3,106 research outputs found
An Algorithm for Probabilistic Alternating Simulation
In probabilistic game structures, probabilistic alternating simulation
(PA-simulation) relations preserve formulas defined in probabilistic
alternating-time temporal logic with respect to the behaviour of a subset of
players. We propose a partition based algorithm for computing the largest
PA-simulation, which is to our knowledge the first such algorithm that works in
polynomial time, by extending the generalised coarsest partition problem (GCPP)
in a game-based setting with mixed strategies. The algorithm has higher
complexities than those in the literature for non-probabilistic simulation and
probabilistic simulation without mixed actions, but slightly improves the
existing result for computing probabilistic simulation with respect to mixed
actions.Comment: We've fixed a problem in the SOFSEM'12 conference versio
Implementation of multigrid methods for solving Navier-Stokes equations on a multiprocessor system
Presented are schemes for implementing multigrid algorithms on message based MIMD multiprocessor systems. To address the various issues involved, a nontrivial problem of solving the 2-D incompressible Navier-Stokes equations is considered as the model problem. Three different multigrid algorithms are considered. Results from implementing these algorithms on an Intel iPSC are presented
A Parallel Solver for Graph Laplacians
Problems from graph drawing, spectral clustering, network flow and graph
partitioning can all be expressed in terms of graph Laplacian matrices. There
are a variety of practical approaches to solving these problems in serial.
However, as problem sizes increase and single core speeds stagnate, parallelism
is essential to solve such problems quickly. We present an unsmoothed
aggregation multigrid method for solving graph Laplacians in a distributed
memory setting. We introduce new parallel aggregation and low degree
elimination algorithms targeted specifically at irregular degree graphs. These
algorithms are expressed in terms of sparse matrix-vector products using
generalized sum and product operations. This formulation is amenable to linear
algebra using arbitrary distributions and allows us to operate on a 2D sparse
matrix distribution, which is necessary for parallel scalability. Our solver
outperforms the natural parallel extension of the current state of the art in
an algorithmic comparison. We demonstrate scalability to 576 processes and
graphs with up to 1.7 billion edges.Comment: PASC '18, Code: https://github.com/ligmg/ligm
Dimension Reduction via Colour Refinement
Colour refinement is a basic algorithmic routine for graph isomorphism
testing, appearing as a subroutine in almost all practical isomorphism solvers.
It partitions the vertices of a graph into "colour classes" in such a way that
all vertices in the same colour class have the same number of neighbours in
every colour class. Tinhofer (Disc. App. Math., 1991), Ramana, Scheinerman, and
Ullman (Disc. Math., 1994) and Godsil (Lin. Alg. and its App., 1997)
established a tight correspondence between colour refinement and fractional
isomorphisms of graphs, which are solutions to the LP relaxation of a natural
ILP formulation of graph isomorphism.
We introduce a version of colour refinement for matrices and extend existing
quasilinear algorithms for computing the colour classes. Then we generalise the
correspondence between colour refinement and fractional automorphisms and
develop a theory of fractional automorphisms and isomorphisms of matrices.
We apply our results to reduce the dimensions of systems of linear equations
and linear programs. Specifically, we show that any given LP L can efficiently
be transformed into a (potentially) smaller LP L' whose number of variables and
constraints is the number of colour classes of the colour refinement algorithm,
applied to a matrix associated with the LP. The transformation is such that we
can easily (by a linear mapping) map both feasible and optimal solutions back
and forth between the two LPs. We demonstrate empirically that colour
refinement can indeed greatly reduce the cost of solving linear programs
h-multigrid agglomeration based solution strategies for discontinuous Galerkin discretizations of incompressible flow problems
In this work we exploit agglomeration based -multigrid preconditioners to
speed-up the iterative solution of discontinuous Galerkin discretizations of
the Stokes and Navier-Stokes equations. As a distinctive feature -coarsened
mesh sequences are generated by recursive agglomeration of a fine grid,
admitting arbitrarily unstructured grids of complex domains, and agglomeration
based discontinuous Galerkin discretizations are employed to deal with
agglomerated elements of coarse levels. Both the expense of building coarse
grid operators and the performance of the resulting multigrid iteration are
investigated. For the sake of efficiency coarse grid operators are inherited
through element-by-element projections, avoiding the cost of numerical
integration over agglomerated elements. Specific care is devoted to the
projection of viscous terms discretized by means of the BR2 dG method. We
demonstrate that enforcing the correct amount of stabilization on coarse grids
levels is mandatory for achieving uniform convergence with respect to the
number of levels. The numerical solution of steady and unsteady, linear and
non-linear problems is considered tackling challenging 2D test cases and 3D
real life computations on parallel architectures. Significant execution time
gains are documented.Comment: 78 pages, 7 figure
An improved method for solving quasilinear convection diffusion problems on a coarse mesh
A method is developed for solving quasilinear convection diffusion problems
starting on a coarse mesh where the data and solution-dependent coefficients
are unresolved, the problem is unstable and approximation properties do not
hold. The Newton-like iterations of the solver are based on the framework of
regularized pseudo-transient continuation where the proposed time integrator is
a variation on the Newmark strategy, designed to introduce controllable
numerical dissipation and to reduce the fluctuation between the iterates in the
coarse mesh regime where the data is rough and the linearized problems are
badly conditioned and possibly indefinite. An algorithm and updated marking
strategy is presented to produce a stable sequence of iterates as boundary and
internal layers in the data are captured by adaptive mesh partitioning. The
method is suitable for use in an adaptive framework making use of local error
indicators to determine mesh refinement and targeted regularization. Derivation
and q-linear local convergence of the method is established, and numerical
examples demonstrate the theory including the predicted rate of convergence of
the iterations.Comment: 21 pages, 8 figures, 1 tabl
- …