scholarly journals Mixed precision path tracking for polynomial homotopy continuation

2021 ◽  
Vol 47 (5) ◽  
Author(s):  
Sascha Timme
Keyword(s):  
Size Control ◽  
Initial Guess ◽  
Path Tracking ◽  
Homotopy Continuation ◽  
Step Size ◽  
Mixed Precision ◽  
Approximate Zero ◽  
Predictor Corrector ◽  
Adaptive Step ◽  
Region Of Convergence

AbstractThis article develops a new predictor-corrector algorithm for numerical path tracking in the context of polynomial homotopy continuation. In the corrector step, it uses a newly developed Newton corrector algorithm which rejects an initial guess if it is not an approximate zero. The algorithm also uses an adaptive step size control that builds on a local understanding of the region of convergence of Newton’s method and the distance to the closest singularity following Telen, Van Barel, and Verschelde. To handle numerically challenging situations, the algorithm uses mixed precision arithmetic. The efficiency and robustness are demonstrated in several numerical examples.

Continuation Power Flow: a Parameterization Technique and Adaptive Step Size Control

2021 ◽  
Author(s):  
Alfredo Bonini Neto ◽  
Luis Roberto Almeida Gabriel Filho ◽  
Dilson Amancio Alves
Keyword(s):  
Power Flow ◽  
Size Control ◽  
Step Size ◽  
Adaptive Step Size ◽  
Continuation Power Flow ◽  
Step Size Control ◽  
Adaptive Step

A new gradient based step size controlled inverse eigen sensitivity algorithm for identification of material and boundary parameters of plates

2015 ◽  
Vol 23 (16) ◽  
pp. 2619-2634
Author(s):  
Sk Abdul Goni ◽  
Subhajit Mondal ◽  
Sushanta Chakraborty
Keyword(s):  
Objective Function ◽  
Material Properties ◽  
Size Control ◽  
Initial Guess ◽  
Mode Shapes ◽  
Orthotropic Plates ◽  
Step Size ◽  
Step Size Control ◽  
Gradient Based ◽  
Size Controlled

The dynamic performance of any structure is function of existing material properties and boundary stiffness parameters which may deteriorate or become more flexible due to prolonged use. These parameters are estimated inversely through optimization of a suitable objective function. The gradient based optimization methods are preferred due to their faster convergence from a set of initial guess points, but suffers mostly from lack of reliable methodology to select appropriate step sizes. Arbitrary selection of step sizes may sometimes work well, depending upon the judgment of the user, but is case specific. The present work describes the estimation of existing material properties and boundary stiffness of isotropic and orthotropic plates from measured frequencies and mode shapes using a new gradient based step size controlled inverse eigensensitivity algorithm. The method takes a strategy that the step sizes automatically become smaller when the change in gradient of objective function is having a high value and similarly, takes larger steps when the gradient is remaining fairly constants in subsequent iterations. The results obtained from the investigations are encouraging, as some convergences could be achieved by this new adaptive step size control only, whereas methods adopting arbitrary or no step size control diverged.

Nonlinear Seismic Torsional Pounding Between an Asymmetric Tower and a Barrier

2009 ◽  
Vol 25 (4) ◽  
pp. 899-925 ◽  
Author(s):  
X. X. Wei ◽  
L. X. Wang ◽  
K. T. Chau
Keyword(s):  
Analytical Solution ◽  
Impact Velocity ◽  
Equations Of Motion ◽  
Size Control ◽  
Separation Distance ◽  
Step Size ◽  
Runge Kutta Method ◽  
Abrupt Jump ◽  
Velocity Spectra ◽  
Adaptive Step

Seismic torsional pounding between an asymmetric single-story tower and a neighboring barrier under harmonic ground excitation is modeled as the nonlinear Hertz contact in this paper. The governing equations of motion are numerically solved using the fourth-order Runge-Kutta method with an adaptive step size control. An analytical solution is obtained for the first time for a special case of periodic torsional pounding. Results of our numerical simulations reveal that torsional pounding tends to be much more complex and unpredictable than translational pounding, and most of them are either multiple or chaotic impacts. The maximum torsional impact velocity tends to be insensitive to the changes of separation distance and eccentricity as long as impact is developed. Although the analytical solution fails to predict the exact impact velocity of multiple torsional pounding, it does provide some useful insights into this complex phenomenon. The analytical solution succeeds in predicting the overall patterns as well as the abrupt jump of torsional impact velocity spectra.

A Coordinating Procedure for Tracing Static Bifurcation Points of Voltage Instability

2011 ◽  
Vol 413 ◽  
pp. 90-94
Author(s):  
Xue Song Zhou ◽  
Hua Xin Zhai ◽  
You Jie Ma
Keyword(s):  
Power Systems ◽  
Bifurcation Theory ◽  
Control Method ◽  
Size Control ◽  
Step Size ◽  
Step Size Control ◽  
Bifurcation Points ◽  
Voltage Instability ◽  
Adaptive Step ◽  
The Mathematical Model

Based on static bifurcation theory, a systematic procedure for tracing the bifurcation points of power system voltage instability was presented. Two main kinds of bifurcations, a common saddle node bifurcation (SNB) and a kind of bifurcation—limit induced bifurcation (LIB), which can cause the voltage collapse, were studied simultaneity and all types of device limits were considered. Firstly, make sure the mathematical model of the system. Secondly, starting from the basic running point of power systems, a predictor scheme was presented to find all the limit points (LP) and identify the LIB points; an adaptive step size control method considering the curvature of the manifold was introduced for tracing the equilibrium manifold and the detection of the SNB points without the existing of LPs. The two schemes were combined to lead a more effective method for SNB, LP, and LIB detection.

Sign in / Sign up

Export Citation Format

Share Document