42,089 research outputs found
Block SOR for Kronecker structured representations
Cataloged from PDF version of article.The Kronecker structure of a hierarchical Markovian model (HMM) induces nested block
partitionings in the transition matrix of its underlying Markov chain. This paper shows how
sparse real Schur factors of certain diagonal blocks of a given partitioning induced by the
Kronecker structure can be constructed from smaller component matrices and their real Schur
factors. Furthermore, it shows how the column approximate minimum degree (COLAMD)
ordering algorithm can be used to reduce fill-in of the remaining diagonal blocks that are
sparse LU factorized. Combining these ideas, the paper proposes three-level block successive
over-relaxation (BSOR) as a competitive steady state solver for HMMs. Finally, on a set of
numerical experiments it demonstrates how these ideas reduce storage required by the factors
of the diagonal blocks and improve solution time compared to an all LU factorization implementation
of the BSOR solver.
© 2004 Elsevier Inc. All rights reserved
High performance interior point methods for three-dimensional finite element limit analysis
The ability to obtain rigorous upper and lower bounds on collapse loads of various structures makes finite element limit analysis an attractive design tool. The increasingly high cost of computing those bounds, however, has limited its application on problems in three dimensions. This work reports on a high-performance homogeneous self-dual primal-dual interior point method developed for three-dimensional finite element limit analysis. This implementation achieves convergence times over 4.5× faster than the leading commercial solver across a set of three-dimensional finite element limit analysis test problems, making investigation of three dimensional limit loads viable. A comparison between a range of iterative linear solvers and direct methods used to determine the search direction is also provided, demonstrating the superiority of direct methods for this application. The components of the interior point solver considered include the elimination of and options for handling remaining free variables, multifrontal and supernodal Cholesky comparison for computing the search direction, differences between approximate minimum degree [1] and nested dissection [13] orderings, dealing with dense columns and fixed variables, and accelerating the linear system solver through parallelization. Each of these areas resulted in an improvement on at least one of the problems in the test set, with many achieving gains across the whole set. The serial implementation achieved runtime performance 1.7× faster than the commercial solver Mosek [5]. Compared with the parallel version of Mosek, the use of parallel BLAS routines in the supernodal solver saw a 1.9× speedup, and with a modified version of the GPU-enabled CHOLMOD [11] and a single NVIDIA Tesla K20c this speedup increased to 4.65×
On the Approximability of Digraph Ordering
Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute
a labeling maximizing the number of forward edges, i.e.
edges (u,v) such that (u) < (v). For different values of k, this
reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work
studies the approximability of Max-k-Ordering and its generalizations,
motivated by their applications to job scheduling with soft precedence
constraints. We give an LP rounding based 2-approximation algorithm for
Max-k-Ordering for any k={2,..., n}, improving on the known
2k/(k-1)-approximation obtained via random assignment. The tightness of this
rounding is shown by proving that for any k={2,..., n} and constant
, Max-k-Ordering has an LP integrality gap of 2 -
for rounds of the
Sherali-Adams hierarchy.
A further generalization of Max-k-Ordering is the restricted maximum acyclic
subgraph problem or RMAS, where each vertex v has a finite set of allowable
labels . We prove an LP rounding based
approximation for it, improving on the
approximation recently given by Grandoni et al.
(Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact,
our approximation algorithm also works for a general version where the
objective counts the edges which go forward by at least a positive offset
specific to each edge.
The minimization formulation of digraph ordering is DAG edge deletion or
DED(k), which requires deleting the minimum number of edges from an n-vertex
directed acyclic graph (DAG) to remove all paths of length k. We show that
both, the LP relaxation and a local ratio approach for DED(k) yield
k-approximation for any .Comment: 21 pages, Conference version to appear in ESA 201
Ranking with Submodular Valuations
We study the problem of ranking with submodular valuations. An instance of
this problem consists of a ground set , and a collection of monotone
submodular set functions , where each .
An additional ingredient of the input is a weight vector . The
objective is to find a linear ordering of the ground set elements that
minimizes the weighted cover time of the functions. The cover time of a
function is the minimal number of elements in the prefix of the linear ordering
that form a set whose corresponding function value is greater than a unit
threshold value.
Our main contribution is an -approximation algorithm
for the problem, where is the smallest non-zero marginal value that
any function may gain from some element. Our algorithm orders the elements
using an adaptive residual updates scheme, which may be of independent
interest. We also prove that the problem is -hard to
approximate, unless P = NP. This implies that the outcome of our algorithm is
optimal up to constant factors.Comment: 16 pages, 3 figure
Matrix-F5 algorithms over finite-precision complete discrete valuation fields
Let be a sequence
of homogeneous polynomials with -adic coefficients. Such system may happen,
for example, in arithmetic geometry. Yet, since is not an
effective field, classical algorithm does not apply.We provide a definition for
an approximate Gr{\"o}bner basis with respect to a monomial order We
design a strategy to compute such a basis, when precision is enough and under
the assumption that the input sequence is regular and the ideals are weakly--ideals. The conjecture of Moreno-Socias
states that for the grevlex ordering, such sequences are generic.Two variants
of that strategy are available, depending on whether one lean more on precision
or time-complexity. For the analysis of these algorithms, we study the loss of
precision of the Gauss row-echelon algorithm, and apply it to an adapted
Matrix-F5 algorithm. Numerical examples are provided.Moreover, the fact that
under such hypotheses, Gr{\"o}bner bases can be computed stably has many
applications. Firstly, the mapping sending to the reduced
Gr{\"o}bner basis of the ideal they span is differentiable, and its
differential can be given explicitly. Secondly, these hypotheses allows to
perform lifting on the Grobner bases, from to
or Finally, asking for the same
hypotheses on the highest-degree homogeneous components of the entry
polynomials allows to extend our strategy to the affine case
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