An improved firefly algorithm for identifying parameters of nonlinear empirical models

The nonlinear model is to describe the vortex-induced resonance of long-span bridges under the action of natural wind. The identification accuracy of its parameters directly affects people's understanding of vortex-induced vibration. People have been trying different algorithms to solve this parameter identification problem, but the efficiency and accuracy of algorithms are not satisfied. In this work, a firefly algorithm based on local chaos search and brightness variant (FACLBV) is proposed. The characteristics of chaos make FACLBV search the widely local scope and improve the accuracy of the solution. FACLBV modifies the fixed initial brightness, discards the absorption coefficient of light intensity, links the initial brightness of every firefly with the position of its solution space, and sets the attraction of every firefly as a simple linear function, which reduces the complexity of the algorithm and improves the efficiency. In order to better verify the superiority of FACLBV, the simulation experiment includes three parts: the comparison between FACLBV and other firefly algorithms, the verification of the parameters identified by FACLBV, and the nonparametric test between FACLBV and other intelligent algorithms. Simulation results show that FACLBV is better than other algorithms in performance.

Maximal spaces for approximation rates in $$\ell ^1$$-regularization

We study Tikhonov regularization for possibly nonlinear inverse problems with weighted $$\ell ^1$$ ℓ 1 -penalization. The forward operator, mapping from a sequence space to an arbitrary Banach space, typically an $$L^2$$ L 2 -space, is assumed to satisfy a two-sided Lipschitz condition with respect to a weighted $$\ell ^2$$ ℓ 2 -norm and the norm of the image space. We show that in this setting approximation rates of arbitrarily high Hölder-type order in the regularization parameter can be achieved, and we characterize maximal subspaces of sequences on which these rates are attained. On these subspaces the method also converges with optimal rates in terms of the noise level with the discrepancy principle as parameter choice rule. Our analysis includes the case that the penalty term is not finite at the exact solution (’oversmoothing’). As a standard example we discuss wavelet regularization in Besov spaces $$B^r_{1,1}$$ B 1 , 1 r . In this setting we demonstrate in numerical simulations for a parameter identification problem in a differential equation that our theoretical results correctly predict improved rates of convergence for piecewise smooth unknown coefficients.

Orthogonal Learning-Based Gray Wolf Optimizer For Identifying The Uncertain Parameters of Various Photovoltaic Models

Determining the optimal parameters for the photovoltaic system (PV) model is essential during the design, evolution, development, estimation, and PV systems analysis. Therefore, it is crucial for the proper advancement of the best parameters of the PV models based on modern computational techniques. Thus, this work suggests a new Orthogonal-Learning-Based Gray Wolf Optimizer (OLBGWO) through a local exploration for estimating the unknown variables of PV cell models. The exploitation and exploration capability of the basic Gray Wolf Optimizer (GWO) is improved by the orthogonal-learning-based (OLB) approach, and this arrangement promotes a highly reliable equilibrium between the exploitation and exploration levels of the algorithm. In OLBGWO, the OLB strategy is used to find the best solution for the poor populations and directs the population to review the potential search area during the iterative process. Also, an exponential decay function is employed to decrease the value of vector a in GWO. The developed algorithm is directly applied to the parameter identification problem of the PV system. The proposed OLBGWO algorithm estimates the unknown parameters of the single-diode model (SDM), double-diode model (DDM), and PV module model. The performance of the OLBGWO is compared with other competitive algorithms to prove its superiority. The simulation results prove that the OLBGWO algorithm can achieve high solution accuracy with high convergence speed.

THE PARAMETER IDENTIFICATION PROBLEM FOR SYSTEM OF DIFFERENTIAL EQUATIONS

In this paper, the parameter identification problem for system of ordinary differential equations is considered. The parameter identification problem for system of ordinary differential equations is investigated by the Dzhumabaev’s parametrization method. At first, conditions for a unique solvability of the parameter identification problem for system of ordinary differential equations are obtained in the term of fundamental matrix of system’s differential part. Further, we establish conditions for a unique solvability of the parameter identification problem for system of ordinary differential equations in the terms of initial data. Algorithm for finding of approximate solution to a unique solvability of the parameter identification problem for system of ordinary differential equations is proposed and the conditions for its convergence are setted. Results this paper can be use for investigating of various problems with parameter and control problems for system of ordinary differential equations. The approach in this paper can be apply to the parameter identification problems for partial differential equations.

Damped Klein-Gordon equation with variable diffusion coefficient

<p style='text-indent:20px;'>We consider a damped Klein-Gordon equation with a variable diffusion coefficient. This problem is challenging because of the equation's unbounded nonlinearity. First, we study the nonlinearity's continuity properties. Then the existence and the uniqueness of the solutions is established. The main result is the continuity of the solution map on the set of admissible parameters. Its application to the parameter identification problem is considered.</p>

Error estimates for the iteratively regularized Newton–Landweber method in Banach spaces under approximate source conditions

In 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method.

Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition

In this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method.

A linear regularization method for a parameter identification problem in heat equation

Recently, Nair and Roy (2017) considered a linear regularization method for a parameter identification problem in an elliptic PDE. In this paper, we consider similar procedure for identifying the diffusion coefficient in the heat equation, modifying the Sobolev spaces involved appropriately. We derive error estimates under appropriate conditions and also consider the finite-dimensional realization of the method, which is essential for practical application. In the analysis of finite-dimensional realization, we give a procedure to obtain finite-dimensional subspaces of an infinite-dimensional Hilbert space {L^{2}(0,T;H^{1}(\Omega))} by doing double discretization, that is, discretization corresponding to both the space and time domain. Also, we analyze the parameter choice strategy and obtain an a posteriori parameter which is order optimal.