758 research outputs found
07071 Abstracts Collection -- Web Information Retrieval and Linear Algebra Algorithms
From 12th to 16th February 2007, the Dagstuhl Seminar 07071 ``Web Information Retrieval and Linear Algebra Algorithms\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
An Extension of Two Conjugate Direction Methods to Markov Chain Problems
Motivated by the recent applications of the conjugate residual method to nonsymmetric linear systems by Sogabe, Sugihara and Zhang [An extension of the conjugate residual method to nonsymmetric linear systems. J. Comput. Appl. Math., Vol. 266, 2009, pp. 103--113], this paper describes two conjugate direction methods, BiCR and BiCG, and attempts to extend their applications to compute the stationary probability distribution for an irreducible Markov chain with the aim of finding an alternative basic solver. Numerical experiments show the feasibility of the BiCR and BiCG to some extent, with applications to several practical Markov chain problems
Kronecker Graphs: An Approach to Modeling Networks
How can we model networks with a mathematically tractable model that allows
for rigorous analysis of network properties? Networks exhibit a long list of
surprising properties: heavy tails for the degree distribution; small
diameters; and densification and shrinking diameters over time. Most present
network models either fail to match several of the above properties, are
complicated to analyze mathematically, or both. In this paper we propose a
generative model for networks that is both mathematically tractable and can
generate networks that have the above mentioned properties. Our main idea is to
use the Kronecker product to generate graphs that we refer to as "Kronecker
graphs".
First, we prove that Kronecker graphs naturally obey common network
properties. We also provide empirical evidence showing that Kronecker graphs
can effectively model the structure of real networks.
We then present KronFit, a fast and scalable algorithm for fitting the
Kronecker graph generation model to large real networks. A naive approach to
fitting would take super- exponential time. In contrast, KronFit takes linear
time, by exploiting the structure of Kronecker matrix multiplication and by
using statistical simulation techniques.
Experiments on large real and synthetic networks show that KronFit finds
accurate parameters that indeed very well mimic the properties of target
networks. Once fitted, the model parameters can be used to gain insights about
the network structure, and the resulting synthetic graphs can be used for null-
models, anonymization, extrapolations, and graph summarization
Algebraic Multigrid for Markov Chains and Tensor Decomposition
The majority of this thesis is concerned with the development of efficient and robust numerical methods based on adaptive algebraic multigrid to compute the stationary distribution of Markov chains. It is shown that classical algebraic multigrid techniques can be applied in an exact interpolation scheme framework to compute the stationary distribution of irreducible, homogeneous Markov chains. A quantitative analysis shows that algebraically smooth multiplicative error is locally constant along strong connections in a scaled system operator, which suggests that classical algebraic multigrid coarsening and interpolation can be applied to the class of nonsymmetric irreducible singular M-matrices with zero column sums. Acceleration schemes based on fine-level iterant recombination, and over-correction of the coarse-grid correction are developed to improve the rate of convergence and scalability of simple adaptive aggregation multigrid methods for Markov chains. Numerical tests over a wide range of challenging nonsymmetric test problems demonstrate the effectiveness of the proposed multilevel method and the acceleration schemes.
This thesis also investigates the application of adaptive algebraic multigrid techniques for computing the canonical decomposition of higher-order tensors. The canonical decomposition is formulated as a least squares optimization problem, for which local minimizers are computed by solving the first-order optimality equations. The proposed multilevel method consists of two phases: an adaptive setup phase that uses a multiplicative correction scheme in conjunction with bootstrap algebraic multigrid interpolation to build the necessary operators on each level, and a solve phase that uses additive correction cycles based on the full approximation scheme to efficiently obtain an accurate solution. The alternating least squares method, which is a standard one-level iterative method for computing the canonical decomposition, is used as the relaxation scheme. Numerical tests show that for certain test problems arising from the discretization of high-dimensional partial differential equations on regular lattices the proposed multilevel method significantly outperforms the standard alternating least squares method when a high level of accuracy is required
A finite state projection algorithm for the stationary solution of the chemical master equation
The chemical master equation (CME) is frequently used in systems biology to
quantify the effects of stochastic fluctuations that arise due to biomolecular
species with low copy numbers. The CME is a system of ordinary differential
equations that describes the evolution of probability density for each
population vector in the state-space of the stochastic reaction dynamics. For
many examples of interest, this state-space is infinite, making it difficult to
obtain exact solutions of the CME. To deal with this problem, the Finite State
Projection (FSP) algorithm was developed by Munsky and Khammash (Jour. Chem.
Phys. 2006), to provide approximate solutions to the CME by truncating the
state-space. The FSP works well for finite time-periods but it cannot be used
for estimating the stationary solutions of CMEs, which are often of interest in
systems biology. The aim of this paper is to develop a version of FSP which we
refer to as the stationary FSP (sFSP) that allows one to obtain accurate
approximations of the stationary solutions of a CME by solving a finite
linear-algebraic system that yields the stationary distribution of a
continuous-time Markov chain over the truncated state-space. We derive bounds
for the approximation error incurred by sFSP and we establish that under
certain stability conditions, these errors can be made arbitrarily small by
appropriately expanding the truncated state-space. We provide several examples
to illustrate our sFSP method and demonstrate its efficiency in estimating the
stationary distributions. In particular, we show that using a quantised tensor
train (QTT) implementation of our sFSP method, problems admitting more than 100
million states can be efficiently solved.Comment: 8 figure
A finite state projection algorithm for the stationary solution of the chemical master equation
The chemical master equation (CME) is frequently used in systems biology to
quantify the effects of stochastic fluctuations that arise due to biomolecular
species with low copy numbers. The CME is a system of ordinary differential
equations that describes the evolution of probability density for each
population vector in the state-space of the stochastic reaction dynamics. For
many examples of interest, this state-space is infinite, making it difficult to
obtain exact solutions of the CME. To deal with this problem, the Finite State
Projection (FSP) algorithm was developed by Munsky and Khammash (Jour. Chem.
Phys. 2006), to provide approximate solutions to the CME by truncating the
state-space. The FSP works well for finite time-periods but it cannot be used
for estimating the stationary solutions of CMEs, which are often of interest in
systems biology. The aim of this paper is to develop a version of FSP which we
refer to as the stationary FSP (sFSP) that allows one to obtain accurate
approximations of the stationary solutions of a CME by solving a finite
linear-algebraic system that yields the stationary distribution of a
continuous-time Markov chain over the truncated state-space. We derive bounds
for the approximation error incurred by sFSP and we establish that under
certain stability conditions, these errors can be made arbitrarily small by
appropriately expanding the truncated state-space. We provide several examples
to illustrate our sFSP method and demonstrate its efficiency in estimating the
stationary distributions. In particular, we show that using a quantised tensor
train (QTT) implementation of our sFSP method, problems admitting more than 100
million states can be efficiently solved.Comment: 8 figure
Clustering in Block Markov Chains
This paper considers cluster detection in Block Markov Chains (BMCs). These
Markov chains are characterized by a block structure in their transition
matrix. More precisely, the possible states are divided into a finite
number of groups or clusters, such that states in the same cluster exhibit
the same transition rates to other states. One observes a trajectory of the
Markov chain, and the objective is to recover, from this observation only, the
(initially unknown) clusters. In this paper we devise a clustering procedure
that accurately, efficiently, and provably detects the clusters. We first
derive a fundamental information-theoretical lower bound on the detection error
rate satisfied under any clustering algorithm. This bound identifies the
parameters of the BMC, and trajectory lengths, for which it is possible to
accurately detect the clusters. We next develop two clustering algorithms that
can together accurately recover the cluster structure from the shortest
possible trajectories, whenever the parameters allow detection. These
algorithms thus reach the fundamental detectability limit, and are optimal in
that sense.Comment: 73 pages, 18 plots, second revisio
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