1,030 research outputs found
A fully-discrete scheme for systems of nonlinear Fokker-Planck-Kolmogorov equations
We consider a system of Fokker-Planck-Kolmogorov (FPK) equations, where the
dependence of the coefficients is nonlinear and nonlocal in time with respect
to the unknowns. We extend the numerical scheme proposed and studied recently
by the authors for a single FPK equation of this type. We analyse the
convergence of the scheme and we study its applicability in two examples. The
first one concerns a population model involving two interacting species and the
second one concerns two populations Mean Field Games
Control Strategies for the Fokker-Planck Equation
Using a projection-based decoupling of the Fokker-Planck equation, control
strategies that allow to speed up the convergence to the stationary
distribution are investigated. By means of an operator theoretic framework for
a bilinear control system, two different feedback control laws are proposed.
Projected Riccati and Lyapunov equations are derived and properties of the
associated solutions are given. The well-posedness of the closed loop systems
is shown and local and global stabilization results, respectively, are
obtained. An essential tool in the construction of the controls is the choice
of appropriate control shape functions. Results for a two dimensional double
well potential illustrate the theoretical findings in a numerical setup
Probability tree algorithm for general diffusion processes
Motivated by path-integral numerical solutions of diffusion processes,
PATHINT, we present a new tree algorithm, PATHTREE, which permits extremely
fast accurate computation of probability distributions of a large class of
general nonlinear diffusion processes
Searching for self-similarity in switching time and turbulent cascades in ion transport through a biochannel. A time delay asymmetry
The process of ion transport through a locust potassium channel is described
by means of the Fokker-Planck equation (FPE). The deterministic and stochastic
components of the process of switching between various conducting states of the
channel are expressed by two coefficients, and , a drift and
a diffusion coefficient, respectively. The FPE leads to a Langevin equation.
This analysis reveals beside the well known deterministic aspects a turbulent,
cascade type of action. The (noisy-like) switching between different conducting
states prevents the channel from staying in one, closed or open state. The
similarity between the hydrodynamic flow in the turbulent regime and
hierarchical switching between conducting states of this biochannel is
discussed. A non-trivial character of and coefficients is
shown, which points to different processes governing the channel's action,
asymetrically depending on the history of the previously conducting states.
Moreover, the Fokker-Planck and Langevin equations provide information on
whether and how the statistics of the channel action change over various time
scales.Comment: submitted to physica A text : 12 pages + 8 figure
Estimation of the parameters of a stochastic logistic growth model
We consider a stochastic logistic growth model involving both birth and death
rates in the drift and diffusion coefficients for which extinction eventually
occurs almost surely. The associated complete Fokker-Planck equation describing
the law of the process is established and studied. We then use its solution to
build a likelihood function for the unknown model parameters, when discretely
sampled data is available. The existing estimation methods need adaptation in
order to deal with the extinction problem. We propose such adaptations, based
on the particular form of the Fokker-Planck equation, and we evaluate their
performances with numerical simulations. In the same time, we explore the
identifiability of the parameters which is a crucial problem for the
corresponding deterministic (noise free) model
Nonparametric Uncertainty Quantification for Stochastic Gradient Flows
This paper presents a nonparametric statistical modeling method for
quantifying uncertainty in stochastic gradient systems with isotropic
diffusion. The central idea is to apply the diffusion maps algorithm to a
training data set to produce a stochastic matrix whose generator is a discrete
approximation to the backward Kolmogorov operator of the underlying dynamics.
The eigenvectors of this stochastic matrix, which we will refer to as the
diffusion coordinates, are discrete approximations to the eigenfunctions of the
Kolmogorov operator and form an orthonormal basis for functions defined on the
data set. Using this basis, we consider the projection of three uncertainty
quantification (UQ) problems (prediction, filtering, and response) into the
diffusion coordinates. In these coordinates, the nonlinear prediction and
response problems reduce to solving systems of infinite-dimensional linear
ordinary differential equations. Similarly, the continuous-time nonlinear
filtering problem reduces to solving a system of infinite-dimensional linear
stochastic differential equations. Solving the UQ problems then reduces to
solving the corresponding truncated linear systems in finitely many diffusion
coordinates. By solving these systems we give a model-free algorithm for UQ on
gradient flow systems with isotropic diffusion. We numerically verify these
algorithms on a 1-dimensional linear gradient flow system where the analytic
solutions of the UQ problems are known. We also apply the algorithm to a
chaotically forced nonlinear gradient flow system which is known to be well
approximated as a stochastically forced gradient flow.Comment: Find the associated videos at: http://personal.psu.edu/thb11
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