Solving the chemical master equation using sliding windows

<p>Abstract</p> <p>Background</p> <p>The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.</p> <p>Results</p> <p>In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.</p> <p>Conclusions</p> <p>The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.</p

Proton transport model in the ionosphere .1. Multistream approach of the transport equations

The suprathermal particles, electrons and protons, coming from the magnetosphere and precipitating into the high-latitude atmosphere are an energy source of the Earth's ionosphere. They interact with ambient thermal gas through inelastic and elastic collisions. The physical quantities perturbed by these precipitations, such as the heating rate, the electron production rate, or the emission intensities, can be provided in solving the kinetic stationary Boltzmann equation. This equation yields particle fluxes as a function of altitude, energy, and pitch angle. While this equation has been solved through different ways for the electron transport and fully tested, the proton transport is more complicated. Because of charge-changing reactions, the latter is a set of two-coupled transport equations that must be solved: one for protons and the other for H atoms. We present here a new approach that solves the multistream proton/hydrogen transport equations encompassing the collision angular redistributions and the magnetic mirroring effect. In order to validate our model we discuss the energy conservation and we compare with another model under the same inputs and with rocket observations. The influence of the angular redistributions is discussed in a forthcoming paper

On the use of State Predictors in Networked Control Systems

International audienceWithout pretending to be exhaustive, the aim of this chapter is to give an overview on the use of the state predictor in the context of time-delay systems, and more particularly for the stabilisation of networked control systems. We show that the stabilisation of a system through a deterministic network can be considered as the stabilisation of a time-delayed system with a delay of known dynamics. The predictor approach is proposed, along with some historical background on its application to time-delayed systems, to solve this problem. Some simulation results are also presented

The ReLPM Exponential Integrator for FE Discretizations of Advection-Diffusion Equations

We implement an exponential integrator for large and sparse systems of ODEs, generated by FE (Finite Element) discretization with mass-lumping of advection-diffusion equations. The relevant exponential-like matrix function is approximated by polynomial interpolation, at a sequence of real Leja points related to the spectrum of the FE matrix (ReLPM, Real Leja Points Method). Application to 2D and 3D advection-dispersion models shows speed-ups of one order of magnitude with respect to a classical variable step-size Crank-Nicolson solver

Efficient tomography of a quantum many-body system

Quantum state tomography (QST) is the gold standard technique for obtaining an estimate for the state of small quantum systems in the laboratory [1]. Its application to systems with more than a few constituents (e.g. particles) soon becomes impractical as the e ff ort required grows exponentially with the number of constituents. Developing more e ffi cient techniques is particularly pressing as precisely-controllable quantum systems that are well beyond the reach of QST are emerging in laboratories. Motivated by this, there is a considerable ongoing e ff ort to develop new state characterisation tools for quantum many-body systems [2–11]. Here we demonstrate Matrix Product State (MPS) tomography [2], which is theoretically proven to allow the states of a broad class of quantum systems to be accurately estimated with an e ff ort that increases e ffi ciently with constituent number. We use the technique to reconstruct the dynamical state of a trapped-ion quantum simulator comprising up to 14 entangled and individually-controlled spins (qubits): a size far beyond the practical limits of QST. Our results reveal the dynamical growth of entanglement and description complexity as correlations spread out during a quench: a necessary condition for future beyond-classical performance. MPS tomography should therefore find widespread use to study large quantum many-body systems and to benchmark and verify quantum simulators and computers