María J. Ibáñez

On near-best discrete quasi-interpolation on a four-directional mesh

Spline quasi-interpolants are practical and effective approximation operators. In this paper, we construct QIs with optimal approximation orders and small infinity norms called near-best discrete quasi-interpolants which are based on @W-splines, i.e. B-splines with octagonal supports on the uniform four-directional mesh of the plane. These quasi-interpolants are exact on some space of polynomials and they minimize an upper bound of their infinity norms depending on a finite number of free parameters. We show that this problem has always a solution, in general nonunique. Concrete examples of such quasi-interpolants are given in the last section.

A general spline differential quadrature method based on quasi-interpolation

The differential quadrature method is a numerical discretization technique for the approximation of derivatives. The classical method is polynomial-based, and there is a natural restriction in the number of grid points involved. A general spline-based method is proposed to avoid this problem. For any degree a Lagrangian spline interpolant is defined having a fundamental function with small support. A quasi-interpolant is used to achieve the optimal approximation order. That two-stage scheme is detailed for the cubic, quartic, quintic and sextic cases and compared with another methods that appear in the literature.

On Chebyshev-type integral quasi-interpolation operators

Spline quasi-interpolants on the real line are approximating splines to given functions with optimal approximation orders. They are called integral quasi-interpolants if the coefficients in the spline series are linear combinations of weighted mean values of the function to be approximated. This paper is devoted to the construction of new integral quasi-interpolants with compactly supported piecewise polynomial weights. The basic idea consists of minimizing an expression appearing in an estimate for the quasi-interpolation error. It depends on how well the quasi-interpolation operator approximates the first non-reproduced monomial. Explicit solutions as well as some numerical tests in the B-spline case are given.

An in-depth study on WENO-based techniques to improve parameter extraction procedures in MOSFET transistors

WENO-based techniques, along with some particular polynomial interpolation procedures, have been employed to improve parameter extraction in Metal Oxide Semiconductor Field Effect Transistors (MOSFETs), in particular for the determination of the threshold voltage. The limitations detected in conventional numerical methods to calculate derivatives of experimental data are overcome with this new application of WENO-based techniques. The numerical noise that comes up in the experimental and simulated data usually employed to characterize MOSFETs transistors is strongly reduced. The need for an accurate determination of the threshold voltage motivates the use of this advanced numerical approach that solves many of the issues that affect the conventional parameter extraction procedures currently in use in the microelectronics industry. In addition, also the influence of DIBL effects on the threshold voltage in short channel MOSFETs has been analyzed with this smart weighted ENO procedure.

Increasing the approximation order of spline quasi-interpolants

Abstract In this paper, we show how by a very simple modification of bivariate spline discrete quasi-interpolants, we can construct a new class of quasi-interpolants which have remarkable properties such as high order of regularity and polynomial reproduction. More precisely, given a spline discrete quasi-interpolation operator Q d , which is exact on the space P m of polynomials of total degree at most m , we first propose a general method to determine a new differential quasi-interpolation operator Q r D which is exact on P m + r . Q r D uses the values of the function to be approximated at the points involved in the linear functional defining Q d as well as the partial derivatives up to the order r at the same points. From this result, we then construct and study a first order differential quasi-interpolant based on the C 1 cubic B-spline on the equilateral triangulation with a hexagonal support. When the derivatives are not available or extremely expensive to compute, we approximate them by appropriate finite differences to derive new discrete quasi-interpolants Q d . We estimate with small constants the quasi-interpolation errors f − Q r D [ f ] and f − Q d [ f ] in the infinity norm. Finally, numerical examples are used to analyze the performance of the method.

Computing quasi-interpolants from the B-form of B-splines

Abstract: In general, for a sufficiently regular function, an expression for the quasi-interpolation error associated with discrete, differential and integral quasi-interpolants can be derived involving a term measuring how well the non-reproduced monomials are approximated. That term depends on some expressions of the coefficients defining the quasi-interpolant, and its minimization has been proposed. However, the resulting problem is rather complex and often requires some computational effort. Thus, for quasi-interpolants defined from a piecewise polynomial function, @f, we propose a simpler minimization problem, based on the Bernstein-Bezier representation of some related piecewise polynomial functions, leading to a new class of quasi-interpolants.

On spline-based differential quadrature

In the paper Barrera et al. (2014), a boolean sum differential quadrature method (DQM) was proposed by combining a spline interpolation operator having a fundamental function with minimal compact support and a spline quasi-interpolation operator reproducing the polynomials in the spline space. It is a general framework that provides results that differ from the ones obtained by defining specific schemes with structures which depend on the degree of the B-spline to be considered. The main drawback of these boolean sum DQMs is that the number of evaluation points increases quickly with the degree of the B-spline due to the use of a quasi-interpolation operator. We propose a different construction avoiding this problem and derive explicit results for low degree B-splines.

Near-best operators based on a C2 quartic spline on the uniform four-directional mesh

We present some results about the construction of quasi-interpolant operators based on a special C^2 quartic B-spline. We show that these operators, called near-best quasi-interpolants, have the best approximation order and small infinity norms. They are obtained by solving a minimization problem that admits always a solution. We give an error bound of these quasi-interpolants and we illustrate our results by a numerical example.

Original article: A comprehensive characterization of the threshold voltage extraction in MOSFETs transistors based on smoothing splines

In this work we propose a method to obtain the MOSFET transistor threshold voltage, which is known to be an essential magnitude from the modeling viewpoint. Generally, there are a large number N of experimental data, and the use of smoothing splines leads to resolution of linear systems of size N+d, where d depends on the degree of the splines used. The computational effort could be reduced by decomposing the original problem into subproblems and using an appropriate method to combine the corresponding solutions. We adopt this idea, considering for simplicity the decomposition into two subsets and using a boolean sum based method for combining the intermediate spline approximants.

Construction techniques for multivariate modified quasi-interpolants with high approximation order

In this paper, we propose several approximations of a multivariate function by quasi-interpolants on non-uniform data and we study their properties. In particular, we characterize those that preserve constants via the partition of unity approach. As one of the main results, we show how by a very simple modification of a given quasi-interpolant it is possible to construct new quasi-interpolants with remarkable properties. We also provide some results regarding bivariate C^2 quintic spline quasi-interpolation. Finally, numerical tests are presented to show the approximation power of these quasi-interpolants.