2,492 research outputs found
Correlation, hierarchies, and networks in financial markets
We discuss some methods to quantitatively investigate the properties of
correlation matrices. Correlation matrices play an important role in portfolio
optimization and in several other quantitative descriptions of asset price
dynamics in financial markets. Specifically, we discuss how to define and
obtain hierarchical trees, correlation based trees and networks from a
correlation matrix. The hierarchical clustering and other procedures performed
on the correlation matrix to detect statistically reliable aspects of the
correlation matrix are seen as filtering procedures of the correlation matrix.
We also discuss a method to associate a hierarchically nested factor model to a
hierarchical tree obtained from a correlation matrix. The information retained
in filtering procedures and its stability with respect to statistical
fluctuations is quantified by using the Kullback-Leibler distance.Comment: 37 pages, 9 figures, 3 table
A Direct Elliptic Solver Based on Hierarchically Low-rank Schur Complements
A parallel fast direct solver for rank-compressible block tridiagonal linear
systems is presented. Algorithmic synergies between Cyclic Reduction and
Hierarchical matrix arithmetic operations result in a solver with arithmetic complexity and memory footprint. We provide a
baseline for performance and applicability by comparing with well known
implementations of the -LU factorization and algebraic multigrid
with a parallel implementation that leverages the concurrency features of the
method. Numerical experiments reveal that this method is comparable with other
fast direct solvers based on Hierarchical Matrices such as -LU and
that it can tackle problems where algebraic multigrid fails to converge
Butterfly Factorization
The paper introduces the butterfly factorization as a data-sparse
approximation for the matrices that satisfy a complementary low-rank property.
The factorization can be constructed efficiently if either fast algorithms for
applying the matrix and its adjoint are available or the entries of the matrix
can be sampled individually. For an matrix, the resulting
factorization is a product of sparse matrices, each with
non-zero entries. Hence, it can be applied rapidly in operations.
Numerical results are provided to demonstrate the effectiveness of the
butterfly factorization and its construction algorithms
A direct solver with O(N) complexity for variable coefficient elliptic PDEs discretized via a high-order composite spectral collocation method
A numerical method for solving elliptic PDEs with variable coefficients on
two-dimensional domains is presented. The method is based on high-order
composite spectral approximations and is designed for problems with smooth
solutions. The resulting system of linear equations is solved using a direct
(as opposed to iterative) solver that has optimal O(N) complexity for all
stages of the computation when applied to problems with non-oscillatory
solutions such as the Laplace and the Stokes equations. Numerical examples
demonstrate that the scheme is capable of computing solutions with relative
accuracy of or better, even for challenging problems such as highly
oscillatory Helmholtz problems and convection-dominated convection diffusion
equations. In terms of speed, it is demonstrated that a problem with a
non-oscillatory solution that was discretized using nodes was solved
in 115 minutes on a personal work-station with two quad-core 3.3GHz CPUs. Since
the solver is direct, and the "solution operator" fits in RAM, any solves
beyond the first are very fast. In the example with unknowns, solves
require only 30 seconds.Comment: arXiv admin note: text overlap with arXiv:1302.599
An efficient multi-core implementation of a novel HSS-structured multifrontal solver using randomized sampling
We present a sparse linear system solver that is based on a multifrontal
variant of Gaussian elimination, and exploits low-rank approximation of the
resulting dense frontal matrices. We use hierarchically semiseparable (HSS)
matrices, which have low-rank off-diagonal blocks, to approximate the frontal
matrices. For HSS matrix construction, a randomized sampling algorithm is used
together with interpolative decompositions. The combination of the randomized
compression with a fast ULV HSS factorization leads to a solver with lower
computational complexity than the standard multifrontal method for many
applications, resulting in speedups up to 7 fold for problems in our test
suite. The implementation targets many-core systems by using task parallelism
with dynamic runtime scheduling. Numerical experiments show performance
improvements over state-of-the-art sparse direct solvers. The implementation
achieves high performance and good scalability on a range of modern shared
memory parallel systems, including the Intel Xeon Phi (MIC). The code is part
of a software package called STRUMPACK -- STRUctured Matrices PACKage, which
also has a distributed memory component for dense rank-structured matrices
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