667 research outputs found
Multimodal Network Alignment
A multimodal network encodes relationships between the same set of nodes in
multiple settings, and network alignment is a powerful tool for transferring
information and insight between a pair of networks. We propose a method for
multimodal network alignment that computes a matrix which indicates the
alignment, but produces the result as a low-rank factorization directly. We
then propose new methods to compute approximate maximum weight matchings of
low-rank matrices to produce an alignment. We evaluate our approach by applying
it on synthetic networks and use it to de-anonymize a multimodal transportation
network.Comment: 14 pages, 6 figures, Siam Data Mining 201
Factorizing the Stochastic Galerkin System
Recent work has explored solver strategies for the linear system of equations
arising from a spectral Galerkin approximation of the solution of PDEs with
parameterized (or stochastic) inputs. We consider the related problem of a
matrix equation whose matrix and right hand side depend on a set of parameters
(e.g. a PDE with stochastic inputs semidiscretized in space) and examine the
linear system arising from a similar Galerkin approximation of the solution. We
derive a useful factorization of this system of equations, which yields bounds
on the eigenvalues, clues to preconditioning, and a flexible implementation
method for a wide array of problems. We complement this analysis with (i) a
numerical study of preconditioners on a standard elliptic PDE test problem and
(ii) a fluids application using existing CFD codes; the MATLAB codes used in
the numerical studies are available online.Comment: 13 pages, 4 figures, 2 table
Direct QR factorizations for tall-and-skinny matrices in MapReduce architectures
The QR factorization and the SVD are two fundamental matrix decompositions
with applications throughout scientific computing and data analysis. For
matrices with many more rows than columns, so-called "tall-and-skinny
matrices," there is a numerically stable, efficient, communication-avoiding
algorithm for computing the QR factorization. It has been used in traditional
high performance computing and grid computing environments. For MapReduce
environments, existing methods to compute the QR decomposition use a
numerically unstable approach that relies on indirectly computing the Q factor.
In the best case, these methods require only two passes over the data. In this
paper, we describe how to compute a stable tall-and-skinny QR factorization on
a MapReduce architecture in only slightly more than 2 passes over the data. We
can compute the SVD with only a small change and no difference in performance.
We present a performance comparison between our new direct TSQR method, a
standard unstable implementation for MapReduce (Cholesky QR), and the classic
stable algorithm implemented for MapReduce (Householder QR). We find that our
new stable method has a large performance advantage over the Householder QR
method. This holds both in a theoretical performance model as well as in an
actual implementation
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