ABS Algorithms for Linear Equations and Linear Least Squares

2006 ◽  
pp. 12-18
Author(s):  
Emilio Spedicato
2012 ◽  
Vol 256-259 ◽  
pp. 2854-2862
Author(s):  
Ling Bai ◽  
Jin Zhao Liu ◽  
Ai Min Xu ◽  
Xing Fang

HHT is widely used to analyze nonlinear and non-stationary signals. But how to extend boundaries of signals in decomposition processes is a key problem of HHT. A new technique based on response surface method (RSM), which establishes the recursive relations between sample points of signals, is presented to deal with this difficult problem. Besides, the boundary extension problem arising from HHT can be described by mathematical least squares problem but traditional gradient algorithms may diverge when the Hessian matrix of the object function of the least squares problem is non-positive. It has been proved that the generalized inverse of the linear equations (derived from the linear least squares problem) by singular value decomposition is the solution of original linear least squares problems. Thereby the divergence problem is also solved. Analysis results with respect to simulation signals and measured signals show that the method with new boundary extension technique performs successfully for HHT.


Sensors ◽  
2020 ◽  
Vol 20 (11) ◽  
pp. 3191 ◽  
Author(s):  
Zilong Zhou ◽  
Yichao Rui ◽  
Xin Cai ◽  
Ruishan Cheng ◽  
Xueming Du ◽  
...  

The location of an acoustic emission (AE) source is crucial for predicting and controlling potential hazards. In this paper, a novel weighted linear least squares location method for AE sources without measuring wave velocity is proposed. First, the governing equations of each sensor are established according to the sensor coordinates and arrival times. Second, a mean reference equation is established by taking the mean of the squared governing equations. Third, the system of linear equations can be obtained based on the mean reference equation, and their residuals are estimated to obtain their weights. Finally, the AE source coordinate is obtained by weighting the linear equations and inserting the parameter constraint. The AE location method is verified by a pencil lead break experiment, and the results show that the locating accuracy of the proposed method is significantly higher than that of traditional methods. Furthermore, the simulation test proves that the proposed method also has a better performance (location accuracy and stability) than the traditional methods under any given scale of arrival errors.


Author(s):  
Nived Chebrolu ◽  
Thomas Labe ◽  
Olga Vysotska ◽  
Jens Behley ◽  
Cyrill Stachniss

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