High-dimensional Stochastic Systems Research Articles

Response and reliability analysis of a high-dimensional stochastic system

The efficiency of the probability density evolution method (PDEM) is improved in this paper by embedding the Kullback–Leibler (K–L) relative sensitivity in the response analysis of a stochastic dynamic system. The response reliability obtained and the probability density function of the response peaks are used for ranking to get a reduced set of random variables for the PDEM analysis. The need of complicated point selection technique with the high-dimensional uncertain variables is therefore alleviated. The proposed method is illustrated with the response analysis of a random crowd-structure system where the load randomness is considered. The acceleration response induced by the presence of the crowd is evaluated with the proposed method. Results obtained highlight the significant improvements in the computation efficiency of the probabilistic response analysis of a high-dimensional dynamic system.

Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part I: Theory

A theory of Ruelle-Pollicott (RP) resonances for stochastic differential systems is presented. These resonances are defined as the eigenvalues of the generator (Kolmogorov operator) of a given stochastic system. By relying on the theory of Markov semigroups, decomposition formulas of correlation functions and power spectral densities (PSDs) in terms of RP resonances are then derived. These formulas describe, for a broad class of stochastic differential equations (SDEs), how the RP resonances characterize the decay of correlations as well as the signal's oscillatory components manifested by peaks in the PSD.It is then shown that a notion reduced RP resonances can be rigorously defined, as soon as the dynamics is partially observed within a reduced state space V . These reduced resonances are obtained from the spectral elements of reduced Markov operators acting on functions of the state space V , and can be estimated from series. They inform us about the spectral elements of some coarse-grained version of the SDE generator. When the time-lag at which the transitions are collected from partial observations in V , is either sufficiently small or large, it is shown that the reduced RP resonances approximate the (weak) RP resonances of the generator of the conditional expectation in V , i.e. the optimal reduced system in V obtained by averaging out the contribution of the unobserved variables. The approach is illustrated on a stochastic slow-fast system for which it is shown that the reduced RP resonances allow for a good reconstruction of the correlation functions and PSDs, even when the time-scale separation is weak.The companions articles, Part II and Part III, deal with further practical aspects of the theory presented in this contribution. One important byproduct consists of the diagnosis usefulness of stochastic dynamics that RP resonances provide. This is illustrated in the case of a stochastic Hopf bifurcation in Part II. There, it is shown that such a bifurcation has a clear manifestation in terms of a geometric organization of the RP resonances along discrete parabolas in the left half plane. Such geometric features formed by (reduced) RP resonances are extractable from time series and allow thus for providing an unambiguous "signature" of nonlinear oscillations embedded within a stochastic background. By relying then on the theory of reduced RP resonances presented in this contribution, Part III addresses the question of detection and characterization of such oscillations in a high-dimensional stochastic system, namely the Cane-Zebiak model of El Ni{\~n}o-Southern Oscillation subject to noise modeling fast atmospheric fluctuations.

Ruelle–Pollicott Resonances of Stochastic Systems in Reduced State Space. Part III: Application to the Cane–Zebiak Model of the El Niño–Southern Oscillation

The response of a low-frequency mode of climate variability, El Nino–Southern Oscillation, to stochastic forcing is studied in a high-dimensional model of intermediate complexity, the fully-coupled Cane–Zebiak model (Zebiak and Cane 1987), from the spectral analysis of Markov operators governing the decay of correlations and resonances in the power spectrum. Noise-induced oscillations excited before a supercritical Hopf bifurcation are examined by means of complex resonances, the reduced Ruelle–Pollicott (RP) resonances, via a numerical application of the reduction approach of the first part of this contribution (Chekroun et al. 2019) to model simulations. The oscillations manifest themselves as peaks in the power spectrum which are associated with RP resonances organized along parabolas, as the bifurcation is neared. These resonances and the associated eigenvectors are furthermore well described by the small-noise expansion formulas obtained by Gaspard (2002) and made explicit in the second part of this contribution (Tantet et al. 2019). Beyond the bifurcation, the spectral gap between the imaginary axis and the real part of the leading resonances quantifies the diffusion of phase of the noise-induced oscillations and can be computed from the linearization of the model and from the diffusion matrix of the noise. In this model, the phase diffusion coefficient thus gives a measure of the predictability of oscillatory events representing ENSO. ENSO events being known to be locked to the seasonal cycle, these results should be extended to the non-autonomous case. More generally, the reduction approach theorized in Chekroun et al. (2019), complemented by our understanding of the spectral properties of reference systems such as the stochastic Hopf bifurcation, provides a promising methodology for the analysis of low-frequency variability in high-dimensional stochastic systems.

NIRK-based Cholesky-factorized square-root accurate continuous-discrete unscented Kalman filters for state estimation in nonlinear continuous-time stochastic models with discrete measurements

This paper further advances the idea of accurate Gaussian filtering towards efficient unscented-type Kalman methods for estimating continuous-time nonlinear stochastic systems with discrete measurements. It implies that the differential equations evolving sigma points utilized in computations of the predicted mean and covariance in time-propagations of the Gaussian distribution are solved accurately, i.e. with negligible error. The latter allows the total error of the unscented Kalman filtering technique to be reduced significantly and gives rise to the novel accurate continuous-discrete unscented Kalman filtering algorithm. At the same time, this algorithm is rather vulnerable to round-off and numerical integration errors committed in each state estimation run because of the need for the Cholesky decomposition of covariance matrices involved. Such a factorization will always fail when the covariance's positivity is lost. This positivity lost issue is commonly resolved with square-root filtering implementations, which propagate not the full covariance matrix but its square root (Cholesky factor) instead. Unfortunately, negative weights encountered in applications of the accurate continuous-discrete unscented Kalman filter to high-dimensional stochastic systems preclude from designing conventional square-root methods. In this paper, we address this problem with low-rank Cholesky factor update procedures or with hyperbolic QR transforms used for yielding J-orthogonal square roots. Our novel square-root algorithms are justified theoretically and examined numerically in an air traffic control scenario.

Square-root accurate continuous-discrete extended-unscented Kalman filtering methods with embedded orthogonal and J-orthogonal QR decompositions for estimation of nonlinear continuous-time stochastic models in radar tracking

This paper presents a number of new state estimation algorithms, which unify the best features of the accurate continuous-discrete extended and unscented Kalman filters in treating nonlinear continuous-time stochastic systems with discrete measurements. In particular, our mixed-type algorithms succeed in estimating continuous-discrete stochastic systems with nonlinear and/or nondifferentiable measurements. The main weakness of these methods is the need for the Cholesky decomposition of predicted covariance matrices. Such a factorization is highly sensitive to numerical integration and round-off errors committed, which may result in losing the covariance’s positivity and, hence, failing the Cholesky decomposition. The latter problem is usually solved in the form of square-root filtering implementations, which propagate not the covariance matrix but its square root (Cholesky factor), only. Unfortunately, negative weights arising in applications of our mixed-type methods to high-dimensional stochastic systems preclude from designing conventional square-root filters. We address the mentioned issue with one-rank Cholesky factor updates or with hyperbolic QR transforms used for yielding J-orthogonal square-root filters. These novel algorithms are justified theoretically and examined and compared numerically to the non-square-root one in severe conditions of tackling a seven-dimensional radar tracking problem, where an aircraft executes a coordinated turn, in the presence of Gaussian or glint noise.

Mean-field Macro Computation in Large-scale Cloud Service Systems with Resource Management and Job Scheduling

Service computing is an emerging and distributed computing mode in cloud service systems, and has become an interesting research direction for both academia and industry. Note that the cloud service systems always display new characteristics, such as stochasticity, large scale, loose coupling, concurrency, non-homogeneity and heterogeneity, thus their load balancing investigation has been more interesting, difficult and challenging until now. By using resource management and job scheduling, this paper proposes an integrated, real-time and dynamic control mechanism for large-scale cloud service systems and their load balancing through combining supermarket models with not only work stealing models but also scheduling of public reserved resource. To this end, this paper provides a novel stochastic model with weak interactions by means of nonlinear Markov processes. To overcome theoretical difficulties growing out of the state explosion in high-dimensional stochastic systems, this paper applies the mean-field theory to develop a macro computational technique in terms of an infinite-dimensional system of mean-field equations. Furthermore, this paper proves the asymptotic independence of the large-scale cloud service system, and show how to compute the fixed point by virtue of an infinite-dimensional system of nonlinear equations. Based on the fixed point, this paper provides effective numerical computation for performance analysis of this system under a high approximate precision. Therefore, we hope that the methodology and results given in this paper can be applicable to the study of more general large-scale cloud service systems.

Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations

Gene regulatory networks, which are complex high-dimensional stochastic dynamical systems, are often subject to evident intrinsic fluctuations. It is deemed reasonable to model the systems by the chemical Langevin equations. Since the mRNA dynamics are faster than the protein dynamics, we have a two-time scales system. In general, the process of protein degradation involves time delays. In this paper, we take the system memory into consideration in which we consider a model with a complete memory represented by an integral delay from \begin{document}$ 0 $\end{document} to \begin{document}$ t $\end{document} . Based on the averaging principle and perturbed test function method, this work examines the weak convergence of the slow-varying process. By treating the fast-varying process as a random noise, under appropriate conditions, it is shown that the slow-varying process converges weakly to the solution of a stochastic differential delay equation whose coefficients are the average of those of the original slow-varying process with respect to the invariant measure of the fast-varying process.

Data-driven closures for stochastic dynamical systems

We develop a new data-driven closure approximation method to compute the statistical properties of quantities of interest in high-dimensional stochastic dynamical systems. The proposed framework relies on estimating system-dependent conditional expectations from sample paths or experimental data, and then using such estimates to compute data-driven solutions to exact probability density function (PDF) equations. We also address the important question of whether enough useful data is being injected into the exact PDF equation for the purpose of computing an accurate numerical solution. Numerical examples are presented and discussed for prototype nonlinear dynamical systems and models of systems biology evolving from random initial states.

Stochastic collocation approach with adaptive mesh refinement for parametric uncertainty analysis

The presence of a high-dimensional stochastic input domain with discontinuities poses major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic collocation method with adaptive mesh refinement (SCAMR) to deal with high dimensional stochastic systems with discontinuities. Specifically, the proposed approach uses generalized polynomial chaos (gPC) expansion with Legendre polynomial basis and solves for the gPC coefficients using the least squares method. It also implements an adaptive mesh (element) refinement strategy which checks for abrupt variations in the output based on a low-order gPC approximation error to track discontinuities or non-smoothness. In addition, the proposed method involves a criterion for checking possible dimensionality reduction and consequently, the decomposition of the original high-dimensional problem to a number of lower-dimensional subproblems. Specifically, this criterion checks all the existing interactions between input parameters of a specific problem based on the high-dimensional model representation (HDMR) method, and therefore automatically provides the subproblems which only involve interacting input parameters. The efficiency of the approach is demonstrated using examples of both smooth and non-smooth problems with number of input parameters up to 500, and the approach is compared against other existing algorithms.

Sparse representations and compressive sampling for enhancing the computational efficiency of the Wiener path integral technique

Abstract The computational efficiency of the Wiener path integral (WPI) technique for determining the stochastic response of diverse dynamical systems is enhanced by exploiting recent developments in the area of sparse representations. Specifically, an appropriate basis for expanding the system joint response probability density function (PDF) is utilized. Next, only very few PDF points are determined based on the localization capabilities of the WPI technique. Further, compressive sampling procedures in conjunction with group sparsity concepts and appropriate optimization algorithms are employed for efficiently determining the coefficients of the system response PDF expansion. It is shown that the herein developed enhancement renders the technique capable of treating readily relatively high-dimensional stochastic systems. Two illustrative numerical examples are considered. The first refers to a single-degree-of-freedom Duffing oscillator exhibiting a bimodal response PDF. In the second example, the 20-variate joint response transition PDF of a 10-degree-of-freedom nonlinear structural system under stochastic excitation is determined. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the enhanced WPI technique.

Scalable Underapproximation for the Stochastic Reach-Avoid Problem for High-Dimensional LTI Systems Using Fourier Transforms

We present a scalable underapproximation of the terminal hitting time stochastic reach-avoid probability at a given initial condition, for verification of high-dimensional stochastic LTI systems. While several approximation techniques have been proposed to alleviate the curse of dimensionality associated with dynamic programming, these techniques cannot handle larger, more realistic systems. We present a scalable method that uses Fourier transforms to compute an underapproximation of the reach-avoid probability for systems with disturbances with arbitrary probability densities. We characterize sufficient conditions for Borel-measurability of the value function. We exploit fixed control sequences parameterized by the initial condition (an open-loop control policy) to generate the underapproximation. For Gaussian disturbances, the underapproximation can be obtained using existing efficient algorithms by solving a convex optimization problem. Our approach produces non-trivial lower bounds and is demonstrated on a 40-D chain of integrators.

Relating high dimensional stochastic complex systems to low-dimensional intermittency

We evaluate the implication and outlook of an unanticipated simplification in the macroscopic behavior of two high-dimensional sto-chastic models: the Replicator Model with Mutations and the Tangled Nature Model (TaNa) of evolutionary ecology. This simplification consists of the apparent display of low-dimensional dynamics in the non-stationary intermittent time evolution of the model on a coarse-grained scale. Evolution on this time scale spans generations of individuals, rather than single reproduction, death or mutation events. While a local one-dimensional map close to a tangent bifurcation can be derived from a mean-field version of the TaNa model, a nonlinear dynamical model consisting of successive tangent bifurcations generates time evolution patterns resembling those of the full TaNa model. To advance the interpretation of this finding, here we consider parallel results on a game-theoretic version of the TaNa model that in discrete time yields a coupled map lattice. This in turn is represented, a la Langevin, by a one-dimensional nonlinear map. Among various kinds of behaviours we obtain intermittent evolution associated with tangent bifurcations. We discuss our results.

ATLAS: A Geometric Approach to Learning High-Dimensional Stochastic Systems Near Manifolds

When simulating multiscale stochastic differential equations (SDEs) in high-dimensions, separation of timescales, stochastic noise, and high-dimensionality can make simulations prohibitively expensive. The computational cost is dictated by microscale properties and interactions of many variables, while the behavior of interest often occurs at the macroscale level and at large timescales, often characterized by few important, but unknown, degrees of freedom. For many problems bridging the gap between the microscale and macroscale by direct simulation is computationally infeasible. In this work we propose a novel approach to automatically learn a reduced model with an associated fast macroscale simulator. Our unsupervised learning algorithm uses short parallelizable microscale simulations to learn provably accurate macroscale SDE models, which are continuous in space and time. The learning algorithm takes as input the microscale simulator, a local distance function, and a homogenization spatial or temporal ...

Tangent map intermittency as an approximate analysis of intermittency in a high dimensional fully stochastic dynamical system: The Tangled Nature model.

It is well known that low-dimensional nonlinear deterministic maps close to a tangent bifurcation exhibit intermittency and this circumstance has been exploited, e.g., by Procaccia and Schuster [Phys. Rev. A 28, 1210 (1983)], to develop a general theory of 1/f spectra. This suggests it is interesting to study the extent to which the behavior of a high-dimensional stochastic system can be described by such tangent maps. The Tangled Nature (TaNa) Model of evolutionary ecology is an ideal candidate for such a study, a significant model as it is capable of reproducing a broad range of the phenomenology of macroevolution and ecosystems. The TaNa model exhibits strong intermittency reminiscent of punctuated equilibrium and, like the fossil record of mass extinction, the intermittency in the model is found to be non-stationary, a feature typical of many complex systems. We derive a mean-field version for the evolution of the likelihood function controlling the reproduction of species and find a local map close to tangency. This mean-field map, by our own local approximation, is able to describe qualitatively only one episode of the intermittent dynamics of the full TaNa model. To complement this result, we construct a complete nonlinear dynamical system model consisting of successive tangent bifurcations that generates time evolution patterns resembling those of the full TaNa model in macroscopic scales. The switch from one tangent bifurcation to the next in the sequences produced in this model is stochastic in nature, based on criteria obtained from the local mean-field approximation, and capable of imitating the changing set of types of species and total population in the TaNa model. The model combines full deterministic dynamics with instantaneous parameter random jumps at stochastically drawn times. In spite of the limitations of our approach, which entails a drastic collapse of degrees of freedom, the description of a high-dimensional model system in terms of a low-dimensional one appears to be illuminating.

Multifidelity Information Fusion Algorithms for High-Dimensional Systems and Massive Data sets

We develop a framework for multifidelity information fusion and predictive inference in high-dimensional input spaces and in the presence of massive data sets. Hence, we tackle simultaneously the “big N" problem for big data and the curse of dimensionality in multivariate parametric problems. The proposed methodology establishes a new paradigm for constructing response surfaces of high-dimensional stochastic dynamical systems, simultaneously accounting for multifidelity in physical models as well as multifidelity in probability space. Scaling to high dimensions is achieved by data-driven dimensionality reduction techniques based on hierarchical functional decompositions and a graph-theoretic approach for encoding custom autocorrelation structure in Gaussian process priors. Multifidelity information fusion is facilitated through stochastic autoregressive schemes and frequency-domain machine learning algorithms that scale linearly with the data. Taking together these new developments leads to linear complexity algorithms as demonstrated in benchmark problems involving deterministic and stochastic fields in up to $10^{5}$ input dimensions and $10^{5}$ training points on a standard desktop computer.

A numerical framework to understand transitions in high-dimensional stochastic dynamical systems

Dynamical systems methodology is a mature complementary approach to forward simulation which can be used to investigate many aspects of climate dynamics. With this paper, a review is given on the methods to analyse deterministic and stochastic climate models and show that these are not restricted to low-dimensional toy models, but that they can be applied to models formulated by stochastic partial differential equations. We sketch the numerical implementation of these methods and illustrate these by showing results for two canonical problems in climate dynamics.

Forecasting systemic transitions in high dimensional stochastic complex systems

We briefly describe a new procedure to monitor and forecast transitions occurring in high dimensional complex systems. The method is illustrated by applications to two model systems: firstly to the Tangled Nature Model of evolutionary ecology and secondly to a stochastic replicator system. The quasi-stable configurations of the stochastic dynamics are taken as input for a stability analysis of the deterministic mean field approximation of the dynamics. We demonstrate that the largest overlap between the observed configuration and the unstable eigendirections serves as a precursor that allows us to forecast transitions with an efficiency of about 80% even if we only know the couplings matrix describing the dynamics to with 10% accuracy.

A Low Dimensional Approximation For Competence In Bacillus Subtilis.

The behaviour of a high dimensional stochastic system described by a chemical master equation (CME) depends on many parameters, rendering explicit simulation an inefficient method for exploring the properties of such models. Capturing their behaviour by low-dimensional models makes analysis of system behaviour tractable. In this paper, we present low dimensional models for the noise-induced excitable dynamics in Bacillus subtilis, whereby a key protein ComK, which drives a complex chain of reactions leading to bacterial competence, gets expressed rapidly in large quantities (competent state) before subsiding to low levels of expression (vegetative state). These rapid reactions suggest the application of an adiabatic approximation of the dynamics of the regulatory model that, however, lead to competence durations that are incorrect by a factor of 2. We apply a modified version of an iterative functional procedure that faithfully approximates the time-course of the trajectories in terms of a two-dimensional model involving proteins ComK and ComS. Furthermore, in order to describe the bimodal bivariate marginal probability distribution obtained from the Gillespie simulations of the CME, we introduce a tunable multiplicative noise term in a two-dimensional Langevin model whose stationary state is described by the time-independent solution of the corresponding Fokker-Planck equation.

Automatic Recognition and Tagging of Topologically Different Regimes in Dynamical Systems

Complex systems are commonly modeled using nonlinear dynamical systems. These models are often high-dimensional and chaotic. An important goal in studying physical systems through the lens of mathematical models is to determine when the system undergoes changes in qualitative behavior. A detailed description of the dynamics can be difficult or impossible to obtain for high-dimensional and chaotic systems. Therefore, a more sensible goal is to recognize and mark transitions of a system between qualitatively different regimes of behavior. In practice, one is interested in developing techniques for detection of such transitions from sparse observations, possibly contaminated by noise. In this paper we develop a framework to accurately tag different regimes of complex systems based on topological features. In particular, our framework works with a high degree of success in picking out a cyclically orbiting regime from a stationary equilibrium regime in high-dimensional stochastic dynamical systems.

Statistical analysis and simulation of random shocks in stochastic Burgers equation

We study the statistical properties of random shock waves in stochastic Burgers equation subject to random space–time perturbations and random initial conditions. By using the response–excitation probability density function (PDF) method and the Mori–Zwanzig (MZ) formulation of irreversible statistical mechanics, we derive exact reduced-order equations for the one-point and two-point PDFs of the solution field. In particular, we compute the statistical properties of random shock waves in the inviscid limit by using an adaptive (shock-capturing) discontinuous Galerkin method in both physical and probability spaces. We consider stochastic flows generated by high-dimensional random initial conditions and random additive forcing terms, yielding multiple interacting shock waves collapsing into clusters and settling down to a similarity state. We also address the question of how random shock waves in space and time manifest themselves in probability space. The proposed new mathematical framework can be applied to different conservation laws, potentially leading to new insights into high-dimensional stochastic dynamical systems and more efficient computational algorithms.