122 research outputs found
LU factorization with panel rank revealing pivoting and its communication avoiding version
We present the LU decomposition with panel rank revealing pivoting (LU_PRRP),
an LU factorization algorithm based on strong rank revealing QR panel
factorization. LU_PRRP is more stable than Gaussian elimination with partial
pivoting (GEPP). Our extensive numerical experiments show that the new
factorization scheme is as numerically stable as GEPP in practice, but it is
more resistant to pathological cases and easily solves the Wilkinson matrix and
the Foster matrix. We also present CALU_PRRP, a communication avoiding version
of LU_PRRP that minimizes communication. CALU_PRRP is based on tournament
pivoting, with the selection of the pivots at each step of the tournament being
performed via strong rank revealing QR factorization. CALU_PRRP is more stable
than CALU, the communication avoiding version of GEPP. CALU_PRRP is also more
stable in practice and is resistant to pathological cases on which GEPP and
CALU fail.Comment: No. RR-7867 (2012
Hybrid Models for Mixed Variables in Bayesian Optimization
This paper presents a new type of hybrid models for Bayesian optimization
(BO) adept at managing mixed variables, encompassing both quantitative
(continuous and integer) and qualitative (categorical) types. Our proposed new
hybrid models merge Monte Carlo Tree Search structure (MCTS) for categorical
variables with Gaussian Processes (GP) for continuous ones. Addressing
efficiency in searching phase, we juxtapose the original (frequentist) upper
confidence bound tree search (UCTS) and the Bayesian Dirichlet search
strategies, showcasing the tree architecture's integration into Bayesian
optimization. Central to our innovation in surrogate modeling phase is online
kernel selection for mixed-variable BO. Our innovations, including dynamic
kernel selection, unique UCTS (hybridM) and Bayesian update strategies
(hybridD), position our hybrid models as an advancement in mixed-variable
surrogate models. Numerical experiments underscore the hybrid models'
superiority, highlighting their potential in Bayesian optimization.Comment: 32 pages, 8 Figure
MEMS-based, phase-shifting interferometer
Provided herein are optical devices fabricated to include a reflective surface, actuators and stress-relieving structures. Systems containing such devices, and methods of manufacturing such devices, are also provided
Dynamic Computation of Network Statistics via Updating Schema
In this paper we derive an updating scheme for calculating some important
network statistics such as degree, clustering coefficient, etc., aiming at
reduce the amount of computation needed to track the evolving behavior of large
networks; and more importantly, to provide efficient methods for potential use
of modeling the evolution of networks. Using the updating scheme, the network
statistics can be computed and updated easily and much faster than
re-calculating each time for large evolving networks. The update formula can
also be used to determine which edge/node will lead to the extremal change of
network statistics, providing a way of predicting or designing evolution rule
of networks.Comment: 17 pages, 6 figure
Approximation de rang faible pour les matrices creuses
In this paper we present an algorithm for computing a low rank approximation of a sparse matrix based on a truncated LU factorization with column and row permutations. We present various approaches for determining the column and row permutations that show a trade-off between speed versus deterministic/probabilistic accuracy. We show that if the permutations are chosen by using tournament pivoting based on QR factorization, then the obtained truncated LU factorization with column/row tournament pivoting, LU\_CRTP, satisfies bounds on the singular values which have similarities with the ones obtained by a communication avoiding rank revealing QR factorization. Experiments on challenging matrices show that LU_CRTP provides a good low rank approximation of the input matrix and it is less expensive than the rank revealing QR factorization in terms of computational and memory usage costs, while also minimizing the communication cost. We also compare the computational complexity of our algorithm with randomizedalgorithms and show that for sparse matrices and high enough but still modest accuracies, our approach is faster.Ce papier introduit un algorithme pour calculer une approximation de rang faible d’une matrice creuse. Cet algorithme est basé sur une factorisation LU avec des permutations de lignes et de colonnes
Write-Avoiding Algorithms
Short version of the technical report available at http://www.eecs.berkeley.edu/Pubs/TechRpts/2015/EECS-2015-163.pdf as Technical Report No. UCB/EECS-2015-163International audienc
Matrix Factorization at Scale: a Comparison of Scientific Data Analytics in Spark and C+MPI Using Three Case Studies
We explore the trade-offs of performing linear algebra using Apache Spark,
compared to traditional C and MPI implementations on HPC platforms. Spark is
designed for data analytics on cluster computing platforms with access to local
disks and is optimized for data-parallel tasks. We examine three widely-used
and important matrix factorizations: NMF (for physical plausability), PCA (for
its ubiquity) and CX (for data interpretability). We apply these methods to
TB-sized problems in particle physics, climate modeling and bioimaging. The
data matrices are tall-and-skinny which enable the algorithms to map
conveniently into Spark's data-parallel model. We perform scaling experiments
on up to 1600 Cray XC40 nodes, describe the sources of slowdowns, and provide
tuning guidance to obtain high performance
The inherent inaccuracy of implicit tridiagonal QR
Recently Demmel and Veselic showed that Jacobi's method has a tighter relative error bound for the computed eigenvalues of a symmetric positive de nite matrix than does QR iteration. Here we show the weaker error bound of QR as implemented in LAPACK's SSTEQR or EISPACK's IMTQL is unavoidable. We do this by presenting a particular symmetric positive de nite tridiagonal matrix for whichQRmust fail, given any reasonable shift strategy
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