15,758 research outputs found
Gene regulatory networks: a coarse-grained, equation-free approach to multiscale computation
We present computer-assisted methods for analyzing stochastic models of gene
regulatory networks. The main idea that underlies this equation-free analysis
is the design and execution of appropriately-initialized short bursts of
stochastic simulations; the results of these are processed to estimate
coarse-grained quantities of interest, such as mesoscopic transport
coefficients. In particular, using a simple model of a genetic toggle switch,
we illustrate the computation of an effective free energy and of a
state-dependent effective diffusion coefficient that characterize an
unavailable effective Fokker-Planck equation. Additionally we illustrate the
linking of equation-free techniques with continuation methods for performing a
form of stochastic "bifurcation analysis"; estimation of mean switching times
in the case of a bistable switch is also implemented in this equation-free
context. The accuracy of our methods is tested by direct comparison with
long-time stochastic simulations. This type of equation-free analysis appears
to be a promising approach to computing features of the long-time,
coarse-grained behavior of certain classes of complex stochastic models of gene
regulatory networks, circumventing the need for long Monte Carlo simulations.Comment: 33 pages, submitted to The Journal of Chemical Physic
Solving the Boltzmann equation in N log N
In [C. Mouhot and L. Pareschi, "Fast algorithms for computing the Boltzmann
collision operator," Math. Comp., to appear; C. Mouhot and L. Pareschi, C. R.
Math. Acad. Sci. Paris, 339 (2004), pp. 71-76], fast deterministic algorithms
based on spectral methods were derived for the Boltzmann collision operator for
a class of interactions including the hard spheres model in dimension three.
These algorithms are implemented for the solution of the Boltzmann equation in
two and three dimension, first for homogeneous solutions, then for general non
homogeneous solutions. The results are compared to explicit solutions, when
available, and to Monte-Carlo methods. In particular, the computational cost
and accuracy are compared to those of Monte-Carlo methods as well as to those
of previous spectral methods. Finally, for inhomogeneous solutions, we take
advantage of the great computational efficiency of the method to show an
oscillation phenomenon of the entropy functional in the trend to equilibrium,
which was suggested in the work [L. Desvillettes and C. Villani, Invent. Math.,
159 (2005), pp. 245-316].Comment: 32 page
Statistical Mechanics of the Quantum K-Satisfiability problem
We study the quantum version of the random -Satisfiability problem in the
presence of the external magnetic field applied in the transverse
direction. We derive the replica-symmetric free energy functional within static
approximation and the saddle-point equation for the order parameter: the
distribution of functions of magnetizations. The order parameter is
interpreted as the histogram of probability distributions of individual
magnetizations. In the limit of zero temperature and small transverse fields,
to leading order in magnetizations become relevant in
addition to purely classical values of . Self-consistency
equations for the order parameter are solved numerically using Quasi Monte
Carlo method for K=3. It is shown that for an arbitrarily small
quantum fluctuations destroy the phase transition present in the classical
limit , replacing it with a smooth crossover transition. The
implications of this result with respect to the expected performance of quantum
optimization algorithms via adiabatic evolution are discussed. The
replica-symmetric solution of the classical random -Satisfiability problem
is briefly revisited. It is shown that the phase transition at T=0 predicted by
the replica-symmetric theory is of continuous type with atypical critical
exponents.Comment: 35 pages, 23 figures; changed abstract, improved discussion in the
introduction, added references, corrected typo
Boundary conditions in local electrostatics algorithms
We study the simulation of charged systems in the presence of general
boundary conditions in a local Monte Carlo algorithm based on a constrained
electric field. We firstly show how to implement constant-potential, Dirichlet,
boundary conditions by introducing extra Monte Carlo moves to the algorithm.
Secondly, we show the interest of the algorithm for studying systems which
require anisotropic electrostatic boundary conditions for simulating planar
geometries such as membranes.Comment: 8 pages, 6 figures, accepted in JC
Polynomial Chaos Expansion of random coefficients and the solution of stochastic partial differential equations in the Tensor Train format
We apply the Tensor Train (TT) decomposition to construct the tensor product
Polynomial Chaos Expansion (PCE) of a random field, to solve the stochastic
elliptic diffusion PDE with the stochastic Galerkin discretization, and to
compute some quantities of interest (mean, variance, exceedance probabilities).
We assume that the random diffusion coefficient is given as a smooth
transformation of a Gaussian random field. In this case, the PCE is delivered
by a complicated formula, which lacks an analytic TT representation. To
construct its TT approximation numerically, we develop the new block TT cross
algorithm, a method that computes the whole TT decomposition from a few
evaluations of the PCE formula. The new method is conceptually similar to the
adaptive cross approximation in the TT format, but is more efficient when
several tensors must be stored in the same TT representation, which is the case
for the PCE. Besides, we demonstrate how to assemble the stochastic Galerkin
matrix and to compute the solution of the elliptic equation and its
post-processing, staying in the TT format.
We compare our technique with the traditional sparse polynomial chaos and the
Monte Carlo approaches. In the tensor product polynomial chaos, the polynomial
degree is bounded for each random variable independently. This provides higher
accuracy than the sparse polynomial set or the Monte Carlo method, but the
cardinality of the tensor product set grows exponentially with the number of
random variables. However, when the PCE coefficients are implicitly
approximated in the TT format, the computations with the full tensor product
polynomial set become possible. In the numerical experiments, we confirm that
the new methodology is competitive in a wide range of parameters, especially
where high accuracy and high polynomial degrees are required.Comment: This is a major revision of the manuscript arXiv:1406.2816 with
significantly extended numerical experiments. Some unused material is remove
Recent developments in Quantum Monte-Carlo simulations with applications for cold gases
This is a review of recent developments in Monte Carlo methods in the field
of ultra cold gases. For bosonic atoms in an optical lattice we discuss path
integral Monte Carlo simulations with worm updates and show the excellent
agreement with cold atom experiments. We also review recent progress in
simulating bosonic systems with long-range interactions, disordered bosons,
mixtures of bosons, and spinful bosonic systems. For repulsive fermionic
systems determinantal methods at half filling are sign free, but in general no
sign-free method exists. We review the developments in diagrammatic Monte Carlo
for the Fermi polaron problem and the Hubbard model, and show the connection
with dynamical mean-field theory. We end the review with diffusion Monte Carlo
for the Stoner problem in cold gases.Comment: 68 pages, 22 figures, review article; replaced with published versio
06391 Abstracts Collection -- Algorithms and Complexity for Continuous Problems
From 24.09.06 to 29.09.06, the Dagstuhl Seminar 06391 ``Algorithms and Complexity for Continuous Problems\u27\u27 was held
in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar
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
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