Iterative Method using the Generalized Hopf Formula: Avoiding Spatial Discretization for Computing Solutions of Hamilton-Jacobi Equations for Nonlinear Systems

2019 ◽  
Author(s):  
Donggun Lee ◽  
Claire J. Tomlin
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method ◽  
Spatial Discretization ◽  
Hopf Formula ◽  
Jacobi Equations

An efficient iterative method with order five for solving nonlinear systems

2015 ◽  
Vol 14 (6) ◽  
pp. 363-372
Author(s):  
Xiao-Feng Wang ◽  
Tie Zhang ◽  
Fu Zheng ◽  
Wei-Yi Qian
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method

scholarly journals An EKF-Based Fixed-Point Iterative Filter for Nonlinear Systems

Sensors ◽  
2019 ◽  
Vol 19 (8) ◽  
pp. 1893
Author(s):  
Feng ◽  
Feng ◽  
Wen
Keyword(s):  
Fixed Point ◽  
Kalman Filter ◽  
Nonlinear Systems ◽  
Iterative Method ◽  
Taylor Series ◽  
Nonlinear Filtering ◽  
Higher Order ◽  
Point Function ◽  
State Estimate ◽  
Linearized System

In this paper, a fixed-point iterative filter developed from the classical extended Kalman filter (EKF) was proposed for general nonlinear systems. As a nonlinear filter developed from EKF, the state estimate was obtained by applying the Kalman filter to the linearized system by discarding the higher-order Taylor series items of the original nonlinear system. In order to reduce the influence of the discarded higher-order Taylor series items and improve the filtering accuracy of the obtained state estimate of the steady-state EKF, a fixed-point function was solved though a nested iterative method, which resulted in a fixed-point iterative filter. The convergence of the fixed-point function is also discussed, which provided the existing conditions of the fixed-point iterative filter. Then, Steffensen’s iterative method is presented to accelerate the solution of the fixed-point function. The final simulation is provided to illustrate the feasibility and the effectiveness of the proposed nonlinear filtering method.

scholarly journals Algorithm for Hamilton–Jacobi Equations in Density Space Via a Generalized Hopf Formula

2019 ◽  
Vol 80 (2) ◽  
pp. 1195-1239 ◽  
Author(s):  
Yat Tin Chow ◽  
Wuchen Li ◽  
Stanley Osher ◽  
Wotao Yin
Keyword(s):  
Hopf Formula ◽  
Jacobi Equations ◽  
Density Space

scholarly journals Increasing the convergence order of an iterative method for nonlinear systems

2012 ◽  
Vol 25 (12) ◽  
pp. 2369-2374 ◽  
Author(s):  
Alicia Cordero ◽  
José L. Hueso ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method ◽  
Convergence Order

scholarly journals Generalized High-Order Classes for Solving Nonlinear Systems and Their Applications

Mathematics ◽  
2019 ◽  
Vol 7 (12) ◽  
pp. 1194 ◽  
Author(s):  
Francisco I. Chicharro ◽  
Alicia Cordero ◽  
Neus Garrido ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Iterative Method ◽  
Fourth Order ◽  
Order Of Convergence ◽  
Divided Difference ◽  
High Order ◽  
Weight Functions ◽  
Systems Of Equations ◽  
Nonlinear Systems Of Equations ◽  
New Methods

A generalized high-order class for approximating the solution of nonlinear systems of equations is introduced. First, from a fourth-order iterative family for solving nonlinear equations, we propose an extension to nonlinear systems of equations holding the same order of convergence but replacing the Jacobian by a divided difference in the weight functions for systems. The proposed GH family of methods is designed from this fourth-order family using both the composition and the weight functions technique. The resulting family has order of convergence 9. The performance of a particular iterative method of both families is analyzed for solving different test systems and also for the Fisher’s problem, showing the good performance of the new methods.

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