Multilevel Solvers for Unstructured Surface Meshes

Parameterization of unstructured surface meshes is of fundamental importance in many applications of digital geometry processing. Such parameterization approaches give rise to large and exceedingly ill-conditioned systems which are difficult or impossible to solve without the use of sophisticated multilevel preconditioning strategies. Since the underlying meshes are very fine to begin with, such multilevel preconditioners require mesh coarsening to build an appropriate hierarchy. In this paper we consider several strategies for the construction of hierarchies using ideas from mesh simplification algorithms used in the computer graphics literature. We introduce two novel hierarchy construction schemes and demonstrate their superior performance when used in conjunction with a multigrid preconditioner

Algebraic multilevel incomplete factorization methods for five-point matrices

Contains fulltext : 265205.pdf (Publisher’s version ) (Open Access)21 p

On a modification of algebraic multilevel iteration method for finite element matrices *

A M S subject classification: 65F10, 65N20 Ларин М.Р. О модификации алгебраического многоуровневого итерационного ме-тода для решения конечно-элементных систем линейных алгебраических уравнений // Сиб. журн. вычисл. математики / РАН. Сиб. отд-ние.--Новосибирск, 2007.--Т. 10, № 1.--С. 101-116. В настоящее время многосеточные и многоуровневые методы очень популярны для решения раз-реженных систем линейных алгебраических уравнений. Они обладают как широкой областью примене-ния, так и эффективностью. В работе [6] был предложен алгебраический многоуровневый итерационный (AMLI) метод для решения конечно-элементных систем линейных алгебраических уравнений. Однако этот метод имеет два ограничения на свойства исходной матрицы, которые могут нарушаться на практи-ке. C целью избежать их и улучшить качество AMLI-предбуславливателя предлагается и анализируется семейство итерационных параметров релаксации. Ключевые слова: алгебраический многоуровневый метод, метод сопряженных градиентов с пре-добуславливателем, системы линейных алгебраических уравнений, метод конечных элементов. Today, multigrids and multilevel methods for solving a sparse linear system of equations are well known. They are both robust and efficient. I

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte

On residual smoothing in ILUM-type preconditioning

Incomplete block factorisations are used to construct flexible preconditioners for iterative linear solvers. In practice these approaches have shown to be very effective and robust. Especially they are more suitable for parallel computer architectures when comparing to classic ILU preconditioning. In this paper we introduce residual smoothing into the forward/backward substitution in order to compensate the element dropping in the Schur complement

Enhanced multi-level block ILU preconditioning strategies for general sparse linear systems

AbstractThis paper introduces several strategies to deal with pivot blocks in multi-level block incomplete LU factorization (BILUM) preconditioning techniques. These techniques are aimed at increasing the robustness and controlling the amount of fill-ins of BILUM for solving large sparse linear systems when large-size blocks are used to form block-independent set. Techniques proposed in this paper include double-dropping strategies, approximate singular-value decomposition, variable size blocks and use of an arrowhead block submatrix. We point out the advantages and disadvantages of these strategies and discuss their efficient implementations. Numerical experiments are conducted to show the usefulness of the new techniques in dealing with hard-to-solve problems arising from computational fluid dynamics. In addition, we discuss the relation between multi-level ILU preconditioning methods and algebraic multi-level methods

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Rev. iberoam. bioecon. cambio clim. Vol.1(1) 2015; 95-114Los cambios medioambientales globales hacen pensar en un aumento futuro de la aridez, por ello es necesario buscar alternativas que permitan un uso más eficiente del agua y reducir su consumo, teniendo en cuenta que es un recurso limitado. En la actualidad, aproximadamente el 59,7% del total de agua planificada para todos los usos en Cuba se utiliza en la agricultura, pero no más del 50% de esa agua se convierte directamente en productos agrícolas. El estudio de las funciones agua-rendimiento y su uso dentro de la planificación del agua para riego es una vía importante para trazar estrategias de manejo que contribuyan al incremento en la producción agrícola. Utilizando los datos de agua aplicada por riego y los rendimientos obtenidos en más de 100 experimentos de campo realizados fundamentalmente en suelo Ferralítico Rojo de la zona sur de La Habana y con ayuda de herramientas de análisis de regresión en este trabajo se estiman las funciones agua aplicada-rendimientos para algunos cultivos agrícolas y se analizan las posibles estrategias de optimización del riego a seguir en función de la disponibilidad de agua. Seleccionar una estrategia de máxima eficiencia del riego puede conducir a reducciones de agua a aplicar entre un 21,6 y 46,8%, incrementos de la productividad del agua entre 17 y 32% y de la relación beneficios/costo estimada de hasta un 3,4%. Lo anterior indica la importancia desde el punto de vista económico que puede llegar a alcanzar el uso de esta estrategia en condiciones de déficit hídrico. El conocimiento de las funciones agua aplicada por riego-rendimiento y el uso de la productividad del agua, resultan parámetros factibles de introducir como indicadores de eficiencia en el planeamiento del uso del agua en la agricultura, con lo cual es posible reducir los volúmenes de agua a aplicar y elevar la relación beneficio-costo actual.Rev. iberoam. bioecon. cambio clim. Vol.1(1) 2015; 95-11

Graph coarsening: From scientific computing to machine learning

The general method of graph coarsening or graph reduction has been a remarkably useful and ubiquitous tool in scientific computing and it is now just starting to have a similar impact in machine learning. The goal of this paper is to take a broad look into coarsening techniques that have been successfully deployed in scientific computing and see how similar principles are finding their way in more recent applications related to machine learning. In scientific computing, coarsening plays a central role in algebraic multigrid methods as well as the related class of multilevel incomplete LU factorizations. In machine learning, graph coarsening goes under various names, e.g., graph downsampling or graph reduction. Its goal in most cases is to replace some original graph by one which has fewer nodes, but whose structure and characteristics are similar to those of the original graph. As will be seen, a common strategy in these methods is to rely on spectral properties to define the coarse graph