39 research outputs found
A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices
The first focus of this paper is the characterization of the spectrum and the
singular values of the coefficient matrix stemming from the discretization with
space-time grid for a parabolic diffusion problem and from the approximation of
distributed order fractional equations. For this purpose we will use the
classical GLT theory and the new concept of GLT momentary symbols. The first
permits to describe the singular value or eigenvalue asymptotic distribution of
the sequence of the coefficient matrices, the latter permits to derive a
function, which describes the singular value or eigenvalue distribution of the
matrix of the sequence, even for small matrix-sizes but under given
assumptions. The note is concluded with a list of open problems, including the
use of our machinery in the study of iteration matrices, especially those
concerning multigrid-type techniques
A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices
The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Preconditioners for Krylov subspace methods: An overview
When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind
High performance implementation of 3D FEM for nonlocal Poisson problem with different ball approximation strategies
Nonlocality brings many challenges to the implementation of finite element
methods (FEM) for nonlocal problems, such as large number of queries and invoke
operations on the meshes. Besides, the interactions are usually limited to
Euclidean balls, so direct numerical integrals often introduce numerical
errors. The issues of interactions between the ball and finite elements have to
be carefully dealt with, such as using ball approximation strategies. In this
paper, an efficient representation and construction methods for approximate
balls are presented based on combinatorial map, and an efficient parallel
algorithm is also designed for assembly of nonlocal linear systems.
Specifically, a new ball approximation method based on Monte Carlo integrals,
i.e., the fullcaps method, is also proposed to compute numerical integrals over
the intersection region of an element with the ball
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