Paco Arandiga - Academia.edu (original) (raw)

Papers by Paco Arandiga

SIAM Journal on Numerical Analysis, 2013

Monotonicity-preserving approximation methods are used in numerous applications. With them we can... more Monotonicity-preserving approximation methods are used in numerous applications. With them we can reconstruct a function from a discrete set of data while preserving its monotonicity properties. In this paper we analyze monotone piecewise cubic Hermite interpolants for uniform and nonuniform grids. We present and analyze two new methods to prescribe the derivatives at the breakpoints that lead to third order approximations even in the case of nonuniform grids. We demonstrate that with these alternatives the monotonicity may only be lost in rather extreme situations. In such cases we propose modifications of the algorithms that guarantee the monotonicity but with local second order accuracy. We also perform several numerical experiments which exemplify the properties of the proposed algorithms and compare them with the technique proposed by Fritsch and Butland, in which the derivatives at the breakpoints are calculated using Brodlie's function. It is available in Netlib (PCHIP.FOR) and it is also used in t...

Applied Mathematics and Computation, Jul 1, 2023

Journal of Computational and Applied Mathematics, Mar 1, 2019

In this paper we extend this result to get order four. We observe the Gibbs phenomenon that is ob... more In this paper we extend this result to get order four. We observe the Gibbs phenomenon that is obtained near discontinuities with its weights. With the weights we propose we obtain approximations of order four in smooth regions and three near discontinuities. We also introduce a rational nonlinear extrapolation which is also of order four in the smooth region of the given function. In the experiments we calculate numerically approximation orders for the different methods described in this paper and see that they coincide with those that have been obtained theoretically. We also present reconstructions in 1d and 2d with which we reach the same conclusions.

Applied Mathematics and Computation, Nov 1, 2023

SIAM Journal on Scientific Computing, 1998

Applied Mathematics and Computation, Apr 1, 2022

Adaptive rational interpolation has been designed in the context of image processing as a new non... more Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to the method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique, however because of the design of the weights in this case is more simple, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are performed to demonstrate our results and to compare them with standard methods.

arXiv (Cornell University), Feb 23, 2021

arXiv (Cornell University), Oct 2, 2020

Adaptive rational interpolation has been designed in the context of image processing as a new non... more Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to the method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique, however because of the design of the weights in this case is more simple, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are performed to demonstrate our results and to compare them with standard methods.

Journal of The Franklin Institute-engineering and Applied Mathematics, Feb 1, 2016

Cell-average multiresolution Harten's algorithms have been satisfactorily used to compress data. ... more Cell-average multiresolution Harten's algorithms have been satisfactorily used to compress data. These schemes are based on two operators: decimation and prediction. The accuracy of the method depends on the prediction operator. In order to design a precise function, local polynomial regression has been used in the last years. This paper is devoted to construct a family of non-separable two-dimensional linear prediction operators approximating the real values with this procedure. Some properties are proved as the order of the scheme and the stability. Some numerical experiments are performed comparing the new methods with the classical linear method.

Birkhäuser Basel eBooks, 2003

Applied Mathematics Letters, Aug 1, 2019

Monotonicity-preserving interpolants are used in several applications as engineering or computer ... more Monotonicity-preserving interpolants are used in several applications as engineering or computer aided design. In last years some new techniques have been developed. In particular, in [Aràndiga, SIAM J. Numer. Anal., 51(5) (2013), pp. 2613-2633] some new methods to design monotone cubic Hermite interpolants for uniform and non-uniform grids are presented and analyzed. They consist on calculating the derivative values introducing the weighted harmonic mean and a non-linear variation. With these changes, the methods obtained are third-order accurate, except in extreme situations. In this paper, a new general mean is used and a third-order interpolant for all cases is gained. We perform several experiments comparing the known techniques as the method proposed by Fritsch and Butland using the Brodlie's function,

Abstract: We explore the use of the singular value decomposition (SVD) in image compression. We l... more Abstract: We explore the use of the singular value decomposition (SVD) in image compression. We link the SVD and the multiresolution algorithms. In [22] it is derived a multiresolution representation of the SVD decomposition, and in [15] the SVD algorithm and Wavelets are linked, proposing a mixed algorithm which roughly consist on applying firstly a discrete Wavelet transform and secondly the SVD algorithm to each subband. We propose a new algorithm, which is carried out in two main steps. Firstly we decompose the data matrix corresponding to the image following a singular value decomposition. Secondly we apply a Harten’s multiresolution decomposition to the singular vectors which are considered significant. We study the compression capabilities of this new algorithm. We also propose a variant of the implementation, where the multiresolution transformation is carried out by blocks. We apply on each block, depending on a selection process, either the algorithm presented or the 2D mu...

Axioms

The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-inter... more The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Applied Numerical Mathematics, Feb 1, 2022

Spline interpolation has been used in several applications due to its favorable properties regard... more Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. In this paper, we study sufficient conditions to obtain monotone cubic splines based on Hermite cubic interpolators and propose different ways to construct them using non-linear formulas. The order of approximation, in each case, is calculated and several numerical experiments are performed to contrast the theoretical results.

Applied Mathematics Letters, Sep 1, 2020

This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Mathematische Zeitschrift, 1992

Applied Mathematics and Computation, 2016

In this talk we present an algorithm for monotonic interpolation to monotone data on a rectangula... more In this talk we present an algorithm for monotonic interpolation to monotone data on a rectangular mesh by piecewise bicubic functions. Carlton and Fritsch develop conditions on the Hermite derivatives that are sufficient for such a function to be monotonic. Here we obtain nonlinear approximations to the first partial and first mixed partial derivatives at the mesh points. We prove that we get a monotone piecewise bicubic interpolant and analize the order of this nonlinear interpolant. We also present some numerical experiments were we compare the results we obtain our algorithm with the obtained using linear techniques.

Total Variation denoising, proposed by Rudin, Osher and Fatemi in (22), is an image processing va... more Total Variation denoising, proposed by Rudin, Osher and Fatemi in (22), is an image processing variational technique that has attracted considerable attention in the past fifteen years. It is an advantageous technique for preserving image edges but tends to sharpen excessively smooth transitions. With the purpose of alleviating this staircase effect some generalizations of Total Variation denoising have been introduced in (17, 18, 19). In this paper we propose a fast and robust algorithm for the solution of the variational problems that generalize Total Variation image denoising (22). This method extends the primal-dual Newton method, proposed by Chan, Golub and Mulet in (7) for total variation restoration, to these variational problems. We perform some experiments for assessing the efficiency of this scheme with respect to the fixed point method that generalizes the lagged diffusivity fixed point method proposed by Vogel and Oman in (24).

Mathematics, 2020

In this work, a model for the simulation of infectious disease outbreaks including mobility data ... more In this work, a model for the simulation of infectious disease outbreaks including mobility data is presented. The model is based on the SAIR compartmental model and includes mobility data terms that model the flow of people between different regions. The aim of the model is to analyze the influence of mobility on the evolution of a disease after a lockdown period and to study the appearance of small epidemic outbreaks due to the so-called imported cases. We apply the model to the simulation of the COVID-19 in the various areas of Spain, for which the authorities made available mobility data based on the position of cell phones. We also introduce a method for the estimation of incomplete mobility data. Some numerical experiments show the importance of data completion and indicate that the model is able to qualitatively simulate the spread tendencies of small outbreaks. This work was motivated by an open call made to the mathematical community in Spain to help predict the spread of t...

Mathematics and Computers in Simulation, 2020

The terms and conditions for the reuse of this version of the manuscript are specified in the pub... more The terms and conditions for the reuse of this version of the manuscript are specified in the publishing policy. For all terms of use and more information see the publisher's website.

SIAM Journal on Numerical Analysis, 2013

Monotonicity-preserving approximation methods are used in numerous applications. With them we can... more Monotonicity-preserving approximation methods are used in numerous applications. With them we can reconstruct a function from a discrete set of data while preserving its monotonicity properties. In this paper we analyze monotone piecewise cubic Hermite interpolants for uniform and nonuniform grids. We present and analyze two new methods to prescribe the derivatives at the breakpoints that lead to third order approximations even in the case of nonuniform grids. We demonstrate that with these alternatives the monotonicity may only be lost in rather extreme situations. In such cases we propose modifications of the algorithms that guarantee the monotonicity but with local second order accuracy. We also perform several numerical experiments which exemplify the properties of the proposed algorithms and compare them with the technique proposed by Fritsch and Butland, in which the derivatives at the breakpoints are calculated using Brodlie's function. It is available in Netlib (PCHIP.FOR) and it is also used in t...

Applied Mathematics and Computation, Jul 1, 2023

Journal of Computational and Applied Mathematics, Mar 1, 2019

In this paper we extend this result to get order four. We observe the Gibbs phenomenon that is ob... more In this paper we extend this result to get order four. We observe the Gibbs phenomenon that is obtained near discontinuities with its weights. With the weights we propose we obtain approximations of order four in smooth regions and three near discontinuities. We also introduce a rational nonlinear extrapolation which is also of order four in the smooth region of the given function. In the experiments we calculate numerically approximation orders for the different methods described in this paper and see that they coincide with those that have been obtained theoretically. We also present reconstructions in 1d and 2d with which we reach the same conclusions.

Applied Mathematics and Computation, Nov 1, 2023

SIAM Journal on Scientific Computing, 1998

Applied Mathematics and Computation, Apr 1, 2022

Adaptive rational interpolation has been designed in the context of image processing as a new non... more Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to the method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique, however because of the design of the weights in this case is more simple, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are performed to demonstrate our results and to compare them with standard methods.

arXiv (Cornell University), Feb 23, 2021

arXiv (Cornell University), Oct 2, 2020

Adaptive rational interpolation has been designed in the context of image processing as a new non... more Adaptive rational interpolation has been designed in the context of image processing as a new nonlinear technique that avoids the Gibbs phenomenon when we approximate a discontinuous function. In this work, we present a generalization to the method giving explicit expressions for all the weights for any order of the algorithm. It has a similar behavior to weighted essentially non oscillatory (WENO) technique, however because of the design of the weights in this case is more simple, we propose a new way to construct them obtaining the maximum order near the discontinuities. Some experiments are performed to demonstrate our results and to compare them with standard methods.

Journal of The Franklin Institute-engineering and Applied Mathematics, Feb 1, 2016

Cell-average multiresolution Harten's algorithms have been satisfactorily used to compress data. ... more Cell-average multiresolution Harten's algorithms have been satisfactorily used to compress data. These schemes are based on two operators: decimation and prediction. The accuracy of the method depends on the prediction operator. In order to design a precise function, local polynomial regression has been used in the last years. This paper is devoted to construct a family of non-separable two-dimensional linear prediction operators approximating the real values with this procedure. Some properties are proved as the order of the scheme and the stability. Some numerical experiments are performed comparing the new methods with the classical linear method.

Birkhäuser Basel eBooks, 2003

Applied Mathematics Letters, Aug 1, 2019

Monotonicity-preserving interpolants are used in several applications as engineering or computer ... more Monotonicity-preserving interpolants are used in several applications as engineering or computer aided design. In last years some new techniques have been developed. In particular, in [Aràndiga, SIAM J. Numer. Anal., 51(5) (2013), pp. 2613-2633] some new methods to design monotone cubic Hermite interpolants for uniform and non-uniform grids are presented and analyzed. They consist on calculating the derivative values introducing the weighted harmonic mean and a non-linear variation. With these changes, the methods obtained are third-order accurate, except in extreme situations. In this paper, a new general mean is used and a third-order interpolant for all cases is gained. We perform several experiments comparing the known techniques as the method proposed by Fritsch and Butland using the Brodlie's function,

Abstract: We explore the use of the singular value decomposition (SVD) in image compression. We l... more Abstract: We explore the use of the singular value decomposition (SVD) in image compression. We link the SVD and the multiresolution algorithms. In [22] it is derived a multiresolution representation of the SVD decomposition, and in [15] the SVD algorithm and Wavelets are linked, proposing a mixed algorithm which roughly consist on applying firstly a discrete Wavelet transform and secondly the SVD algorithm to each subband. We propose a new algorithm, which is carried out in two main steps. Firstly we decompose the data matrix corresponding to the image following a singular value decomposition. Secondly we apply a Harten’s multiresolution decomposition to the singular vectors which are considered significant. We study the compression capabilities of this new algorithm. We also propose a variant of the implementation, where the multiresolution transformation is carried out by blocks. We apply on each block, depending on a selection process, either the algorithm presented or the 2D mu...

Axioms

The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-inter... more The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.

Applied Numerical Mathematics, Feb 1, 2022

Spline interpolation has been used in several applications due to its favorable properties regard... more Spline interpolation has been used in several applications due to its favorable properties regarding smoothness and accuracy of the interpolant. However, when there exists a discontinuity or a steep gradient in the data, some artifacts can appear due to the Gibbs phenomenon. Also, preservation of data monotonicity is a requirement in some applications, and that property is not automatically verified by the interpolator. In this paper, we study sufficient conditions to obtain monotone cubic splines based on Hermite cubic interpolators and propose different ways to construct them using non-linear formulas. The order of approximation, in each case, is calculated and several numerical experiments are performed to contrast the theoretical results.

Applied Mathematics Letters, Sep 1, 2020

This is a PDF file of an article that has undergone enhancements after acceptance, such as the ad... more This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Mathematische Zeitschrift, 1992

Applied Mathematics and Computation, 2016

In this talk we present an algorithm for monotonic interpolation to monotone data on a rectangula... more In this talk we present an algorithm for monotonic interpolation to monotone data on a rectangular mesh by piecewise bicubic functions. Carlton and Fritsch develop conditions on the Hermite derivatives that are sufficient for such a function to be monotonic. Here we obtain nonlinear approximations to the first partial and first mixed partial derivatives at the mesh points. We prove that we get a monotone piecewise bicubic interpolant and analize the order of this nonlinear interpolant. We also present some numerical experiments were we compare the results we obtain our algorithm with the obtained using linear techniques.

Total Variation denoising, proposed by Rudin, Osher and Fatemi in (22), is an image processing va... more Total Variation denoising, proposed by Rudin, Osher and Fatemi in (22), is an image processing variational technique that has attracted considerable attention in the past fifteen years. It is an advantageous technique for preserving image edges but tends to sharpen excessively smooth transitions. With the purpose of alleviating this staircase effect some generalizations of Total Variation denoising have been introduced in (17, 18, 19). In this paper we propose a fast and robust algorithm for the solution of the variational problems that generalize Total Variation image denoising (22). This method extends the primal-dual Newton method, proposed by Chan, Golub and Mulet in (7) for total variation restoration, to these variational problems. We perform some experiments for assessing the efficiency of this scheme with respect to the fixed point method that generalizes the lagged diffusivity fixed point method proposed by Vogel and Oman in (24).

Mathematics, 2020

In this work, a model for the simulation of infectious disease outbreaks including mobility data ... more In this work, a model for the simulation of infectious disease outbreaks including mobility data is presented. The model is based on the SAIR compartmental model and includes mobility data terms that model the flow of people between different regions. The aim of the model is to analyze the influence of mobility on the evolution of a disease after a lockdown period and to study the appearance of small epidemic outbreaks due to the so-called imported cases. We apply the model to the simulation of the COVID-19 in the various areas of Spain, for which the authorities made available mobility data based on the position of cell phones. We also introduce a method for the estimation of incomplete mobility data. Some numerical experiments show the importance of data completion and indicate that the model is able to qualitatively simulate the spread tendencies of small outbreaks. This work was motivated by an open call made to the mathematical community in Spain to help predict the spread of t...

Mathematics and Computers in Simulation, 2020

The terms and conditions for the reuse of this version of the manuscript are specified in the pub... more The terms and conditions for the reuse of this version of the manuscript are specified in the publishing policy. For all terms of use and more information see the publisher's website.