work some links between all these ideas are established.

Sampling associated with resolvent-type kernels and lagrange-type interpolation series

In this paper a new class of Kramer kernels is introduced, motivated by the resolvent of a symmetric operator with compact resolvent. The article gives a necessary and sufficient condition to ensure that the associ- ated sampling formula can be expressed as a Lagrange-type interpolation series. Finally, an illustrative example, taken from the Hamburger moment problem theory, is included.

Influence of the Operational Constraints in the Active Transmission Losses Allocation via Lagrange Multipliers

This paper presents an approach for the active transmission losses allocation between the agents of the system. The approach uses the primal and dual variable information of the Optimal Power Flow in the losses allocation strategy. The allocation coefficients are determined via Lagrange multipliers. The paper emphasizes the necessity to consider the operational constraints and parameters of the systems in the problem solution. An example, for a 3-bus system is presented in details, as well as a comparative test with the main allocation methods. Case studies on the IEEE 14-bus systems are carried out to verify the influence of the constraints and parameters of the system in the losses allocation.

Soil CO(2) emission estimated by different interpolation techniques

Soil CO(2) emissions are highly variable, both spatially and across time, with significant changes even during a one-day period. The objective of this study was to compare predictions of the diurnal soil CO(2) emissions in an agricultural field when estimated by ordinary kriging and sequential Gaussian simulation. The dataset consisted of 64 measurements taken in the morning and in the afternoon on bare soil in southern Brazil. The mean soil CO(2) emissions were significantly different between the morning (4.54 mu mol m(-2) s(-1)) and afternoon (6.24 mu mol m(-2) s(-1)) measurements. However, the spatial variability structures were similar, as the models were spherical and had close range values of 40.1 and 40.0 m for the morning and afternoon semivariograms. In both periods, the sequential Gaussian simulation maps were more efficient for the estimations of emission than ordinary kriging. We believe that sequential Gaussian simulation can improve estimations of soil CO(2) emissions in the field, as this property is usually highly non-Gaussian distributed.

A consistent point-searching algorithm for solution interpolation in unstructured meshes consisting of 4-node bilinear quadrilateral elements

To translate and transfer solution data between two totally different meshes (i.e. mesh 1 and mesh 2), a consistent point-searching algorithm for solution interpolation in unstructured meshes consisting of 4-node bilinear quadrilateral elements is presented in this paper. The proposed algorithm has the following significant advantages: (1) The use of a point-searching strategy allows a point in one mesh to be accurately related to an element (containing this point) in another mesh. Thus, to translate/transfer the solution of any particular point from mesh 2 td mesh 1, only one element in mesh 2 needs to be inversely mapped. This certainly minimizes the number of elements, to which the inverse mapping is applied. In this regard, the present algorithm is very effective and efficient. (2) Analytical solutions to the local co ordinates of any point in a four-node quadrilateral element, which are derived in a rigorous mathematical manner in the context of this paper, make it possible to carry out an inverse mapping process very effectively and efficiently. (3) The use of consistent interpolation enables the interpolated solution to be compatible with an original solution and, therefore guarantees the interpolated solution of extremely high accuracy. After the mathematical formulations of the algorithm are presented, the algorithm is tested and validated through a challenging problem. The related results from the test problem have demonstrated the generality, accuracy, effectiveness, efficiency and robustness of the proposed consistent point-searching algorithm. Copyright (C) 1999 John Wiley & Sons, Ltd.

Compression Efficiency Analysis of Wyner-Ziv Video Coding with Motion Compensated Side Information Interpolation

The Wyner-Ziv video coding (WZVC) rate distortion performance is highly dependent on the quality of the side information, an estimation of the original frame, created at the decoder. This paper, characterizes the WZVC efficiency when motion compensated frame interpolation (MCFI) techniques are used to generate the side information, a difficult problem in WZVC especially because the decoder only has available some reference decoded frames. The proposed WZVC compression efficiency rate model relates the power spectral of the estimation error to the accuracy of the MCFI motion field. Then, some interesting conclusions may be derived related to the impact of the motion field smoothness and the correlation to the true motion trajectories on the compression performance.

Image upsampling algorithms applied on Nuclear Medicine: comparision of hqnx and nxSaI algorithms families with two interpolation algorithms - nearest neighbour and bicubic interpolation

Introduction: Image resizing is a normal feature incorporated into the Nuclear Medicine digital imaging. Upsampling is done by manufacturers to adequately fit more the acquired images on the display screen and it is applied when there is a need to increase - or decrease - the total number of pixels. This paper pretends to compare the “hqnx” and the “nxSaI” magnification algorithms with two interpolation algorithms – “nearest neighbor” and “bicubic interpolation” – in the image upsampling operations. Material and Methods: Three distinct Nuclear Medicine images were enlarged 2 and 4 times with the different digital image resizing algorithms (nearest neighbor, bicubic interpolation nxSaI and hqnx). To evaluate the pixel’s changes between the different output images, 3D whole image plot profiles and surface plots were used as an addition to the visual approach in the 4x upsampled images. Results: In the 2x enlarged images the visual differences were not so noteworthy. Although, it was clearly noticed that bicubic interpolation presented the best results. In the 4x enlarged images the differences were significant, with the bicubic interpolated images presenting the best results. Hqnx resized images presented better quality than 4xSaI and nearest neighbor interpolated images, however, its intense “halo effect” affects greatly the definition and boundaries of the image contents. Conclusion: The hqnx and the nxSaI algorithms were designed for images with clear edges and so its use in Nuclear Medicine images is obviously inadequate. Bicubic interpolation seems, from the algorithms studied, the most suitable and its each day wider applications seem to show it, being assumed as a multi-image type efficient algorithm.