Robust Averaged Control of Vibrations for the Bernoulli–Euler Beam Equation

Stochastic collocation and stochastic Galerkin methods for linear differential algebraic equations

Annals of operations research (vol. 28 - 1991) Markov decision processes — Part I Annals of operations research (vol. 29 - 1991) Markov decision processes — Part II: O. Hernandez-Lerma and J.B. Lasserre (eds.) Baltzer AG, Basel, 1991, 649 pages, ISSN 0254 5330

In this paper, the Stochastic Collocation Method is introduced into EMC uncertainty analysis. It is demonstrated to be accurate, with high computational efficiency, compared with the Method of Moments, the perturbation method and the Monte Carlo method. The Stochastic Collocation Method requires no modification to the original solver, and it is fit for being introduced into EMC simulation software.

Fast and accurate evaluation of system failure rate is performed using stochastic collocation methods. First, we demonstrate that variability in I/O performance, such as driven voltage and slew rate, will impact failure probability of the link. For instance, higher slew rates will lead to increased levels of crosstalk between signals. Crosstalk above a certain threshold can be an indicator of system failure. Then, we demonstrate that by defining an upper bound for failure probability, an optimally relaxed set of I/O performance metrics can be generated corresponding to a stochastic interconnect model. This can be achieved by minimizing a cost function formulated with some performance metric, if these metrics are random variables. Lastly, we demonstrate that, if more information about the packaging design becomes available, the randomness of the interconnect model can be reduced, leading to a more relaxed set of I/O performance metrics. The method proposed in [1] is then be used to verify that the packaging design will meet the tolerated failure probability.

This paper is concerned with spherically invariant or elliptically contoured vector random fields in space and/or time, which are formulated as scale mixtures of vector Gaussian random fields. While a spherically invariant vector random field may or may not have second-order moments, a spherically invariant second-order vector random field is determined by its mean and covariance matrix functions, just like the Gaussian one. This paper explores basic properties of spherically invariant second-order vector random fields, and proposes an efficient approach to develop covariance matrix functions for such vector random fields.

Our ultimate goal is a topology optimization for a permanent-magnet (PM) machine, while including material uncertainties. The uncertainties in the output data are, e.g., due to measurement errors in the non-/linear material laws. In the resulting stochastic forward problem, these uncertainties are stochastically modeled by random fields. The solution of the underlying PDE, which describes magnetostatics, is represented using the generalized polynomial chaos expansion. As crucial ingredient we exploit the stochastic collocation method (SCM). Eventually, this leads to a random-dependent bi-objective cost functional, which is comprised of the expectation and the variance. Subject to the optimization of the PM machine are the shapes of the rotor poles, which are described by zero-level sets. Thus, the optimization will be done by redistributing the iron and magnet material over the design domain, which allows to attain an innovative low cogging torque design of an electric machine. For this purpose, the gradient directions are evaluated by using the continuous design sensitivity analysis in conjunction with the SCM. In the end, our numerical result for the optimization of a two-dimensional model demonstrates that the proposed approach is robust and effective.

Uncertainty investigations in nonlinear aeroelastic systems

In this paper we propose and analyze a stochastic collocation method to solve elliptic partial differential equations with random coefficients and forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables. The method consists in a Galerkin approximation in space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space and naturally leads to the solution of uncoupled deterministic problems as in the Monte Carlo approach. It can be seen as a generalization of the stochastic Galerkin method proposed in [I. Babuška, R. Tempone, and G. E. Zouraris, SIAM J. Numer. Anal., 42 (2004), pp. 800–825] and allows one to treat easily a wider range of situations, such as input data that depend nonlinearly on the random variables, diffusivity coefficients with unbounded second moments, and random variables that are correlated or even unbounded. We provide a rigorous convergence analysis and demonstrate exponential convergence of the “probability error” with respect to the number of Gauss points in each direction in the probability space, under some regularity assumptions on the random input data. Numerical examples show the effectiveness of the method.

There have been many challenges in studying multiscale kinetic models with uncertainties. Quantifying uncertainties in the models such as arisen from collision kernels, initial or boundary data, forcing terms, and developing efficient computational methods have become an important task in industrial applications. In this article, we will report some of the recent progress on multiscale kinetic models with random inputs. We will discuss from the aspects of both mathematical theory, by using the hypocoercivity of kinetic operators, and numerical computation. Two categories of intrusive and non-intrusive methods will be addressed, in particular the stochastic Galerkin method for problems with relatively low-dimensional random inputs and stochastic collocation method in a multi-fidelity setting for more challenging, higher-dimensional problems. Numerical experiments for several kinetic models such as the Boltzmann, bipolar Boltzmann-Poisson, linear transport and epidemic kinetic equations will be shown as examples.

In this work we focus on the numerical approximation of the solution u of a linear elliptic PDE with stochastic coefficients. The problem is rewritten as a parametric PDE and the functional dependence of the solution on the parameters is approximated by multivariate polynomials. We first consider the stochastic Galerkin method, and rely on sharp estimates for the decay of the Fourier coefficients of the spectral expansion of u on an orthogonal polynomial basis to build a sequence of polynomial subspaces that features better convergence properties, in terms of error versus number of degrees of freedom, than standard choices such as Total Degree or Tensor Product subspaces. We consider then the Stochastic Collocation method, and use the previous estimates to introduce a new class of Sparse Grids, based on the idea of selecting a priori the most profitable hierarchical surpluses, that, again, features better convergence properties compared to standard Smolyak or tensor product grids. Numerical results show the effectiveness of the newly introduced polynomial spaces and sparse grids.

This paper presents an uncertainty propagation analysis of viscoelastic sandwich laminated structures using a stochastic collocation method based on Smolyak’s sparse grids approach. Shear deformation in the viscoelastic layers is accurately captured by means of layer-wise plate finite elements. Modal properties are determined by solving nonlinear eigenvalue problems due to the frequency-dependent nature of the considered viscoelastic materials. Variations in material properties and layer thicknesses are both treated in a probabilistic way. The variabilities of natural frequencies and modal loss factors of two different viscoelastic laminated plate structures (plane and cylindrical) are studied. Results show that the present method converges quickly with small sample sizes when compared to the classical Monte-Carlo method. The influence of the variability of several parameters on viscoelastic structure modal properties is commented.

This article introduces an ‐adaptive multi‐element stochastic collocation method, which additionally allows to re‐use existing model evaluations during either ‐ or ‐refinement. The collocation method is based on weighted Leja nodes. After ‐refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub‐element in a hierarchical manner. For ‐refinement, the local polynomial approximations are based on total‐degree or dimension‐adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non‐smooth or strongly localized response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.

The stochastic collocation method (Babuska et al. in SIAM J Numer Anal 45(3):1005---1034, 2007; Nobile et al. in SIAM J Numer Anal 46(5):2411---2442, 2008a; SIAM J Numer Anal 46(5):2309---2345, 2008b; Xiu and Hesthaven in SIAM J Sci Comput 27(3):1118---1139, 2005) has recently been applied to stochastic problems that can be transformed into parametric systems. Meanwhile, the reduced basis method (Maday et al. in Comptes Rendus Mathematique 335(3):289---294, 2002; Patera and Rozza in Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations Version 1.0. Copyright MIT, http://augustine.mit.edu , 2007; Rozza et al. in Arch Comput Methods Eng 15(3):229---275, 2008), primarily developed for solving parametric systems, has been recently used to deal with stochastic problems (Boyaval et al. in Comput Methods Appl Mech Eng 198(41---44):3187---3206, 2009; Arch Comput Methods Eng 17:435---454, 2010). In this work, we aim at comparing the performance of the two methods when applied to the solution of linear stochastic elliptic problems. Two important comparison criteria are considered: (1), convergence results of the approximation error; (2), computational costs for both offline construction and online evaluation. Numerical experiments are performed for problems from low dimensions $$O(1)$$ O ( 1 ) to moderate dimensions $$O(10)$$ O ( 10 ) and to high dimensions $$O(100)$$ O ( 100 ) . The main result stemming from our comparison is that the reduced basis method converges better in theory and faster in practice than the stochastic collocation method for smooth problems, and is more suitable for large scale and high dimensional stochastic problems when considering computational costs.

In this study, the asymptotic performance analysis for target detection-identification through Bayesian hypothesis testing in infrared images is presented. In the problem, probabilistic representations in terms of Bayesian pattern-theoretic framework is used. The infrared clutter is modelled as a second-order random field. The targets are represented as rigid CAD models. Their infinite variety of pose is modelled as transformations on the templates. For the template matching in hypothesis testing, a metric distance, based on empirical covariance, is used. The asymptotic performance of ATR algorithm under this metric and Euclidian metric is compared. The receiver operating characteristic (ROC) curves indicate that using the empirical covariance metric improves the performance significantly. These curves are also compared with the curves based on analytical expressions. The analytical results predict the experimental results quite well.

Data Assimilation for Marine Ecosystem Models