51 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
Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing
Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM
Multilevel Combinatorial Optimization Across Quantum Architectures
Emerging quantum processors provide an opportunity to explore new approaches
for solving traditional problems in the post Moore's law supercomputing era.
However, the limited number of qubits makes it infeasible to tackle massive
real-world datasets directly in the near future, leading to new challenges in
utilizing these quantum processors for practical purposes. Hybrid
quantum-classical algorithms that leverage both quantum and classical types of
devices are considered as one of the main strategies to apply quantum computing
to large-scale problems. In this paper, we advocate the use of multilevel
frameworks for combinatorial optimization as a promising general paradigm for
designing hybrid quantum-classical algorithms. In order to demonstrate this
approach, we apply this method to two well-known combinatorial optimization
problems, namely, the Graph Partitioning Problem, and the Community Detection
Problem. We develop hybrid multilevel solvers with quantum local search on
D-Wave's quantum annealer and IBM's gate-model based quantum processor. We
carry out experiments on graphs that are orders of magnitudes larger than the
current quantum hardware size, and we observe results comparable to
state-of-the-art solvers in terms of quality of the solution
On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation
In many applications, it is desired to obtain extreme eigenvalues and
eigenvectors of large Hermitian matrices by efficient and compact algorithms.
In particular, orthogonalization-free methods are preferred for large-scale
problems for finding eigenspaces of extreme eigenvalues without explicitly
computing orthogonal vectors in each iteration. For the top eigenvalues,
the simplest orthogonalization-free method is to find the best rank-
approximation to a positive semi-definite Hermitian matrix by algorithms
solving the unconstrained Burer-Monteiro formulation. We show that the
nonlinear conjugate gradient method for the unconstrained Burer-Monteiro
formulation is equivalent to a Riemannian conjugate gradient method on a
quotient manifold with the Bures-Wasserstein metric, thus its global
convergence to a stationary point can be proven. Numerical tests suggest that
it is efficient for computing the largest eigenvalues for large-scale
matrices if the largest eigenvalues are nearly distributed uniformly
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