1,720,985 research outputs found
Analysis of upwind method for piecewise deterministic Markov processes
No full textIn this paper we investigate the behaviour of the numerical solution of the Liouville---Master Equation (LME) for time dependent distributions, that arises by the statistical characterisation of a class of piecewise deterministic stochastic processes (PDP). This is a system of linear hyperbolic PDEs with non-constant coefficients. The numerical solution is found by means of upwind and forward Euler scheme. We find a Courant---Friedrichs---Lewy condition ensuring both convergence and monotonicity of the numerical solution. In particular, the global error is shown to be bounded by a linear increasing in the integration time, under an appropriate norm. Some numerical tests for known analytical solutions of practical problems verify the theoretical findings
A finite difference method for piecewise deterministic processes with memory II
Full text linkWe deal with the numerical scheme for the Liouville Master Equation (LME) of a kind
of Piecewise Deterministic Processes (PDP) with memory, analysed in [2]. The LME is
a linear system of hyperbolic PDEs, written in non?conservative form, with non-local
boundary conditions. The solutions of that equation are time dependent marginal
distribution functions whose sum satisfies the total probability conservation law.
In [2] the convergence of the numerical scheme, based on the Courant-Isaacson-Rees
jointly with a direct quadrature, has been proved under a Courant-Friedrichs-Lewy
like (CFL) condition. Here we show that the numerical solution is monotonic under
a similar CFL condition. Moreover, we evaluate the conservativity of the total
probability for the calculated solution. Finally, an implementation of a parallel
algorithm by using the MPI library is described and the results of some performance
tests are presented
A Finite Difference Method for Piecewise Deterministic Processes with Memory
No full textIn this paper the numerical approximation of solutions of Liouvill-Master Equation
for time-dependent distribution functions of Piecewise Deterministic Processes with
memory is considered. These equations are linear hyperbolic PDEs with non-constant
coefficients. and boundary conditions that depend on integrals over the interior of
the integration domain. We construct a finite difference method of the first order,
by a combination of the upwind method, for PDEs, and by a direct quadrature, for the
boundary condition. We analyse convergence of the numerical solution for distribution
functions evolving towards an equilibrium. Numerical results for two problems,
whose analytical solutions are known in closed form, illustrate the theoretical finding
Non-gaussian equilibrium distribution arising from the Langevin equation
No full textWe study the Langevin equation of a point particle driven by random noise, modeled as a two-state Markov process. The corresponding master equation differs from the Fokker-Planck equation. In equilibrium, the velocity of the particle is distributed according to a binomial power law. We discuss transient (i.e., nonequilibrium) behavior, and the consequences of non-Markovian noise statistics
On the Action of a Semi-Markov Process on a System of Differential Equations
Full text linkWe deal with a model equation for stochastic processes that results from the action of a semi-Markov process on a system of ordinary differential equations. The resulting stochastic process is deterministic in pieces, with random changes of the motion at random time epochs. By using classical methods of probability calculus, we first build and discuss the fundamental equation for the statistical analysis, i.e. a Liouville Master Equation for the distribution functions, that is a system of hyperbolic PDE with non-local boundary conditions. Then, as the main contribute to this paper, by using the characteristics’ method we recast it to a system of Volterra integral equations with space fluxes, and prove existence and uniqueness of the solution. A numerical experiment for a case of practical application is performed
Calibration of Lévy processes using optimal control of Kolmogorov equations with periodic boundary conditions
Full text linkWe present an optimal control approach to the problem of model calibration for Lévy processes based on an non-parametric estimation procedure of the measure. The optimization problem is related to the maximum likelihood theory of sieves [25] and is formulated with the Fokker-Planck-Kolmogorov approach [3, 4].
We use a generic spline discretization of the Lévy jump measure and select an adequate size of the spline basis using the Akaike Information Criterion (AIC) [12]. The first order necessary optimality conditions are derived based on the Lagrange multiplier technique in a functional space. The resulting Partial Integral-Differential Equations (PIDE) are discretized, numerically solved using a scheme composed of Chang-Cooper, BDF2 and direct quadrature methods, jointly to a non-linear conjugate gradient method. For the numerical solver of the Kolmogorov's forward equation we prove conditions for non-negativity and stability in the L1 norm of the discrete solution
Stochastic versus dynamic approach to Lévy statistics in the presence of an external perturbation
No full textWe study the influence of a dissipation process on diffusion dynamics triggered by slow fluctuations. We study both strong- and weak-friction regimes. When the latter regime applies, the system is attracted by the basin of either Gauss or Lévy statistics according to whether the fluctuation correlation function is integrable or not. We analyze with a numerical calculation the border between the two basins of attraction
Fast solvers of Fredholm optimal control problems
No full textThe formulation of optimal control problems governed by Fredholm integral equations
of second kind and an efficient computational framework for solving these control
problems is presented. Existence and uniqueness of optimal solutions is proved.
A collective Gauss-Seidel scheme and a multigrid scheme are discussed. Optimal
computational performance of these iterative schemes is proved by local Fourier
analysis and demonstrated by results of numerical experiments
Fokker-Planck-based control of a two level open quantum system
No full textThe control of a two-level open quantum system subject to dissipation due to environment
interaction is considered. The evolution of this system is governed by a
Lindblad master equation which is augmented by a stochastic term to model the
effect of time-continuous measurements. In order to control this stochastic
master equation model, a Fokker-Planck control framework is investigated.
Within this strategy, the control objectives are defined based on the probability density
functions of the two-level stochastic process and the controls are computed as
minimizers of these objectives subject to the constraints represented by the
Fokker-Planck equation. This minimization problem is characterized by an
optimality system including the Fokker-Planck equation and its adjoint. This
optimality system is approximated by a second-order accurate, stable,
conservative, and positive preserving discretization scheme. The implementation
of the resulting open-loop controls is realized with a receding-horizon algorithm
over a sequence of time windows. Results of numerical experiments demonstrate
the effectiveness of the proposed approach
Optimal control of a class of piecewise deterministic processes
No full textA new control strategy for a class of piecewise deterministic processes (PDP) is presented. In this class, PDP stochastic processes consist of ordinary differential equations that are subject to random switches corresponding to a discrete Markov process. The proposed strategy aims at controlling the probability density function (PDF) of the PDP. The optimal control formulation is based on the hyperbolic Fokker–Planck system that governs the time evolution of the PDF of the PDP and on tracking objectives of terminal configuration with a target PDF. The corresponding optimization problems are formulated as a sequence of open-loop hyperbolic optimality systems following a model predictive control framework. These systems are discretized by first-order schemes that guarantee positivity and conservativeness of the numerical PDF solution. The effectiveness of the proposed computational control framework is validated considering PDP with dichotomic noise
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