Modeling Biological Systems in Stochastic Concurrent Constraint Programming

We present an application of stochastic Concurrent Constraint Programming (sCCP) for modeling biological systems. We provide a library of sCCP processes that can be used to describe straightforwardly biological networks. In the meanwhile, we show that sCCP proves to be a general and extensible framework, allowing to describe a wide class of dynamical behaviours and kinetic laws.     

Stochastic Concurrent Constraint Programming and Differential Equations

We tackle the problem of relating models of systems (mainly biological systems) based on stochastic process algebras (SPA) with models based on differential equations. We define a syntactic procedure that translates programs written in stochastic Concurrent Constraint Programming (sCCP) into a set of Ordinary Differential Equations (ODE), and also the inverse procedure translating ODE's into sCCP programs. For the class of biochemical reactions, we show that the translation is correct w.r.t. the intended rate semantics of the models. Finally, we show that the translation does not generally preserve the dynamical behavior, giving a list of open research problems in this direction.     

The maximum principle for stochastic differential systems with general cost functional

In this paper, under the framework of Fréchet derivatives, we study a stochastic optimal control problem driven by a stochastic differential equation with general cost functional. By constructing a series of first-order and second-order adjoint equations, we establish the stochastic maximum principle and get the related Hamilton systems.     

Manoeuvring target tracking with coordinated-turn motion using stochastic non-linear filter

Allowing for perturbations in speed and turn rate of a target moving in a coordinated turn obeys a non-linear stochastic differential equation. Existing algorithms for coordinated turn tracking avoid this problem by ignoring perturbations in the continuous time model and adding process noise only after discretisation. The dynamic model used here adds small perturbations, modelled as independent Brownian motion processes, to the speed and turn rate. The target state is to be recursively estimated from noisy discrete-time measurements of the target's range and bearing. In particular, this paper examines the effect of the perturbations in speed and turn rate on the coordinated turn motion of the aircraft, and subsequently the stochastic algorithm is developed by deriving the evolutions of conditional means and variances for estimating the state of the aircraft. By linearizing the stochastic differential equations about the mean of the state vector using first-order approximation, the mean trajectory of the resulting first-order approximated stochastic differential model does not preserve the perturbation effect felt by the moving target; only the variance trajectory includes the perturbation effect. For this reason, the effectiveness of the perturbed model is examined on the basis of the second-order approximations of the system non-linearity. The theory of the non-linear filter of this paper is developed using the Kolmogorov forward equation ‘between the observation’ and a functional difference equation for the conditional probability density ‘at the observation’. The effectiveness of the second-order non-linear filter is examined on the basis of its ability to preserve perturbation effect felt by the aircraft. The Kolmogorov forward equation, however, is not appropriate for numerical simulations, since it is the equation for the evolution of the conditional probability density. Instead of the Kolmogorov equation, one derives the evolutions for the moments of the state vector, which in our case consists of positions, velocities and turn rate of the manoeuvring aircraft. Even these equations are not appropriate for the numerical simulations, since they are not closed in the sense that computing the evolution of a given moment involves the knowledge of higher-order moments. Hence we consider the approximations to these moment evolution equations. Simulation results are introduced to demonstrate the usefulness of an analytic theory developed in this paper.     

Comparison of the approximate solutions of a noisy non-linear system based on Monte Carlo simulation

This paper is concerned with the comparison of the approximate methods in order to evaluate the values of state variables of a noisy nonlinear system by Monte Carlo simulation. The approximate methods considered here are (1) the stochastic linearisation, (2) the first-order approximation and (3) the second-order approximation. It is shown that the order of the accuracies of the covariance based on the approximate equation of Riccati type does not always coincide with the order of the practical error covariance of the original system which is obtained by using Monte Carlo simulation. Therefore the comparison of the accuracies should not be carried out by the solution obtained from the covariance equation but by the practical error covariance of the original system directly. This result is demonstrated with numerical examples. The characteristics of the random numbers which are used for Monte Carlo simulation are examined by various kinds of tests.     

Semi-discrete approximations for stochastic differential equations and applications

In this paper, we propose a new point of view in numerical approximation of stochastic differential equations. By using Ito–Taylor expansions, we expand only a part of the stochastic differential equation. Thus, in each step, we have again a stochastic differential equation which we solve explicitly or by using another method or a finer mesh. We call our approach as a semi-discrete approximation. We give two applications of this approach. Using the semi-discrete approach, we can produce numerical schemes which preserves monotonicity so in our first application, we prove that the semi-discrete Euler scheme converge in the mean square sense even when the drift coefficient is only continuous, using monotonicity arguments. In our second application, we study the square root process which appears in financial mathematics. We observe that a semi-discrete scheme behaves well producing non-negative values.     

Approximate solution of the fractional advection-dispersion equation

In this paper, we consider practical numerical method to solve a space-time fractional advection-dispersion equation with variable coefficients on a finite domain. The equation is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative, and the first-order and second-order space derivatives by the Riemann-Liouville fractional derivative, respectively. Here, a new method for solving this equation is proposed in the reproducing kernel space. The representation of solution is given by the form of series and the n-term approximation solution is obtained by truncating the series. The method is easy to implement and the numerical results show the accuracy of the method.     

A phase-space formulation and Gaussian approximation of the filtering equations for nonlinear quantum stochastic systems

This paper is concerned with a filtering problem for a class of nonlinear quantum stochastic systems with multichannel nondemolition measurements. The system-observation dynamics are governed by a Markovian Hudson-Parthasarathy quantum stochastic differential equation driven by quantum Wiener processes of bosonic fields in vacuum state. The Hamiltonian and system-field coupling operators, as functions of the system variables, are assumed to be represented in a Weyl quantization form. Using the Wigner-Moyal phase-space framework, we obtain a stochastic integro-differential equation for the posterior quasi-characteristic function (QCF) of the system conditioned on the measurements. This equation is a spatial Fourier domain representation of the Belavkin-Kushner-Stratonovich stochastic master equation driven by the innovation process associated with the measurements. We discuss a specific form of the posterior QCF dynamics in the case of linear system-field coupling and outline a Gaussian approximation of the posterior quantum state.     

Gibbs Field Approaches in Image Processing Problems

In this paper, we address the problem of image denoising using a stochastic differential equation approach. Proposed stochastic dynamics schemes are based on the property of diffusion dynamics to converge to a distribution on global minima of the energy function of the model, under a special cooling schedule (the annealing procedure). To derive algorithms for computer simulations, we consider discrete-time approximations of the stochastic differential equation. We study convergence of the corresponding Markov chains to the diffusion process. We give conditions for the ergodicity of the Euler approximation scheme. In the conclusion, we compare results of computer simulations using the diffusion dynamics algorithms and the standard Metropolis–Hasting algorithm. Results are shown on synthetic and real data.