Wavelet and Multiscale Methods

Various scientific models demand finer and finer resolutions of relevant features. Paradoxically, increasing computational power serves to even heighten this demand. Namely, the wealth of available data itself becomes a major obstruction. Extracting essential information from complex structures and developing rigorous models to quantify the quality of information leads to tasks that are not tractable by standard numerical techniques. The last decade has seen the emergence of several new computational methodologies to address this situation. Their common features are the nonlinearity of the solution methods as well as the ability of separating solution characteristics living on different length scales. Perhaps the most prominent examples lie in multigrid methods and adaptive grid solvers for partial differential equations. These have substantially advanced the frontiers of computability for certain problem classes in numerical analysis. Other highly visible examples are: regression techniques in nonparametric statistical estimation, the design of universal estimators in the context of mathematical learning theory and machine learning; the investigation of greedy algorithms in complexity theory, compression techniques and encoding in signal and image processing; the solution of global operator equations through the compression of fully populated matrices arising from boundary integral equations with the aid of multipole expansions and hierarchical matrices; attacking problems in high spatial dimensions by sparse grid or hyperbolic wavelet concepts. This workshop proposed to deepen the understanding of the underlying mathematical concepts that drive this new evolution of computation and to promote the exchange of ideas emerging in various disciplines

Adaptive low and high-order hybridized methods for unsteady incompressible flow simulations

Tesi en modalitat de cotutela: Universitat Politècnica de Catalunya i Università degli Studi di PaviaSimulations of incompressible flows are performed on a daily basis to solve problems of practical and industrial interest in several fields of engineering, including automotive, aeronautical, mechanical and biomedical applications. Although finite volume (FV) methods are still the preferred choice by the industry due to their efficiency and robustness, sensitivity to mesh quality and limited accuracy represent two main bottlenecks of these approaches. This is especially critical in the context of transient phenomena, in which FV methods show excessive numerical diffusion. In this context, there has been a growing interest towards high-order discretisation strategies in last decades. In this PhD thesis, a high-order adaptive hybidisable discontinuous Galerkin (HDG) method is proposed for the approximation of steady and unsteady laminar incompressible Navier-Stokes equations. Voigt notation for symmetric second-order tensors is exploited to devise an HDG method for the Cauchy formulation of the momentum equation with optimal convergence properties, even when low-order polynomial degrees of approximation are considered. In addition, a postprocessing strategy accounting for rigid translational and rotational modes is proposed to construct an element-by-element superconvergent velocity field. The discrepancy between the computed and postprocessed velocities is utilised to define a local error indicator to drive degree adaptivity procedures and accurately capture localised features of the flow. The resulting HDG solver is thus extended to the case of transient problems via high-order time integration schemes, namely the explicit singly diagonal implicit Runge-Kutta (ESDIRK) schemes. In this context, the embedded explicit step is exploited to define an inexpensive estimate of the temporal error to devise an efficient timestep control strategy. Finally, in order to efficiently solve the global problem arising from the HDG discretisation, a preconditioned iterative solver is proposed. This is critical in the context of high-order approximations in three-dimensional domains leading to large-scale problems, especially in transient simulations. A block diagonal preconditioner coupled with an inexpensive approximation of the Schur complement of the matrix is proposed to reduce the computational cost of the overall HDG solver. Extensive numerical validation of two and three-dimensional steady and unsteady benchmark tests of viscous laminar incompressible flows is performed to validate the proposed methodology.Simulaciones de flujo incompresible se emplean a diario para resolver problemas de interés práctico e industrial en varios campos de la ingeniería, p.ej. en aplicaciones automovilísticas, aeronáuticas, mecánicas y biomédicas. Aunque los métodos de volúmenes finitos (FV) siguen siendo la opción preferida por la industria debido a su eficiencia y robustez, la sensibilidad a la calidad de la malla y la baja precisión representan dos limitaciones importantes para estas técnicas. Estas limitaciones son todavía más críticas en el contexto de simulaciones de fenómenos transitorios, donde los FV están penalizados por su excesiva difusión numérica. En este contexto, las estrategias de discretización de alto orden han ganado una popularidad creciente en las últimas décadas para problemas transitorios dónde se necesitan soluciones precisas. Esta tesis propone un método de Galerkin discontinuo híbrido (HDG), de alto orden y adaptativo para la aproximación de las ecuaciones de Navier-Stokes incomprensible laminar, en el caso estacionario y transitorio en el entorno de aplicaciones ingenieriles. Para ello, la notación de Voigt para tensores simétricos de segundo orden (habituales en mecánica de los medios continuos) permite introducir un método HDG para la formulación de Cauchy de la ecuación de momento. La novedad de este resultado reside en la convergencia óptima alcanzada por el método, incluso para aproximaciones de orden polinómico bajo. Además, se desarrolla una estrategia de post-proceso local para construir elemento a elemento un campo de velocidad súper-convergente, tomando en cuenta los modos rígidos de traslación y rotación. La discrepancia entre el campo de velocidad calculado y el súper-convergente, obtenido a través del post-proceso, permite definir un indicador del error local. De esta forma, se desarrolla una estrategia para realizar adecuar elemento a elemento el grado de la aproximación polinómica y así mejorar la precisión adaptándose a las características localizadas del flujo. Seguidamente, se extiende el método HDG propuesto al tratamiento de problemas dependientes del tiempo. Más concretamente, se consideran los esquemas de integración temporal de alto orden explicit singly diagonal implicit Runge-Kutta (ESDIRK). En este contexto, se utiliza el paso explícito embedded para calcular una estimación computacionalmente eficiente del error temporal y definir una estrategia de adaptividad del paso de tiempo. Finalmente, se desarrolla un precondicionador adaptado a la estrategia HDG que acelera la convergencia del método iterativo empleado y, de esta forma, obtener resoluciones eficaces del problema global surgido de la discretización HDG. Es importante resaltar la importancia de una herramienta de resolución eficiente para problemas de gran escala en el contexto de aproximaciones de alto orden y en dominios tridimensionales. Estas herramientas se hacen aún más criticas en simulaciones transitorias. Más concretamente, se proponen un precondicionador diagonal por bloques y una aproximación eficiente del complemento Schur de la matriz para reducir el coste computacional del método HDG. Para validar la metodología propuesta, se realizan varias simulaciones numéricas de flujo incompresible laminar viscoso, para problemas estacionarios y transitorios, en dos y tres dimensiones.Postprint (published version

Adaptive Discontinuous Galerkin Finite Element Methods for a Diffuse Interface Model of Biological Growth

This PhD dissertation concentrates on the development and application of adaptive Discontinuous Galerkin Finite Element (DG-FE) methods for the numerical solution of a Cahn-Hilliard-type diffuse interface model for biological growth. Models of this type have become popular for studying cancerous tumor progression in vivo. The work in this dissertation advances the state-of-the-art in the following ways: To our knowledge the work here contains the first primitive-variable, completely discontinuous numerical implementations of a 2D scheme for the Cahn-Hilliard equation as well as a diffuse interface model of cancer growth. We provide numerical evidence that the schemes above are convergent, with the optimal order. The efficiency of the numerical algorithms depends largely on the implementation of fast solvers for the systems of equations resulting from the DG-FE discretizations. We have developed such capabilities based on multigrid and sparse direct solver techniques. We demonstrate proof-of-concept regarding the implementation of a practical spatially adaptive meshing algorithm for the numerical schemes just mentioned and th1 effective use of a very simple, but powerful, marking strategy based on an inverse estimate. We demonstrate proof-of-concept for a novel simplified diffuse interface model of tumor growth. This model is essentially the Cahn-Hilliard equation with an added source term that is specialized for the context of cancerous tumor progression. We devise and analyze a mixed DG-FE scheme of convex splitting (CS) type for the Cahn-Hilliard equation in any space dimension. Specifically, we prove that our scheme is unconditionally energy stable and unconditionally uniquely solvable. Likewise, we devise and analyze a CS, mixed DG-FE scheme for our diffuse interface cancer model. This scheme is energy stable for any (positive) time step size and for any (positive) space step size that is sufficiently small

Multigrid methods for nonlinear second order partial differential operators

This thesis is concerned with the efficient numerical solution of nonlinear partial differential equations (PDEs) of elliptic and parabolic type. Such PDEs arise frequently in models used to describe many physical phenomena, from the diffusion of a toxin in soil to the flow of viscous fluids. The main focus of this research is to better understand the implementation and performance of nonlinear multigrid methods for the solution of elliptic and parabolic PDEs, following their discretisation. For the most part finite element discretisations are considered, but other techniques are also discussed. Following discretisation of a PDE the two most frequently used nonlinear multigrid methods are Newton-Multigrid and the Full Approximation Scheme (FAS). These are both very efficient algorithms, and have the advantage that when they are applied to practical problems, their execution times scale linearly with the size of the problem being solved. Even though this has yet to be proved in theory for most problems, these methods have been widely adopted in practice in order to solve highly complex nonlinear (systems of) PDEs. Many research groups use either Newton-MG or FAS without much consideration as to which should be preferred, since both algorithms perform satisfactorily. In this thesis we address the question as to which method is likely to be more computationally efficient in practice. As part of this investigation the implementation of the algorithms is considered in a framework which allows the direct comparison of the computational effort of the two iterations. As well as this, the convergence properties of the methods are considered, applied to a variety of model problems. Extensive results are presented in the comparison, which are explained by available theory whenever possible. The strength and range of results presented allows us to confidently conclude that for a practical problem, discretised using a finite element discretisation, an improved efficiency and stability of a Newton-MG iteration, compared to an FAS iteration, is likely to be observed. The relative advantage of a Newton-MG method is likely to be larger the more complex the problem being solved becomes

Hybridizable compatible finite element discretizations for numerical weather prediction: implementation and analysis

There is a current explosion of interest in new numerical methods for atmospheric modeling. A driving force behind this is the need to be able to simulate, with high efficiency, large-scale geophysical flows on increasingly more parallel computer systems. Many current operational models, including that of the UK Met Office, depend on orthogonal meshes, such as the latitude-longitude grid. This facilitates the development of finite difference discretizations with favorable numerical properties. However, such methods suffer from the ``pole problem," which prohibits the model to make efficient use of a large number of computing processors due to excessive concentration of grid-points at the poles. Recently developed finite element discretizations, known as ``compatible" finite elements, avoid this issue while maintaining the key numerical properties essential for accurate geophysical simulations. Moreover, these properties can be obtained on arbitrary, non-orthogonal meshes. However, the efficient solution of the resulting discrete systems depend on transforming the mixed velocity-pressure (or velocity-pressure-buoyancy) system into an elliptic problem for the pressure. This is not so straightforward within the compatible finite element framework due to inter-element coupling. This thesis supports the proposition that systems arising from compatible finite element discretizations can be solved efficiently using a technique known as ``hybridization." Hybridization removes inter-element coupling while maintaining the desired numerical properties. This permits the construction of sparse, elliptic problems, for which fast solver algorithms are known, using localized algebra. We first introduce the technique for compatible finite element discretizations of simplified atmospheric models. We then develop a general software abstraction for the rapid implementation and composition of hybridization methods, with an emphasis on preconditioning. Finally, we extend the technique for a new compatible method for the full, compressible atmospheric equations used in operational models.Open Acces

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Applications and Experiences of Quality Control

The rich palette of topics set out in this book provides a sufficiently broad overview of the developments in the field of quality control. By providing detailed information on various aspects of quality control, this book can serve as a basis for starting interdisciplinary cooperation, which has increasingly become an integral part of scientific and applied research