420 research outputs found
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
A literature survey of low-rank tensor approximation techniques
During the last years, low-rank tensor approximation has been established as
a new tool in scientific computing to address large-scale linear and
multilinear algebra problems, which would be intractable by classical
techniques. This survey attempts to give a literature overview of current
developments in this area, with an emphasis on function-related tensors
Nearest-Neighbor Interaction Systems in the Tensor-Train Format
Low-rank tensor approximation approaches have become an important tool in the
scientific computing community. The aim is to enable the simulation and
analysis of high-dimensional problems which cannot be solved using conventional
methods anymore due to the so-called curse of dimensionality. This requires
techniques to handle linear operators defined on extremely large state spaces
and to solve the resulting systems of linear equations or eigenvalue problems.
In this paper, we present a systematic tensor-train decomposition for
nearest-neighbor interaction systems which is applicable to a host of different
problems. With the aid of this decomposition, it is possible to reduce the
memory consumption as well as the computational costs significantly.
Furthermore, it can be shown that in some cases the rank of the tensor
decomposition does not depend on the network size. The format is thus feasible
even for high-dimensional systems. We will illustrate the results with several
guiding examples such as the Ising model, a system of coupled oscillators, and
a CO oxidation model
Tensor Networks for Big Data Analytics and Large-Scale Optimization Problems
In this paper we review basic and emerging models and associated algorithms
for large-scale tensor networks, especially Tensor Train (TT) decompositions
using novel mathematical and graphical representations. We discus the concept
of tensorization (i.e., creating very high-order tensors from lower-order
original data) and super compression of data achieved via quantized tensor
train (QTT) networks. The purpose of a tensorization and quantization is to
achieve, via low-rank tensor approximations "super" compression, and
meaningful, compact representation of structured data. The main objective of
this paper is to show how tensor networks can be used to solve a wide class of
big data optimization problems (that are far from tractable by classical
numerical methods) by applying tensorization and performing all operations
using relatively small size matrices and tensors and applying iteratively
optimized and approximative tensor contractions.
Keywords: Tensor networks, tensor train (TT) decompositions, matrix product
states (MPS), matrix product operators (MPO), basic tensor operations,
tensorization, distributed representation od data optimization problems for
very large-scale problems: generalized eigenvalue decomposition (GEVD),
PCA/SVD, canonical correlation analysis (CCA).Comment: arXiv admin note: text overlap with arXiv:1403.204
Tensor product approach to modelling epidemics on networks
To improve mathematical models of epidemics it is essential to move beyond
the traditional assumption of homogeneous well--mixed population and involve
more precise information on the network of contacts and transport links by
which a stochastic process of the epidemics spreads. In general, the number of
states of the network grows exponentially with its size, and a master equation
description suffers from the curse of dimensionality. Almost all methods widely
used in practice are versions of the stochastic simulation algorithm (SSA),
which is notoriously known for its slow convergence. In this paper we
numerically solve the chemical master equation for an SIR model on a general
network using recently proposed tensor product algorithms. In numerical
experiments we show that tensor product algorithms converge much faster than
SSA and deliver more accurate results, which becomes particularly important for
uncovering the probabilities of rare events, e.g. for number of infected people
to exceed a (high) threshold
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