4,182 research outputs found
Differentiating the multipoint Expected Improvement for optimal batch design
This work deals with parallel optimization of expensive objective functions
which are modeled as sample realizations of Gaussian processes. The study is
formalized as a Bayesian optimization problem, or continuous multi-armed bandit
problem, where a batch of q > 0 arms is pulled in parallel at each iteration.
Several algorithms have been developed for choosing batches by trading off
exploitation and exploration. As of today, the maximum Expected Improvement
(EI) and Upper Confidence Bound (UCB) selection rules appear as the most
prominent approaches for batch selection. Here, we build upon recent work on
the multipoint Expected Improvement criterion, for which an analytic expansion
relying on Tallis' formula was recently established. The computational burden
of this selection rule being still an issue in application, we derive a
closed-form expression for the gradient of the multipoint Expected Improvement,
which aims at facilitating its maximization using gradient-based ascent
algorithms. Substantial computational savings are shown in application. In
addition, our algorithms are tested numerically and compared to
state-of-the-art UCB-based batch-sequential algorithms. Combining starting
designs relying on UCB with gradient-based EI local optimization finally
appears as a sound option for batch design in distributed Gaussian Process
optimization
An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics
We present a three-point iterative method without memory for solving
nonlinear equations in one variable. The proposed method provides convergence
order eight with four function evaluations per iteration. Hence, it possesses a
very high computational efficiency and supports Kung and Traub's conjecture.
The construction, the convergence analysis, and the numerical implementation of
the method will be presented. Using several test problems, the proposed method
will be compared with existing methods of convergence order eight concerning
accuracy and basin of attraction. Furthermore, some measures are used to judge
methods with respect to their performance in finding the basin of attraction.Comment: arXiv admin note: substantial text overlap with arXiv:1508.0174
Computing Real Roots of Real Polynomials
Computing the roots of a univariate polynomial is a fundamental and
long-studied problem of computational algebra with applications in mathematics,
engineering, computer science, and the natural sciences. For isolating as well
as for approximating all complex roots, the best algorithm known is based on an
almost optimal method for approximate polynomial factorization, introduced by
Pan in 2002. Pan's factorization algorithm goes back to the splitting circle
method from Schoenhage in 1982. The main drawbacks of Pan's method are that it
is quite involved and that all roots have to be computed at the same time. For
the important special case, where only the real roots have to be computed, much
simpler methods are used in practice; however, they considerably lag behind
Pan's method with respect to complexity.
In this paper, we resolve this discrepancy by introducing a hybrid of the
Descartes method and Newton iteration, denoted ANEWDSC, which is simpler than
Pan's method, but achieves a run-time comparable to it. Our algorithm computes
isolating intervals for the real roots of any real square-free polynomial,
given by an oracle that provides arbitrary good approximations of the
polynomial's coefficients. ANEWDSC can also be used to only isolate the roots
in a given interval and to refine the isolating intervals to an arbitrary small
size; it achieves near optimal complexity for the latter task.Comment: to appear in the Journal of Symbolic Computatio
History Matching Using Principal Component Analysis
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