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

Abstract

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.

Online parameter identification methods for oscillatory systems: Estimation of changes in stiffness properties

“…If the nonlinearity of the underlying system is too large, however, it may fail (cf. Augustin et al., 2013).…”

“…To overcome some of these limitations, various extensions have been introduced, notably the extended Kalman filter (EKF) (Anderson and Moore, 1979;Chui and Chen, 1999;Welch and Bishop, 2006) and the unscented Kalman filter (Julier et al, 2000;Kandepu et al, 2008) which have, for example, successfully been applied to state estimation problems for induction machines (Loron and Laliberte, 1993;Kandepu et al, 2008). A comparison of these nonlinear filtering techniques is given in Augustin et al (2013). Other approaches for nonlinear state estimation include particle filters and moving horizon estimators (Rawlings and Bakshi, 2006;Rawlings and Mayne, 2015).…”

Sensitivity of Noninvasive Cardiac Electrophysiological Imaging to Variations in Personalized Anatomical Modeling

“…Particularly, it uses optimization methods to find a minimal number of samples that can preserve the statistical moments of a given distribution up to a desired order [23]. Compared to SC, in the presence of a high-dimensional input variable x, UT requires substantially less number of samples to accurately estimate statistics of output variable up to the second moment [24]. …”

A new network approach to Bayesian inference in partial differential equations

“…Popular versions are the Metropolis–Hastings algorithm and Gibbs sampling . If the data come sequentially and one wishes for real‐time inference, sequential Monte Carlo methods or Kalman filters can be applied; in recent years, a multitude of methods have been developed, e.g., variants of Kalman filters for nonlinear models and unscented filters , to name but a few.…”

“…Popular versions are the Metropolis-Hastings algorithm [17] and Gibbs sampling [6]. If the data come sequentially and one wishes for real-time inference, sequential Monte Carlo methods [18] or Kalman filters [19] can be applied; in recent years, a multitude of methods have been developed, e.g., variants of Kalman filters for nonlinear models and unscented filters [20][21][22], to name but a few.The computational bottleneck for all these methods is that the potentially complex forward model in the likelihood has to be evaluated over and over again. Improvements use surrogate forward models based on polynomial chaos and Karhunen-Loève expansions [23,24].…”