Stochastic Ordinary Differential Equations Research Articles

Data-driven surrogates for high dimensional models using Gaussian process regression on the Grassmann manifold

This paper introduces a surrogate modeling scheme based on Grassmannian manifold learning to be used for cost-efficient predictions of high-dimensional stochastic systems. The method exploits subspace-structured features of each solution by projecting it onto a Grassmann manifold. This point-wise linear dimensionality reduction harnesses the structural information to assess the similarity between solutions at different points in the input parameter space. The method utilizes a solution clustering approach in order to identify regions of the parameter space over which solutions are sufficiently similarly such that they can be interpolated on the Grassmannian. In this clustering, the reduced-order solutions are partitioned into disjoint clusters on the Grassmann manifold using the eigen-structure of properly defined Grassmannian kernels and, the Karcher mean of each cluster is estimated. Then, the points in each cluster are projected onto the tangent space with origin at the corresponding Karcher mean using the exponential mapping. For each cluster, a Gaussian process regression model is trained that maps the input parameters of the system to the reduced solution points of the corresponding cluster projected onto the tangent space. Using this Gaussian process model, the full-field solution can be efficiently predicted at any new point in the parameter space. In certain cases, the solution clusters will span disjoint regions of the parameter space. In such cases, for each of the solution clusters we utilize a second, density-based spatial clustering to group their corresponding input parameter points in the Euclidean space. The proposed method is applied to two numerical examples. The first is a nonlinear stochastic ordinary differential equation with uncertain initial conditions where the surrogate is used to predict the time history solution. The second involves modeling of plastic deformation in a model amorphous solid using the Shear Transformation Zone theory of plasticity, where the proposed surrogate is used to predict the full strain field of a material specimen under large shear strains.

Particle Velocity Fluctuations in Viscous Gas with Random Velocity as the Sum of Two Correlated Color Noises

The article focuses on methods for studying the phenomenon of two-phase turbulent flows. The turbulence effect on the movement of solid particles in a viscous gas is under study. Dynamics of particles movement in a gas is written in the Stokes approximation, which allows us to suppose the dynamic relaxation time to be a constant value.The random gas velocity is modeled by the sum of two correlated random noises. It is shown that this approach makes it possible to model noise of any structural complexity. The paper describes two research methods based on fundamentally different Euler and Lagrange approaches to the description of a continuous medium. The first approach uses a well-known generalization of the spectral analysis technique for random processes, a popular method for studying turbulence. The second approach implementation is based on the modern generalizations of the theory of numerical algorithms for solving stochastic ordinary differential equations. The spectral method is used to obtain analytical expressions of correlation functions and variance of random processes describing the velocity of gas and solid particles. The qualitative difference between the correlation of fluctuations of modulated random velocities and the behavior of correlations in the case of a single-component gas velocity composition is analyzed. A method of direct numerical simulation for studied processes based on the numerical solution of a stochastic ordinary differential equations system is proposed and analyzed in detail. An array of statistical data obtained as a result of direct numerical modeling is collected and processed. Analytical results are compared qualitatively with numerical results. The influence of input parameters on the character of turbulent flow is studied. The dynamic relaxation time has a significant effect on the complexity of the autocorrelation function of the particle velocity and the response function of particles to gas velocity fluctuations. It is shown that the obtained functions tend to the known results of the standard theory. The considered methods for describing two-phase turbulent flows hold promise for further research.

Simulation of thermal explosion on the base of solution of the equation for the probability density function of particle temperature fluctuations

The influence of temperature fluctuations of gas on the thermal explosion of particles with internal exothermic reactions is considered. The dependence of the rate of a chemical reaction on the concentration of a gas reagent is taken into account. Based on a rigorous system of heat and mass transfer equations, equations for the average temperature and concentration of the reagent inside the particle are obtained. A study of the dynamics of changes in the concentration of reagents and temperature inside the particle in a random field of temperature is carried out on the basis of the probability density function. Based on the solution of the systems of stochastic ordinary differential equations, a method of direct numerical simulation of random temperatures and reagent concentration in a particle is developed. A method for solving the equation for the probability density function is created. The results of comparing the empirical probability density function obtained by direct numerical simulation and the results of solving the equation for the probability density function are presented.

Numerical methods for the deterministic second moment equation of parabolic stochastic PDEs

Numerical methods for stochastic partial differential equations typically estimate moments of the solution from sampled paths. Instead, we shall directly target the deterministic equations satisfied by the mean and the spatio-temporal covariance structure of the solution process. In the first part, we focus on stochastic ordinary differential equations. For the canonical examples with additive noise (Ornstein–Uhlenbeck process) or multiplicative noise (geometric Brownian motion) we derive these deterministic equations in variational form and discuss their well-posedness in detail. Notably, the second moment equation in the multiplicative case is naturally posed on projective–injective tensor product spaces as trial–test spaces. We then propose numerical approximations based on Petrov–Galerkin discretizations with tensor product piecewise polynomials and analyze their stability and convergence in the natural tensor norms. In the second part, we proceed with parabolic stochastic partial differential equations with affine multiplicative noise. We prove well-posedness of the deterministic variational problem for the second moment, improving an earlier result. We then propose conforming space-time Petrov–Galerkin discretizations, which we show to be stable and quasi-optimal. In both parts, the outcomes are validated by numerical examples.

Computer simulating of nanoprocesses: Thermal jumps over a low barrier in the overdamped regime

We model numerically the process of a thermally activated decay of a metastable state over a low barrier in the overdamped regime. This process is typical for the recent nanoscale experiments reported in the literature. Within the framework of a novel unified computer code, two ways of modeling are applied: (i) the partial differential equation (Smoluchowski equation) and (ii) the stochastic ordinary differential equation (Langevin equation for the generalized coordinate). The advantages and disadvantages of both approaches are discussed and analyzed. Special attention is paid to the transient stage of the process yet the quasistationary stage is considered as well.

Data-based importance sampling estimates for extreme events

An accurate, efficient, and conceptually simple method is developed to estimate distributions of maxima of solutions of stochastic equations, i.e., ordinary or partial differential equations with random entries. The method is data-based. It constructs importance sampling (IS) or biasing measures from samples of surrogates of full model solutions of stochastic equations and uses these measures and mixtures of surrogate and full model samples to estimate probabilities of extreme events. Numerical examples are presented to illustrate the implementations of the proposed method and demonstrate numerically its performance.

A criterion on a repeller being a null set of any limit measure for stochastic differential equations

This paper studies limit behaviors of stationary measures for stochastic ordinary differential equations with nondegenerate noise and presents a criterion to guarantee that a repeller with zero Lebesgue measure is a null set of any limit measure. Using this criterion, we first provide a series of nontrivial concrete examples to show that their repelling limit cycles or quasi-periodic orbits are null sets for all limit measures, which deduces that all their limit measures are concentrated on stable equilibria and stable limit cycles or quasi-periodic orbits, and saddles. Interesting open questions on exact supports of limit measures are proposed.

An Ultra-Weak Discontinuous Galerkin Method with Implicit–Explicit Time-Marching for Generalized Stochastic KdV Equations

In this paper, an ultra-weak discontinuous Galerkin (DG) method is developed to solve the generalized stochastic Korteweg–de Vries (KdV) equations driven by a multiplicative temporal noise. This method is an extension of the DG method for purely hyperbolic equations and shares the advantage and flexibility of the DG method. Stability is analyzed for the general nonlinear equations. The ultra-weak DG method is shown to admit the optimal error of order $$k+1$$ in the sense of the spatial $$L^2(0,2\pi )$$-norm for semi-linear stochastic equations, when polynomials of degree $$k\ge 2$$ are used in the spatial discretization. A second order implicit–explicit derivative-free time discretization scheme is also proposed for the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples using Monte Carlo simulation are provided to illustrate the theoretical results.

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with nonglobally monotone coefficients

We develop a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $L^{p}$-distance between the solution process of an SDE and an arbitrary Itô process, which we view as a perturbation of the solution process of the SDE, by the $L^{q}$-distances of the differences of the local characteristics for suitable $p,q>0$. As one application of the developed perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with nonglobally monotone coefficients. As another application of the developed perturbation theory, we prove strong convergence rates for spatial spectral Galerkin approximations of solutions of semilinear SPDEs with nonglobally monotone nonlinearities including Cahn–Hilliard–Cook-type equations and stochastic Burgers equations. Further applications of the developed perturbation theory include regularity analyses of solutions of SDEs with respect to their initial values as well as small-noise analyses for ordinary and partial differential equations.

A Discontinuous Galerkin Method for Stochastic Conservation Laws

In this paper we present a discontinuous Galerkin (DG) method to approximate stochastic conservation laws, which is an efficient high-order scheme. We study the stability for the semidiscrete DG methods for fully nonlinear stochastic equations. Error estimates are obtained for smooth solutions of semilinear stochastic equations with variable coefficients. We also establish a derivative-free second-order time discretization scheme for matrix-valued stochastic ordinary differential equations. Numerical experiments are performed to confirm the analytical results.

A Backward Technique for Demographic Noise in Biological Ordinary Differential Equation Models

Physical systems described by deterministic differential equations represent idealized situations since they ignore stochastic effects. In the context of biomathematical modeling, we distinguish between environmental or extrinsic noise and demographic or intrinsic noise, for which it is assumed that the variation over time is due to demographic variation of two or more interacting populations (birth, death, immigration, and emigration). The modeling and simulation of demographic noise as a stochastic process affecting units of populations involved in the model is well known in the literature, resulting in discrete stochastic systems or, when the population sizes are large, in continuous stochastic ordinary differential equations and, if noise is ignored, in continuous ordinary differential equation models. The inverse process, i.e., inferring the effects of demographic noise on a natural system described by a set of ordinary differential equations, is still an issue to be addressed. With this paper, we provide a technique to model and simulate demographic noise going backward from a deterministic continuous differential system to its underlying discrete stochastic process, based on the framework of chemical kinetics, since demographic noise is nothing but the biological or ecological counterpart of intrinsic noise in genetic regulation. Our method can, thus, be applied to ordinary differential systems describing any kind of phenomena when intrinsic noise is of interest.

The discontinuous Galerkin method for stochastic differential equations driven by additive noises

In this paper, we present a discontinuous Galerkin (DG) finite element method for stochastic ordinary differential equations (SDEs) driven by additive noises. First, we construct a new approximate SDE whose solution converges to the solution of the original SDE in the mean-square sense. The new approximate SDE is obtained from the original SDE by approximating the Wiener process with a piecewise constant random process. The new approximate SDE is shown to have better regularity which facilitates the convergence proof for the proposed scheme. We then apply the DG method for deterministic ordinary differential equations (ODEs) to approximate the solution of the new SDE. Convergence analysis is presented for the numerical solution based on the standard DG method for ODEs. The orders of convergence are proved to be one in the mean-square sense, when p-degree piecewise polynomials are used. Finally, we present several numerical examples to validate the theoretical results. Unlike the Monte Carlo method, the proposed scheme requires fewer sample paths to reach a desired accuracy level.

Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

This work provides a finite dimensional reducing and a smooth approximating for a class of stochastic partial differential equations with an additive white noise. Using the invariant random cone to show the asymptotical completion, this stochastic partial differential equation is reduced to a stochastic ordinary differential equation on a random invariant manifold. Furthermore, after deriving the finite dimensional reducing for another stochastic partial differential equation driven by a Wong-Zakai scheme via a smooth colored noise, it is proved that when the smooth colored noise tends to the white noise, the solution and the finite dimensional reducing of the approximate system converge pathwisely to those of the original system.

Effects of Random Migration on the Growth of the Population of a Biological System

The problem under consideration is an example of the anomalous behavior of systems with an explosive pattern of behavior in a random medium. The growth is modeled of a population of microorganisms as a result of their migration into a closed region with favorable conditions for regeneration. The initial number of microorganisms inside the region under consideration can be zero and the number of organisms outside the region is a random quantity. The rate of growth of the population as a result of reproduction is proportional to the squared size of the population, and the rate of extinction of the population is proportional to the population size. The microorganisms arrive inside the region due to diffusion, and the condition for mass transfer is assigned at the region′s boundary. As a result of averaging over the region′s volume, an equation has been obtained for the average size of the population. For a stationary number of microorganisms outside the region, a new class has been obtained of analytical solutions describing the change in the size of the population inside the region depending on the biological characteristics of the system and parameters of migration. The effects of random fluctuations in the number of microorganisms outside the region under consideration are studied on the basis of two approaches. In the first case, a closed-form equation was obtained for the probability density function of a random number of microorganisms inside the region. The effects of random migration on the population′s growth dynamics have been investigated on the basis of a closed system for the first and second moments of the fluctuation in the number of microorganisms. In the second case, direct modeling was implemented of the behavior of a random size of the population inside the region on the basis of solution of a stochastic ordinary differential equation. It has been shown that random fluctuations may result in anomalous growth of the microbiological population size compared to the determinate case.

Chaotic analysis of Kelvin–Voigt viscoelastic plates under combined transverse periodic and white noise excitation: an analytic approach

Due to the use of materials with high structural damping in new applications and the role of plates as one of the basic elements of many engineering and interdisciplinary structures, the study of the stability of viscoelastic plates is very important in real conditions. In this article, the border curves of instability for a nonlinear Kelvin–Voigt viscoelastic plate under combined lateral periodic and white noise excitation are obtained analytically. Firstly, the governing equation of the plate is derived and then transformed into a nonlinear stochastic ordinary differential equation using Galerkin’s method. Secondly, Melnikov’s equation and its modified version in a stochastic sense are evaluated. At last, the border curves of instability for many types of plates and different variations of plate parameters and external excitations are obtained and drawn. The results show how and when the chaotic behavior changes by varying the plate parameters or fluctuating the intensities, magnitudes or frequencies of the external loads. It is shown that in the presence of white noise excitation the chaotic area become larger and this effect is larger at the frequencies far from the natural frequency of the corresponding linear system. It is also shown that under a combination of periodic and white noise excitations the chaotic behavior at low damping values might be completely different from the case there is only a periodic force.

Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics

We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov’s formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators.

On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity

The limiting behavior of stochastic evolution processes with small noise intensity ϵ is investigated in distribution-based approaches. Let μϵ be a stationary measure for stochastic process Xϵ with small ϵ and X0 be a semiflow on a Polish space. Assume that {μϵ: 0 < ϵ ⩽ ϵ0} is tight. Then all their limits in the weak sense are X0-invariant and their supports are contained in the Birkhoff center of X0. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, and stochastic functional differential equations driven by Brownian motion or Levy processes.

A Stochastic Multiscale Model of Cardiac Thin Filament Activation Using Brownian-Langevin Dynamics

We use Brownian-Langevin dynamics principles to derive a coarse-graining multiscale myofilament model that can describe the thin-filament activation process during contraction. The model links atomistic molecular simulations of protein-protein interactions in the thin-filament regulatory unit to sarcomere-level activation dynamics. We first calculate the molecular interaction energy between tropomyosin and actin surface using Brownian dynamics simulations. This energy profile is then generalized to account for the observed tropomyosin transitions between its regulatory stable states. The generalized energy landscape then served as a basis for developing a filament-scale model using Langevin dynamics. This integrated analysis, spanning molecular to thin-filament scales, is capable of tracking the events of the tropomyosin conformational changes as it moves over the actin surface. The tropomyosin coil with flexible overlap regions between adjacent tropomyosins is represented in the model as a system of coupled stochastic ordinary differential equations. The proposed multiscale approach provides a more detailed molecular connection between tropomyosin dynamics, the trompomyosin-actin interaction-energy landscape, and the generated force by the sarcomere.

Stochastic Differential Equations for the Variability of Atmospheric Convection Fluctuating Around the Equilibrium

AbstractMost convection parameterization schemes used within current atmospheric models make a convective equilibrium assumption, which breaks at the resolutions currently used by many numerical weather prediction models. To account for fluctuations of the cloud base mass flux about its equilibrium value, stochastic convection schemes have been developed for both deep (Plant &amp; Craig, 2008, https://doi.org/10.1175/2007JAS2263.1) and shallow (Sakradzija et al., 2015, https://doi.org/10.5194/npg-22-65-2015) convection. Due to the need to explicitly track individual clouds in each grid box, these schemes can be computationally expensive. Motivated by the above considerations, the present study demonstrates how the machinery of the above schemes can be reduced to the solution of two ordinary stochastic differential equations for the cloud number and the total cloud base mass flux. The properties of the resulting stochastic processes for the cloud number and the total mass flux are not exactly equivalent to those of the original schemes but recover them to a very good approximation.

Mean-Square Exponential Input-to-State Stability of Stochastic Gene Regulatory Networks with Multiple Time Delays

This paper is concerned with the input-to-state stability of stochastic gene regulatory networks with multiple time delays. It is well acknowledged that stochastic systems can accurately describe some complex systems with random disturbances. So it is significant that stochastic systems are applied to model gene regulatory networks because of the complex relationship between genes and proteins from a micro perspective. Considering the differences between stochastic differential equations and ordinary differential equations, we introduce the new stability criterion which is different from the general stability criteria. Making use of Lyapunov functionals, It$$\hat{o}$$ formula and Dynkin formula, we present sufficient conditions to guarantee that the proposed system is mean-square exponentially input-to-state stable. Moreover, numerical examples are given to illustrate validity and feasibility of the obtained results.