Iteration Homotopy Perturbation Research Articles

A projected homotopy perturbation method for nonlinear inverse problems in Banach spaces

Abstract In this paper, we propose and analyze a projected homotopy perturbation method based on sequential Bregman projections for nonlinear inverse problems in Banach spaces. To expedite convergence, the approach uses two search directions given by homotopy perturbation iteration, and the new iteration is calculated as the projection of the current iteration onto the intersection of stripes decided by above directions. The method allows to use L 1 {L^{1}} -like penalty terms, which is significant to reconstruct sparsity solutions. Under reasonable conditions, we establish the convergence and regularization properties of the method. Finally, two parameter identification problems are presented to indicate the effectiveness of capturing the property of the sparsity solutions and the acceleration effect of the proposed method.

Approximation Solution for Fuzzy Fractional-Order Partial Differential Equations

In this article, the authors study the comparison of the generalization differential transform method (DTM) and fuzzy variational iteration method (VIM) applied to determining the approximate analytic solutions of fuzzy fractional KdV, K(2,2) and mKdV equations. Furthermore, we establish the approximation solution two-and three-dimensional fuzzy time-fractional telegraphic equations via the fuzzy reduced differential transform method (RDTM). Finding an exact or closed-approximation solution to a differential equation is possible via fuzzy RDTM. Finally, we present the fuzzy fractional variational homotopy perturbation iteration method (VHPIM) with a modified Riemann-Liouville derivative to solve the fuzzy fractional diffusion equation (FDE). Using this approach achieves a rapidly convergent sequence that approaches the exact solution of the equation. The proposed methods are investigated based on fuzzy fractional derivatives with some illustrative examples. The results reveal that the schemes are highly effective for obtaining the solutions to fuzzy fractional partial differential equations.

An accelerated homotopy perturbation iteration for nonlinear ill-posed problems in Banach spaces with uniformly convex penalty

We propose and analyze an accelerated homotopy perturbation iteration based on Nesterov strategy for nonlinear inverse problems in Banach spaces. The method allows to use L 1-like and total variation-like penalty terms. Therefore, it can cope with the situation where the sought solution is sparse and piece-wise constant. Under some standard assumptions, we show the convergence and regularization properties of the method. Numerical experiments are presented to show the effectiveness as well as the acceleration effect of the proposed method.

Accelerated homotopy perturbation iteration method for a non-smooth nonlinear ill-posed problem

In this paper, a fast iterative algorithm based on J-order homotopy perturbation method is proposed for the nonlinear ill-posed problem whose forward operator is not Gâteaux differentiable. The Bouligand subderivative of the forward operator is utilized to replace the Fréchet derivative in iteration system. The sequential subspace optimization technique is introduced to accelerate the convergence speed, which regards the correction term of homotopy perturbation as multiple search directions to update the new iterate. To this end, the current iteration is sequentially projected to the stripes whose width is determined by search directions, the nonlinearity of the forward operator and noise level. We present the convergence analysis based on the asymptotic stability estimates and a generalized tangential cone condition. Numerical experiments are performed to illustrate the effectiveness of the proposed method.

Application to the Fuzzy Fractional Diffusion Equation by using Fuzzy Fractional Variational Homotopy Perturbation Iteration Method

Application to the Fuzzy Fractional Diffusion Equation by using Fuzzy Fractional Variational Homotopy Perturbation Iteration Method

An accelerated Kaczmarz type method for nonlinear inverse problems in Banach spaces with uniformly convex penalty

In this paper, we propose and analyze a novel Kaczmarz type method for solving inverse problems which can be written as systems of nonlinear operator equations in Banach spaces. The proposed method is formulated by combining homotopy perturbation iteration and Kaczmarz approach with uniformly convex penalty terms. The penalty term is allowed to be non-smooth, including the L1 and the total variation like penalty functionals, to reconstruct special features of solutions such as sparsity and piecewise constancy. To accelerate the iteration, we introduce a sophisticated rule to determine the step sizes per iteration. Under certain conditions, we present the convergence result of the proposed method in the exact data case. When the data is given approximately, together with a suitable stopping rule, we establish the stability and regularization properties of the method. Finally, some numerical experiments on parameter identification in partial differential equations by boundary as well as interior measurements are provided to validate the effectiveness of the proposed method.

An accelerated sequential subspace optimization method based on homotopy perturbation iteration for nonlinear ill-posed problems

Homotopy perturbation iteration is an effective and fast method for solving nonlinear ill-posed problems. It only needs approximately half the computation time of Landweber iteration to reach the similar recovery precision. In this paper, a Nesterov-type accelerated sequential subspace optimization method based on homotopy perturbation iteration is proposed for solving nonlinear inverse problems. The convergence analysis is provided under the general assumptions for iterative regularization methods. The numerical experiments on inverse potential problem indicate that the proposed method dramatically reduces the total number of iterations and time consumption to obtain satisfying approximations, especially for the problems with costly solution of forward calculation.

A new Kaczmarz-type method and its acceleration for nonlinear ill-posed problems

In this paper, we first introduce a new Kaczmarz-type method for solving inverse problems that can be written as a system of a finite number of nonlinear equations. The proposed homotopy perturbation Kaczmarz (HPK) iteration is seen as a hybrid method between the homotopy perturbation iteration and the Kaczmarz strategy. Furthermore, an accelerated homotopy perturbation Kaczmarz (AHPK) method is presented based on the general case of Nesterov’s acceleration scheme. Under the classical assumptions for iterative regularization methods, we provide the corresponding convergence analysis for HPK and AHPK, respectively. The HPK iteration is shown to have faster calculation speed and less time consumption than the Landweber–Kaczmarz iteration through some numerical experiments on inverse potential problem. Besides, the significantly reduced computation cost and much better reconstruction quality indicate a remarkable acceleration effect for AHPK.

Homotopy Decomposition Method for Solving Higher-Order Time- Fractional Diffusion Equation via Modified Beta Derivative

In this paper, the homotopy decomposition method with a modified definition of beta fractional derivative is adopted to find approximate solutions of higher-dimensional time-fractional diffusion equations. To apply this method, we find the modified beta integral for both sides of a fractional differential equation first, then using homotopy decomposition method we can obtain the solution of the integral equation in a series form. We compare the solutions obtained by the proposed method with the exact solutions obtained using fractional variational homotopy perturbation iteration method via modified Riemann-Liouville derivative. The comparison shows that the results are in a good agreement.

A comparative analysis for the solution of nonlinear Burgers' equation.

In this work, a comparative study of seven well-known mathematical techniques for the coupled Burgers' equations is reported. The techniques involve in this comparison are as follows: Laplace transform Adomian decomposition method, Laplace transform homotopy perturbation method, Variational iteration method, Variational iteration decomposition method, Variational iteration homotopy perturbation method, the optimal homotopy asymptotic method, and OHAM with Daftardar-Jafari polynomial. Here we considered a practical example which consists of coupled Burgers' equations with the kinematic viscosity ε=1. Convergence and stability analysis is a major part of this analysis. After a careful observation, it is found that the variational iteration method has faster convergence than all the remaining methods. Adomian decomposition method and Homotopy perturbation method show weaker stability in comparison with other involved techniques.

Numerical Solution of High-Order Fractional Volterra Integro-Differential Equations by Variational Homotopy Perturbation Iteration Method

In this paper, variational homotopy perturbation iteration method (VHPIM) has been applied along with Caputo derivative to solve high-order fractional Volterra integro-differential equations (FVIDEs). The “VHPIM” is present in all two steps. In order to indicate the efficiency and simplicity of the proposed method, we have presented some examples. All of the numerical computations in this study have been done on a personal computer applying some programs written in Maple18.

Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation

In this paper, we use the fractional variational homotopy perturbation iteration method (FVHPIM) with modified Riemann–Liouville derivative to solve a time-fractional diffusion equation. Using this method, a rapid convergent sequence tending to the exact solution of the equation can be obtained. To show the efficiency of the considered method, some numerical examples are presented.