Multiscale Dynamical Systems Research Articles

Particle Filters in a Multiscale Environment: Homogenized Hybrid Particle Filter

State estimation of random dynamical systems with noisy observations has been an important problem in many areas of science and engineering. Efficient new algorithms to estimate the present and future state of a dynamic signal based upon corrupted, distorted, and possibly partial observations of the signal are required. Since the true state is usually hidden and evolves according to its own dynamics, the objective of this work is to get an optimal estimation of the true state via noisy observations. The theory of filtering provides a recursive procedure for estimating an evolving signal or state from a noisy observation process. We consider a particle filter approach for nonlinear filtering in multiscale dynamical systems. Particle filters represent the posterior conditional distribution of the state variables by a system of particles, which evolves and adapts recursively as new information becomes available. Particle filters suffer from computational inefficiency when applied to high dimensional problems. In practice, large numbers of particles may be required to provide adequate approximations in higher dimensional poblems. In several high dimensional applications, after a sequence of updates, the particle system will often collapse to a single point. With the help of rigorous dimensional reduction methods, particle filters could regain their versatility. Based on our theoretical developments (Park, J. H., Sri Namachchivaya, N., and Sowers, R. B., 2008, “A Problem in Stochastic Averaging of Nonlinear Filters,” Stochastics Dyn., 8(3), pp. 543–560; Park, J. H., Sowers, R. B., and Sri Namachchivaya, N., 2010, “Dimensional Reduction in Nonlinear Filtering,” Nonlinearity, 23(2), pp. 305–324), we devise an efficient particle filter algorithm, which is applicable to high dimensional multiscale nonlinear filtering problems. In this paper, we present the homogenized hybrid particle filtering method that combines homogenization of random dynamical systems, reduced order nonlinear filtering, and particle methods.

Joint online recognition of speech and speaker using bayesian inference for multi-scale nonlinear dynamical systems

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Scalable Bayesian Reduced-Order Models for Simulating High-Dimensional Multiscale Dynamical Systems

While existing mathematical descriptions can accurately account for phenomena at microscopic scales (e.g., molecular dynamics), these are often high-dimensional, stochastic, and their applicability over macroscopic time scales of physical interest is computationally infeasible or impractical. In complex systems, with limited physical insight on the coherent behavior of their constituents, the only available information is data obtained from simulations of the trajectories of huge numbers of degrees of freedom over microscopic time scales. The analysis of these large amounts of data hinges upon the ability to efficiently extract meaningful latent properties and to discover reduced, predictive descriptions. This paper discusses a Bayesian approach to deriving probabilistic coarse-grained models that simultaneously addresses the problems of identifying appropriate reduced coordinates and the effective dynamics in this lower-dimensional representation. At the core of the models proposed lie simple, low-dimensional dynamical systems which serve as the building blocks of the global model. These approximate the latent generating sources and parametrize the reduced-order dynamics. On their own, each of these simple models would be unable to explain and predict the various complexities encountered in multiscale dynamics of physical interest. Similar to the way one would synthesize opinions from various experts in order to reach a conclusion, we propose probabilistic models that combine the predictions of all these building blocks in order to obtain an integrated model that provides a good global approximation. We discuss parallelizable, online inference and learning algorithms that employ sequential Monte Carlo samplers and scale linearly with the dimensionality of the observed dynamics. We propose a Bayesian adaptive time-integration scheme that utilizes probabilistic predictive estimates and enables rigorous concurrent simulation over macroscopic time scales. The data-driven perspective advocated assimilates computational and experimental data and thus can materialize data-model fusion. It can deal with applications that lack a mathematical description and where only observational data is available. Furthermore, it makes nonintrusive use of existing computational models.

Methodological Bridges for Multi-Level Systems

“All models are wrong, but some are useful.” George Box, statistician Nowdays since modeling is growing of importance, this sentence would reflect many things: the continuous improvement of developing new models in all scientific fields, the different level of abstractions that a model could express and our difficulties in modelling multiscale dynamical systems. If the scientific progress relies on asking the right questions, we observe that different modelling methodologies often suggest slightly different questions and provide similar or different answers. In the figure on the right, a point in parameter space, given by a set of parameter values, defines a dynamics on the state space. If the system is prepared in an initial condition, then the dynamics typically lead to an attractor, pictured here as a star. We believe that a better understanding of the compositional framework of different modelling will bring easiness in making sense of large amount of heterogenesous data gathered on varison scales and also in robust parameter estimation and reverse engineering properties of various types of networks and parameters relationship. We found that when the system has a multiscale structure, formal methods help in guiding the exploration across dimensionality. In particular we report that Shape Calculus can effectively interface with a wide range of methodologies for example: Hybrid Automata, Stochastic Simulations (implemented as Gillespie algorithm or software Agent) and Model Checking, Bayesian inference and Model Checking, ODE (Ordinary Differential Equation) and PDE. We can imagine Shape Calculus as “high order” set of levers that pull the low order procedures implemented. Here we investigate the emerging behavior of connecting the Shape high levers (the tissue, the social network) with the low lever types of gear-bevel, spur-lever, helical gear constituents, etc (the cells, the individuals). Multi-Method Multi-Level approach

Strong convergence of principle of averaging for multiscale stochastic dynamical systems

Strong convergence of principle of averaging for multiscale stochastic dynamical systems

Identification of multiscale spatio-temporal dynamical systems using a wavelet multiresolution analysis

In this article, a new algorithm for the multiscale identification of spatio-temporal dynamical systems is derived. It is shown that the input and output observations can be represented in a multiscale manner based on a wavelet multiresolution analysis. The system dynamics at some specific scale of interest can then be identified using an orthogonal forward least-squares algorithm. This model can then be converted between different scales to produce predictions of the system outputs at different scales. The method can be applied to both multiscale and conventional spatio-temporal dynamical systems. For multiscale systems, the method can generate a parsimonious and effective model at a coarser scale while considering the effects from finer scales. Additionally, the proposed method can be used to improve the performance of the identification when the measurements are noisy. Numerical examples are provided to demonstrate the application of the proposed new approach.

Identification of partial differential equation models for a class of multiscale spatio-temporal dynamical systems

In this article, the identification of a class of multiscale spatio-temporal dynamical systems, which incorporate multiple spatial scales, from observations is studied. The proposed approach is a combination of Adams integration and an orthogonal least squares algorithm, in which the multiscale operators are expanded, using polynomials as basis functions, and the spatial derivatives are estimated by finite difference methods. The coefficients of the polynomials can vary with respect to the space domain to represent the feature of multiple scales involved in the system dynamics and are approximated using a B-spline wavelet multi-resolution analysis. The resulting identified models of the spatio-temporal evolution form a system of partial differential equations with different spatial scales. Examples are provided to demonstrate the efficiency of the proposed method.

Anomalous Diffusion in Folding Dynamics of Minimalist Protein Landscape

A novel method is proposed to quantify collectivity at different space and time scales in multiscale dynamics of proteins. This is based on the combination of the principal component (PC) and the concept recently developed for multiscale dynamical systems called the finite size Lyapunov exponent. The method can differentiate the well-known apparent correlation along the low-indexed PCs in multidimensional Brownian systems from the correlated motion inherent to the system. As an illustration, we apply the method to a model protein of 46 amino beads with three different types of residues. We show how the motion of the model protein changes depending on the space scales and the choices of degrees of freedom. In particular, anomalous superdiffusion is revealed along the low-indexed PC in the unfolded state. The implication of superdiffusion in the process of folding is also discussed.

Numerical Techniques for Multi-Scale Dynamical Systems with Stochastic Effects

Numerical schemes are presented for dynamical systems with multiple time-scales. Two classes of mehtods are discussed, depending on the time interval which the evolution of the slow variables in the system is sought. On rather short time intervals, the slow variables satisfy ordinary differential equations. On longer time intervals, however, fluctuations become important, and stochastic differential equations are obtained. In both cases, the numerical methods compute the evolution of the slow variables without having to derive explicitly the effective equations beforehand; rather, the coefficients entering these equations are obtained on the fly using simulations of appropriate auxiliary systems.