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Über dieses Buch

The first part of this volume gathers the lecture notes of the courses of the “XVII Escuela Hispano-Francesa”, held in Gijón, Spain, in June 2016. Each chapter is devoted to an advanced topic and presents state-of-the-art research in a didactic and self-contained way. Young researchers will find a complete guide to beginning advanced work in fields such as High Performance Computing, Numerical Linear Algebra, Optimal Control of Partial Differential Equations and Quantum Mechanics Simulation, while experts in these areas will find a comprehensive reference guide, including some previously unpublished results, and teachers may find these chapters useful as textbooks in graduate courses.
The second part features the extended abstracts of selected research work presented by the students during the School. It highlights new results and applications in Computational Algebra, Fluid Mechanics, Chemical Kinetics and Biomedicine, among others, offering interested researchers a convenient reference guide to these latest advances.

Inhaltsverzeichnis

Frontmatter

Theory

Frontmatter

Optimal Control of Partial Differential Equations

In this chapter, we present an introduction to the optimal control of partial differential equations. After explaining what an optimal control problem is and the goals of the analysis of these problems, we focus the study on a model example. We consider an optimal control problem governed by a semilinear elliptic equation, the control being subject to bound constraints. Then we explain the methods to prove the existence of a solution; to derive the first and second order optimality conditions; to approximate the control problem by discrete problems; to prove the convergence of the discretization and to get some error estimates. Finally we present a numerical algorithm to solve the discrete problem and we provide some numerical results. Though the whole analysis is done for an elliptic control problem, with distributed controls, some other control problems are formulated, which show the scope of the field of control theory and the variety of mathematical methods necessary for the analysis. Among these problems, we consider the case of evolution equations, Neumann or Dirichlet boundary controls, and state constraints.
Eduardo Casas, Mariano Mateos

Introduction to First-Principle Simulation of Molecular Systems

First-principle molecular simulation aims at computing the physical and chemical properties of a molecule, or more generally of a material system, from the fundamental laws of quantum mechanics. It is widely used in various application fields ranging from quantum chemistry to materials science and molecular biology, and is the source of many very interesting and challenging mathematical and numerical problems. This chapter is an elementary introduction to this field, covering some modeling, mathematical, and numerical aspects.
Eric Cancès

Accurate Computations and Applications of Some Classes of Matrices

Performing an algorithm with high relative accuracy is a very desirable goal. High relative accuracy means that the relative errors of the computations are of the order of machine precision, independently of the size of the condition number. This goal is difficult to assure although in recent years there have been some advances, in particular in the field of Numerical Linear Algebra. Up to now, computations with high relative accuracy are guaranteed only for a few classes of matrices, mainly for some subclasses of M-matrices and for some subclasses of totally positive matrices. Previously, a reparametrization of the matrices is needed. We review this procedure related with the high relative accuracy computations of these matrices. We also present some recent applications of the two classes of matrices mentioned previously. On the one hand, applications of M-matrices to the linear complementarity problem. On the other hand, applications of totally positive matrices to Computer Aided Geometric Design.
J. M. Peña

Introduction to Communication Avoiding Algorithms for Direct Methods of Factorization in Linear Algebra

Modern, massively parallel computers play a fundamental role in a large and rapidly growing number of academic and industrial applications. However, extremely complex hardware architectures, which these computers feature, effectively prevent most of the existing algorithms to scale up to a large number of processors. Part of the reason behind this is the exponentially increasing divide between the time required to communicate a floating-point number between two processors and the time needed to perform a single floating point operation by one of the processors. Previous investigations have typically aimed at overlapping as much as possible communication with computation. While this is important, the improvement achieved by such an approach is not sufficient. The communication problem needs to be addressed also directly at the mathematical formulation and the algorithmic design level. This requires a shift in the way the numerical algorithms are devised, which now need to reduce, or even minimize when possible, the number of communication instances. Communication avoiding algorithms provide such a perspective on designing algorithms that minimize communication in numerical linear algebra. In this document we describe some of the novel numerical schemes employed by those communication avoiding algorithms, with a particular focus on direct methods of factorization.
Laura Grigori

Applications

Frontmatter

Singular Traveling Waves and Non-linear Reaction-Diffusion Equations

We review some recent results on singular traveling waves arising as solutions to reaction-diffusion equations combining flux saturation mechanisms and porous media type terms. These can be regarded as toy models in connection with some difficulties arising on the mathematical modelization of several scenarios in Developmental Biology, exemplified by pattern formation in the neural tube of chick’s embryo.
Juan Calvo

Numerical Simulation of Flows Involving Singularities

Many interesting fluid interface problems involve singular events, as breaking-up or merging of the physical domain. In particular, wave propagation and breaking, droplet and bubble break-up, electro-jetting, rain drops, etc. are good examples of such processes. All these mentioned problems can be modeled using the potential flow assumptions, in which an interface needs to be advanced by a velocity determined by the solution of a surface partial differential equation posed on this moving boundary. The standard approach, the Lagrangian-Eulerian formulation together with some sort of front tracking method, is prone to fail when break-up or merging processes appear. The embedded formulation using level sets seamlessly allows topological breakup or merging of the fluid domain. In this work we present the numerical approximation of the embedded model and some computational results regarding electrohydrodynamic applications.
Maria Garzon, James A. Sethian, August Johansson

A Projection Hybrid Finite Volume-ADER/Finite Element Method for Turbulent Navier-Stokes

We present a second order finite volume/finite element projection method for low-Mach number flows. Moreover, transport of species law is also considered and turbulent regime is solved using a kɛ standard model. Starting with a 3D tetrahedral finite element mesh of the computational domain, the momentum equation is discretized by a finite volume method associated with a dual finite volume mesh where the nodes of the volumes are the barycenter of the faces of the initial tetrahedra. The resolution of Navier-Stokes equations coupled with a kɛ turbulence model requires the use of a high order scheme. The ADER methodology is extended to compute the flux terms with second order accuracy in time and space. Finally, the order of convergence is analysed by means of academic problems and some numerical results are presented.
A. Bermúdez, S. Busto, J. L. Ferrín, L. Saavedra, E. F. Toro, M. E. Vázquez-Cendón

Stable Discontinuous Galerkin Approximations for the Hydrostatic Stokes Equations

We propose a Discontinuous Galerkin scheme for the numerical solution of the Anisotropic (in particular, Hydrostatic) Stokes equations in Oceanography. The key is the introduction of interior penalties into the usual Stokes bilinear forms and, moreover, in the anisotropy (with respect to the horizontal and vertical directions) of these forms. Using \(\mathbb{P}_{k}\) discontinuous finite elements for velocity and pressure, we obtain discrete inf-sup stability independently on the ratio ɛ between the horizontal and vertical domain scales. Numerical tests are provided.
F. Guillén-González, M. V. Redondo-Neble, J. R. Rodríguez-Galván

A Two-Step Model Identification for Stirred Tank Reactors: Incremental and Integral Methods

In this work we present a new methodology for solving an inverse identification problem with application in chemistry, using two approaches in cascade. More precisely, we are interested in the identification of kinetic models and their corresponding parameters in stirred tank reactors, using a set of experimental data and the reactions taking place. A catalogue of kinetic models containing the parameters to be identified will be provided too. In order to solve it, we use a combination of an incremental and an integral method.
A. Bermúdez, E. Carrizosa, Ó. Crego, N. Esteban, J. F. Rodríguez-Calo

Variance Reduction Result for a Projected Adaptive Biasing Force Method

This paper is committed to investigate an extension of the classical adaptive biasing force method, which is used to compute the free energy related to the Boltzmann-Gibbs measure and a reaction coordinate function. The issue of this technique is that the approximated gradient of the free energy, called biasing force, is not a gradient. The commitment to this field is to project the estimated biasing force on a gradient using the Helmholtz decomposition. The variance of the biasing force is reduced using this technique, which makes the algorithm more efficient than the standard ABF method. We prove exponential convergence to equilibrium of the estimated free energy, with a precise rate of convergence in function of Logarithmic Sobolev inequality constants.
Houssam AlRachid, Tony Lelièvre

Modeling Chemical Kinetics in Solid State Reactions

This work deals with the kinetics of thermally stimulated processes which take place in the solid state phases. The activation energy of the solid is calculated using several methods of different families of isoconversional methods (differential, integral and incremental). A model of the kinetics is obtained by a method independent from the procedure used to compute the activation energy and it is analysed in three theoretical simulations as well as the thermal degradation of FeNH4(HPO4)2. The reconstructed αT curves of the simulations and the experimental case indicates that the model works properly.
J. A. Huidobro, I. Iglesias, B. F. Alfonso, C. Trobajo, J. R. Garcia

ASSR Matrices and Some Particular Cases

A real matrix is said Almost Strictly Sign Regular (ASSR) if all its nontrivial minors of the same order have the same strict sign. In this research, nonsingular ASSR matrices are characterized through the Neville elimination (NE). In addition, the algorithm is simplified for two important subclases: almost strictly totally negative (ASTN) matrices and Jacobi (tridiagonals) ASSR matrices.
P. Alonso, J. M. Peña, M. L. Serrano

A Computational Approach to Verbal Width in Alternating Groups

We know that every element in an Alternating group A n , n ≥ 5, can be written as a Engel word of length two (Carracedo, Extracta Math. 30(2), 251–262, 2015 and J. Algebra Appl. 16(2), 1750021, 10 p., 2017). There is a conjecture that every element in an Alternating group A n , n ≥ 5, can be written as an Engel word of arbitrary length. We give here a computational approach to this problem, what allows to prove the conjecture for 5 ≤ n ≤ 14.
Jorge Martínez Carracedo, Consuelo Martínez López

Improvements in Resampling Techniques for Phenotype Prediction: Applications to Neurodegenerative Diseases

Searching for new biomarkers, biological networks and pathways is crucial in the solution of neurodegenerative diseases. In this research we have compared three different algorithms and resampling techniques to find possible genetic causes in patients with Alzheimer’s and Parkinson’s diseases, providing some interesting insights about the main causes involved in these diseases.
Juan Carlos Beltrán Vargas, Enrique J. deAndrés-Galiana, Ana Cernea, Juan Luis Fernández-Martínez

An Aortic Root Geometric Model, Based on Transesophageal Echocardiographic Image Sequences (TEE), for Biomechanical Simulation

Aortic valve (AoV) stenosis is one of the most common valvular diseases. Assessing the aortic valve function could provide crucial information towards a better understanding of the disease, where numerical simulation will have an important role to play. The main scope of this work is to find an aortic root (AR) patient specific geometric model, which could be used for simulation purposes. Several models were followed to obtain an AR geometry implementing them in open source tools. Necessary parameters were obtained from 2D echo images. In order to test the obtained AR geometry, a finite element study was performed solving a fixed mesh fluid structure interaction (FSI) model. The fluid was supposed to be laminar and the tissues were modeled as St. Venant-Kirchhoff materials. Obtained results for the 1-way FSI study are compared with the published ones for structural and 2-way FSI studies showing similar results. An AR geometric reconstruction from clinic data is suited for numerical simulation.
Marcos Loureiro-Ga, Maria F. Garcia, Cesar Veiga, G. Fdez-Manin, Emilio Paredes, Victor Jimenez, Francisco Calvo-Iglesias, Andrés Iñiguez
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