5 research outputs found
Aggregation-based aggressive coarsening with polynomial smoothing
This paper develops an algebraic multigrid preconditioner for the graph
Laplacian. The proposed approach uses aggressive coarsening based on the
aggregation framework in the setup phase and a polynomial smoother with
sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration
as a preconditioner to the flexible Conjugate Gradient iteration in the solve
phase. We show that by combining these techniques it is possible to design a
simple and scalable algorithm. Results of the algorithm applied to graph
Laplacian systems arising from the standard linear finite element
discretization of the scalar Poisson problem are reported
Parallel Unsmoothed Aggregation Algebraic Multigrid Algorithms on GPUs
We design and implement a parallel algebraic multigrid method for isotropic
graph Laplacian problems on multicore Graphical Processing Units (GPUs). The
proposed AMG method is based on the aggregation framework. The setup phase of
the algorithm uses a parallel maximal independent set algorithm in forming
aggregates and the resulting coarse level hierarchy is then used in a K-cycle
iteration solve phase with a -Jacobi smoother. Numerical tests of a
parallel implementation of the method for graphics processors are presented to
demonstrate its effectiveness.Comment: 18 pages, 3 figure
HAZniCS -- Software Components for Multiphysics Problems
We introduce the software toolbox HAZniCS for solving interface-coupled
multiphysics problems. HAZniCS is a suite of modules that combines the
well-known FEniCS framework for finite element discretization with solver and
graph library HAZmath. The focus of the paper is on the design and
implementation of a pool of robust and efficient solver algorithms which tackle
issues related to the complex interfacial coupling of the physical problems
often encountered in applications in brain biomechanics. The robustness and
efficiency of the numerical algorithms and methods is shown in several
numerical examples, namely the Darcy-Stokes equations that model flow of
cerebrospinal fluid in the human brain and the mixed-dimensional model of
electrodiffusion in the brain tissue
Boundary Treatment and Multigrid Preconditioning for Semi-Lagrangian Schemes Applied to Hamilton-Jacobi-Bellman Equations
We analyse two practical aspects that arise in the numerical solution of
Hamilton-Jacobi-Bellman (HJB) equations by a particular class of monotone
approximation schemes known as semi-Lagrangian schemes. These schemes make use
of a wide stencil to achieve convergence and result in discretization matrices
that are less sparse and less local than those coming from standard finite
difference schemes. This leads to computational difficulties not encountered
there. In particular, we consider the overstepping of the domain boundary and
analyse the accuracy and stability of stencil truncation. This truncation
imposes a stricter CFL condition for explicit schemes in the vicinity of
boundaries than in the interior, such that implicit schemes become attractive.
We then study the use of geometric, algebraic and aggregation-based multigrid
preconditioners to solve the resulting discretised systems from implicit time
stepping schemes efficiently. Finally, we illustrate the performance of these
techniques numerically for benchmark test cases from the literature