scholarly journals Constructions of general polynomial lattice rules based on the weighted star discrepancy

2007 ◽  
Vol 13 (4) ◽  
pp. 1045-1070 ◽  
Author(s):  
Josef Dick ◽  
Peter Kritzer ◽  
Gunther Leobacher ◽  
Friedrich Pillichshammer
Keyword(s):  
Lattice Rules ◽  
General Polynomial ◽  
Star Discrepancy

scholarly journals Constructions of general polynomial lattices for multivariate integration

2007 ◽  
Vol 76 (1) ◽  
pp. 93-110 ◽  
Author(s):  
Peter Kritzer ◽  
Friedrich Pillichshammer
Keyword(s):  
Hilbert Spaces ◽  
Lattice Rules ◽  
Irreducible Polynomials ◽  
Worst Case ◽  
Multivariate Integration ◽  
Construction Algorithm ◽  
General Polynomial ◽  
Construction Algorithms ◽  
Digital Nets

We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed moduloarbitrarypolynomials.

scholarly journals Good lattice rules based on the general weighted star discrepancy

2006 ◽  
Vol 76 (258) ◽  
pp. 989-1004 ◽  
Author(s):  
Vasile Sinescu ◽  
Stephen Joe
Keyword(s):  
Lattice Rules ◽  
Good Lattice ◽  
Star Discrepancy

scholarly journals Iterative stability analysis for general polynomial control systems

2021 ◽  
Author(s):  
Bo Xiao ◽  
Hak-Keung Lam ◽  
Zhixiong Zhong
Keyword(s):  
Stability Analysis ◽  
Lyapunov Function ◽  
Control Systems ◽  
Polynomial System ◽  
Stability Conditions ◽  
General Polynomial ◽  
Main Challenge ◽  
Detailed Simulation ◽  
Feasible Solutions ◽  
The Stability

AbstractThe main challenge of the stability analysis for general polynomial control systems is that non-convex terms exist in the stability conditions, which hinders solving the stability conditions numerically. Most approaches in the literature impose constraints on the Lyapunov function candidates or the non-convex related terms to circumvent this problem. Motivated by this difficulty, in this paper, we confront the non-convex problem directly and present an iterative stability analysis to address the long-standing problem in general polynomial control systems. Different from the existing methods, no constraints are imposed on the polynomial Lyapunov function candidates. Therefore, the limitations on the Lyapunov function candidate and non-convex terms are eliminated from the proposed analysis, which makes the proposed method more general than the state-of-the-art. In the proposed approach, the stability for the general polynomial model is analyzed and the original non-convex stability conditions are developed. To solve the non-convex stability conditions through the sum-of-squares programming, the iterative stability analysis is presented. The feasible solutions are verified by the original non-convex stability conditions to guarantee the asymptotic stability of the general polynomial system. The detailed simulation example is provided to verify the effectiveness of the proposed approach. The simulation results show that the proposed approach is more capable to find feasible solutions for the general polynomial control systems when compared with the existing ones.

Sign in / Sign up

Export Citation Format

Share Document