Random Ordinary Differential Equations Research Articles

GPU Optimization of Pseudo Random Number Generators for Random Ordinary Differential Equations

Solving differential equations with stochastic terms involves a massive use of pseudo random numbers. We present an application for the simulation of wireframe buildings under stochastic earthquake excitation. The inherent potential for vectorization of the application is used to its full extent on GPU accelerator hardware. A representative set of pseudo random number gen- erators for uniformly and normally distributed pseudo random numbers has been implemented, optimized, and benchmarked. The resulting optimized variants outperform standard library implementations on GPUs. The techniques and improvements shown in this contribution using the Kanai-Tajimi model can be generalized to other random differential equations or stochastic models as well as other accelerators.

A Lamperti-type representation of continuous-state branching processes with immigration

Guided by the relationship between the breadth-first walk of a rooted tree and its sequence of generation sizes, we are able to include immigration in the Lamperti representation of continuous-state branching processes. We provide a representation of continuous-state branching processes with immigration by solving a random ordinary differential equation driven by a pair of independent Levy processes. Stability of the solutions is studied and gives, in particular, limit theorems (of a type previously studied by Grimvall, Kawazu and Watanabe and by Li) and a simulation scheme for continuous-state branching processes with immigration. We further apply our stability analysis to extend Pitman's limit theorem concerning Galton-Watson processes conditioned on total population size to more general offspring laws.

Stable Integration of Stiff Random Ordinary Differential Equations

Implicit one-step numerical schemes are presented for the pathwise simulation of stiff random ordinary differential equations (RODEs), specifically an implicit averaged Euler scheme and an implicit averaged midpoint scheme. These involve averaging the noise inside the vector field and lead to integrals of the noise over discretization subintervals. Distributions of such integrals for Wiener and Ornstein-Uhlenbeck processes are determined and enable their efficient simulation. Convergence and B-stability proofs are presented and the schemes are tested and compared for a three-compartment system arising in the modelling and treatment of the hepatitis C virus.

Random Field Sampling for a Simplified Model of Melt-Blowing Considering Turbulent Velocity Fluctuations

In melt-blowing very thin liquid fiber jets are spun due to high-velocity air streams. In literature there is a clear, unsolved discrepancy between the measured and computed jet attenuation (thinning). In this paper we will verify numerically that the turbulent velocity fluctuations causing a random aerodynamic drag on the fiber jets—that has been neglected so far—are the crucial effect to close this gap. For this purpose, we model the velocity fluctuations as vector Gaussian random fields on top of a k–ϵ turbulence description and develop an efficient sampling procedure. Taking advantage of the special covariance structure the effort of the sampling is linear in the discretization and makes the realization possible. Numerical results are discussed for a simplified melt-blowing model consisting of a system of random ordinary differential equations.

An accuracy comparison of polynomial chaos type methods for the propagation of uncertainties

In (Augustin et al. in European J. Appl. Math. 19:149-190, 2008) we considered the Polynomial Chaos Expansion for the treatment of uncertainties in industrial applications. For many applications the method has been proven to be a computationally superior alternative to Monte Carlo evaluations. In the current overview we compare the accuracy of Polynomial Chaos type methods for the propagation of uncertainties in nonlinear problems and verify them on two examples relevant for industry. For weakly nonlinear time-dependent models, the generalized Kalman filter equations define an efficient method, yielding good approximations if the quantities of interest are restricted to the first two moments of the solution. Secondly, stochastic collocation is discussed. The method is applied to delay differential equations and random ordinary differential equations. Finally, a generalized PC method is discussed which is based on a subdivision of the random space. This approach is even suitable for highly nonlinear models.

Analytic-Numerical Solution of Random Boundary Value Heat Problems in a Semi-Infinite Bar

This paper deals with the analytic-numerical solution of random heat problems for the temperature distribution in a semi-infinite bar with different boundary value conditions. We apply a random Fourier sine and cosine transform mean square approach. Random operational mean square calculus is developed for the introduced transforms. Using previous results about random ordinary differential equations, a closed form solution stochastic process is firstly obtained. Then, expectation and variance are computed. Illustrative numerical examples are included.

Solving Continuous Models with Dependent Uncertainty: A Computational Approach

This paper presents a computational study on a quasi-Galerkin projection-based method to deal with a class of systems of random ordinary differential equations (r.o.d.e.’s) which is assumed to depend on a finite number of random variables (r.v.’s). This class of systems of r.o.d.e.’s appears in different areas, particularly in epidemiology modelling. In contrast with the other available Galerkin-based techniques, such as the generalized Polynomial Chaos, the proposed method expands the solution directly in terms of the random inputs rather than auxiliary r.v.’s. Theoretically, Galerkin projection-based methods take advantage of orthogonality with the aim of simplifying the involved computations when solving r.o.d.e.’s, which means to compute both the solution and its main statistical functions such as the expectation and the standard deviation. This approach requires the previous determination of an orthonormal basis which, in practice, could become computationally burden and, as a consequence, could ruin the method. Motivated by this fact, we present a technique to deal with r.o.d.e.’s that avoids constructing an orthogonal basis and keeps computationally competitive even assuming statistical dependence among the random input parameters. Through a wide range of examples, including a classical epidemiologic model, we show the ability of the method to solve r.o.d.e.’s.

Error analysis of generalized polynomial chaos for nonlinear random ordinary differential equations

This paper deals with the error analysis of generalized polynomial chaos (gPC) for nonlinear random ordinary differential equations. The analysis shows that the global error mainly relies on the projection error and the numerical error. For the deterministic systems obtained from the gPC method, a kind of numerical approach with error analysis is given. At last, a numerical experiment is carried out to support the theoretical results.

Stochastic Galerkin techniques for random ordinary differential equations

Over the last decade the stochastic Galerkin method has become an established method to solve differential equations involving uncertain parameters. It is based on the generalized Wiener expansion of square integrable random variables. Although there exist very sophisticated variants of the stochastic Galerkin method (wavelet basis, multi-element approach) convergence for random ordinary differential equations has rarely been considered analytically. In this work we develop an asymptotic upper boundary for the L 2-error of the stochastic Galerkin method. Furthermore, we prove convergence of a local application of the stochastic Galerkin method and confirm convergence of the multi-element approach within this context.

Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations

We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion.

SYNCHRONIZATION OF COUPLED STOCHASTIC SYSTEMS WITH MULTIPLICATIVE NOISE

We consider the synchronization of solutions to coupled systems of the conjugate random ordinary differential equations (RODEs) for the N-Stratronovich stochastic ordinary differential equations (SODEs) with linear multiplicative noise (N ∈ ℕ). We consider the synchronization between two solutions and among different components of solutions under one-sided dissipative Lipschitz conditions. We first show that the random dynamical system generated by the solution of the coupled RODEs has a singleton sets random attractor which implies the synchronization of any two solutions. Moreover, the singleton sets random attractor determines a stationary stochastic solution of the equivalently coupled SODEs. Then we show that any solution of the RODEs converge to a solution of the averaged RODE within any finite time interval as the coupled coefficient tends to infinity. Our results generalize the work of two Stratronovich SODEs in [9].

Pathwise Taylor schemes for random ordinary differential equations

Random ordinary differential equations (RODEs) are ordinary differential equations which contain a stochastic process in their vector fields. They can be analyzed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable. Traditional numerical schemes for ordinary differential equations thus do not achieve their usual order of convergence when applied to RODEs. Nevertheless, deterministic calculus can still be used to derive higher order numerical schemes for RODEs by means of a new kind of integral Taylor expansion. The theory is developed systematically here, applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes and compared with other numerical schemes for RODEs in the literature.

SYNCHRONIZATION OF SYSTEMS WITH MULTIPLICATIVE NOISE

The synchronization of Stratonovich stochastic differential equations (SDE) with a one-sided dissipative Lipschitz drift and linear multiplicative noise is investigated by transforming the SDE to random ordinary differential equations (RODE) and synchronizing their dynamics. In terms of the original SDE, this gives synchronization only when the driving noises are the same. Otherwise, the synchronization is modulo exponential factors involving Ornstein–Uhlenbeck processes corresponding to the driving noises. Moreover, this occurs no matter how large the intensity coefficients of the noise.

Pathwise convergent higher order numerical schemes for random ordinary differential equations

Random ordinary differential equations (RODEs) are ordinary differential equations (ODEs) with a stochastic process in their vector field. They can be analysed pathwise using deterministic calculus, but since the driving stochastic process is usually only Hölder continuous in time, the vector field is not differentiable in the time variable, so traditional numerical schemes for ODEs do not achieve their usual order of convergence when applied to RODEs. Nevertheless deterministic calculus can still be used to derive higher order numerical schemes for RODEs via integral versions of implicit Taylor-like expansions. The theory is developed systematically here and applied to illustrative examples involving Brownian motion and fractional Brownian motion as the driving processes.

Volatility time and properties of option prices

We use a notion of stochastic time, here called volatility time, to show convexity of option prices in the underlying asset if the contract function is convex as well as continuity and monotonicity of the option price in the volatility. The volatility time is obtained as the almost surely unique stopping time solution to a random ordinary differential equation related to volatility. This enables us to write price processes, or processes modeled by local martingales, as Brownian motions with respect to volatility time. The results are shown under very weak assumptions and are of independent interest in the study of stochastic differential equations. Options on several underlying assets are also studied and we prove that if the volatility matrix is independent of time, then the option prices decay with time if the contract function is convex. However, the option prices are no longer necessarily convex in the underlying assets and the option prices do not necessarily decay with time, if a time-dependent volatility is allowed.

Pathwise Approximation of Random Ordinary Differential Equations

Standard error estimates for one-step numerical schemes for nonautonomous ordinary differential equations usually assume appropriate smoothness in both time and state variables and thus are not suitable for the pathwise approximation of random ordinary differential equations which are typically at most continuous or Holder continuous in the time variable. Here it is shown that the usual higher order of convergence can be retained if one first averages the time dependence over each discretization subinterval.

Motion in a Gaussian incompressible flow

We prove that the solution of a system of random ordinary differential equations $d\mathbf{X}(t)/dt = \mathbf{V}(t, \mathbf{X}(t))$ with diffusive scaling, $\mathbf{X}_{\varepsilon}(t) = \varepsilon \mathbf{X}(t/ \varepsilon^2)$, converges weakly to a Brownian motion when $\varepsilon \downarrow 0$. We assume that $\mathbf{V}(t, \mathbf{x}), t \epsilon R, \mathbf{x} \epsilon R^d$ is a d-dimensional, random, incompressible, stationary Gaussian field which has mean zero and decorrelates in finite time.

Approximate solution of random ordinary differential equations

This is a largely expository paper on approximate methods of solving random ordinary differential equations, with an emphasis on direct numerical methods. Two methods are discussed in some detail and several others are mentioned briefly.

Approximate solution of random ordinary differential equations

This is a largely expository paper on approximate methods of solving random ordinary differential equations, with an emphasis on direct numerical methods. Two methods are discussed in some detail and several others are mentioned briefly.

Random ordinary differential equations

Random ordinary differential equations