On the Power of Standard Information for Tractability for L2- Approximation in the Randomized Setting

We study the approximation of multivariate functions from a separable Hilbert space in the randomized setting with the error measured in the weighted L2 norm. We consider algorithms that use standard information Λstd consisting of function values or general linear information Λall consisting of arbitrary linear functionals. We investigate the equivalences of various notions of algebraic and exponential tractability in the randomized setting for Λstd and Λall for the normalized or absolute error criterion. For the normalized error criterion, we show that the power of Λstd is the same as that of Λall for all notions of exponential tractability and some notions of algebraic tractability without any condition. For the absolute error criterion, we show that the power of Λstd is the same as that of Λall for all notions of algebraic and exponential tractability without any condition. Specifically, we solve Open Problems 98, 101, 102 and almost solve Open Problem 100 as posed by E.Novak and H.Wo´zniakowski in the book: Tractability of Multivariate Problems, Volume III: Standard Information for Operators, EMS Tracts in Mathematics, Zürich, 2012.

Download Full-text

Optimization of sliding mode control with PID surface for robot manipulator by Evolutionary Algorithms

Open Computer Science ◽

10.1515/comp-2020-0144 ◽

2020 ◽

Vol 10 (1) ◽

pp. 396-407

Author(s):

Fatiha Loucif ◽

Sihem Kechida

Keyword(s):

Sliding Mode ◽

Robot Manipulator ◽

Absolute Error ◽

Grey Wolf Optimizer ◽

Optimal Parameters ◽

Error Criterion ◽

Trajectory Tracking Control ◽

Integral Time ◽

Sine Cosine Algorithm ◽

Antlion Optimization

AbstractIn this paper, a sliding mode controller (SMC) with PID surface is designed for the trajectory tracking control of a robot manipulator using different optimization algorithms such as, Antlion Optimization Algorithm (ALO) Sine Cosine Algorithm (SCA) Grey Wolf Optimizer (GWO) and Whale Optimizer Algorithm (WOA). The aim of this work is to introduce a novel SMC-PID-ALO to control nonlinear systems, especially the position of two of the joints of a 2DOF robot manipulator. The basic idea is to determinate four optimal parameters (Kp, Ki, Kd and lamda) ensuring the best performance of a robot manipulator system, minimizing the integral time absolute error criterion (ITAE) and the integral time square error criterion (ISTE). The robot manipulator is modeled in Simulink and the control is implemented using the MATLAB environment. The obtained simulation results prove the robustness of ALO in comparison with other algorithms.

Download Full-text

Fixed Points- Noncontractive Functions

Optimal Solution of Nonlinear Equations ◽

10.1093/oso/9780195106909.003.0007 ◽

2001 ◽

Author(s):

Krzysztof A. Sikorski

Keyword(s):

Fixed Points ◽

Absolute Error ◽

Small Error ◽

Homotopy Continuation ◽

Average Case Analysis ◽

Error Criterion ◽

Worst Case ◽

Average Case ◽

Worst Case Complexity ◽

Exponential Complexity

In this chapter we consider the approximation of fixed points of noncontractive functions with respect to the absolute error criterion. In this case the functions may have multiple and/or whole manifolds of fixed points. We analyze methods based on sequential function evaluations as information. The simple iteration usually does not converge in this case, and the problem becomes much more difficult to solve. We prove that even in the two-dimensional case the problem has infinite worst case complexity. This means that no methods exist that solve the problem with arbitrarily small error tolerance for some “bad” functions. In the univariate case the problem is solvable, and a bisection envelope method is optimal. These results are in contrast with the solution under the residual error criterion. The problem then becomes solvable, although with exponential complexity, as outlined in the annotations. Therefore, simplicial and/or homotopy continuation and all methods based on function evaluations exhibit exponential worst case cost for solving the problem in the residual sense. These results indicate the need of average case analysis, since for many test functions the existing algorithms computed ε-approximations with polynomial in 1/ε cost.

Download Full-text