7,734 research outputs found
Convex Optimal Uncertainty Quantification
Optimal uncertainty quantification (OUQ) is a framework for numerical
extreme-case analysis of stochastic systems with imperfect knowledge of the
underlying probability distribution. This paper presents sufficient conditions
under which an OUQ problem can be reformulated as a finite-dimensional convex
optimization problem, for which efficient numerical solutions can be obtained.
The sufficient conditions include that the objective function is piecewise
concave and the constraints are piecewise convex. In particular, we show that
piecewise concave objective functions may appear in applications where the
objective is defined by the optimal value of a parameterized linear program.Comment: Accepted for publication in SIAM Journal on Optimizatio
On the computation of Gaussian quadrature rules for Chebyshev sets of linearly independent functions
We consider the computation of quadrature rules that are exact for a
Chebyshev set of linearly independent functions on an interval . A
general theory of Chebyshev sets guarantees the existence of rules with a
Gaussian property, in the sense that basis functions can be integrated
exactly with just points and weights. Moreover, all weights are positive
and the points lie inside the interval . However, the points are not the
roots of an orthogonal polynomial or any other known special function as in the
case of regular Gaussian quadrature. The rules are characterized by a nonlinear
system of equations, and earlier numerical methods have mostly focused on
finding suitable starting values for a Newton iteration to solve this system.
In this paper we describe an alternative scheme that is robust and generally
applicable for so-called complete Chebyshev sets. These are ordered Chebyshev
sets where the first elements also form a Chebyshev set for each . The
points of the quadrature rule are computed one by one, increasing exactness of
the rule in each step. Each step reduces to finding the unique root of a
univariate and monotonic function. As such, the scheme of this paper is
guaranteed to succeed. The quadrature rules are of interest for integrals with
non-smooth integrands that are not well approximated by polynomials
Boundary integral methods in high frequency scattering
In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources
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