Linear approximation

About: Linear approximation is a research topic. Over the lifetime, 3901 publications have been published within this topic receiving 74764 citations.

Improved bridge constructs for stochastic differential equations

Abstract: We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an Ito stochastic differential equation conditional on an observation taken at a fixed future time-point. Such realisations are typically termed diffusion bridges. Since, in general, no closed form expression exists for the transition densities of the process of interest, a widely adopted solution works with the Euler-Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. However, the density of the conditioned discrete-time process remains intractable, necessitating the use of computationally intensive methods such as Markov chain Monte Carlo. Designing an efficient proposal mechanism which can be applied to a noisy and partially observed system that exhibits nonlinear dynamics is a challenging problem, and is the focus of this paper. By partitioning the process into two parts, one that accounts for nonlinear dynamics in a deterministic way, and another as a residual stochastic process, we develop a class of novel constructs that bridge the residual process via a linear approximation. In addition, we adapt a recently proposed construct to a partial and noisy observation regime. We compare the performance of each new construct with a number of existing approaches, using three applications.

Gyroscopic stability and its loss in systems with two essential coordinates

Abstract: Cyclic systems with two essential coordinates are studied. The particular case is investigated, when a steady motion of the system is neutrally stable in linear approximation due to gyroscopic effects on an originally unstable equilibrium. Conditions are given to ensure the preservation of stability for the non-linear conservative system. In addition, a universal loss of stability is proved as a result of non-total dissipation or acceleration acting on the system.

Analytical Standard Uncertainty Evaluation Using Mellin Transform

Abstract: Uncertainty evaluation plays an important role in ensuring that a designed system can indeed achieve its desired performance. There are three standard methods to evaluate the propagation of uncertainty: 1) analytic linear approximation; 2) Monte Carlo (MC) simulation; and 3) analytical methods using mathematical representation of the probability density function (pdf). The analytic linear approximation method is inaccurate for highly nonlinear systems, which limits its application. The MC simulation approach is the most widely used technique, as it is accurate, versatile, and applicable to highly nonlinear systems. However, it does not define the uncertainty of the output in terms of those of its inputs. Therefore, designers who use this method need to resimulate their systems repeatedly for different combinations of input parameters. The most accurate solution can be attained using the analytical method based on pdf. However, it is unfortunately too complex to employ. This paper introduces the use of an analytical standard uncertainty evaluation (ASUE) toolbox that automatically performs the analytical method for multivariate polynomial systems. The backbone of the toolbox is a proposed ASUE framework. This framework enables the analytical process to be automated by replacing the complex mathematical steps in the analytical method with a Mellin transform lookup table and a set of algebraic operations. The ASUE toolbox was specifically designed for engineers and designers and is, therefore, simple to use. It provides the exact solution obtainable using the MC simulation, but with an additional output uncertainty expression as a function of its input parameters. This paper goes on to show how this expression can be used to prevent overdesign and/or suboptimal design solutions. The ASUE framework and toolbox substantially extend current analytical techniques to a much wider range of applications.