1,176 research outputs found
Stochastic collocation on unstructured multivariate meshes
Collocation has become a standard tool for approximation of parameterized
systems in the uncertainty quantification (UQ) community. Techniques for
least-squares regularization, compressive sampling recovery, and interpolatory
reconstruction are becoming standard tools used in a variety of applications.
Selection of a collocation mesh is frequently a challenge, but methods that
construct geometrically "unstructured" collocation meshes have shown great
potential due to attractive theoretical properties and direct, simple
generation and implementation. We investigate properties of these meshes,
presenting stability and accuracy results that can be used as guides for
generating stochastic collocation grids in multiple dimensions.Comment: 29 pages, 6 figure
Accuracy and Stability of Computing High-Order Derivatives of Analytic Functions by Cauchy Integrals
High-order derivatives of analytic functions are expressible as Cauchy
integrals over circular contours, which can very effectively be approximated,
e.g., by trapezoidal sums. Whereas analytically each radius r up to the radius
of convergence is equal, numerical stability strongly depends on r. We give a
comprehensive study of this effect; in particular we show that there is a
unique radius that minimizes the loss of accuracy caused by round-off errors.
For large classes of functions, though not for all, this radius actually gives
about full accuracy; a remarkable fact that we explain by the theory of Hardy
spaces, by the Wiman-Valiron and Levin-Pfluger theory of entire functions, and
by the saddle-point method of asymptotic analysis. Many examples and
non-trivial applications are discussed in detail.Comment: Version 4 has some references and a discussion of other quadrature
rules added; 57 pages, 7 figures, 6 tables; to appear in Found. Comput. Mat
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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
Minimizing Finite Sums with the Stochastic Average Gradient
We propose the stochastic average gradient (SAG) method for optimizing the
sum of a finite number of smooth convex functions. Like stochastic gradient
(SG) methods, the SAG method's iteration cost is independent of the number of
terms in the sum. However, by incorporating a memory of previous gradient
values the SAG method achieves a faster convergence rate than black-box SG
methods. The convergence rate is improved from O(1/k^{1/2}) to O(1/k) in
general, and when the sum is strongly-convex the convergence rate is improved
from the sub-linear O(1/k) to a linear convergence rate of the form O(p^k) for
p \textless{} 1. Further, in many cases the convergence rate of the new method
is also faster than black-box deterministic gradient methods, in terms of the
number of gradient evaluations. Numerical experiments indicate that the new
algorithm often dramatically outperforms existing SG and deterministic gradient
methods, and that the performance may be further improved through the use of
non-uniform sampling strategies.Comment: Revision from January 2015 submission. Major changes: updated
literature follow and discussion of subsequent work, additional Lemma showing
the validity of one of the formulas, somewhat simplified presentation of
Lyapunov bound, included code needed for checking proofs rather than the
polynomials generated by the code, added error regions to the numerical
experiment
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