244 research outputs found
CECM: A continuous empirical cubature method with application to the dimensional hyperreduction of parameterized finite element models
We present the Continuous Empirical Cubature Method (CECM), a novel algorithm
for empirically devising efficient integration rules. The CECM aims to improve
existing cubature methods by producing rules that are close to the optimal,
featuring far less points than the number of functions to integrate.
The CECM consists on a two-stage strategy. First, a point selection strategy
is applied for obtaining an initial approximation to the cubature rule,
featuring as many points as functions to integrate. The second stage consists
in a sparsification strategy in which, alongside the indexes and corresponding
weights, the spatial coordinates of the points are also considered as design
variables. The positions of the initially selected points are changed to render
their associated weights to zero, and in this way, the minimum number of points
is achieved.
Although originally conceived within the framework of hyper-reduced order
models (HROMs), we present the method's formulation in terms of generic
vector-valued functions, thereby accentuating its versatility across various
problem domains. To demonstrate the extensive applicability of the method, we
conduct numerical validations using univariate and multivariate Lagrange
polynomials. In these cases, we show the method's capacity to retrieve the
optimal Gaussian rule. We also asses the method for an arbitrary
exponential-sinusoidal function in a 3D domain, and finally consider an example
of the application of the method to the hyperreduction of a multiscale finite
element model, showcasing notable computational performance gains.
A secondary contribution of the current paper is the Sequential Randomized
SVD (SRSVD) approach for computing the Singular Value Decomposition (SVD) in a
column-partitioned format. The SRSVD is particularly advantageous when matrix
sizes approach memory limitations
Is Gauss quadrature better than Clenshaw-Curtis?
We consider the question of whether Gauss quadrature, which is very famous, is more powerful than the much simpler Clenshaw-Curtis quadrature, which is less well-known. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following Elliott and O'Hara and Smith in the 1960s, the phenomenon is explained as a consequence of aliasing of coefficients in Chebyshev expansions. Then another explanation is offered based on the interpretation of a quadrature formula as a rational approximation of in the complex plane. Gauss quadrature corresponds to Pad\'e approximation at . Clenshaw-Curtis quadrature corresponds to an approximation whose order of accuracy at is only half as high, but which is nevertheless equally accurate near
High-Dimensional Stochastic Design Optimization by Adaptive-Sparse Polynomial Dimensional Decomposition
This paper presents a novel adaptive-sparse polynomial dimensional
decomposition (PDD) method for stochastic design optimization of complex
systems. The method entails an adaptive-sparse PDD approximation of a
high-dimensional stochastic response for statistical moment and reliability
analyses; a novel integration of the adaptive-sparse PDD approximation and
score functions for estimating the first-order design sensitivities of the
statistical moments and failure probability; and standard gradient-based
optimization algorithms. New analytical formulae are presented for the design
sensitivities that are simultaneously determined along with the moments or the
failure probability. Numerical results stemming from mathematical functions
indicate that the new method provides more computationally efficient design
solutions than the existing methods. Finally, stochastic shape optimization of
a jet engine bracket with 79 variables was performed, demonstrating the power
of the new method to tackle practical engineering problems.Comment: 18 pages, 2 figures, to appear in Sparse Grids and
Applications--Stuttgart 2014, Lecture Notes in Computational Science and
Engineering 109, edited by J. Garcke and D. Pfl\"{u}ger, Springer
International Publishing, 201
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