6,351 research outputs found
A Computational Framework for the Mixing Times in the QBD Processes with Infinitely-Many Levels
In this paper, we develop some matrix Poisson's equations satisfied by the
mean and variance of the mixing time in an irreducible positive-recurrent
discrete-time Markov chain with infinitely-many levels, and provide a
computational framework for the solution to the matrix Poisson's equations by
means of the UL-type of -factorization as well as the generalized inverses.
In an important special case: the level-dependent QBD processes, we provide a
detailed computation for the mean and variance of the mixing time. Based on
this, we give new highlight on computation of the mixing time in the
block-structured Markov chains with infinitely-many levels through the
matrix-analytic method
A Web Aggregation Approach for Distributed Randomized PageRank Algorithms
The PageRank algorithm employed at Google assigns a measure of importance to
each web page for rankings in search results. In our recent papers, we have
proposed a distributed randomized approach for this algorithm, where web pages
are treated as agents computing their own PageRank by communicating with linked
pages. This paper builds upon this approach to reduce the computation and
communication loads for the algorithms. In particular, we develop a method to
systematically aggregate the web pages into groups by exploiting the sparsity
inherent in the web. For each group, an aggregated PageRank value is computed,
which can then be distributed among the group members. We provide a distributed
update scheme for the aggregated PageRank along with an analysis on its
convergence properties. The method is especially motivated by results on
singular perturbation techniques for large-scale Markov chains and multi-agent
consensus.Comment: To appear in the IEEE Transactions on Automatic Control, 201
A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs
A self-learning algebraic multigrid method for dominant and minimal singular
triplets and eigenpairs is described. The method consists of two multilevel
phases. In the first, multiplicative phase (setup phase), tentative singular
triplets are calculated along with a multigrid hierarchy of interpolation
operators that approximately fit the tentative singular vectors in a collective
and self-learning manner, using multiplicative update formulas. In the second,
additive phase (solve phase), the tentative singular triplets are improved up
to the desired accuracy by using an additive correction scheme with fixed
interpolation operators, combined with a Ritz update. A suitable generalization
of the singular value decomposition is formulated that applies to the coarse
levels of the multilevel cycles. The proposed algorithm combines and extends
two existing multigrid approaches for symmetric positive definite eigenvalue
problems to the case of dominant and minimal singular triplets. Numerical tests
on model problems from different areas show that the algorithm converges to
high accuracy in a modest number of iterations, and is flexible enough to deal
with a variety of problems due to its self-learning properties.Comment: 29 page
Sensitivity of Markov chains for wireless protocols
Network communication protocols such as the IEEE 802.11 wireless protocol are currently best modelled as Markov chains. In these situations we have some protocol parameters , and a transition matrix from which we can compute the steady state (equilibrium) distribution and hence final desired quantities , which might be for example the throughput of the protocol. Typically the chain will have thousands of states, and a particular example of interest is the Bianchi chain defined later. Generally we want to optimise , perhaps subject to some constraints that also depend on the Markov chain. To do this efficiently we need the gradient of with respect to , and therefore need the gradient of and other properties of the chain with respect to . The matrix formulas available for this involve the so-called fundamental matrix, but are there approximate gradients available which are faster and still sufficiently accurate? In some cases BT would like to do the whole calculation in computer algebra, and get a series expansion of the equilibrium with respect to a parameter in . In addition to the steady state , the same questions arise for the mixing time and the mean hitting times. Two qualitative features that were brought to the Study Group’s attention were:
* the transition matrix is large, but sparse.
* the systems of linear equations to be solved are generally singular and need some additional normalisation condition, such as is provided by using the fundamental matrix.
We also note a third highly important property regarding applications of numerical linear algebra:
* the transition matrix is asymmetric.
A realistic dimension for the matrix in the Bianchi model described below is 8064×8064, but on average there are only a few nonzero entries per column. Merely storing such a large matrix in dense form would require nearly 0.5GBytes using 64-bit floating point numbers, and computing its LU factorisation takes around 80 seconds on a modern microprocessor. It is thus highly desirable to employ specialised algorithms for sparse matrices. These algorithms are generally divided between those only applicable to symmetric matrices, the most prominent being the conjugate-gradient (CG) algorithm for solving linear equations, and those applicable to general matrices. A similar division is present in the literature on numerical eigenvalue problems
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