14,017 research outputs found
Fourth Order Gradient Symplectic Integrator Methods for Solving the Time-Dependent Schr\"odinger Equation
We show that the method of splitting the operator
to fourth order with purely positive coefficients produces excellent algorithms
for solving the time-dependent Schr\"odinger equation. These algorithms require
knowing the potential and the gradient of the potential. One 4th order
algorithm only requires four Fast Fourier Transformations per iteration. In a
one dimensional scattering problem, the 4th order error coefficients of these
new algorithms are roughly 500 times smaller than fourth order algorithms with
negative coefficient, such as those based on the traditional Ruth-Forest
symplectic integrator. These algorithms can produce converged results of
conventional second or fourth order algorithms using time steps 5 to 10 times
as large. Iterating these positive coefficient algorithms to 6th order also
produced better converged algorithms than iterating the Ruth-Forest algorithm
to 6th order or using Yoshida's 6th order algorithm A directly.Comment: 11 pages, 2 figures, submitted to J. Chem. Phy
Optimizing 0/1 Loss for Perceptrons by Random Coordinate Descent
The 0/1 loss is an important cost function for perceptrons. Nevertheless it cannot be easily minimized by most existing perceptron learning algorithms. In this paper, we propose a family of random coordinate descent algorithms to directly minimize the 0/1 loss for perceptrons, and prove their convergence. Our algorithms are computationally efficient, and usually achieve the lowest 0/1 loss compared with other algorithms. Such advantages make them favorable for nonseparable real-world problems. Experiments show that our algorithms are especially useful for ensemble learning, and could achieve the lowest test error for many complex data sets when coupled with AdaBoost
Performance Limits of Stochastic Sub-Gradient Learning, Part II: Multi-Agent Case
The analysis in Part I revealed interesting properties for subgradient
learning algorithms in the context of stochastic optimization when gradient
noise is present. These algorithms are used when the risk functions are
non-smooth and involve non-differentiable components. They have been long
recognized as being slow converging methods. However, it was revealed in Part I
that the rate of convergence becomes linear for stochastic optimization
problems, with the error iterate converging at an exponential rate
to within an neighborhood of the optimizer, for some and small step-size . The conclusion was established under weaker
assumptions than the prior literature and, moreover, several important problems
(such as LASSO, SVM, and Total Variation) were shown to satisfy these weaker
assumptions automatically (but not the previously used conditions from the
literature). These results revealed that sub-gradient learning methods have
more favorable behavior than originally thought when used to enable continuous
adaptation and learning. The results of Part I were exclusive to single-agent
adaptation. The purpose of the current Part II is to examine the implications
of these discoveries when a collection of networked agents employs subgradient
learning as their cooperative mechanism. The analysis will show that, despite
the coupled dynamics that arises in a networked scenario, the agents are still
able to attain linear convergence in the stochastic case; they are also able to
reach agreement within of the optimizer
Performance Limits of Stochastic Sub-Gradient Learning, Part II: Multi-Agent Case
The analysis in Part I revealed interesting properties for subgradient
learning algorithms in the context of stochastic optimization when gradient
noise is present. These algorithms are used when the risk functions are
non-smooth and involve non-differentiable components. They have been long
recognized as being slow converging methods. However, it was revealed in Part I
that the rate of convergence becomes linear for stochastic optimization
problems, with the error iterate converging at an exponential rate
to within an neighborhood of the optimizer, for some and small step-size . The conclusion was established under weaker
assumptions than the prior literature and, moreover, several important problems
(such as LASSO, SVM, and Total Variation) were shown to satisfy these weaker
assumptions automatically (but not the previously used conditions from the
literature). These results revealed that sub-gradient learning methods have
more favorable behavior than originally thought when used to enable continuous
adaptation and learning. The results of Part I were exclusive to single-agent
adaptation. The purpose of the current Part II is to examine the implications
of these discoveries when a collection of networked agents employs subgradient
learning as their cooperative mechanism. The analysis will show that, despite
the coupled dynamics that arises in a networked scenario, the agents are still
able to attain linear convergence in the stochastic case; they are also able to
reach agreement within of the optimizer
FE tool for drape modelling and resin pocket prediction of fully embedded optical fiber sensor system
This work highlights some of the achievements obtained within the EU FP7 SmartFiber project, aiming to develop a fully embeddable optical fiber sensor system including the interrogator chip. The focus is on resolving issues holding back the industrial uptake of optical sensing technology. In a first section, the development of a placement head for automated lay-down of an optical sensor line (including the SmartFiber interrogator system) during composite manufacturing is discussed. In a second section, the attention is shifted to the occurrence of resin pockets surrounding inclusions such as the SmartFiber interrogator. A computationally efficient F.E. approach is presented capable of accurately predicting resin pocket geometries. Both small (i.e. optical fiber sensors) and large (i.e. the SmartFiber interrogator) inclusions are considered, and the F.E. predictions are validated with experimental observations
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