El objetivo de esta tesis doctoral es desarrollar nuevos métodos basados en splines para la resolución de sistemas de ecuaciones diferenciales del tipo Y'(x)=f(x,Y(x)) , a<x<b Y(a)=Y_a (1) donde Y_a, Y(x) son matrices rxq, comenzando con splines de tipo cúbico y con un algoritmo similar al propuesto por Loscalzo y Talbot en el caso escalar [20], intentando poder aumentar el orden del spline, lo que con el método dado en [20] no puede hacerse de forma convergente. Trataremos también de aplicar dicho método al problema Y''(x)=f(x,Y(x),Y'(x)) , a<x<b Y(a)=Y_a Y'(a)=Y_b (2) sin aumentar la dimensión del problema para evitar el sobrecoste computacional. Los métodos presentados se compararán con los existentes en la literatura y serán implementados en algoritmos para ponerlos, debidamente documentados, a disposición de la comunidad científica. / Tung, MM. (2013). Spline approximations for systems of ordinary differential equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/31658 / Premios Extraordinarios de tesis doctorales

Compressed sensing and matrix completion are two new data acquisition techniques whose efficiency is achieved by exploring low dimensional structures in high dimensional data. Despite the combinatorial nature of compressed sensing and matrix completion, there has been significant development of computationally efficient algorithms which can produce accurate desired solutions to these problems. In this thesis, we are concerned with the development of low per iteration computational complexity algorithms for compressed sensing and matrix completion. First, we derive a locally optimal stepsize selection rule for the simplest iterative hard thresholding algorithm for matrix completion, and obtain a simple yet efficient algorithm. It is observed to have average case performance superior in some aspects to other matrix completion algorithms. To balance the fast convergence rates of more sophisticated recovery algorithms with the low per iteration computational cost of simple line-search algorithms, we introduce a family of conjugate gradient iterative hard thresholding algorithms for both compressed sensing and matrix completion. The theoretical results establish recovery guarantees for the restarted and projected variants of the algorithms, while the empirical performance comparisons establish significant computational advantages of the proposed methods over other hard thresholding algorithms. Finally, we introduce an alternating steepest descent method and a scaled variant especially designed for the matrix completion problem based on a simple factorization model of the low rank matrix. The computational efficacy of this method is achieved by reducing the high per iteration computational cost of the second order method and fully exploring the numerical linear algebra structure in the algorithm. Empirical evaluations establish the effectiveness of the proposed algorithms, compared with other state-of-the-art algorithms.

A graph is a mathematical structure for modelling the pairwise relations between objects. This thesis studies two types of graphs, namely, similarity-based graphs and evolving graphs. We look at ways to traverse an evolving graph. In particular, we examine the influence of temporal information on node centrality. In the process, we develop EvolvingGraphs.jl, a software package for analyzing time-dependent networks. We develop Etymo, a search system for discovering interesting research papers. Etymo utilizes both similarity-based graphs and evolving graphs to build a knowledge graph of research articles in order to help users to track the development of ideas. We construct content similarity-based graphs using the full text of research papers. And we extract key concepts from research papers and exploit the temporal information in research papers to construct a concepts evolving graph.

In this thesis, we consider the numerical solution of a large eigenvalue problem in which the integral operator comes from a radiative transfer problem. It is considered the use of hierarchical matrices, an efficient data-sparse representation of matrices, especially useful for large dimensional problems. It consists on low-rank subblocks leading to low memory requirements as well as cheap computational costs. We discuss the use of the hierarchical matrix technique in the numerical solution of a large scale eigenvalue problem arising from a finite rank discretization of an integral operator. The operator is of convolution type, it is defined through the first exponential-integral function and hence it is weakly singular. We access HLIB (Hierarchical matrices LIBrary) that provides, among others, routines for the construction of hierarchical matrix structures and arithmetic algorithms to perform approximative matrix operations. Moreover, it is incorporated the matrix-vector multiply routines from HLIB, as well as LU factorization for preconditioning, into SLEPc (Scalable Library for Eigenvalue Problem Computations) in order to exploit the available algorithms to solve eigenvalue problems. It is also developed analytical expressions for the approximate degenerate kernels and deducted error upper bounds for these approximations. The numerical results obtained with other approaches to solve the problem are used to compare with the ones obtained with this technique, illustrating the efficiency of the techniques developed and implemented in this work

Aerodynamic shape optimization and multidisciplinary optimization algorithms have the potential not only to improve conventional aircraft, but also to enable the design of novel configurations. By their very nature, these algorithms generate and analyze a large number of unique shapes, resulting in high computational costs. In order to improve their efficiency and enable their use in the early stages of the design process, a fast and robust flow solution algorithm is necessary. This thesis presents an efficient parallel Newton-Krylov-Schur flow solution algorithm for the three-dimensional Navier-Stokes equations coupled with the Spalart-Allmaras one-equation turbulence model. The algorithm employs second-order summation-by-parts (SBP) operators on multi-block structured grids with simultaneous approximation terms (SATs) to enforce block interface coupling and boundary conditions. The discrete equations are solved iteratively with an inexact-Newton method, while the linear system at each Newton iteration is solved using the flexible Krylov subspace iterative method GMRES with an approximate-Schur parallel preconditioner. The algorithm is thoroughly verified and validated, highlighting the correspondence of the current algorithm with several established flow solvers. The solution for a transonic flow over a wing on a mesh of medium density (15 million nodes) shows good agreement with experimental results. Using 128 processors, deep convergence is obtained in under 90 minutes. The solution of transonic flow over the Common Research Model wing-body geometry with grids with up to 150 million nodes exhibits the expected grid convergence behavior. This case was completed as part of the Fifth AIAA Drag Prediction Workshop, with the algorithm producing solutions that compare favourably with several widely used flow solvers. The algorithm is shown to scale well on over 6000 processors. The results demonstrate the effectiveness of the SBP-SAT spatial discretization, which can be readily extended to high order, in combination with the Newton-Krylov-Schur iterative method to produce a powerful parallel algorithm for the numerical solution of the Reynolds-averaged Navier-Stokes equations. The algorithm can efficiently solve the flow over a range of clean geometries, making it suitable for use at the core of an optimization algorithm.

Aerodynamic shape optimization and multidisciplinary optimization algorithms have the potential not only to improve conventional aircraft, but also to enable the design of novel configurations. By their very nature, these algorithms generate and analyze a large number of unique shapes, resulting in high computational costs. In order to improve their efficiency and enable their use in the early stages of the design process, a fast and robust flow solution algorithm is necessary. This thesis presents an efficient parallel Newton-Krylov-Schur flow solution algorithm for the three-dimensional Navier-Stokes equations coupled with the Spalart-Allmaras one-equation turbulence model. The algorithm employs second-order summation-by-parts (SBP) operators on multi-block structured grids with simultaneous approximation terms (SATs) to enforce block interface coupling and boundary conditions. The discrete equations are solved iteratively with an inexact-Newton method, while the linear system at each Newton iteration is solved using the flexible Krylov subspace iterative method GMRES with an approximate-Schur parallel preconditioner. The algorithm is thoroughly verified and validated, highlighting the correspondence of the current algorithm with several established flow solvers. The solution for a transonic flow over a wing on a mesh of medium density (15 million nodes) shows good agreement with experimental results. Using 128 processors, deep convergence is obtained in under 90 minutes. The solution of transonic flow over the Common Research Model wing-body geometry with grids with up to 150 million nodes exhibits the expected grid convergence behavior. This case was completed as part of the Fifth AIAA Drag Prediction Workshop, with the algorithm producing solutions that compare favourably with several widely used flow solvers. The algorithm is shown to scale well on over 6000 processors. The results demonstrate the effectiveness of the SBP-SAT spatial discretization, which can be readily extended to high order, in combination with the Newton-Krylov-Schur iterative method to produce a powerful parallel algorithm for the numerical solution of the Reynolds-averaged Navier-Stokes equations. The algorithm can efficiently solve the flow over a range of clean geometries, making it suitable for use at the core of an optimization algorithm.

Flows of fluids and solid particles through microchannels have a very wide range of applications in biological and medical science and engineering. Understanding the mechanism of microflows will help to improve the development of the devices and systems in those applications. The aim of this study is to develop a sophisticated simulation and analysis technique for the study of fluid-particle flow through microchannels. This work involves construction of mathematical models, development of analytical methods and numerical algorithms, and numerical investigation and analysis. / The study consists of three parts. The first part of the research focuses on the transient flow of an incompressible Newtonian fluid through a micro-annual with a slip boundary. The flow of the fluid is governed by the continuity equation and the Navier-Stokes equations, and is driven by the pressure field with a timevarying pressure gradient. By using the Fourier series expansion in time and Bessel functions in space, an exact solution is derived for the velocity field. The velocity solution is then used to obtain the exact solutions for the flow rate and the stress field. Based on the exact solutions, the influence of the slip parameter on the flow behaviour is then investigated. / The second part of the research focuses on the particle-fluid flow in microchannels. The transport of fluid in the vessel is governed by the continuity equation and the transient Navier-Stokes equations, while the motion of the particles is governed by Newton’s laws. The particle-wall and particle-particle interactions are modelled by the interacting forces, while the particle-fluid interaction is described by the fluid drag force. A numerical scheme based on the finite element method and the Arbitary Lagrangian-Eulerian method is developed to simulate the motion of the particles and the fluid flow in the vessels. The influence of boundary slip on the velocity field in the fluid is also investigated numerically. / Based on the work in the second part, the third part of the research focuses onthe control of the movement of particles in the fluid by applying an external magneticfield to the system. Maxwell’s equations are used to model the magnetic fieldgenerated by the external magnetic source, and a finite element based numericalscheme is developed to solve the underlying boundary value problem for the magneticflux density generated. From the computed flux density and magnetic vectorpotential, the magnetic forces acting on the particles are determined. These magneticforces together with the drag force and the particle-particle interacting forcesdominate the behaviour of the particle motion. A numerical scheme, similar to thatfor the second part of the research, is then developed to study the fluid-particle flowin microchannels under magnetic forces, followed by a numerical investigation onthe influence of the magnetic forces on the particle flow behaviour.

The purpose of this paper is to supply a summary of library subroutines and functions for parallel MIMD computers. The subroutines have been developed and continously extended at the University of Chemnitz since the end of the eighties. In detail, they are concerned with vector operations, inter-processor communication and simple graphic output to workstations. One of the most valuable features is the machine-independence of the communication subroutines proposed in this paper for a hypercube topology of the parallel processors (excepting a kernel of only two primitive system-dependend operations). They were implemented and tested for different hardware and operating systems including PARIX for transputers and PowerPC, nCube, PVM, MPI. The vector subroutines are optimized by the use of C language and unrolled loops (BLAS1-like). Hardware-optimized BLAS1 routines may be integrated. The paper includes hints for programmers how to use the libraries with both Fortran and C programs.

This paper deals with the visualization of numerical results as a very convenient method to understand and evaluate a solution which has been calculated as a set of millions of numerical values. One of the central research fields of the Chemnitz SFB 393 is the analysis of parallel numerical algorithms for large systems of linear equations arising from differential equations (e.g. in solid and fluid mechanics). Solving large problems on massively parallel computers makes it more and more impossible to store numerical data from the distributed memory of the parallel computer to the disk for later postprocessing. However, the developer of algorithms is interested in an on-line response of his algorithms. Both visual and numerical response of the running program may be evaluated by the user for a decision how to switch or adjust interactively certain parameters that may influence the solution process. The paper gives a survey of current programmer and user interfaces that are used in our various 2D and 3D parallel finite element programs for the visualization of the solution.

A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order elliptic PDE on the parameter space. The chosen elliptic PDE is solved analytically, even in the case of a very general complex trim, allowing the design process to be carried out interactively in real time. To demonstrate the capability for this technique we discuss a series of examples where trimmed PDE surfaces may be applicable.