scholarly journals Shuttle Ascent Trajectory Optimization with Function Space Quasi-Newton Techniques

AIAA Journal ◽  
1976 ◽  
Vol 14 (10) ◽  
pp. 1369-1376 ◽  
Author(s):  
Ernest R. Edge ◽  
William F. Powers
Keyword(s):  
Function Space ◽  
Trajectory Optimization ◽  
Quasi Newton

Function Space BFGS Quasi-Newton Learning Algorithm for Time-Varying Recurrent Neural Networks

1996 ◽  
Vol 118 (1) ◽  
pp. 132-138
Author(s):  
Lilai Yan ◽  
C. James Li ◽  
Tung-Yung Huang
Keyword(s):  
Neural Networks ◽  
Boundary Value Problem ◽  
Function Space ◽  
Recurrent Neural Networks ◽  
Necessary Conditions ◽  
Learning Algorithm ◽  
Boundary Value ◽  
Time Varying ◽  
Point Boundary ◽  
Quasi Newton

This paper describes a new learning algorithm for time-varying recurrent neural networks whose weights are functions of time instead of scalars. First, an objective functional that is a function of the weight functions quantifying the discrepancies between the desired outputs and the network’s outputs is formulated. Then, dynamical optimization is used to derive the necessary conditions for the extreme of the functional. These necessary conditions result in a two-point boundary-value problem. This two-point boundary-value problem is subsequently solved by the Hilbert function space BFGS quasi-Newton algorithm, which is obtained by using the dyadic operator to extend the Euclidean space BFGS method into an infinite-dimensional, real Hilbert space. Finally, the ability of the network and the learning algorithm is demonstrated in the identification of three simulated nonlinear systems and a resistance spot welding process.

On the Use of POCS for Restoring the “Missing Wedge” in Electron Tomography

1990 ◽  
Vol 48 (1) ◽  
pp. 520-521
Author(s):  
Neng-Yu Zhang ◽  
Bruce F. McEwen ◽  
Joachim Frank
Keyword(s):  
Function Space ◽  
Convex Sets ◽  
Electron Tomography ◽  
Spinal Ganglion ◽  
Fourier Space ◽  
Missing Information ◽  
Voltage Electron ◽  
High Voltage Electron Microscope ◽  
Energy Bound ◽  
Spatial Boundedness

Reconstructions of asymmetric objects computed by electron tomography are distorted due to the absence of information, usually in an angular range from 60 to 90°, which produces a “missing wedge” in Fourier space. These distortions often interfere with the interpretation of results and thus limit biological ultrastructural information which can be obtained. We have attempted to use the Method of Projections Onto Convex Sets (POCS) for restoring the missing information. In POCS, use is made of the fact that known constraints such as positivity, spatial boundedness or an upper energy bound define convex sets in function space. Enforcement of such constraints takes place by iterating a sequence of function-space projections, starting from the original reconstruction, onto the convex sets, until a function in the intersection of all sets is found. First applications of this technique in the field of electron microscopy have been promising.To test POCS on experimental data, we have artificially reduced the range of an existing projection set of a selectively stained Golgi apparatus from ±60° to ±50°, and computed the reconstruction from the reduced set (51 projections). The specimen was prepared from a bull frog spinal ganglion as described by Lindsey and Ellisman and imaged in the high-voltage electron microscope.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

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