We present an uncertainty propagation method for computing the expected value and variance of a quantity of interest (QoI), which can then be used in a robust design optimization. To avoid intractable costs due to high-dimensional integrals, we use the Hessian of the QoI to identify the dominant nonlinear directions. Specifically, the dominant Hessian eigenmodes provide the dimensions along which the QoI is integrated in stochastic space. Explicit computation of the Hessian is avoided by using Arnoldi's method to estimate the eigenmodes. The method is applied to multi-dimensional quadratic functions and its accuracy is examined for synthetic eigenmodes.
We present an extensible Julia-based solver for the Euler equations that uses a summationby-parts (SBP) discretization on unstructured triangular grids. While SBP operators have been used for tensor-product discretizations for some time, they have only recently been extended to simplices. Here we investigate the accuracy and stability properties of simplexbased SBP discretizations of the Euler equations. Non-linear stabilization is a particular concern in this context, because SBP operators are nearly skew-symmetric. We consider an edge-based stabilization method, which has previously been used for advection-diffusionreaction problems and the Oseen equations, and apply it to the Euler equations. Additionally, we discuss how the development of our software has been facilitated by the use of Julia, a new, fast, dynamic programming language designed for technical computing. By taking advantage of Julia's unique capabilities, code that is both efficient and generic can be written, enhancing the extensibility of the solver.
This paper proposes a family of exponential estimators for estimating the finite population variance using auxiliary information in simple random sampling. Expressions for bias, mean squared error and its minimum values have been obtained. The comparisons have been made with the usual unbiased estimator, Isaki (
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