397 research outputs found
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
Recursively accelerated multilevel aggregation for markov chains
Abstract. A recursive acceleration method is proposed for multiplicative multilevel aggregation algorithms that calculate the stationary probability vector of large, sparse, and irreducible Markov chains. Pairs of consecutive iterates at all branches and levels of a multigrid W cycle with simple, nonoverlapping aggregation are recombined to produce improved iterates at those levels. This is achieved by solving quadratic programming problems with inequality constraints: the linear combination of the two iterates is sought that has a minimal two-norm residual, under the constraint that all vector components are nonnegative. It is shown how the two-dimensional quadratic programming problems can be solved explicitly in an efficient way. The method is further enhanced by windowed top-level acceleration of the W cycles using the same constrained quadratic programming approach. Recursive acceleration is an attractive alternative to smoothing the restriction and interpolation operators, since the operator complexity is better controlled and the probabilistic interpretation of coarse-level operators is maintained on all levels. Numerical results are presented showing that the resulting recursively accelerated multilevel aggregation cycles for Markov chains, combined with top-level acceleration, converge significantly faster than W cycles and lead to close-to-linear computational complexity for challenging test problems
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
A new level-dependent coarsegrid correction scheme for indefinite Helmholtz problems
In this paper we construct and analyse a level-dependent coarsegrid
correction scheme for indefinite Helmholtz problems. This adapted multigrid
method is capable of solving the Helmholtz equation on the finest grid using a
series of multigrid cycles with a grid-dependent complex shift, leading to a
stable correction scheme on all levels. It is rigourously shown that the
adaptation of the complex shift throughout the multigrid cycle maintains the
functionality of the two-grid correction scheme, as no smooth modes are
amplified in or added to the error. In addition, a sufficiently smoothing
relaxation scheme should be applied to ensure damping of the oscillatory error
components. Numerical experiments on various benchmark problems show the method
to be competitive with or even outperform the current state-of-the-art
multigrid-preconditioned Krylov methods, like e.g. CSL-preconditioned GMRES or
BiCGStab.Comment: 21 page
Shifted Laplacian multigrid for the elastic Helmholtz equation
The shifted Laplacian multigrid method is a well known approach for
preconditioning the indefinite linear system arising from the discretization of
the acoustic Helmholtz equation. This equation is used to model wave
propagation in the frequency domain. However, in some cases the acoustic
equation is not sufficient for modeling the physics of the wave propagation,
and one has to consider the elastic Helmholtz equation. Such a case arises in
geophysical seismic imaging applications, where the earth's subsurface is the
elastic medium. The elastic Helmholtz equation is much harder to solve than its
acoustic counterpart, partially because it is three times larger, and partially
because it models more complicated physics. Despite this, there are very few
solvers available for the elastic equation compared to the array of solvers
that are available for the acoustic one. In this work we extend the shifted
Laplacian approach to the elastic Helmholtz equation, by combining the complex
shift idea with approaches for linear elasticity. We demonstrate the efficiency
and properties of our solver using numerical experiments for problems with
heterogeneous media in two and three dimensions
Adaptive Aggregation Based Domain Decomposition Multigrid for the Lattice Wilson Dirac Operator
In lattice QCD computations a substantial amount of work is spent in solving
discretized versions of the Dirac equation. Conventional Krylov solvers show
critical slowing down for large system sizes and physically interesting
parameter regions. We present a domain decomposition adaptive algebraic
multigrid method used as a precondtioner to solve the "clover improved" Wilson
discretization of the Dirac equation. This approach combines and improves two
approaches, namely domain decomposition and adaptive algebraic multigrid, that
have been used seperately in lattice QCD before. We show in extensive numerical
test conducted with a parallel production code implementation that considerable
speed-up over conventional Krylov subspace methods, domain decomposition
methods and other hierarchical approaches for realistic system sizes can be
achieved.Comment: Additional comparison to method of arXiv:1011.2775 and to
mixed-precision odd-even preconditioned BiCGStab. Results of numerical
experiments changed slightly due to more systematic use of odd-even
preconditionin
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