105,209 research outputs found
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
Software for Exact Integration of Polynomials over Polyhedra
We are interested in the fast computation of the exact value of integrals of
polynomial functions over convex polyhedra. We present speed ups and extensions
of the algorithms presented in previous work. We present the new software
implementation and provide benchmark computations. The computation of integrals
of polynomials over polyhedral regions has many applications; here we
demonstrate our algorithmic tools solving a challenge from combinatorial voting
theory.Comment: Major updat
Symmetry groups, semidefinite programs, and sums of squares
We investigate the representation of symmetric polynomials as a sum of
squares. Since this task is solved using semidefinite programming tools we
explore the geometric, algebraic, and computational implications of the
presence of discrete symmetries in semidefinite programs. It is shown that
symmetry exploitation allows a significant reduction in both matrix size and
number of decision variables. This result is applied to semidefinite programs
arising from the computation of sum of squares decompositions for multivariate
polynomials. The results, reinterpreted from an invariant-theoretic viewpoint,
provide a novel representation of a class of nonnegative symmetric polynomials.
The main theorem states that an invariant sum of squares polynomial is a sum of
inner products of pairs of matrices, whose entries are invariant polynomials.
In these pairs, one of the matrices is computed based on the real irreducible
representations of the group, and the other is a sum of squares matrix. The
reduction techniques enable the numerical solution of large-scale instances,
otherwise computationally infeasible to solve.Comment: 38 pages, submitte
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