60 research outputs found
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
Generalizing Reduction-Based Algebraic Multigrid
Algebraic Multigrid (AMG) methods are often robust and effective solvers for
solving the large and sparse linear systems that arise from discretized PDEs
and other problems, relying on heuristic graph algorithms to achieve their
performance. Reduction-based AMG (AMGr) algorithms attempt to formalize these
heuristics by providing two-level convergence bounds that depend concretely on
properties of the partitioning of the given matrix into its fine- and
coarse-grid degrees of freedom. MacLachlan and Saad (SISC 2007) proved that the
AMGr method yields provably robust two-level convergence for symmetric and
positive-definite matrices that are diagonally dominant, with a convergence
factor bounded as a function of a coarsening parameter. However, when applying
AMGr algorithms to matrices that are not diagonally dominant, not only do the
convergence factor bounds not hold, but measured performance is notably
degraded. Here, we present modifications to the classical AMGr algorithm that
improve its performance on matrices that are not diagonally dominant, making
use of strength of connection, sparse approximate inverse (SPAI) techniques,
and interpolation truncation and rescaling, to improve robustness while
maintaining control of the algorithmic costs. We present numerical results
demonstrating the robustness of this approach for both classical isotropic
diffusion problems and for non-diagonally dominant systems coming from
anisotropic diffusion
Multiscale approach for the network compression-friendly ordering
We present a fast multiscale approach for the network minimum logarithmic
arrangement problem. This type of arrangement plays an important role in a
network compression and fast node/link access operations. The algorithm is of
linear complexity and exhibits good scalability which makes it practical and
attractive for using on large-scale instances. Its effectiveness is
demonstrated on a large set of real-life networks. These networks with
corresponding best-known minimization results are suggested as an open
benchmark for a research community to evaluate new methods for this problem
On local Fourier analysis of multigrid methods for PDEs with jumping and random coefficients
In this paper, we propose a novel non-standard Local Fourier Analysis (LFA)
variant for accurately predicting the multigrid convergence of problems with
random and jumping coefficients. This LFA method is based on a specific basis
of the Fourier space rather than the commonly used Fourier modes. To show the
utility of this analysis, we consider, as an example, a simple cell-centered
multigrid method for solving a steady-state single phase flow problem in a
random porous medium. We successfully demonstrate the prediction capability of
the proposed LFA using a number of challenging benchmark problems. The
information provided by this analysis helps us to estimate a-priori the time
needed for solving certain uncertainty quantification problems by means of a
multigrid multilevel Monte Carlo method
Adaptive Near-Optimal Rank Tensor Approximation for High-Dimensional Operator Equations
We consider a framework for the construction of iterative schemes for
operator equations that combine low-rank approximation in tensor formats and
adaptive approximation in a basis. Under fairly general assumptions, we obtain
a rigorous convergence analysis, where all parameters required for the
execution of the methods depend only on the underlying infinite-dimensional
problem, but not on a concrete discretization. Under certain assumptions on the
rates for the involved low-rank approximations and basis expansions, we can
also give bounds on the computational complexity of the iteration as a function
of the prescribed target error. Our theoretical findings are illustrated and
supported by computational experiments. These demonstrate that problems in very
high dimensions can be treated with controlled solution accuracy.Comment: 51 page
- …