A Girsanov particle filter in nonlinear engineering dynamics

In this Letter, we propose a novel variant of the particle filter (PF) for state and parameter estimations of nonlinear engineering dynamical systems, modelled through stochastic differential equations (SDEs). The aim is to address a possible loss of accuracy in the estimates due to the discretization errors, which are inevitable during numerical integration of the SDEs. In particular, we adopt an explicit local linearization of the governing nonlinear SDEs and the resulting linearization errors in the estimates are corrected using Girsanov transformation of measures. Indeed, the linearization scheme via transformation of measures provides a weak framework for computing moments and this fits in well with any stochastic filtering strategy wherein estimates are themselves statistical moments. We presently implement the strategy using a bootstrap PF and numerically illustrate its performance for state and parameter estimations of the Duffing oscillator with linear and nonlinear measurement equations.     

Simulation of Stochastic Differential Equations Through the Local Linearization Method. A Comparative Study

A new local linearization (LL) scheme for the numerical integration of nonautonomous multidimensional stochastic differential equations (SDEs) with additive noise is introduced. The numerical scheme is based on the local linearization of the SDE's drift coefficient by means of a truncated Ito–Taylor expansion. A comparative study with the other LL schemes is presented which shows some advantanges of the new scheme over other ones.     

On a Stochastic Trotter Formula with Application to Spontaneous Localization Models

We consider the relation between so called continuous localization models—i.e. non-linear stochastic Schrödinger evolutions—and the discrete GRW-model of wave function collapse. The former can be understood as scaling limit of the GRW process. The proof relies on a stochastic Trotter formula, which is of interest in its own right. Our Trotter formula also allows to complement results on existence theory of stochastic Schrödinger evolutions by Holevo and Mora/Rebolledo.     

A Homomorphism Theorem and a Trotter Product Formula for Quantum Stochastic Flows with Unbounded Coefficients

We give a new method for proving the homomorphic property of a quantum stochastic flow satisfying a quantum stochastic differential equation with unbounded coefficients, under some further hypotheses. As an application, we prove a Trotter product formula for quantum stochastic flows and obtain quantum stochastic dilations of a class of quantum dynamical semigroups generalizing results of Goswami et al. (Inst H Poincare Probab Stat 41:505–522, 2005).     

Phonon-induced decoherence and dissipation in donor-based charge qubits

We investigate the phonon-induced decoherence and dissipation in a donor-based charge quantum bit realized by the orbital states of an electron shared by two dopant ions which are implanted in a silicon host crystal. The dopant ions are taken from the group-V elements Bi, As, P, Sb. The excess electron is coupled to deformation potential acoustic phonons which dominate in the Si host. The particular geometry tailors a non-monotonous frequency distribution of the phonon modes. We determine the exact qubit dynamics under the influence of the phonons by employing the numerically exact quasi-adiabatic propagator path integral scheme thereby taking into account all bath-induced correlations. In particular, we have improved the scheme by completely eliminating the Trotter discretization error by a Hirsch-Fye extrapolation. By comparing the exact results to those of a Born-Markov approximation we find that the latter yields appropriate estimates for the decoherence and relaxation rates. However, noticeable quantitative corrections due to non-Markovian contributions appear.     

On the motion of classical three-body system with consideration of quantum fluctuations

We obtained the systemof stochastic differential equations which describes the classicalmotion of the three-body system under influence of quantum fluctuations. Using SDEs, for the joint probability distribution of the total momentum of bodies system were obtained the partial differential equation of the second order. It is shown, that the equation for the probability distribution is solved jointly by classical equations, which in turn are responsible for the topological peculiarities of tubes of quantum currents, transitions between asymptotic channels and, respectively for arising of quantum chaos.     

Integration of Langevin equations with multiplicative noise and the viability of field theories for absorbing phase transitions

Efficient and accurate integration of stochastic (partial) differential equations with multiplicative noise can be obtained through a split-step scheme, which separates the integration of the deterministic part from that of the stochastic part, the latter being performed by sampling exactly the solution of the associated Fokker-Planck equation. We demonstrate the computational power of this method by applying it to the most absorbing phase transitions for which Langevin equations have been proposed. This provides precise estimates of the associated scaling exponents, clarifying the classification of these nonequilibrium problems, and confirms or refutes some existing theories.     

Is there a fundamental limitation on the measurement of spatial coherence for highly incoherent fields?

We investigate a fundamental limitation on the measurement of spatial coherence for highly incoherent fields. We model the near-field detection scheme, required for such a measurement, with pointlike induced dipoles. We find that this fully vector model sets a characteristic length scale beyond which the spatial coherence of an optical field cannot be accurately measured. This length scale forms an uncertainty relationship with the photodetector integration time.     

ISSDE: A Monte Carlo implicit simulation code based on Stratonovich SDE approach of Coulomb collision

A Monte Carlo implicit simulation program, Implicit Stratonovich Stochastic Differential Equations(ISSDE), is developed for solving stochastic differential equations(SDEs) that describe plasmas with Coulomb collision. The basic idea of the program is the stochastic equivalence between the Fokker–Planck equation and the Stratonovich SDEs. The splitting method is used to increase the numerical stability of the algorithm for dynamics of charged particles with Coulomb collision. The cases of Lorentzian plasma, Maxwellian plasma and arbitrary distribution function of background plasma have been considered. The adoption of the implicit midpoint method guarantees exactly the energy conservation for the diffusion term and thus improves the numerical stability compared with conventional Runge–Kutta methods. ISSDE is built with C++ and has standard interfaces and extensible modules. The slowing down processes of electron beams in unmagnetized plasma and relaxation process in magnetized plasma are studied using the ISSDE, which shows its correctness and reliability.     

An Analysis of Polynomial Chaos Approximations for Modeling Single-Fluid-Phase Flow in Porous Medium Systems

We examine a variety of polynomial-chaos-motivated approximations to a stochastic form of a steady state groundwater flow model. We consider approaches for truncating the infinite dimensional problem and producing decoupled systems. We discuss conditions under which such decoupling is possible and show that to generalize the known decoupling by numerical cubature, it would be necessary to find new multivariate cubature rules. Finally, we use the acceleration of Monte Carlo to compare the quality of polynomial models obtained for all approaches and find that in general the methods considered are more efficient than Monte Carlo for the relatively small domains considered in this work. A curse of dimensionality in the series expansion of the log-normal stochastic random field used to represent hydraulic conductivity provides a significant impediment to efficient approximations for large domains for all methods considered in this work, other than the Monte Carlo method.     

Editorial – Topical Review

Starting from a family of equations of motion for the dynamics of extended systems whose trajectories sample constant pressure and temperature ensemble distributions (Ferrario, M., 1993, in Computer Simulation in Chemical Physics, edited by M. P. Allen and D. J. Tildesley (Dordrecht: Kluwer)), explicit time reversible integration schemes are derived through a straightforward Trotter factorization of the dynamic Liouville propagator, along the lines first described by Tuckerman, M., Martyna, G. J., and Berne, B. J., 1992, J. chem. Phys., 97, 1990. The original Andersen's constant-pressure dynamics are recovered in the limit of zero coupling with the Nose thermostat. Reversible integration schemes are derived as a generalization of the velocity Verlet algorithm, with direct handling of the velocity dependent forces in such a way that both predictions and relative iterative corrections are not required. For the sake of clarity both the equations of motion and the Trotter factorization are kept to the basic level. The proposed structure can accommodate easily, when needed, complications such as multiple timesteps and more effective thermostats (Nose-Hoover-chain). Finally, an application is made to a model molecular system subjected to holonomic constraints by means of the SHAKE algorithm. In the constant pressure case it is no longer possible to avoid using a prediction for the constraint contribution to the volume acceleration; however, recourse to a minimal iteration scheme still achieves excellent overall behaviour for the proposed integration algorithm, with no perceptible difference from the unconstrained case.     

Weak second-order splitting schemes for Lagrangian Monte Carlo particle methods for the composition PDF/FDF transport equations

We study a class of methods for the numerical solution of the system of stochastic differential equations (SDEs) that arises in the modeling of turbulent combustion, specifically in the Monte Carlo particle method for the solution of the model equations for the composition probability density function (PDF) and the filtered density function (FDF). This system consists of an SDE for particle position and a random differential equation for particle composition. The numerical methods considered advance the solution in time with (weak) second-order accuracy with respect to the time step size. The four primary contributions of the paper are: (i) establishing that the coefficients in the particle equations can be frozen at the mid-time (while preserving second-order accuracy), (ii) examining the performance of three existing schemes for integrating the SDEs, (iii) developing and evaluating different splitting schemes (which treat particle motion, reaction and mixing on different sub-steps), and (iv) developing the method of manufactured solutions (MMS) to assess the convergence of Monte Carlo particle methods. Tests using MMS confirm the second-order accuracy of the schemes. In general, the use of frozen coefficients reduces the numerical errors. Otherwise no significant differences are observed in the performance of the different SDE schemes and splitting schemes.     

Parametric Estimation of Stationary Stochastic Processes Under Indirect Observability

For many natural turbulent dynamic systems, observed high dimensional dynamic data can be approximated at slow time scales by a process X _t driven by a systems of stochastic differential equations (SDEs). When one tries to estimate the parameters of this unobservable SDEs systems, there is a clear mismatch between the available data and the SDEs dynamics to be parametrized. Here, we formalize this Indirect Observability framework as follows.     

Quantum Memory with Natural Inhomogeneous Broadening in an Optical Cavity

We propose an efficient quantum memory scheme with natural inhomogeneous broadening in an asymmetric optical cavity. The scheme uses the strong rephasing pulses like traditional photon echo techniques rather than spectral holeburning, which enables us to have the potential implementation in a much broader range of material systems. In the condition of impedance matching to an optical cavity, we find that the input light pulse can be completely absorbed by an atomic ensemble. We also show that the quantum memory efficiency can be equal to unity even for a small optical depth of the atomic system.     

Variational quantum eigensolvers by variance minimization

The original variational quantum eigensolver (VQE) typically minimizes energy with hybrid quantum-classical optimization that aims to find the ground state. Here, we propose a VQE based on minimizing energy variance and call it the variance-VQE, which treats the ground state and excited states on the same footing, since an arbitrary eigenstate for a Hamiltonian should have zero energy variance. We demonstrate the properties of the variance-VQE for solving a set of excited states in quantum chemistry problems. Remarkably, we show that optimization of a combination of energy and variance may be more efficient to find low-energy excited states than those of minimizing energy or variance alone. We further reveal that the optimization can be boosted with stochastic gradient descent by Hamiltonian sampling, which uses only a few terms of the Hamiltonian and thus significantly reduces the quantum resource for evaluating variance and its gradients.     

The random replicator model at nonzero temperature

The random replicator model with interspecies coupling strengths prescribed by the competitive exclusion principle – the Hebb rule – is studied analytically in the presence of fast noise that describes the flow of migrants between the ecosystem and the outer world. The stochastic dynamics leads to stationary states distributed according to the Gibbs distribution permitting thus an equilibrium statistical mechanics analysis. We find that a discontinuous phase transition separates a regime of strong competition, and consequently of low diversity, from more cooperative regimes. The statistical analysis is carried out for the annealed scheme, for which the evolutionary and ecological timescales coincide, as well as for the quenched scheme, for which the features that identify the species are fixed.     

求解对流-扩散-反应问题的改进有限积分法

将求解偏微分方程的有限积分法应用于对流-扩散-反应问题,发现对于非对流占优的对流扩散问题,有限积分法的精度比QUICK法高一个数量级,比传统的有限体积法高两个数量级.处理对流占优的对流-扩散-反应问题时,对流项的离散时引进加权参数,通过调节该参数反映输运的方向性.结果表明这种改进的有限积分法的精度比传统的有限体积法至少高四个数量级,同时明显改进了原来的有限积分法的精度和稳定性.对于对流占优的对流-扩散-反应问题,即使采用粗网格,计算结果也未出现非物理振荡现象,表明改进的有限积分法具有很好的稳定性.     

A Mesoscopic Simulation Approach for Modeling Intracellular Reactions

Reactions in the intracellular medium occur in a highly organized and heterogenous environment rendering invalid modeling approaches based on the law of mass action or its stochastic counter-part. This has led to the recent development of a variety of stochastic microscopic approaches based on lattice-gas automata or Brownian dynamics. The main disadvantage of these methods is that they are computationally intensive. We propose a mesoscopic method which permits the efficient simulation of reactions occurring in the complex geometries typical of intracellular environments. This approach is used to model the transport of a substrate through a pore in a semi-permeable membrane, in which its Michaelis–Menten enzyme is embedded. We find that the temporal evolution of the substrate is a sensitive function of the spatial heterogeneity of the environment. The spatial organization and heterogeneities of the intracellular medium seem to be playing an important role in the regulation of biochemical reactions.     

Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models

In this paper, we describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.     

Linear theory for control of nonlinear stochastic systems

We address the role of noise and the issue of efficient computation in stochastic optimal control problems. We consider a class of nonlinear control problems that can be formulated as a path integral and where the noise plays the role of temperature. The path integral displays symmetry breaking and there exists a critical noise value that separates regimes where optimal control yields qualitatively different solutions. The path integral can be computed efficiently by Monte Carlo integration or by a Laplace approximation, and can therefore be used to solve high dimensional stochastic control problems.