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An equivalence result for a general class of models when some variables are missing at random is established. The initial model and the missingness mechanism could be equivalently defined under the form of a system of moment equations. The equivalence means the same sets of probability measures for the observed variables. This type of equivalence greatly simplifies the analysis of the initial model under the missingness mechanism.

Significance Many biochemical processes in living systems take place in compartmentalized environments, where individual compartments can interact with each other and undergo dynamic remodeling. Studying such processes through mathematical models poses formidable challenges because the underlying dynamics involve a large number of states, which evolve stochastically with time. Here we propose a mathematical framework to study stochastic biochemical networks in compartmentalized environments. We develop a generic population model, which tracks individual compartments and their molecular composition. We then show how the time evolution of this system can be studied effectively through a small number of differential equations, which track the statistics of the population. Our approach is versatile and renders an important class of biological systems computationally accessible., Compartmentalization of biochemical processes underlies all biological systems, from the organelle to the tissue scale. Theoretical models to study the interplay between noisy reaction dynamics and compartmentalization are sparse, and typically very challenging to analyze computationally. Recent studies have made progress toward addressing this problem in the context of specific biological systems, but a general and sufficiently effective approach remains lacking. In this work, we propose a mathematical framework based on counting processes that allows us to study dynamic compartment populations with arbitrary interactions and internal biochemistry. We derive an efficient description of the dynamics in terms of differential equations which capture the statistics of the population. We demonstrate the relevance of our approach by analyzing models inspired by different biological processes, including subcellular compartmentalization and tissue homeostasis.

The Fletcher-Ponnambalam (FP) reservoir optimization method has been successfully applied to several case studies in the literature. As an explicit stochastic optimization problem, it requi...

The Ensemble Kalman Filter (EnKF) belongs to the class of iterative particle filtering methods and can be used for solving control--to--observable inverse problems. In this context, the EnKF is known as Ensemble Kalman Inversion (EKI). In recent years several continuous limits in the number of iteration and particles have been performed in order to study properties of the method. In particular, a one--dimensional linear stability analysis reveals possible drawbacks in the phase space of moments provided by the continuous limits of the EKI, but observed also in the multi--dimensional setting. In this work we address this issue by introducing a stabilization of the dynamics which leads to a method with globally asymptotically stable solutions. We illustrate the performance of the stabilized version by using test inverse problems from the literature and comparing it with the classical continuous limit formulation of the method.

The moment method is a successful way to approximate the solution of the Boltzmann equation in reasonable runtime with relatively few unknowns while allowing for non-equilibrium effects of the gas. However, the convergence of the moment method with respect to an increasing number of moments is typically slow. This paper aims to improve the convergence of the moment method by introducing filtered hyperbolic moment equations that result in virtually no additional computational overhead while significantly reducing the error. The filter approach is based on a careful study of averaging solutions of two adjacent moment systems and the reformulation of the averaging using an artificial collision method that naturally gives rise to the filter. We study the properties of the filter and show numerical test cases of one-dimensional problems that demonstrate the superior quality of the new filtered moment method.

We propose a systematic methodology to derive the regularized thirteen-moment equations in the rarefied gas dynamics for a general class of linearized collision models. Detailed expressions of the moment equations are written down for all inverse power law models as well as the hard-sphere model. By linear analysis, we show that the equations are stable near the equilibrium. The models are tested for shock structure problems to show its capability to capture the correct flow structure in strong nonequilibirum.

With Shape Aware Quadratures (SAQ), for a given set of quadrature nodes, order, and domain of integration, the quadrature weights are obtained by solving a system of suitable moment fitting equations in least square sense. The moments in the moment equations are approximated over a simplified domain ( Ω 0 ) that is a reasonable approximation of the original domain (Ω) that are then corrected for the deviation of the shape of Ω 0 from Ω via shape correction factors. This idea was already successfully utilized in the Adaptively Weighted Numerical Integration (AW) method [1] , [2] , [3] , where moments were computed over simplified but homeomorphic domain and then corrected using first-order shape sensitivity, allowing efficient and accurate integration of integrable functions over arbitrary domains. With SAQ, more general shape correction factors can be derived based on a variety of sensitivity analysis techniques such as shape sensitivity (SSA) and topological sensitivity (TSA). Using the right kind/order of shape correction factors for moment approximation enables the resulting quadrature (from the moment fitting equations) to efficiently adapt to the shape of the original domain even in the presence of thousands of small features. We derive shape correction factors based on SSA and TSA of moment integrals. We also demonstrate the efficacy of SAQ in integrating bivariate/trivariate polynomials over several 2D/3D domains in the presence of small features.

We present a new method for calculating the time-transient behavior of stochastic reaction networks. We first derive the set of equations for the moments of the master probability distribution. We then linearize these equations calculating the Jacobian matrix around the stationary probability distribution. In order to demonstrate the method, we present examples of stochastic reaction networks and compute their dynamic behavior. We find that the calculations are accurate and significantly more efficient than stochastic simulation algorithms based on Gillespie’s algorithms.

The paper investigates the navigation problem of following a moving target, using a mathematical model described by a system of differential equations with random parameters. The differential equations, which employ controls for following the target, are solved by a new approach using moment equations. Simulations are presented to test effectiveness of the approach.

Hyperbolic moment equations based on Burnett’s expansion of the distribution function are derived for the Boltzmann equation with linearized collision operator. Boundary conditions are equipped for these models, and it is proven that the number of boundary conditions is correct for a large class of moment models. A new second-order numerical scheme is proposed for solving these moment equations, and the new method is suitable for both ordered- and full-moment theories. Numerical experiments are carried out for both one- and two-dimensional problems to show the performance of the moment methods.

In this article, we consider the estimation problem of continuous-time stochastic systems with discrete measurements, having linear drift and nonlinear diffusion term. We build the infinite-dimensional linear system equivalent to this class of systems by means of a Carleman linearization approach. Based on this embedding, we investigate the properties of the moment equations of the original system, and we show that it is possible to write the optimal linear filter, for which a finite-dimensional approximation can be implemented. We validate the approach by showing that the resulting algorithm may outperform widely used continuous-discrete filters without increasing the computational burden.

We introduce Sim.DiffProc, an R package for symbolic and numerical computations on scalar and multivariate systems of stochastic differential equations (SDEs). It provides users with a wide range of tools to simulate, estimate, analyze, and visualize the dynamics of these systems in both forms, Itô and Stratonovich. One of Sim.DiffProc key features is to implement the Monte Carlo method for the iterative evaluation and approximation of an interesting quantity at a fixed time on SDEs with parallel computing, on multiple processors on a single machine or a cluster of computers, which is an important tool to improve capacity and speed-up calculations. We also provide an easy-to-use interface for symbolic calculation and numerical approximation of the first and central second-order moments of SDEs (i.e., mean, variance and covariance), by solving a system of ordinary differential equations, which yields insights into the dynamics of stochastic systems. The final result object of Monte Carlo and moment equations can be derived and presented in terms of LATEX math expressions and visualized in terms of LATEX tables. Furthermore, we illustrate various features of the package by proposing a general bivariate nonlinear dynamic system of Haken-Zwanzig, driven by additive, linear and nonlinear multiplicative noises. In addition, we consider the particular case of a scalar SDE driven by three independent Wiener processes. The Monte Carlo simulation thereof is obtained through a transformation to a system of three equations. We also study some important applications of SDEs in different fields.

We describe an analog stochastic switch that exhibits three distinct phases as its parameters change. The phases are classified by the mean and variance of the switch’s output. A phase change appears if the mean or the variance tends to a finite value or to infinity. The switch can be embedded in a large gene regulatory network for which the moment equations naturally close at the second order. This switch was used to model the response of a heat-shock system.We describe an analog stochastic switch that exhibits three distinct phases as its parameters change. The phases are classified by the mean and variance of the switch’s output. A phase change appears if the mean or the variance tends to a finite value or to infinity. The switch can be embedded in a large gene regulatory network for which the moment equations naturally close at the second order. This switch was used to model the response of a heat-shock system.

We develop a moment-equation-copula-closure method for the inexpensive approximation of the steady state statistical structure of strongly nonlinear systems which are subjected to correlated excitations. Our approach relies on the derivation of moment equations that describe the dynamics governing the two-time statistics. These are combined with a non-Gaussian pdf representation for the joint response-excitation statistics, based on copula functions that has (i) single time statistical structure consistent with the analytical solutions of the Fokker–Planck equation, and (ii) two-time statistical structure with Gaussian characteristics. Through the adopted pdf representation, we derive a closure scheme which we formulate in terms of a consistency condition involving the second order statistics of the response, the closure constraint. A similar condition, the dynamics constraint, is also derived directly through the moment equations. These two constraints are formulated as a low-dimensional minimization problem with respect to the unknown parameters of the representation, the minimization of which imposes an interplay between the dynamics and the adopted closure. The new method allows for the semi-analytical representation of the two-time, non-Gaussian structure of the solution as well as the joint statistical structure of the response-excitation over different time instants. We demonstrate its effectiveness through the application on bistable nonlinear single-degree-of-freedom energy harvesters with mechanical and electromagnetic damping, and we show that the results compare favorably with direct Monte-Carlo simulations., Samsung Scholarship Foundation, MIT Energy Initiative

The moment method is not only a modeling tool that gives macroscopic fluid equations by reducing kinetic equations, but also a numerical method for solving kinetic equations. It has been the subject of rapid development, and in recent years has acquired widespread applications. In this paper, we review and summarize the research development of moment methods in kinetic theory from the aspects of modeling, numerical methods, and applications. First, we discuss the deficiencies of moment methods and summarize the remedy, where in particular the regularized moment method and globally hyperbolic moment method are introduced, owing to the wide interest regarding them. Subsequently, we investigate various numerical methods for solving moment equations, and highlight the numerical regularized method for moment equations of arbitrary orders. In addition, this paper reviews the applications of moment methods in the fields of rarefied gases, microflows, electron transport, plasma, and density functionals and presents an outlook regarding the future development of moment methods.

The shock structures of a 13 moment generalized hydrodynamics system of rarefied gases are simulated. These are first order hyperbolic equations derived from the Boltzmann equation. The investigated moment system stands out due to having an entropy evolution. In addition, a particular interest arises from the fact that the equations not only contain nonconservative products, but also provide the key to solving this mathematical and numerical issue by means of a simple substitution utilizing the physical entropy evolution. The apparent success of this method warrants investigation and provides a new perspective and starting point for finding general approaches to nonconservative products and irreversible processes. Furthermore, the system shows physically accurate results for low Mach numbers and is able to reveal the nonequilibrium entropy profile across a shock wave., Journal of Non-Equilibrium Thermodynamics, 44 (1), ISSN:0340-0204, ISSN:1437-4358

The moment equations are derived for a system of linear difference equations with Markov switching. One example illustrates the use of such systems for modeling the security of threats in cyberspace.

The hydrodynamics of nematic liquid crystals in two dimensions are considered, and the existence of global strong solutions of the system is obtained in the article. Some nonlinear stress terms in the moment equations and transport terms in the director equations, together with the Dirichlet boundary condition, induce some difficulties in the analysis. We introduce an elliptic operator for such coupled system to overcome the problems.

Hyperbolicity is a necessary property of model equations for the solution of the BGK equation to achieve stable and physical solutions. However, the standard approach for velocity space discretization developed by Grad only yields locally hyperbolic equations. The method has recently been improved, and several new globally hyperbolic model systems have been derived such as the Hyperbolic Moment Equations (HME) and the Quadrature-Based Moment Equations (QBME). We will describe the derivation and properties of a new model system called Simplified Hyperbolic Moment Equations (SHME) which inherits hyperbolicity from the other models but is simpler to implement and to solve. First simulation results show a good accuracy of the new SHME model in comparison with the existing models.

The paper proposes an approximate closure procedure for hierarchies of macroscopic equations for rarefied gases, derived as moment equations from the Boltzmann equation in kinetic theory of gases. The procedure is based upon application of the maximum entropy principle. If the exact minimizer is exploited, moments of the distribution function may diverge, unless the restriction on the structure of the moments is introduced. In this paper, a perturbative approach is proposed by restricting the set of admissible functions in the variational problem. This leads to an approximate minimizer, but the procedure can be applied to an arbitrary choice of the moments.

We study the mixed formulation of the stochastic Hodge-Laplace problem dened on a $n$-dimensional domain $D (n ≥ 1)$, with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three dimensional case. We derive and analyze the moment equations, that is the deterministic equations solved by the $m-th$ moment $(m ≥ 1$) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order of convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element spaces.

International audience; This article addresses issues from applied stochastic analysis for solving parameter identification problems in interaction networks. Purely discontinuous Markov processes are the most appropriate choice for modelling these systems and it is of paramount importance to estimate the unknown characteristics of the model given the measured data. The model induces a Fokker–Planck–Kolmogorov equation along with moment equations, and achieving parametric identification based on direct solutions of these equations has remained elusive. We propose a novel approach which utilizes stochastic analysis of continuous-time Markov processes to lift the curse of dimensionality in parametric identification. We illustrate through a case study from ecological engineering.

The generalized method of moments (GMM) is a conceptually simple and flexible estimation method that has come to play an increasingly prominent role in empirical research in economics over the last 30 years. Application of GMM requires the availability of so-called moment equations or moment conditions. There should be at least as many moment equations as there are parameters to be estimated. If this condition is satisfied (plus some regularity conditions), application of GMM is in principle straightforward and delivers estimators for the parameters that are consistent and asymptotically normal. If desired, the estimators can in addition be made asymptotically efficient given the available moment equations, that is, have the lowest achievable variance or highest precision asymptotically.

We use an example to present in exhaustive detail the algorithmic steps of the zero-information (ZI) closure scheme (Smadbeck and Kaznessis, Proc Natl Acad Sci USA 110:14261–14265, 2013). ZI-closure is a method for solving the chemical master equation (CME) of stochastic chemical reaction networks.

In this chapter the methods used to generate own cubature rules are developed. The algorithm presented in the first chapter is based on a least squares approach to the multidimensional moment equations and is used to generate cubature rules with approximate exactness m = 5 and m = 7 which possess optimal stability. In the second section, three new kinds of cubature rules will be presented which are based on the Smolyak algorithm. The developed procedure allows the computation of Smolyak cubature rules of high stability.

The fundamental aspects of the Taylor-series expansion method of moment (TEMOM) model proposed to model the aerosol population balance equation due to Brownian coagulation in the continuum regime is shown in this study, such as the choice of the expansion pointu, the relationship between asymptotic behavior and analytical solution, and the error of the high-order moment equations. All these analyses will contribute to the buildup of the theoretical system of the TEMOM model.

Perturbation-based moment equations for advection-diffusion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to the same equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches, using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, and prove that the resulting moment equations are in fact hyperbolic for the special case of pure advection b...

The paper develops a mathematical model of foreign currency exchange market in the form of a stochastic linear differential equation with coefficients depending on a semi-Markov process. The boundaries of the domain of its instability is determined by using moment equations.

PurposeThis paper intends to present a hydrodynamical model which describes the hole motion in silicon and couples holes and electrons.Design/methodology/approachThe model is based on the moment method and the closure of the system of moment equations is obtained by using the maximum entropy principle (hereafter MEP). The heavy, light and split‐off valence bands are considered. The first two are described by taking into account their warped shape, while for the split‐off band a parabolic approximation is used.FindingsThe model for holes is coupled with an analogous one for electrons, so obtaining a complete description of charge transport in silicon. Numerical simulations are performed both for bulk silicon and a p‐n junction.Research limitations/implicationsThe model uses a linear approximation of the maximum entropy distribution in order to close the system of moment equations. Furthermore, the non‐parabolicity of the heavy and light bands is neglected. This implies an approximation on the high field results. This issue is under current investigation.Practical implicationsThe paper improves the previous hydrodynamical models on holes and furnishes a complete model which couples electrons and holes. It can be useful in simulations of bipolar devices.Originality/valueThe results of the paper are new since a better approximation of the band structure is used and a description of both electron and hole behavior is present, therefore the results are of a certain relevance for the theory of charge transport in semiconductors.

Efficient modelling of gas microflows requires accurate, yet fast to solve, models. For finite but moderate Knudsen numbers, extended macroscopic transport equations offer an alternative to the Boltzmann equation, from which they are derived. Classical and modern approaches for the derivation of these models are reviewed, and the resulting equations are compared for their ability to describe the multitude of known rarefaction phenomena. Among the equations discussed are the Burnett and super-Burnett equations, Grad’s 13 moment equations, and the regularized 13 and 26 moment equations.

We compute, for the first time, high-order entropy-based ($M_N$) models for a linear transport equation on a one-dimensional, slab geometry. We simulate two test problems from the literature: the two-beam instability and the plane-source problem. In the former case we compute solutions for systems up to order $N=5$ ; in the latter, up to $N=15$. The most notable outcome of these results is the existence of shocks in the steady-state profiles of the two-beam instability for all odd values of $N$.

Moment methods are classical approaches that approximate the mesoscopic radiative transfer equation by a system of macroscopic moment equations. An expansion in the angular variables transforms the original equation into a system of infinitely many moments. The truncation of this infinite system is the moment closure problem. Many types of closures have been presented in the literature. In this note, we demonstrate that optimal prediction, an approach originally developed to approximate the mean solution of systems of nonlinear ordinary differential equations, can be used to derive moment closures. To that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can be re-derived, such as $P_N$, diffusion, and diffusion correction closures. This provides a new perspective on several approximations done in the process and gives rise to ideas for modifications to existing closures., Comment: 15 pages; version 4: sections removed, major reformulations

Accurate prediction of flows that exist between the traditional continuum regime and the free-molecular regime have proven difficult to obtain. Current methods are either inaccurate in this regime or prohibitively expensive for practical problems. Moment closures have long held the promise of providing new, affordable, accurate methods in this regime. The maximum-entropy hierarchy of closures seems to offer particularly attractive physical and mathematical properties. Unfortunately, several difficulties render the practical implementation of maximum-entropy closures very difficult. This work examines the use of simple approximations to these maximum-entropy closures and shows that physical accuracy that is vastly improved over continuum methods can be obtained without a significant increase in computational cost. Initially the technique is demonstrated for a simple one-dimensional gas. It is then extended to the full three-dimensional setting. The resulting moment equations are used for the numerical solu...

This paper studies an alternative bias correction for the M-estimator, which is obtained by correcting the moment equation in the spirit of Firth (1993). In particular, this paper compares the stochastic expansions of the analytically bias-corrected estimator and the alternative estimator and finds that the third-order stochastic expansions of these two estimators are identical. This implies that at least in terms of the third order stochastic expansion, we cannot improve on the simple one-step bias correction by using the bias correction of moment equations. Though the result in this paper is for a fixed number of parameters, our intuition may extend to the analytical bias correction of the panel data models with individual specific effects. Noting the M-estimation can nest many kinds of estimators including IV, 2SLS, MLE, GMM, and GEL, our finding is a rather strong result.

How should we model evolution in spatially structured populations? Here, I review an evolutionary ecology approach based on the technique of spatial moment equations. I first provide a mathematical underpinning to the derivation of equations for the densities of various spatial configurations in network-based models. I then show how this spatial ecological framework can be coupled with an adaptive dynamics approach to compute the invasion fitness of a rare mutant in a resident population at equilibrium. Under the additional assumption that mutations have small phenotypic effects, I show that the selection gradient can be expressed as a function of neutral measures of genetic and demographic structure. I discuss the connections between this approach and inclusive fitness theory, as well as the applicability and limits of this technique. My main message is that spatial moment equations can be used as a means to obtain compact qualitative arguments about the evolution of life-history traits for a variety of life cycles.

The focus of this chapter is to assess, improve and extend ODE constrained mixture modeling (ODE-MM) (Hasenauer et al., 2014b), a method for studying dynamics and structures of subpopulations. In Section 3.1, we introduce the underlying method and formulate the problems that are subsequently addressed in the following sections. In Section 3.2, we apply ODE-MMs to novel single-cell snapshot data for NGF-induced Erk signaling.

The phenomenon of innovation diffusion, modeled as a nonlinear birth process, leads to a hierarchy of moment equations. For gaining insight into the evolution of moments of a number of adopters in innovation diffusion, truncation procedures based on point distributions are proposed for truncating the hierarchy of moment equations. It is found that the results obtained by employing new truncation procedures are in close agreement with analytical results based on system size expansion for large population size.

We present a population density and moment-based description of the stochastic dynamics of domain [Formula: see text]-mediated inactivation of L-type [Formula: see text] channels. Our approach accounts for the effect of heterogeneity of local [Formula: see text] signals on whole cell [Formula: see text] currents; however, in contrast with prior work, e.g., Sherman et al. (Biophys J 58(4):985-995, 1990), we do not assume that [Formula: see text] domain formation and collapse are fast compared to channel gating. We demonstrate the population density and moment-based modeling approaches using a 12-state Markov chain model of an L-type [Formula: see text] channel introduced by Greenstein and Winslow (Biophys J 83(6):2918-2945, 2002). Simulated whole cell voltage clamp responses yield an inactivation function for the whole cell [Formula: see text] current that agrees with the traditional approach when domain dynamics are fast. We analyze the voltage-dependence of [Formula: see text] inactivation that may occur via slow heterogeneous domain [[Formula: see text]]. Next, we find that when channel permeability is held constant, [Formula: see text]-mediated inactivation of L-type channels increases as the domain time constant increases, because a slow domain collapse rate leads to increased mean domain [[Formula: see text]] near open channels; conversely, when the maximum domain [[Formula: see text]] is held constant, inactivation decreases as the domain time constant increases. Comparison of simulation results using population densities and moment equations confirms the computational efficiency of the moment-based approach, and enables the validation of two distinct methods of truncating and closing the open system of moment equations. In general, a slow domain time constant requires higher order moment truncation for agreement between moment-based and population density simulations.

The expansion of moments technique for generating short time expansions for the moments with the distribution function is used to solve the reduced Vlasov Fokker–Planck equation. The obtained results are compared with those found by other theories such as the operator technique. The results obtained by expansion of moments confirm the correctness of those obtained by the operator method. The method is straightforward and concise, and its applications are promising and can be applied to other moment equations arising in physics.

The paper presents a general mathematical model for disperse systems in which direct interactions between particles are taken into consideration. The model is formulated in terms of transition measures, introduced on the basis of conditional Markov processes. The population balance equation, describing the behaviour of interactive populations, is developed in a general form of continuous and discontinuous terms. Moment equations are presented and analysed for the axial dispersion model with interparticle exchange processes of heat and mass. The applicability of the model is illustrated by applying it for describing a process of heating particles by gas with interparticle heat transfer, and a mass exchange process between fluid particles.

Time delayed linear stochastic systems with state feedback is investigated. An extension of the semi-discretization method to stochastic systems is developed. Mappings of the second order moments are formulated by using the exact drift mapping and direct Ito integration. It is found that the exact drift mapping yields better results compared to the direct Ito integration. Stability boundaries and steady state second order moments are studied with the proposed method. The results are compared with the known exact solutions or the simulations. Good agreement among the results is found, which strongly validates the proposed method.

A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics is given in the framework of Grad's moment equations. Grad's systems are considered as the minimal kinetic models where the Chapman-Enskog method can be studied exactly, thereby providing the basis to compare various approximations in extending the hydrodynamic description beyond the Navier-Stokes approximation. Various techniques, such as the method of partial summation, Pade approximants, and invariance principle are compared both in linear and nonlinear situations., Comment: 61 pages. Slightly edited and corrected version. It follows mostly Chapter 8 of the book: Gorban, A.N. and Karlin, I.V., Invariant Manifolds for Physical and Chemical Kinetics, Lect. Notes Phys. 660, Springer, Berlin, Heidelberg, 2005

The BGK model for an ideal gas with an internal degree of freedom introduces two characteristic times into the Boltzmann equation, accounting for the times between elastic and inelastic collisions, respectively. Moment equations are derived from the Boltzmann equation and the influence of the characteristic times is studied for some processes, such as heating, sound waves and shocks.

MAGES is a software developed at INRS-Eau (Canada) for forecasting pollution hazards in groundwater. The transport model uses stationary truncated temporal moment equations instead of the classical time dependent advection-dispersion equation. The aim of this work is to propose a numerical validation of the method by comparison with both analytical solutions in homogeneous medium and a sophisticated lagrangian model of transport in heterogeneous medium. It is shown that the temporal moment equations perform well, while saving on computation. This gives MAGES some abilities in water management problems which compensate for the poor knowledge of the transport parameters through numerous stochastic simulations.

Closure schemes for the infinite hierarchy of moment equations describing the time evolution of averages of observables over a non-equilibrium ensemble of initial conditions for the logistic map model are derived with the aid the Frobenius-Perron equation in the long-time regime. For the growth parameter r = 4 the approximations compare well with exact results derived for specific initial probability densities.

MAGES is software for forecasting pollution hazards of groundwater which is in the process of development at INRS-Eau (Canada). The main distinctive feature of the model is the use of stationary truncated temporal moment equations instead of the classical time dependent advection-dispersion equation to solve the transport of contaminants. The aim of this work is to describe the theory of truncated temporal moment equations and to show how the curves of the concentration versus time can be calculated from temporal moments. The discrete method used to solve the equations and its stability is also discussed.

Multivalued solutions with a limited number of branches of the inviscid Burgers equation can be obtained by solving closed systems of moment equations. For this purpose, a suitable concept of entropy multivalued solutions with K branches is introduced.

Traditional Ensemble Kalman Filter (EnKF) data assimilation requires computationally intensive Monte Carlo (MC) sampling, which suffers from filter inbreeding unless the number of simulations is large. Recently we proposed an alternative EnKF groundwater-data assimilation method that obviates the need for sampling and is free of inbreeding issues. In our new approach, theoretical ensemble moments are approximated directly by solving a system of corresponding stochastic groundwater flow equations. Like MC-based EnKF, our moment equations (ME) approach allows Bayesian updating of system states and parameters in real-time as new data become available. Here we compare the performances and accuracies of the two approaches on two-dimensional transient groundwater flow toward a well pumping water in a synthetic, randomly heterogeneous confined aquifer subject to prescribed head and flux boundary conditions.

A systematic nonperturbaiive derivation is presented of a whole hierarchy of closed systems of moment equations corresponding to any classical kinetic theory that describes a gas of identical particles under the influence of an external potential. The closure has two steps. The first ensures that every member of the hierarchy is hyperbolic, has an entropy, and formally recovers the Euler limit The second modifies the collision operator so that members of the hierarchy beyond the second also recover the correct Navier-Stokes behavior. This is achieved through a generalization of the BGK collision operator.