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El éxito del desarrollo de la inteligencia artificial durante los últimos 15 años ha revolucionado todas las áreas del conocimiento. Debido a la ((maldición de la dimensión)), que hace referencia al desafío que surge al tratar de resolver sistemas de ecuaciones diferenciales con un gran número de variables, su aplicación a la aproximación numérica de ecuaciones diferenciales juega un papel muy importante en problemas de alta dimensión. En este trabajo se realizará una introducción a los conceptos básicos de las redes neuronales generales. Se estudiará el algoritmo de diferenciación automática (backpropagation), clave de su eficiencia, y se demostrará el Teorema de aproximación universal, base matemática que justifica la utilidad de las redes neuronales en la aproximación de funciones. Se utilizarán las redes neuronales denominadas como Physical Informed Neural Networks (PINN) para la resolución de ecuaciones y sistemas diferenciales y el ajuste de parámetros en ellos. Además, se implementarán utilizando Python y sus librerías en problemas concretos., The successful development of artificial intelligence over the last 15 years has revolutionised all areas of knowledge. Due to the ((curse of dimensionality)), which refers to the challenge that arises when trying to solve systems of differential equations with a large number of variables, its application to the numerical approximation of differential equations plays a very important role in high-dimensional problems. In this work, an introduction to the basic concepts of general neural networks will be given. The automatic differentiation algorithm (backpropagation), the key to its efficiency, will be studied and the universal approximation theorem, the mathematical basis that justifies the usefulness of neural networks in the approximation of functions, will be demonstrated. Neural networks known as Physically Informed Neural Networks (PINN) will be used to solve differential equations and systems and to adjust parameters in them. Furthermore, they will be implemented using Python and its libraries in concrete problems.

In this work, the phenomenon of chemotaxis is modeled, mainly by focusing on the Keller-Segel model, as well as other variants of it. Furthermore, we carry out approximations of finite differences in time and different schemes in space, such as: finite differences (FD), finite elements (FE) and finite volumes (FV). Subsequently, the schemes are particularized to a one-dimensional spatial domain and numerical simulations are performed. Analyzing the results, spurious oscillations are observed in the centered approximations and upwind schemes are proposed in each of the approximations in space: FD, FE and FV. It is proved that the three upwind schemes are equivalent in 1D and they correct the numerical oscillations of the centered schemes. In addition, the upwind scheme is proven to be solvable, and preserves the properties of conservation of total cell number, and non-negativity of cells and chemical substance, these properties are corroborated with numerical simulations. Finally, the discretization techniques developed in this work are applied to other models with chemotaxis., En este Trabajo de Fin de Grado se modeliza el fenómeno de la quimiotaxis principalmente centrándonos en el modelo de Keller-Segel, además de otras variantes del mismo. A su vez, realizamos aproximaciones numéricas del tipo diferencias finitas en tiempo y distintos esquemas en espacio, tales como: diferencias finitas (DF), elementos finitos (EF) y volúmenes finitos (VF). Posteriormente, se particularizan los esquemas a un dominio espacial unidimensional y se realizan simulaciones numéricas. Analizando los resultados, se observan oscilaciones espúreas en las aproximaciones centradas, y se proponen esquemas descentrados en cada una de las aproximaciones en espacio: DF, EF y VF. Se prueba que los tres esquemas descentrados son equivalentes en 1D y corrigen las oscilaciones numéricas de los esquemas centrados. Además, se prueba que el esquema descentrado tiene solución, y preserva las propiedades de conservación de la cantidad total de células y de no negatividad de células y química, estas propiedades se corroboran con las simulaciones numéricas. Finalmente se aplican las técnicas de discretización desarrolladas en este trabajo a otros modelos con quimiotaxis.

In this work we analyse a PDE-ODE problem modelling the evolution of a Glioblastoma, which includes chemotaxis term directed to vasculature. First, we obtain some a priori estimates for the (possible) solutions of the model. In particular, under some conditions on the parameters, we obtain that the system does not develop blow-up at finite time. In addition, we design a fully discrete finite element scheme for the model which preserves some pointwise estimates of the continuous problem. Later, we make an adimensional study in order to reduce the number of parameters. Finally, we detect the main parameters determining different width of the ring formed by proliferative and necrotic cells and different regular/irregular behaviour of the tumor surface.

Mathematical modelling and numerical optimization are applications of Mathematics with a huge potencial, being able to help a lot of disciplines appart from Mathematics. In this work, the model of Chemotaxis will be studied, which is related with the movement and pattern formation of cells and bacterias. In particular, our base will be the Keller-Segel model. After a breaf theorical study of this model, the fundamental purpose of this work will be making numerical simulations using Matlab software and studying the possibility of controlate this system. For getting numerical simulations, numerical schemes used will be based on the discretization of one of the independependent variables of our problem (space and time), and the other one will be solved by previously defined Matlab functions. To make the different controls showed in this work (space, time, and space-time controls) two different techniques will be proposed: the first one is calculating the minimum of the functional directly by a numerical optimization algorithm, and the second one is solving an optimality system. Finally, the results obtained will be analyzed and compared., La modelización matemática y la optimización numérica son aplicaciones de las Matemáticas con un gran potencial, siendo capaces de ayudar e impulsar muchas disciplinas aparte de las mismas Matemáticas. En este trabajo se estudiará el modelo de la Quimiotaxis, que está relacionado con el movimiento y la formación de patrones de células y bacterias. En particular, nuestra base será el modelo de Keller-Segel. Después de un breve estudio teórico de dicho modelo, el propósito de este trabajo será hacer simulaciones numéricas usando el software Matlab y estudiar la posibilidad de controlar este sistema. Para realizar las simulaciones numéricas, se usarán esquemas numéricos que se basarán en la discretización en una de las dos variables independientes de nuestro problema (espacio y tiempo), y en la otra se utilizarán funciones que están previamente definidas con Matlab. Para realizar los distintos tipos de controles que se mostrarán en este trabajo (control en tiempo, control en espacio, y control en espacio y en tiempo) se propondrán dos técnicas diferentes: la primera será calcular directamente el mínimo del funcional con el que se esté trabajando a través de un algoritmo de optimización numérica, y la segunda será resolviendo un sistema de optimalidad. Finalmente, los resultados obtenidos serán analizados y comparados entre sí.

The main aim of this project is to study the so-called soliton ratchet phenomena in a FrenkelKontorova model, by making use of the collective coordinate approximation. From a mathematical point of view, the discrete double sine-Gordon equation will be considered. This system, which consists of N coupled second-order ordinary differential equations (ODEs), will be reduced to a system of just two coupled second-order ODEs, in terms of the collective coordinates, that is, the center of mass X(t) and the width l(t) of the topological soliton, or kink, solution. Once these equations are obtained, the aim is to look at whether the different motion regimes that this reduced system show adequately approximate the full system., El objetivo principal de este proyecto es estudiar el llamado fen´omeno ratchet de solitones en un modelo de Frenkel-Kontorova, haciendo uso de la aproximaci´on de coordenadas colectivas. Se estudiar´a la ecuaci´on discreta de doble sine-Gordon desde un punto de vista matem´atico. Este sistema, que est´a formado por N ecuaciones diferenciales ordinarias (EDO) acopladas de segundo orden, se reducir´a a un sistema de solo dos EDOs acopladas de segundo orden en t´erminos de las coordenadas colectivas, es decir, el centro de masa X(t) y la anchura l(t) de la soluci´on de solit´on topol´ogico, tambi´en llamada kink. Una vez obtenidas estas ecuaciones, el objetivo es ver si los diferentes reg´ımenes de movimiento que muestra este sistema reducido se aproximan adecuadamente al sistema completo.

We consider a Beris-Edwards system modeling incompressible liquid crystal flows of nematic type. This system couples a Navier–Stokes system for the fluid velocity with a time-dependent system for the Q-tensor variable, whose spectral decomposition is related to the directors of liquid crystal molecules. The long-time behavior for global weak solutions is studied, proving that each whole trajectory converges to a single equilibrium whenever a regularity hypothesis is satisfied by the energy of the weak solution.

The design of numerical approximations of the Cahn-Hilliard model preserving the maximum principle is a challenging problem, even more if considering additional transport terms. In this work, we present a new upwind discontinuous Galerkin scheme for the convective Cahn-Hilliard model with degenerate mobility which preserves the maximum principle and prevents non-physical spurious oscillations. Furthermore, we show some numerical experiments in agreement with the previous theoretical results. Finally, numerical comparisons with other schemes found in the literature are also carried out.

This thesis is dedicated to modeling and analyzing mathematically the development of Glioblastoma. Thanks to considering the vasculature as an additional variable, it is possible to obtain more realistic mathematical models from the biological point of view and, in addition, to introduce the possibility of different types of tumor cell movement such as non-linear diffusion or chemotaxis related to the vasculature. In the Introduction (Chapter 1), we present the problem that we will studied in this thesis. We begin explaining the biological characteristics of Glioblastoma and we mention some studies made with real data and using mathematical models. Later, we design a general PDE-ODE model with nonlinear diffusion and chemotaxis detailing the modeling of the Glioblastoma effects. In addition, we present three models obtained from the general model, which we study in the different Chapters, with their main results. Finally, we will discuss the differences between the nonlinear diffusion and chemotaxis models and show some open problems. In Chapter 2, we study the PDE-ODE system with linear diffusion (and without chemotaxis) obtained as a simplification of the general Glioblastoma model introduced in Chapter 1. Mainly, we prove the existence and uniqueness of the global classical solution in time using a fixed point argument. Furthermore, we show some long-term behaviour results of the solution depending on some conditions in the parameters which appear in the model. In Chapter 3, we analyse a PDE-ODE model derived form the general one, which includes a nonlinear anisotropic diffusion term with a diffusion rate that increases relative to the vasculature and without chemotaxis. First, we prove the existence of global strong-weak solutions in time using a regularization technique through artificial diffusion in the ODE system and a fixed point argument. Furthermore, the long-term behaviour results of the critical points are given under some constraints on the parameters., Esta tesis está dedicada a modelar y analizar matemáticamente el desarrollo del Glioblastoma. Gracias a considerar la vasculatura como una variable adicional, es posible obtener modelos matemáticos más realistas desde el punto de vista biológico y, además, introducir la posibilidad de diferentes tipos de movimiento de las células tumorales como la difusión no lineal o la quimiotaxis relacionadas con la vasculatura. En la Introducci´on (Capítulo 1), presentamos el problema que se estudiará a lo largo de la tesis. Comenzaremos explicando las características biológicas del Glioblastoma y mencionaremos algunos estudios realizados a trav´es de datos reales y usando modelos matemáticos. Posteriormente, diseñaremos un modelo general PDE-ODE con difusión no lineal y quimiotaxis detallando en la modelización, los efectos del Glioblastoma. Además, presentaremos tres modelos obtenidos a partir del modelo general, que estudiaremos en los diferentes Capítulos, con sus principales resultados. Por último, comentaremos las diferencias entre los modelos con difusiión no lineal y con quimiotaxis y mostraremos algunos problemas abiertos. En el Capítulo 2, estudiamos el sistema PDE-ODE con difusión lineal (y sin quimiotaxis) obtenido como una simplificación del modelo general de Glioblastoma introducido en el Capítulo 1. Principalmente, probamos la existencia y unicidad de la solución clásica global en el tiempo utilizando un argumento de punto fijo. Además, mostramos algunos resultados de comportamiento a largo plazo de la solución dependiendo de algunas condiciones sobre los par´ametros que aparecen en el modelo. En el Cap´ıtulo 3, analizamos un modelo PDE-ODE obtenido del general, que incluye un término de difusi´on anisotr´opica no lineal con una velocidad de difusión que aumenta con respecto a la vasculatura y no presenta quimiotaxis. Primero, probamos la existencia de soluciones débiles-fuertes globales en el tiempo utilizando una técnica de regularización a través de una difusió

The dynamic response of modern slender footbridges is usually sensitive to both the pedestrian actions and the uncertainties associated with their inherent structural behavior. Thus, tuned mass dampers have been widely integrated in the design of these structures to guarantee the fulfillment of the vibration serviceability limit state during their overall life cycle. Three different techniques of tuned mass dampers (active, semi-active and passive) are usually considered for this purpose. Although there are algorithms for the robust design of each particular technique, however, this specificity makes difficult the implementation of all these techniques in practical en gineering applications. Herein, the motion-based design method under uncertainty conditions is proposed and further implemented to create a common framework for the robust design of all these techniques when they are employed to mitigate pedestrian-induced vibrations in slender footbridges. According to this method, the design problem may be transformed into the combination of two sequential sub-problems: (i) a reliability multi objective optimization sub-problem; and (ii) a decision-making sub-problem. Subsequently, the performance of this proposal has been validated through a numerical case study in which the dynamic response of a steel footbridge has been controlled by three different tuned mass damper techniques designed according to the proposed common framework.

In this paper we propose two fully discrete Finite Elements (FE) schemes for a repulsive chemotaxis model with a quadratic production term. The first one, which corresponds to the backward Euler in time with FE in space, is energy-stable in the primitive variables of the model, under a “compatibility” condition on the FE spaces. The second one, which is obtained modifying the scheme proposed in [13] by applying a regularization procedure, has an “approximated positivity” property which is obtained from discrete energy estimates and an additional estimate for a singular functional. These properties are not available in previous approaches. Additionally, we study the well-posedness and the long time behaviour of the schemes, obtaining exponential convergence to constant states as in the continuous problem. Finally, we compare the numerical schemes throughout several numerical simulations, which are in agreement with the theoretical results.

We consider the following repulsive-productive chemotaxis model: find 0, the cell density, and 0, the chemical concentration, satisfying 0 in 0 in 0 (1) with 1 2 , a bounded domain ( 1 2 3), endowed with non-flux boundary conditions. By using a regularization technique, we prove the existence of global in time weak solutions of (1) which is regular and unique for 1 2. Moreover, we propose two fully discrete Finite Element (FE) nonlinear schemes, the first one defined in the variables under structured meshes, and the second one by using the auxiliary variable and defined in general meshes. We prove some unconditional properties for both schemes, such as mass-conservation, solvability, energy-stability and approximated positivity. Finally, we compare the behavior of these schemes with respect to the classical FE backward Euler scheme throughout several numerical simulations and give some conclusions.

In this paper, we study a bilinear optimal control problem associated to a chemo-repulsion model with linear production term in a bidimensional domain. The existence, uniqueness and regularity of strong solutions of this model are deduced, proving the existence of a global optimal solution. Afterwards, we derive first-order optimality conditions by using a Lagrange multipliers theorem.

Models to represent the interactions of liquid crystalline phases with vesicles mem- branes have been widely studied in recent times due to its connection with biological materials. In this work we propose a new model to represent a vesicle membrane with internal nematic order whose equilibrium states depend on the competition between the bending, elastic and anchoring energies. Moreover we present a new unconditionally energy stable numerical scheme to approximate the model. Additionally, we present several numerical simulations in order to illustrate the good performance of the proposed scheme and the influence of the nematic order in the dynamics of the system.

Models of flows containing vesicles membranes with liquid crystalline phases have been widely studied in recent times due to its connection with biological applications. In this work we propose a new model to represent the interaction between flows and vesicle membranes with internal nematic order and preferential orientation of their molecules in the membrane. In fact, the dynamics of the system is determined by the dissipation of an energy that regulates the competition between different effects, through the kinetic, bending, elastic and anchoring energies. Moreover we introduce a new numerical scheme to approximate the model, that is unconditionally energy stable. Additionally, we present several numerical results in order to show the well behavior of the proposed scheme and the dynamics of this type of vesicle membranes

In this paper, a bilinear optimal control problem associated to a 3D chemo-repulsion model with linear production is studied. The existence of weak solutions is proved, and a regularity criterion to get global-in-time strong solutions is established. As a consequence, the existence of a global optimal solution with bilinear control is deduced, and using a Lagrange multiplier theorem, first-order optimality conditions for local optimal solutions are derived.

In this paper we deal with the numerical solution of a Hele{Shaw-like system via a cell model with active motion. Convergence of approximations is established for well-posed initial data. These data are chosen in such a way the time derivate is positive at the initial time. The numerical method is constructed by means of a nite element procedure together with the use of a closed-nodal integration. This gives rise to an algorithm which preserves positivity whenever a right-angled triangulation is considered. As a result, uniform-in-time a priori estimates are proven which allows us to pass to limit towards a solution to the Hele{Shaw problem.

In this paper, we introduce a model describing the dynamic of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the NavierStokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn-Hilliard phase-field equation associated to a bending energy plus a penalization term related to the area conservation. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz-Simon’s result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.

Tanto la mejora de las propiedades resistentes de los materiales constructivos como los mayores requerimientos estéticos de las sociedades modernas han originado un claro incremento de la esbeltez de las estructuras de ingeniería civil. Dicha elevada esbeltez es la principal causa tanto de la tendencia que presentan dichas estructuras a sufrir problemas vibratorios inducidos por las acciones exteriores como de que su repuesta sea cada vez más sensible a la variabilidad de los condicionantes operacionales y ambientales. Tal es la influencia de estos factores sobre el comportamiento de dichas estructuras, que sus efectos deben ser considerados como aspecto clave durante el diseño conceptual de las mismas. Una forma habitual de controlar las vibraciones en estructuras de ingeniería civil es instalar en las mismas algún tipo de sistema de disipación de energía (sistemas de control). Dichos sistemas se clasifican en uno de estos tres grupos en función de la fuente de energía exterior que necesiten para garantizar su adecuado funcionamiento: (i) sistemas de control pasivo; (ii) sistemas de control semi-activo; y (iii) sistemas de control activo. De entre los sistemas de control, los denominados sistemas de masa inercial han sido ampliamente utilizados en estructuras de ingeniería civil. Dichos sistemas de masa inercial pueden presentar un comportamiento activo, semi-activo o pasivo mediante una leve modificación de sus parámetros constituyentes. Los algoritmos de diseño existentes para caracterizar los parámetros constituyentes de dichos sistemas inerciales presentan tres grandes problemas: (i) su formulación difiere en función del comportamiento requerido; (ii) su implementación en proyectos reales es difícil debido a su complejidad; y (iii) no se ha analizado en detalle su rendimiento bajo condiciones de incertidumbre asociadas a la variabilidad ambiental y operacional. Al objeto de dar solución a dichas limitaciones se presenta, formula e implementa en este trabajo un, The improvements of the strength properties of the construction material together with the higher aesthetic requirements of the modern societies have caused a clear increase of the slenderness of the civil engineering structures. This high slenderness is the main cause of both the tendency of these structures to suffer from vibratory problems induced by external actions and that their response is more sensitive to the variability of the operational and environmental conditions. Such is the influence of these factors in the behaviour of these structures that their effects must be considered as key aspect during their conceptual design phase. A usual way to control the vibrations in civil engineering structures is to install on them some type of energy dissipation system (control system). These systems can be classified in one of these groups in terms of the external power source needed to guarantee an adequate performance: (i) passive control systems; (ii) semi-active control systems; and (iii) active control systems. Among these control systems, the so-called inertial mass dampers have been widely used in civil engineering structures. These inertial mass dampers can show an active, semi-active or passive behaviour via a slight variation of their constituent parameters. The existing design algorithms, to characterize the constituent parameters of these inertial systems, present three main limitations: (i) their formulation differ in terms of considered behaviour: (ii) its implementation in real project is difficult due to their complexity; and (iii) their performance have not been analysed in detail under stochastic conditions associated with the operational and environmental variability. In order to shed light to these limitations, a motion-based design algorithm has been presented, formulated and implemented herein. According to this algorithm the design problem is transformed into an optimization problem. This optimization problem may be defined as: (i) a multi-ob

This book contains the main results of the talks given at the workshop “Recent Advances in PDEs: Analysis, Numerics and Control”, which took place in Sevilla (Spain) on January 25-27, 2017. The work comprises 12 contributions given by high-level researchers in the partial differential equation (PDE) area to celebrate the 60th anniversary of Enrique Fernández-Cara (University of Sevilla). The main topics covered here are: Control and inverse problems, Analysis of Fluid mechanics and Numerical Analysis. The work is devoted to researchers in these fields.

We study a Q-tensor problem modeling the dynamic of nematic liquid crystals in 3D domains. The system consists of the Navier-Stokes equations, with an extra stress tensor depending on the elastic forces of the liquid crystal, coupled with an Allen-Cahn system for the Q-tensor variable. This problem has a dissipative in time free-energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a Lojasiewicz-Simon's result, we prove the convergence as time goes to infinity of the whole trajectory to a single equilibrium.

A numerical method is developed for solving a system of partial differential equations modeling the flow of a nematic liquid crystal fluid with stretching effect, which takes into account the geometrical shape of its molecules. This system couples the velocity vector, the scalar pressure and the director vector representing the direction along which the molecules are oriented. The scheme is designed by using finite elements in space and a time-splitting algorithm to uncouple the calculation of the variables: the velocity and pressure are computed by using a projection-based algorithm and the director is computed jointly to an auxiliary variable. Moreover, the computation of this auxiliary variable can be avoided at the discrete level by using piecewise constant finite elements in its approximation. Finally, we use a pressure stabilization technique allowing a stable equal-order interpolation for the velocity and the pressure. Numerical experiments concerning annihilation of singularities are presented to show the stability and efficiency of the scheme.

Two-phase flows composed of fluids exhibiting different microscopic structure are an important class of engineering materials. The dynamics of these flows are determined by the coupling among three different length scales: microscopic inside each component, mesoscopic interfacial morphology and macroscopic hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (newtonian fluid) and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due to anchoring effects of the liquid crystal molecules on the interface. Firstly, we have introduced a new differential problem to model Nematic-Isotropic mixtures, taking into account viscous, mixing, nematic and anchoring effects and reformulating the corresponding stress tensors in order to derive a dissipative energy law. Then, we provide two new linear unconditionally energy-stable splitting schemes. Moreover, we present several numerical simulations in order to show the efficiency of the proposed numerical schemes and the influence of the different types of anchoring effects in the dynamics of the system.

This paper studies the stability of velocity-pressure mixed approximations of the Stokes problem when different finite element (FE) spaces for each component of the velocity field are considered. We consider some new combinations of continuous FE reducing the number of degrees of freedom in some velocity components. Although the resulting FE combinations are not stable in general, by using the Stenberg’s macro-element technique, we show their stability in a wide family of meshes (namely, in uniformly unstructured meshes). Moreover, a post-processing is given in order to convert any mesh family in an uniformly unstructured mesh family. Finally, some 2D and 3D numerical simulations are provided agree with the previous analysis.

Two-phase flows composed of fluids exhibiting different microscopic structure are an important class of engineering materials. The dynamics of these flows are determined by the coupling among three different length scales: microscopic inside each component, mesoscopic interfacial morphology, and macroscopic hydrodynamics. Moreover, in the case of complex fluids composed by the mixture between isotropic (Newtonian fluid) and nematic (liquid crystal) flows, its interfaces exhibit novel dynamics due to anchoring effects of the liquid crystal molecules on the interface. Firstly, we have introduced a new differential problem to model nematic–isotropic mixtures, taking into account viscous, mixing, nematic, and anchoring effects and reformulating the corresponding stress tensors in order to derive a dissipative energy law. Then, we provide two new linear unconditionally energy-stable splitting schemes. Moreover, we present several numerical simulations in order to show the efficiency of the proposed numerical schemes and the influence of the different types of anchoring effects in the dynamics of the system.