Adaptive sparse polynomial chaos expansion based on least angle regression

Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.

Calculation of Gauss quadrature rules.

http://www.ukm.my/kamal3/psm/pfpstochastics.pdf

The stochastic finite element method: Past, present and future

High resolution schemes for hyperbolic conservation laws

The DAKOTA (Design Analysis Kit for Optimization and Terascale Applications) toolkit provides a flexible and extensible interface between simulation codes and iterative analysis methods. DAKOTA contains algorithms for optimization with gradient and nongradient-based methods; uncertainty quantification with sampling, reliability, and stochastic finite element methods; parameter estimation with nonlinear least squares methods; and sensitivity/variance analysis with design of experiments and parameter study methods. These capabilities may be used on their own or as components within advanced strategies such as surrogate-based optimization, mixed integer nonlinear programming, or optimization under uncertainty. By employing object-oriented design to implement abstractions of the key components required for iterative systems analyses, the DAKOTA toolkit provides a flexible and extensible problem-solving environment for design and performance analysis of computational models on high performance computers. This report serves as a reference manual for the commands specification for the DAKOTA software, providing input overviews, option descriptions, and example specifications. DAKOTA Version 5.0 Reference Manual generated on May 7, 2010

/pdf/dakota-a-multilevel-parallel-object-oriented-framework-for-52psgais7x.pdf

DAKOTA : a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis. Version 5.0, user's manual.

Complex o w and uid-structure interaction simulations require ecien t uncertainty quantication methods. In addition, a good property of an uncertainty quantication method is non-intrusiveness, meaning that existing deterministic solvers can be used for uncertainty quantication. In this paper the Probabilistic Collocation method is introduced, which combines the non-intrusiveness of the Stochastic Collocation method with the exponential convergence for arbitrary probability distributions of the Galerkin Polynomial Chaos method. Due to the non-intrusiveness and exponential convergence, the Probabilistic Collocation method requires only a few collocation points for a high accuracy. For the one dimensional piston problem the eciency of the Probabilistic Collocation method is compared with the Galerkin Polynomial Chaos method, the Non-Intrusive Polynomial Chaos method and the Stochastic Collocation method. The strength of the Probabilistic Collocation method is demonstrated by solving steady o w around a NACA0012 airfoil with an uncertain free stream velocity using a commercial o w solver. Dieren t possibilities of representing the stochastic solution are demonstrated to show the potential use of uncertainty quantication.

Probabilistic Collocation: An Efficient Non-Intrusive Approach for Arbitrarily Distributed Parametric Uncertainties

Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos

Modeling Arbitrary Uncertainties Using Gram-Schmidt Polynomial Chaos

Stochastic collocation (SC) methods for uncertainty quantification (UQ) in computational problems are usually limited to hypercube probability spaces due to the structured grid of their quadrature rules. Nonhypercube probability spaces with an irregular shape of the parameter domain do, however, occur in practical engineering problems. For example, production tolerances and other geometrical uncertainties can lead to correlated random inputs on nonhypercube domains. In this paper, a simplex stochastic collocation (SSC) method is introduced, as a multielement UQ method based on simplex elements, that can efficiently discretize nonhypercube probability spaces. It combines the Delaunay triangulation of randomized sampling at adaptive element refinements with polynomial extrapolation to the boundaries of the probability domain. The robustness of the extrapolation is quantified by the definition of the essentially extremum diminishing (EED) robustness principle. Numerical examples show that the resulting SSC-EED method achieves superlinear convergence and a linear increase of the initial number of samples with increasing dimensionality. These properties are demonstrated for uniform and nonuniform distributions, and correlated and uncorrelated parameters in problems with 15 dimensions and discontinuous responses.

Simplex Stochastic Collocation with Random Sampling and Extrapolation for Nonhypercube Probability Spaces

Mesh deformation algorithms usually require solving a system of equations which can be computationally intensive for complex three-dimensional applications. Here an explicit mesh deformation method is proposed based on Inverse Distance Weighting (IDW) interpolation of the boundary node displacements to the interior of the flow domain. The point-by-point mesh deformation algorithm results in a straightforward implementation and parallelization. Fluid-structure interaction applications to a two-dimensional airfoil and the three-dimensional AGARD 445.6 wing benchmark show a reduction of the computational costs up to a factor 100 with respect to Radial Basis Function (RBF) mesh deformation for a comparable simulation accuracy.

Jeroen A. S. Witteveen

Papers

Probabilistic Collocation: An Efficient Non-Intrusive Approach for Arbitrarily Distributed Parametric Uncertainties

Modeling physical uncertainties in dynamic stall induced fluid-structure interaction of turbine blades using arbitrary polynomial chaos

Modeling Arbitrary Uncertainties Using Gram-Schmidt Polynomial Chaos

Simplex Stochastic Collocation with Random Sampling and Extrapolation for Nonhypercube Probability Spaces

Explicit Mesh Deformation Using Inverse Distance Weighting Interpolation