61,454 research outputs found
On the ordering of sparse linear systems
AbstractIn this paper we consider the algorithms for transforming an n × n sparse matrix A into another matrix B such that Gaussian elimination applied to B takes time asymptotically less than n3. These algorithms take the sparse matrix A as input, and return a pair of permutation matrices P, Q such that B = PAQ has a small bandwidth, or some other desirable form. We study the average effectiveness of these algorithms by using random matrices with Θ(n) nonzero elements. We prove that with high probability these algorithms cannot produce a reduction of the asymptotic cost of the standard Gaussian elimination algorithm.We also study the effectiveness of these algorithms for ordering very sparse matrices. We show that there exist matrices with 3n nonzeros for which reordering rows and columns does not reduce the asymptotic cost of Gaussian elimination. We also prove that each matrix with at most two nonzeros in each row and in each column, can be transformed into a banded matrix with bandwidth five
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Preordering saddle-point systems for sparse LDLT factorization without pivoting
This paper focuses on efficiently solving large sparse symmetric indefinite systems of linear equations in saddle‐point form using a fill‐reducing ordering technique with a direct solver. Row and column permutations partition the saddle‐point matrix into a block structure constituting a priori pivots of order 1 and 2. The partitioned matrix is compressed by treating each nonzero block as a single entry, and a fill‐reducing ordering is applied to the corresponding compressed graph. It is shown that, provided the saddle‐point matrix satisfies certain criteria, a block LDLT factorization can be computed using the resulting pivot sequence without modification. Numerical results for a range of problems from practical applications using a modern sparse direct solver are presented to illustrate the effectiveness of the approach
Sparse Gr\"obner Bases: the Unmixed Case
Toric (or sparse) elimination theory is a framework developped during the
last decades to exploit monomial structures in systems of Laurent polynomials.
Roughly speaking, this amounts to computing in a \emph{semigroup algebra},
\emph{i.e.} an algebra generated by a subset of Laurent monomials. In order to
solve symbolically sparse systems, we introduce \emph{sparse Gr\"obner bases},
an analog of classical Gr\"obner bases for semigroup algebras, and we propose
sparse variants of the and FGLM algorithms to compute them. Our prototype
"proof-of-concept" implementation shows large speed-ups (more than 100 for some
examples) compared to optimized (classical) Gr\"obner bases software. Moreover,
in the case where the generating subset of monomials corresponds to the points
with integer coordinates in a normal lattice polytope and under regularity assumptions, we prove complexity bounds which depend
on the combinatorial properties of . These bounds yield new
estimates on the complexity of solving -dim systems where all polynomials
share the same Newton polytope (\emph{unmixed case}). For instance, we
generalize the bound on the maximal degree in a Gr\"obner
basis of a -dim. bilinear system with blocks of variables of sizes
to the multilinear case: . We also propose
a variant of Fr\"oberg's conjecture which allows us to estimate the complexity
of solving overdetermined sparse systems.Comment: 20 pages, Corollary 6.1 has been corrected, ISSAC 2014, Kobe : Japan
(2014
Iterative solutions to the steady state density matrix for optomechanical systems
We present a sparse matrix permutation from graph theory that gives stable
incomplete Lower-Upper (LU) preconditioners necessary for iterative solutions
to the steady state density matrix for quantum optomechanical systems. This
reordering is efficient, adding little overhead to the computation, and results
in a marked reduction in both memory and runtime requirements compared to other
solution methods, with performance gains increasing with system size. Either of
these benchmarks can be tuned via the preconditioner accuracy and solution
tolerance. This reordering optimizes the condition number of the approximate
inverse, and is the only method found to be stable at large Hilbert space
dimensions. This allows for steady state solutions to otherwise intractable
quantum optomechanical systems.Comment: 10 pages, 5 figure
Complexity Analysis and Efficient Measurement Selection Primitives for High-Rate Graph SLAM
Sparsity has been widely recognized as crucial for efficient optimization in
graph-based SLAM. Because the sparsity and structure of the SLAM graph reflect
the set of incorporated measurements, many methods for sparsification have been
proposed in hopes of reducing computation. These methods often focus narrowly
on reducing edge count without regard for structure at a global level. Such
structurally-naive techniques can fail to produce significant computational
savings, even after aggressive pruning. In contrast, simple heuristics such as
measurement decimation and keyframing are known empirically to produce
significant computation reductions. To demonstrate why, we propose a
quantitative metric called elimination complexity (EC) that bridges the
existing analytic gap between graph structure and computation. EC quantifies
the complexity of the primary computational bottleneck: the factorization step
of a Gauss-Newton iteration. Using this metric, we show rigorously that
decimation and keyframing impose favorable global structures and therefore
achieve computation reductions on the order of and , respectively,
where is the pruning rate. We additionally present numerical results
showing EC provides a good approximation of computation in both batch and
incremental (iSAM2) optimization and demonstrate that pruning methods promoting
globally-efficient structure outperform those that do not.Comment: Pre-print accepted to ICRA 201
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