Stochastic Differential Algebraic Equation Research Articles

СТОХАСТИЧНА ДЕСКРИПТОРНА ГРА ПЕРЕСЛІДУВАННЯ

A differential pursuit game in a stochastic descriptor linear system is analyzed. The system dynamics is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the formula of variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct a pursuer control bringing the dynamic vector of the system to a terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transient states in a radio technical filter with random perturbations in the form of white noise. Keywords: stochastic differential algebraic equation, Wiener process, descriptor system, differential game, radio technical filter, white noise.

Central exponents of linear stochastic differential-algebraic equations of index 1

Our aim in this paper is to establish a concept of central exponents of a stochastic differential-algebraic equation (SDAE) of index 1. For this purpose, we introduce the inherent regular stochastic differential equation (SDE) associated with a SDAE of index 1. Then, the central exponents are defined samplewise via the induced two-parameter stochastic flow generated by the inherent regular SDE. We prove that the central exponents are nonrandom and greater than or equal to the Lyapunov exponents.

Numerical simulations of wave propagation in a stochastic partial differential equation model for tumor–immune interactions

Abstract In this paper, we introduce a stochastic partial differential equation model for the spatial dynamic of tumor–immune interactions. We perform numerical simulations in order to investigate the propagation of traveling waves in model system under the influence of random space-time fluctuations. One of methods is to solve a stochastic partial differential equation system for tumor–immune cell densities. The second method is to solve a stochastic partial differential algebraic equation system in order to assess the wave behavior of the solution in comparison with the deterministic approach. Finally, we discuss the implications of the model results.

Rapid sampling of stochastic displacements in Brownian dynamics simulations with stresslet constraints.

Brownian Dynamics simulations are an important tool for modeling the dynamics of soft matter. However, accurate and rapid computations of the hydrodynamic interactions between suspended, microscopic components in a soft material are a significant computational challenge. Here, we present a new method for Brownian dynamics simulations of suspended colloidal scale particles such as colloids, polymers, surfactants, and proteins subject to a particular and important class of hydrodynamic constraints. The total computational cost of the algorithm is practically linear with the number of particles modeled and can be further optimized when the characteristic mass fractal dimension of the suspended particles is known. Specifically, we consider the so-called "stresslet" constraint for which suspended particles resist local deformation. This acts to produce a symmetric force dipole in the fluid and imparts rigidity to the particles. The presented method is an extension of the recently reported positively split formulation for Ewald summation of the Rotne-Prager-Yamakawa mobility tensor to higher order terms in the hydrodynamic scattering series accounting for force dipoles [A. M. Fiore et al., J. Chem. Phys. 146(12), 124116 (2017)]. The hydrodynamic mobility tensor, which is proportional to the covariance of particle Brownian displacements, is constructed as an Ewald sum in a novel way which guarantees that the real-space and wave-space contributions to the sum are independently symmetric and positive-definite for all possible particle configurations. This property of the Ewald sum is leveraged to rapidly sample the Brownian displacements from a superposition of statistically independent processes with the wave-space and real-space contributions as respective covariances. The cost of computing the Brownian displacements in this way is comparable to the cost of computing the deterministic displacements. The addition of a stresslet constraint to the over-damped particle equations of motion leads to a stochastic differential algebraic equation (SDAE) of index 1, which is integrated forward in time using a mid-point integration scheme that implicitly produces stochastic displacements consistent with the fluctuation-dissipation theorem for the constrained system. Calculations for hard sphere dispersions are illustrated and used to explore the performance of the algorithm. An open source, high-performance implementation on graphics processing units capable of dynamic simulations of millions of particles and integrated with the software package HOOMD-blue is used for benchmarking and made freely available in the supplementary material.

Adjoint equation and Lyapunov regularity for linear stochastic differential algebraic equations of index 1

We introduce a concept of adjoint equation and Lyapunov regularity of a stochastic differential algebraic Equation (SDAE) of index 1. The notion of adjoint SDAE is introduced in a similar way as in the deterministic differential algebraic equation case. We prove a multiplicative ergodic theorem for the adjoint SDAE and the adjoint Lyapunov spectrum. Employing the notion of adjoint equation and Lyapunov spectrum of an SDAE, we are able to define Lyapunov regularity of SDAEs. Some properties and an example of a metal oxide semiconductor field-effect transistor ring oscillator under thermal noise are discussed.

LYAPUNOV SPECTRUM OF NONAUTONOMOUS LINEAR STOCHASTIC DIFFERENTIAL ALGEBRAIC EQUATIONS OF INDEX-1

We introduce a concept of Lyapunov exponents and Lyapunov spectrum of a stochastic differential algebraic equation (SDAE) of index-1. The Lyapunov exponents are defined samplewise via the induced two-parameter stochastic flow generated by inherent regular stochastic differential equations. We prove that Lyapunov exponents are nonrandom.

Computing Stochastic Traveling Waves

The aim of this paper is to investigate new numerical methods for computing traveling wave solutions and new ways for estimating characteristic properties such as wave speed for stochastically forced partial differential equations. As a particular example we consider the Nagumo equation with multiplicative noise which we mainly consider in the Stratonovich sense. A standard approach for determining the position and hence speed of a wave is to compute the evolution of a level set. We compare this approach against an alternative where the wave position is found by minimizing the $L^2$ norm against a fixed reference profile. This approach can be used to freeze (or stop) the wave and obtain a stochastic partial differential algebraic equation that we then discretize and solve. Although attractive because it leads to a smaller domain size, it can be numerically unstable due to large convection terms. We compare numerically the different approaches for estimating the wave speed. We use these techniques to investigate the effect of both Itô and Stratonovich noise on the Nagumo equation as correlation length and noise intensity increase.

On parameter and state estimation for linear differential–algebraic equations

The current demand for more complex models has initiated a shift away from state-space models towards models described by differential–algebraic equations (DAEs). These models arise as the natural product of object-oriented modeling languages, such as Modelica. However, the mathematics of DAEs is somewhat more involved than the standard state-space theory. The aim of this work is to present a well-posed description of a linear stochastic differential–algebraic equation and more importantly explain how well-posed estimation problems can be formed. We will consider both the system identification problem and the state estimation problem. Besides providing the necessary theory we will also explain how the procedures can be implemented by means of efficient numerical methods.