observation of symmetry breaking state selection and sensitivity in a noisy electronic system abstract measurements of the time evolution of the statistical density as a control parameter is swept through a bifurcation point in the presence of additive noise are presented we have used these densities to study branch selectivity which results from a small additive interaction a universal time scale for all such processes is established the results are in good agreement with a recent theory

refined large deviations asymptotics for markov modulated infinite server systems many networking related settings can be modeled by markov modulated infinite server systems in such models the customers arrival rates and service rates are modulated by a markovian background process additionally there are infinitely many servers and consequently the resulting model is often used as a proxy for the corresponding many server model the markov modulated infinite server model hardly allows any explicit analysis apart from results in terms of systems of ordinary or partial differential equations for the underlying probability generating functions and recursions to obtain all moments as a consequence recent research efforts have pursued an asymptotic analysis in various limiting regimes notably the central limit regime describing fluctuations around the average behavior and the large deviations regime focusing on rare events many of these results use the property that the number of customers in the system obeys a poisson distribution with a random parameter the objective of this paper is to develop techniques to accurately approximate tail probabilities in the large deviations regime we consider the scaling in which the arrival rates are inflated by a factor n and we are interested in the probability that the number of customers exceeds a given level na where earlier contributions focused on so called logarithmic asymptotics of this exceedance probability which are inherently imprecise the present paper improves upon those results in that exact asymptotics are established these are found in two steps first the distribution of the random parameter of the poisson distribution is characterized and then this knowledge is used to identify the exact asymptotics the paper is concluded by a set of numerical experiments in which the accuracy of the asymptotic results is assessed

nonconventional large deviations theorems we obtain large deviations theorems for both discrete time expressions of the form sum n 1 nf big x q 1 n ldots x q ell n big and similar expressions of the form int 0 tf big x q 1 t ldots x q ell t big dt in continuous time here x n n ge 0 or x t t ge 0 is a markov process satisfying doeblin s condition f is a bounded continuous function and q i n in for i le k while for i k they are positive functions taking on integer values on integers with some growth conditions which are satisfied for instance when q i s are polynomials of increasing degrees applications to some types of dynamical systems such as mixing subshifts of finite type and hyperbolic and expanding transformations will be obtained as well

large deviations for two scaled diffusions we formulate large deviations principle ldp for diffusion pair x x t t where first component has a small diffusion parameter while the second is ergodic markovian process with fast time more exactly the ldp is established for x with dt dz being an occupation type measure corresponding to t in some sense we obtain a combination of freidlin wentzell s and donsker varadhan s results our approach relies on the concept of the exponential tightness and puhalskii s theorem

a reduced order model based preconditioner for the efficient solution of transient diffusion equations this paper presents a novel class of preconditioners for the iterative solution of the sequence of symmetric positive definite linear systems arising from the numerical discretization of transient parabolic and selfadjoint partial differential equations the preconditioners are obtained by nesting appropriate projections of reduced order models into the classical iteration of the preconditioned conjugate gradient pcg the main idea is to employ the reduced order solver to project the residual associated with the conjugate gradient iterations onto the space spanned by the reduced bases this approach is particularly appealing for transient systems where the full model solution has to be computed at each time step in these cases the natural reduced space is the one generated by full model solutions at previous time steps when increasing the size of the projection space the proposed methodology highly reduces the system conditioning number and the number of pcg iterations at every time step the cost of the application of the preconditioner linearly increases with the size of the projection basis and a trade off must be found to effectively reduce the pcg computational cost the quality and efficiency of the proposed approach is finally tested in the solution of groundwater flow models c 2016 the authors international journal for numerical methods in engineering published by john wiley sons ltd

two scale large deviations for chemical reaction kinetics through second quantization path integral motivated by the study of rare events for a typical genetic switching model in systems biology in this paper we aim to establish the general two scale large deviations for chemical reaction systems we build a formal approach to explicitly obtain the large deviation rate functionals for the considered two scale processes based upon the second quantization path integral technique we get three important types of large deviation results when the underlying two timescales are in three different regimes this is realized by singular perturbation analysis to the rate functionals obtained by the path integral we find that the three regimes possess the same deterministic mean field limit but completely different chemical langevin approximations the obtained results are natural extensions of the classical large volume limit for chemical reactions we also discuss its implication on the single molecule michaelis menten kinetics our framework and results can be applied to understand general multi scale systems including diffusion processes

balanced model reduction via the proper orthogonal decomposition a new method for performing a balanced reduction of a high order linear system is presented the technique combines the proper orthogonal decomposition and concepts from balanced realization theory the method of snapshotsisused to obtainlow rank reduced rangeapproximationsto thesystemcontrollability and observability grammiansineitherthetimeorfrequencydomain theapproximationsarethenusedtoobtainabalancedreducedorder model the method is particularly effective when a small number of outputs is of interest it is demonstrated for a linearized high order system that models unsteady motion of a two dimensional airfoil computation of the exact grammians would be impractical for such a large system for this problem very accurate reducedorder models are obtained that capture the required dynamics with just three states the new models exhibit far superiorperformancethanthosederived using a conventionalproperorthogonal decomposition although further development is necessary the concept also extends to nonlinear systems

a reduced order model for groundwater flow equation with random hydraulic conductivity application to monte carlo methods 1 we present a model order reduction technique that overcomes the computational burden associated with the application of monte carlo methods to the solution of the groundwater flow equation with random hydraulic conductivity the method is based on the galerkin projection of the high dimensional model equations onto a subspace approximated by a small number of pseudo optimally chosen basis functions principal components to obtain an efficient reduced order model we develop an offline algorithm for the computation of the parameter independent principal components our algorithm combines a greedy algorithm for the snapshot selection in the parameter space and an optimal distribution of the snapshots in time moreover we introduce a residual based estimation of the error associated with the reduced model this estimation allows a considerable reduction of the number of full system model solutions required for the computation of principal components we demonstrate the robustness of our methodology by way of numerical examples comparing the empirical statistics of the ensemble of the numerical solutions obtained using the traditional monte carlo method and our reduced model the numerical results show that our methodology significantly reduces the computational requirements cpu time and storage for the solution of the monte carlo simulation ensuring a good approximation of the mean and variance of the head the analysis of the empirical probability density functions at the observation wells suggests that our reduced model produces good results and is most accurate in the regions with large drawdown

experimental design for estimating unknown groundwater pumping using genetic algorithm and reduced order model 1 an optimal experimental design algorithm is developed to select locations for a network of observation wells that provide maximum information about unknown groundwater pumping in a confined anisotropic aquifer the design uses a maximal information criterion that chooses among competing designs the design that maximizes the sum of squared sensitivities while conforming to specified design constraints the formulated optimization problem is non convex and contains integer variables necessitating a combinatorial search given a realistic large scale model the size of the combinatorial search required can make the problem difficult if not impossible to solve using traditional mathematical programming techniques genetic algorithms gas can be used to perform the global search however because a ga requires a large number of calls to a groundwater model the formulated optimization problem still may be infeasible to solve as a result proper orthogonal decomposition pod is applied to the groundwater model to reduce its dimensionality then the information matrix in the full model space can be searched without solving the full model results from a small scale test case show identical optimal solutions among the ga integer programming and exhaustive search methods this demonstrates the ga s ability to determine the optimal solution in addition the results show that a ga with pod model reduction is several orders of magnitude faster in finding the optimal solution than a ga using the full model the proposed experimental design algorithm is applied to a realistic two dimensional large scale groundwater problem the ga converged to a solution for this large scale problem

reduced order parameter estimation using quasilinearization and quadratic programming 1 the ability of a particular model to accurately predict how a system responds to forcing is predicated on various model parameters that must be appropriately identified there are many algorithms whose purpose is to solve this inverse problem which is often computationally intensive in this study we propose a new algorithm that significantly reduces the computational burden associated with parameter identification the algorithm is an extension of the quasilinearization approach where the governing system of differential equations is linearized with respect to the parameters the resulting inverse problem therefore becomes a linear regression or quadratic programming problem qp for minimizing the sum of squared residuals the solution becomes an update on the parameter set this process of linearization and regression is repeated until convergence takes place this algorithm has not received much attention as the qps can become quite large often infeasible for real world systems to alleviate this drawback proper orthogonal decomposition is applied to reduce the size of the linearized model thereby reducing the computational burden of solving each qp in fact this study shows that the snapshots need only be calculated once at the very beginning of the algorithm after which no further calculations of the reduced model subspace are required the proposed algorithm therefore only requires one linearized full model run per parameter at the first iteration followed by a series of reduced order qps the method is applied to a groundwater model with about 30 000 computation nodes where as many as 15 zones of hydraulic conductivity are estimated

averaging principle of sde with small diffusion moderate deviations consider the following stochastic differential equation in rd dxetxe0 b xet t e dt e a xet t e dwt x0 where the random environment t is an exponentially ergodic markov process independent of the wiener process wt with invariant probability measure and e is some small parameter in this paper we prove the moderate deviations for the averaging principle of xe that is deviations of xet around its limit averaging system x t given by d dx t b x t dt and x 0 x0 where b x e b x more precisely we obtain the large deviation estimation about et xet x te h e t 0 1 in the space of continuous trajectories as e decreases to 0 where h e is some deviation scale satisfying 1 h e e 1 2 our strategy will be first to show the exponential tightness and then the local moderate deviation principle which relies on some new method involving a conditional schilder s theorem and a moderate deviation principle for inhomogeneous integral functionals of markov processes previously established by the author

stochastic modelling replacing fast degrees of freedom by noise in systems with timescale separation where the fast degrees of freedom exhibit chaotic motion the latter are replaced by suitable stochastic processes a projection technique is employed to derive equations of motion for the phase space density of the slow variables by eliminating the fast ones the resulting equations can be approximated in a controlled way by fokker planck equations or equivalently by stochastic differential equations for slow degrees of freedom we discuss some model situations and explore the accuracy of the approximations by numerical simulations

large deviations averaging and periodic orbits of dynamical systems the paper deals with large deviation bounds for the proportion of periodic orbits with irregular behavior for expansive dynamical systems with specification in particular we obtain estimates for large deviations from the equidistribution for closed geodesics on negatively curved manifolds we derive also large deviation bounds in the averaging principle when the fast motion is the shift along periodic orbits

a comparison of several methods of solving nonlinear regression groundwater flow problems computational efficiency and computer memory requirements for four methods of minimizing functions were compared for four test nonlinear regression steady state groundwater flow problems the fastest methods were the marquardt and quasi linearization methods which required almost identical computer times and numbers of iterations the next fastest was the quasi newton method and last was the fletcher reeves method which did not converge in 100 iterations for two of the problems the fastest method per iteration was the fletcher reeves method and this was followed closely by the quasi newton method the marquardt and quasi linearization methods were slower for all four methods the speed per iteration was directly related to the number of parameters in the model however this effect was much more pronounced for the marquardt and quasi linearization methods than for the other two hence the quasi newton and perhaps fletcher reeves method might be more efficient than either the marquardt or quasi linearization methods if the number of parameters in a particular model were large although this remains to be proven the marquardt method required somewhat less central memory than the quasi linearization metilod for three of the four problems for all four problems the quasi newton method required roughly two thirds to three quarters of the memory required by the marquardt method and the fletcher reeves method required slightly less memory than the quasi newton method memory requirements were not excessive for any of the four methods

passage through a barrier with a slowly increasing control parameter a bistable system subjected to noise and a slow increase of the parameter controlling the instability is studied with emphasis on the kinetics of the transitions across the barrier separating the stable states it is shown that the presence of a ramp in the control parameter may considerably affect the distribution of probability mass on the two sides of the barrier as compared to the predictions of the classical kramers theory

direct evaluation of dynamical large deviation rate functions using a variational ansatz we describe a simple form of importance sampling designed to bound and compute large deviation rate functions for time extensive dynamical observables in continuous time markov chains we start with a model defined by a set of rates and a time extensive dynamical observable we construct a reference model a variational ansatz for the behavior of the original model conditioned on atypical values of the observable direct simulation of the reference model provides an upper bound on the large deviation rate function associated with the original model an estimate of the tightness of the bound and if the ansatz is chosen well the exact rate function the exact rare behavior of the original model does not need to be known in advance we use this method to calculate rate functions for currents and counting observables in a set of network and lattice models taken from the literature straightforward ansatze yield bounds that are tighter than bounds obtained from level 2 5 of large deviations via approximations that involve uniform scalings of rates we show how to correct these bounds in order to recover the rate functions exactly our approach is complementary to more specialized methods and offers a physically transparent framework for approximating and calculating the likelihood of dynamical large deviations

large deviations in fast slow systems the incidence of rare events in fast slow systems is investigated via analysis of the large deviation principle ldp that characterizes the likelihood and pathway of large fluctuations of the slow variables away from their mean behavior such fluctuations are rare on short time scales but become ubiquitous eventually classical results prove that this ldp involves an hamilton jacobi equation whose hamiltonian is related to the leading eigenvalue of the generator of the fast process and is typically non quadratic in the momenta in other words the ldp for the slow variables in fast slow systems is different in general from that of any stochastic differential equation sde one would write for the slow variables alone it is shown here that the eigenvalue problem for the hamiltonian can be reduced to a simpler algebraic equation for this hamiltonian for a specific class of systems in which the fast variables satisfy a linear equation whose coefficients depend nonlinearly on the slow variables and the fast variables enter quadratically the equation for the slow variables these results are illustrated via examples inspired by kinetic theories of turbulent flows and plasma in which the quasipotential characterizing the long time behavior of the system is calculated and shown again to be different from that of an sde

averaging in dynamical systems and large deviations the paper treats ordinary differential equations of the form frac dz varepsilon left t right dt varepsilon b z varepsilon t f t y wheref t is a hyperbolic flow large deviations bounds for the averaging principle are obtained here in the form appeared previously in f1 f2 for the case when the flowf t is replaced by a markov process

continuous time stochastic averaging on the infinite interval for locally lipschitz systems we investigate stochastic averaging on the infinite time interval for a class of continuous time nonlinear systems with stochastic perturbation and remove or weaken several restrictions present in existing results global lipschitzness of the nonlinear vector field equilibrium preservation under the stochastic perturbation global exponential stability of the average system and compactness of the state space of the perturbation process if an equilibrium of the average system is exponentially stable we show that the original system is exponentially practically stable in probability if in addition the original system has the same equilibrium as the average system then the equilibrium of the original system is locally asymptotically stable in probability these results extend the deterministic general averaging for aperiodic functions to the stochastic case

the geometric minimum action method a least action principle on the space of curves freidlin wentzell theory of large deviations for the description of the effect of small random perturbations on dynamical systems is exploited as a numerical tool specifically a numerical algorithm is proposed to compute the quasi potential in the theory which is the key object to quantify the dynamics on long time scales when the effect of the noise becomes ubiquitous the equilibrium distribution of the system the pathways of transition between metastable states and their rate etc can all be expressed in terms of the quasi potential we propose an algorithm to compute these quantities called the geometric minimum action method gmam which is a blend of the original minimum action method mam and the string method it is based on a reformulation of the large deviations action functional on the space of curves that allows one to easily perform the double minimization of the original action required to compute the quasi potential the theoretical background of the gmam in the context of large deviations theory is discussed in detail as well as the algorithmic aspects of the method the gmam is then illustrated on several examples a finite dimensional system displaying bistability and modeled by a nongradient stochastic ordinary differential equation an infinite dimensional analogue of this system modeled by a stochastic partial differential equation and an example of a bistable genetic switch modeled by a markov jump process

decision making in noisy bistable systems with time dependent asymmetry our work draws special attention to the importance of the effects of time dependent parameters on decision making in bistable systems here we extend previous studies of the mechanism known as speed dependent cellular decision making in genetic circuits by performing an analytical treatment of the canonical supercritical pitchfork bifurcation problem with an additional time dependent asymmetry and control parameter this model has an analogous behavior to the genetic switch in the presence of transient asymmetries and fluctuations slow passage through the critical region in both systems increases substantially the probability of specific decision outcomes we also study the relevance for attractor selection of reaching maximum values for the external asymmetry before and after the critical region overall maximum asymmetries should be reached at an instant where the position of the critical point allows for compensation of the detrimental effects of noise in retaining memory of the transient asymmetries

statistical properties of mostly contracting fast slow partially hyperbolic systems we consider a class of mathcal c 4 partially hyperbolic systems on mathbb t 2 described by maps f varepsilon x theta f x theta theta varepsilon omega x theta where f cdot theta are expanding maps of the circle for sufficiently small varepsilon and omega generic in an open set we precisely classify the srb measures for f varepsilon and their statistical properties including exponential decay of correlation for holder observables with explicit and nearly optimal bounds on the decay rate

limits of random differential equations on manifolds consider a family of random ordinary differential equations on a manifold driven by vector fields of the form sum ky k alpha k z t epsilon omega where y k are vector fields epsilon is a positive number z t epsilon is a 1 over epsilon mathcal l 0 diffusion process taking values in possibly a different manifold alpha k are annihilators of mathrm ker mathcal l 0 under hormander type conditions on mathcal l 0 we prove that as epsilon approaches zero the stochastic processes y t over epsilon epsilon converge weakly and in the wasserstein topologies we describe this limit and give an upper bound for the rate of the convergence

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

connection between nonlinear energy optimization and instantons how systems transit between different stable states under external perturbation is an important practical issue we discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems in the context of the one dimensional swift hohenberg equation which has multiple stable equilibria we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances minimal seeds for transition to specific attractors from the ground state then after generalizing the technique to consider multiple equally spaced in time perturbations it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations importantly we find that the key features of the instanton can be captured by a low number of discrete perturbations typically one perturbation per basin of attraction crossed this suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

global normal form and asymptotic spectral gap for open partially expanding maps we consider a mathbb r extension of one dimensional uniformly expanding open dynamical systems and prove a new explicit estimate for the asymptotic spectral gap to get these results we use a new application of a global normal form for the dynamical system a semiclassical expression beyond the ehrenfest time that expresses the transfer operator at large time as a sum over rank one operators each is associated to one orbit in this paper we establish the validity of the so called diagonal approximation up to twice the local ehrenfest time

deviations of trajectory averages and the defect in pesin s formula for anosov diffeomorphisms a large deviation theorem at the donsker varadhan level three is obtained for the convergence of trajectory averages of anosov diffeomorphisms it is possible to provide an explicit description of the rate function

fast chaos versus white noise entropy analysis and a fokker planck model for the slow dynamics abstract we show in detail how methods of time series analysis such as dimension and entropy estimates support the idea that fast low dimensional chaos can be modelled properly by noise motivated by this observation we derive a formalism by which fast chaotic degrees of freedom in systems with time scale separation can be replaced by a suitable stochastic process a fokker planck equation for the phase space density of the slow degrees of freedom is derived from the full deterministic system by a projection operator technique together with a perturbation expansion we compare two different projection strategies and illustrate the resulting equations by specifying them for a simple model system

inverse modeling of groundwater flow using model reduction numerical groundwater flow models often have a very high number of model cells greater than a million such models are computationally very demanding which is disadvantageous for inverse modeling this paper describes a low dimensional formulation for groundwater flow that reduces the computational burden necessary for inverse modeling the formulation is a projection of the original groundwater flow equation on a set of orthogonal patterns i e a galerkin projection the patterns empirical orthogonal functions are computed by a decomposition of the covariance matrix over an ensemble of model solutions those solutions represent the behavior of the model as a result of model impulses and the influence of a chosen set of parameter values for an interchangeable set of parameter values the patterns yield a low dimensional model as the number of patterns is often small an advantage of this model is that the adjoint is easily available and most accurate for inverse modeling for several synthetical cases the low dimensional model was able to find the global minimum efficiently and the result was comparable to that of the original model for several cases our model even converged where the original model failed our results demonstrate that the proposed procedure results in a 60 time reduction to solve the groundwater flow inverse problem greater efficiencies can be expected in practice for large scale models with a large number of grid cells that are used to compute transient simulations

geostatistical reduced order models in underdetermined inverse problems 1 reduced order models roms approximate the high dimensional state of a dynamic system with a low dimensional approximation in a subspace of the state space properly constructed they are used to significantly reduce the computational cost associated with the simulation of complex dynamic systems such as flow and transport in the subsurface a key component in model reduction is to construct the subspace where we look for approximate solutions in this work we apply model reduction in inverse modeling and use the solution parameter space of underdetermined geostatistical inverse problems to construct the subspace in which we seek approximate solutions for any given parameters needed in the inversion process the subspace is constructed by collecting state variable e g pressure distributions in the flow domain each of the distributions called snapshots contains the result of full forward model simulation for a given test with a basis vector in the solution parameter space as input parameters we then use linear combinations of the snapshots to approximate the forward model solution for any parameters needed in inverse modeling in geostatistical inverse modeling the solution parameter space is spanned by the cross covariance of measurements and parameters hence we name the rom as the geostatistical reduced order model grom we also show that with minor loss of accuracy in the forward model the accuracy in parameter estimation is still high and the saving in computational cost is significant especially for large scale inverse problems where the number of unknowns is enormous

stochastic processes in random graphs we study the asymptotics of large moderate and normal deviations for the connected components of the sparse random graph by the method of stochastic processes we obtain the logarithmic asymptotics of large deviations of the joint distribution of the number of connected components of the sizes of the giant components and of the numbers of the excess edges of the giant components for the supercritical case we obtain the asymptotics of normal deviations and the logarithmic asymptotics of large and moderate deviations of the joint distribution of the number of components of the size of the largest component and of the number of the excess edges of the largest component for the critical case we obtain the logarithmic asymptotics of moderate deviations of the joint distribution of the sizes of connected components and of the numbers of the excess edges some related asymptotics are also established the proofs of the large and moderate deviation asymptotics employ methods of idempotent probability theory as a byproduct of the results we provide some additional insight into the nature of phase transitions in sparse random graphs