Research on certain properties of an adapted nonlinear reconstruction operator on nonuniform grids

[SPA] Esta tesis doctoral se presenta bajo la modalidad de compendio de publicaciones. Los esquemas de subdivisión y multiresolucion se han utilizado en las últimas décadas en muchas aplicaciones que requieren del diseño geométrico. Estas aplicaciones son numerosas en la industria, por ejemplo, para la fabricación de coches y barcos, y también en la industria cinematográfica para generar diferentes formas tanto en 2D como en 3D: Los esquemas de subdivisión se basan en un proceso de refinamiento sucesivo de un conjunto inicial de datos discretos. Se genera un nuevo conjunto de datos más denso de acuerdo con algunas reglas específicas. A su vez, este nuevo conjunto se refinará aún más. En este punto surgen diversas cuestiones matemáticas importantes, y que van desde asegurar la convergencia de los esquemas a estudiar la suavidad de la función límite, la estabilidad de los esquemas de subdivisión, el orden de aproximación y los requisitos necesarios para su aplicabilidad en problemas de la vida real. En particular, es importante el análisis de las capacidades de preservación de los esquemas para algunas propiedades cruciales que podrán estar presentes en el conjunto inicial de datos, tal como la convexidad. Los esquemas de subdivisión generan algoritmos rápidos para la fácil construcción de curvas y superficies [26], [29]. Todas estas cualidades los convierten en una herramienta interesante para diversas aplicaciones industriales. Además, su estrecha relación con esquemas de multirresolución abre la puerta a más aplicaciones en el campo del procesamiento de datos y señales. Los procesos de compresión y eliminación de ruido son fáciles de implementar mediante el uso de esquemas de multiresolución y se ha comprobado que son bastante eficientes. Véase, por ejemplo [35], [5], [2]. Una cuestión principal a la hora de elegir un esquema de subdivisión adecuado es la propiedad de conservación de la convexidad, porque muchas aplicaciones la requieren. Se han hecho muchos esfuerzos en este sentido, véase por ejemplo [27], [32], [33], [37]. La estabilidad es también un problema principal en las aplicaciones de la vida real, ya que los diseños finales se generan mediante el refinamiento de un conjunto inicial de puntos que suele estar afectado por algún error. Por lo tanto, es esencial hacer un seguimiento del error y mantenerlo por debajo de una tolerancia prescrita. Algunas referencias recomendadas sobre la estabilidad de los esquemas de subdivisión y multiresolución pueden consultarse en [24], [9], [11], [1], [3], [15]. Harten derivó una teoría que conecta estrechamente los operadores de reconstrucción con los esquemas de subdivisión y multiresolución [35], [5]. Las reconstrucciones no lineales aparecen como una buena opción para minimizar los efectos adversos de las posibles singularidades y para mejorar la adaptación a los datos dados. Esta teoría no es tan fácil de estudiar como para el caso lineal. Los operadores de reconstrucción no lineales dan lugar a esquemas de subdivisión y multiresolución no lineales. Para dejar claro el tipo de dificultades que se pueden encontrar, mencionamos por ejemplo el caso del análisis de estabilidad. A este respecto, se ha demostrado que todos los esquemas de subdivisión y multiresolución lineales son estables, mientras que se necesita un análisis particular para cada esquema no lineal concreto. Los esquemas de multiresolución están profundamente conectados con los esquemas de subdivisión y heredan muchas de sus propiedades. Para más información sobre estas herramientas se puede consultar [5] como primera referencia. En [6] se introdujo una reconstrucción no lineal denominada PPH y se estudio el esquema de subdivisión asociado. Esta reconstrucción se definió con el fin de adaptarse a la presencia de potenciales singularidades. Consiste en una modificación ingeniosa de la interpolación centrada de cuarto orden de Lagrange a trozos. Para implementar la adaptación, la reconstrucción se realiza localmente en un intervalo [xj ; xj+1] usando los valores disponibles de la función en las cuatro abscisas centradas fxj1; xj ; xj+1; xj+2g; y teniendo en cuenta dos aspectos principales. El primer aspecto es que la modificación en un área donde la función subyacente es suave debe hacerse de tal manera que las cantidades alteradas no cambien significativamente, de modo que la modificación siga siendo O(h4); donde h representa el espaciado del mallado. El segundo aspecto es que, en los intervalos adyacentes a una singularidad, pero que no la contienen, la reconstrucción conserve cierto orden de aproximación, de hecho O(h2); al contrario de lo que ocurre con su homólogo lineal que pierde completamente el orden de aproximación. Esta tesis se dedica principalmente al estudio del operador de reconstrucción no lineal PPH en mallados no uniformes. En algunos casos y para demostrar determinados resultados teóricos haremos uso de mallados σ cuasi uniformes, que no son otra cosa que un tipo de mallados no uniformes que aparecen en casi todas las aplicaciones prácticas. La definición exacta se da más adelante.[ENG] This doctoral dissertation has been presented in the form of thesis by publication. Subdivision and multiresolution schemes have been used in the last few decades in many ap- plications that require from geometrical design. These applications are numerous in industry, for example for car and ship manufacturing, and also in the film industry in order to generate different shapes as much in 2D as in 3D. Subdivision schemes are based on a process of successive refine- ment of a given initial discrete data set. A new denser set of data is generated according to some specific rules. In turn, this new set will be further refined. A bunch of important mathematical questions arise at this point, and range from ensuring the convergence of the schemes, studying the smoothness of the limit function, the stability of the subdivision schemes and the order of approxi- mation and the necessary requirements for their applicability in real life problems. In particular, it is important the analysis of the preservation capabilities of the schemes for some crucial properties which might be present in the initial set of data such as it could be the convexity. Subdivision schemes generate fast algorithms to the easy construction of curves and surfaces [26], [29]. All these qualities make them an interesting tool for several industrial applications. Also, their close relation to multiresolution schemes opens the door to more applications in the fields of data and signal processing. Compression and denoising processes are easy to implement by using multiresolution schemes and they have been tested to be quite efficient. See for example [35], [5], [2]. A chief issue in choosing an adequate subdivision scheme is the property of convexity preser- vation, because many application require it. Many efforts have been done in this sense, see for example [27], [32], [33], [37]. Stability is also a main issue in real life applications, since the final designs are generated through the refinement of an initial set of points which usually is affected by some error. Therefore, keeping track of the error and maintaining it under a prescribed tolerance is essential. Some recommended references about stability of subdivision and multiresolution schemes can be consulted in [24], [9], [11], [1], [3], [15]. Harten derived a theory which closely connects reconstruction operators with subdivision and multiresolution schemes [35], [5]. Nonlinear reconstructions appear as a good option to minimize the adverse effects of potential singularities and to improve the adaptation to the given data. This theory is not as easy to study as for the linear case. Nonlinear reconstruction operators give rise to nonlinear subdivision and multiresolution schemes. In order to let clear the kind of difficulties to be encountered, we mention for example the case of stability analysis. In what stability issues regards, all linear subdivision and multiresolution schemes are proved to be stable, while a particular analysis is needed for each particular nonlinear scheme. Multiresolution schemes are deeply connected with subdivision schemes and they inherit many of their properties. For more information about these useful schemes one can consult [5] as a first reference. In [6] a nonlinear reconstruction called PPH was introduced, and the associated subdivision scheme was studied. This reconstruction was built in order to get adapted to the presence of potential singularities. It consist on a witty modification of the centered fourth order piecewise Lagrange interpolation. In order to implement the adaptation, the reconstruction is built also locally using a stencil of four centered data, but keeping in mind two main concerns. The first concern is that the modification in an area where the underlying function is smooth must be done in such a way that the modified quantities are not significatively changed, so that the modification remains O(h4), where h stands for the grid size. The second concern is that in the intervals adjacent to a singularity, but not containing it, the reconstruction retains some order of approximation, in fact O(h2), on the contrary to what happens with its linear counterpart that loses completely the approximation order. [ENG] Subdivision and multiresolution schemes have been used in the last few decades in many ap- plications that require from geometrical design. These applications are numerous in industry, for example for car and ship manufacturing, and also in the film industry in order to generate different shapes as much in 2D as in 3D. Subdivision schemes are based on a process of successive refinement of a given initial discrete data set. A new denser set of data is generated according to some specific rules. In turn, this new set will be further refined. A bunch of important mathematical questions arise at this point, and range from ensuring the convergence of the schemes, studying the smoothness of the limit function, the stability of the subdivision schemes and the order of approximation and the necessary requirements for their applicability in real life problems. In particular, it is important the analysis of the preservation capabilities of the schemes for some crucial properties which might be present in the initial set of data such as it could be the convexity. Subdivision schemes generate fast algorithms to the easy construction of curves and surfaces [26], [29]. All these qualities make them an interesting tool for several industrial applications. Also, their close relation to multiresolution schemes opens the door to more applications in the fields of data and signal processing. Compression and denoising processes are easy to implement by using multiresolution schemes and they have been tested to be quite efficient. See for example [35], [5], [2]. A chief issue in choosing an adequate subdivision scheme is the property of convexity preservation, because many application require it. Many efforts have been done in this sense, see for example [27], [32], [33], [37]. Stability is also a main issue in real life applications, since the final designs are generated through the refinement of an initial set of points which usually is affected by some error. Therefore, keeping track of the error and maintaining it under a prescribed tolerance is essential. Some recommended references about stability of subdivision and multiresolution schemes can be consulted in [24], [9], [11], [1], [3], [15]. Harten derived a theory which closely connects reconstruction operators with subdivision and multiresolution schemes [35], [5]. Nonlinear reconstructions appear as a good option to minimize the adverse effects of potential singularities and to improve the adaptation to the given data. This theory is not as easy to study as for the linear case. Nonlinear reconstruction operators give rise to nonlinear subdivision and multiresolution schemes. In order to let clear the kind of difficulties to be encountered, we mention for example the case of stability analysis. In what stability issues regards, all linear subdivision and multiresolution schemes are proved to be stable, while a particular analysis is needed for each particular nonlinear scheme. Multiresolution schemes are deeply connected with subdivision schemes and they inherit many of their properties. For more information about these useful schemes one can consult [5] as a first reference. In [6] a nonlinear reconstruction called PPH was introduced, and the associated subdivision scheme was studied. This reconstruction was built in order to get adapted to the presence of potential singularities. It consists on a witty modification of the centered fourth order piecewise Lagrange interpolation. In order to implement the adaptation, the reconstruction is built also locally using a stencil of four centered data but keeping in mind two main concerns. The first concern is that the modification in an area where the underlying function is smooth must be done in such a way that the modified quantities are not significatively changed, so that the modification remains O(h4), where h stands for the grid size. The second concern is that in the intervals adjacent to a singularity, but not containing it, the reconstruction retains some order of approximation, in fact O(h2), on the contrary to what happens with its linear counterpart that loses completely the approximation order.Esta tesis doctoral se presenta bajo la modalidad de compendio de publicaciones. Está formada por estos siete artículos: 1.Jiménez, I.; Ortiz, P.; Ruiz, J.; Trillo, J. C.; Yañez, D. F. Improving the stability bound for the PPH nonlinear subdivision scheme for data coming from strictly convex functions. Applied Mathematics and Computations. 2021, https://doi.org/10.1016/j.amc.2021.126042. 2. Ortiz, P.; Trillo, J.C. On the Convexity Preservation of a Quasi C3 Nonlinear Interpolatory Reconstruction Operator on σ Quasi-Uniform Grids. Mathematics. 2021, 9(4), 310. https://doi.org/10.3390/math9040310. 3. Ortiz, P.; Trillo, J.C. A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids{Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon. Mathematics. 2021, 9, 335. https://doi.org/10.3390/math9040335. 4. Amat, S.; Ortiz, P.; Ruiz, J.; Trillo, J. C.; Yanez, D. F. Improving the approximation order around inection points of the PPH nonlinear interpolatory reconstruction operator on nonuniform grids. 5. Ortiz, P.; Trillo, J.C. On certain inequalities associated to curvature properties of the nonlinear PPH reconstruction operator. Journal of Inequalities and Applications. 2019, Paper No. 8, 13 pp, https://doi.org/10.1186/s13660-019-1959-0. 6. Ortiz, P.; Trillo, J.C. Analysis of a New Nonlinear Interpolatory subdivision scheme on σ quasi-uniform grids. Mathematics. 2021, 9, 1320. https://doi.org/10.3390/math9121320. 7. Amat, S. ; Ortiz, P.; Ruiz, J.; Trillo, J.C. ; Yáñez, D.F. Graphical interpretation of the weighted harmonic mean of n positive values and applications.Escuela Internacional de Doctorado de la Universidad Politécnica de CartagenaUniversidad Politécnica de CartagenaPrograma de Doctorado en Tecnologías Industriale

Analysis of a new nonlinear interpolatory subdivision scheme on σ quasi-uniform grids

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. The novelty of this work lies in the generalization of the already existing PPH subdivision scheme to the nonuniform case. We define the corresponding subdivision scheme and study some important issues related to subdivision schemes such as convergence, smoothness of the limit function, and preservation of convexity. In order to obtain general results, we consider s quasi-uniform grids. We also perform some numerical experiments to reinforce the theoretical results.This research was funded by the Fundación Séneca, Agencia de Ciencia y Technología de la Región de Murcia grant number 20928/PI/18 and by the Spanish national research project PID2019-108336GB-I00

A piecewise polynomial harmonic nonlinear interpolatory reconstruction operator on non uniform grids—adaptation around jump discontinuities and elimination of gibbs phenomenon

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to σ quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes.This research was funded by the FUNDACIÓN SÉNECA, AGENCIA DE CIENCIA Y TECNOLOGÍA DE LA REGIÓN DE MURCIA grant number 20928/PI/18, and by the Spanish national research project PID2019-108336GB-I00

On the convexity preservation of a quasi C3 nonlinear interpolatory reconstruction operator on σ quasi-uniform grids

This paper is devoted to introducing a nonlinear reconstruction operator, the piecewise polynomial harmonic (PPH), on nonuniform grids. We define this operator and we study its main properties, such as its reproduction of second-degree polynomials, approximation order, and conditions for convexity preservation. In particular, for σ quasi-uniform grids with σ≤4, we get a quasi C3 reconstruction that maintains the convexity properties of the initial data. We give some numerical experiments regarding the approximation order and the convexity preservation.This research was funded by the FUNDACIÓN SÉNECA, AGENCIA DE CIENCIA Y TECNOLOGÍA DE LA REGIÓN DE MURCIA grant number 20928/PI/18, and by the Spanish national research project PID2019-108336GB-I00

Incidence and clearance of anal high-risk human papillomavirus in HIV-positive men who have sex with men: Estimates and risk factors

Background: To estimate incidence and clearance of high-risk human papillomavirus (HR-HPV), and their risk factors, in men who have sex with men (MSM) recently infected by HIV in Spain; 2007-2013. Methods: Multicenter cohort. HR-HPV infection was determined and genotyped with linear array. Two-state Markov models and Poisson regression were used. Results: We analysed 1570 HR-HPV measurements of 612 MSM over 13 608 person-months (p-m) of follow-up. Median (mean) number of measurements was 2 (2.6), median time interval between measurements was 1.1 years (interquartile range: 0.89-1.4). Incidence ranged from 9.0 [95% confidence interval (CI) 6.8-11.8] per 1000 p-m for HPV59 to 15.9 (11.7-21.8) per 1000 p-m for HPV51. HPV16 and HPV18 had slightly above average incidence: 11.9/1000 p-m and 12.8/1000 p-m. HPV16 showed the lowest clearance for both 'prevalent positive' (15.7/1000 p-m; 95% CI 12.0-20.5) and 'incident positive' infections (22.1/1000 p-m; 95% CI 11.8-41.1). More sexual partners increased HR-HPV incidence, although it was not statistically significant. Age had a strong effect on clearance (P-value < 0.001) due to the elevated rate in MSM under age 25; the effect of HIV-RNA viral load was more gradual, with clearance rate decreasing at higher HIV-RNA viral load (P-value 0.008). Conclusion: No large variation in incidence by HR-HPV type was seen. The most common incident types were HPV51, HPV52, HPV31, HPV18 and HPV16. No major variation in clearance by type was observed, with the exception of HPV16 which had the highest persistence and potentially, the strongest oncogenic capacity. Those aged below 25 or with low HIV-RNA- viral load had the highest clearanceThis work was supported by grants from the Fondo de Investigacio´n Sanitaria [PI06/0372, PS09/2181], Red de Investigacio´n en SIDA (RIS) [RD06/006/0026 and RD12/0017/0018 to C.G.] and CIBERESP [group 54A-CB06/02/1009

Clonal chromosomal mosaicism and loss of chromosome Y in elderly men increase vulnerability for SARS-CoV-2

The pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2, COVID-19) had an estimated overall case fatality ratio of 1.38% (pre-vaccination), being 53% higher in males and increasing exponentially with age. Among 9578 individuals diagnosed with COVID-19 in the SCOURGE study, we found 133 cases (1.42%) with detectable clonal mosaicism for chromosome alterations (mCA) and 226 males (5.08%) with acquired loss of chromosome Y (LOY). Individuals with clonal mosaic events (mCA and/or LOY) showed a 54% increase in the risk of COVID-19 lethality. LOY is associated with transcriptomic biomarkers of immune dysfunction, pro-coagulation activity and cardiovascular risk. Interferon-induced genes involved in the initial immune response to SARS-CoV-2 are also down-regulated in LOY. Thus, mCA and LOY underlie at least part of the sex-biased severity and mortality of COVID-19 in aging patients. Given its potential therapeutic and prognostic relevance, evaluation of clonal mosaicism should be implemented as biomarker of COVID-19 severity in elderly people. Among 9578 individuals diagnosed with COVID-19 in the SCOURGE study, individuals with clonal mosaic events (clonal mosaicism for chromosome alterations and/or loss of chromosome Y) showed an increased risk of COVID-19 lethality

Geodivulgar: Geología y Sociedad 2018

Depto. de Geodinámica, Estratigrafía y PaleontologíaFac. de Ciencias GeológicasFALSEsubmitte

Geodivulgar: Geología y Sociedad

Con el lema “Geología para todos” el proyecto Geodivulgar: Geología y Sociedad apuesta por la divulgación de la Geología a todo tipo de público, incidiendo en la importancia de realizar simultáneamente una acción de integración social entre estudiantes y profesores de centros universitarios, de enseñanza infantil, primaria, de educación especial y un acercamiento con público con diversidad funcional

Las anomalías magnéticas al sur del mar de Bransfield.

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