In this paper, we propose a discretized Predictor-Corrector iterative Tikhonov regularization (DPC-ITR) method, which integrates multiscale Galerkin projection for discretization, iterative Tikhonov regularization for inversion, and the Modified Euler Method for time stepping. The proposed DPC-ITR method significantly accelerates computation for ill-posed inverse problems compared to standard iterative Tikhonov methods, while preserving the same order of numerical accuracy. Under specific regularity conditions, we rigorously derive a priori error estimates for the approximate solutions generated by the DPC-ITR method. Furthermore, we propose a novel heuristic parameter choice rule for the DPC-ITR method when applied to linear ill-posed integral equations. The proposed parameter choice rule, under certain conditions, enables the DPC-ITR method to generate approximate solutions that converge at order-optimal rates, as rigorously proven in our analysis. Our numerical experiments confirm that the DPC-ITR method equipped with the proposed heuristic parameter choice rule achieves the theoretically predicted convergence rates while demonstrating improved computational efficiency compared to conventional regularization approaches.

In this paper, we propose two parameter choice rules for the discretizing Tikhonov regularization via multiscale Galerkin projection for solving linear ill-posed integral equations. In contrast to previous theoretical analyses, we introduce a new concept called the projection noise level to obtain error estimates for the approximate solutions. This concept allows us to assess how noise levels change during projection. The balance principle and Hanke-Raus rule are modified by incorporating the error estimates of the projection noise level. We demonstrate the convergence rate of these two modified parameter choice rules through rigorous proof. In addition, we find that the error between the approximate solution and the exact solution improves as the noise frequency increases. Finally, numerical experiments are provided to illustrate the theoretical findings presented in this paper.

A simplified dynamical systems method for solving the nonlinear equation F（u）= f is studied in this paper.Under certain conditions of the operator F and the exact solution y,the error estimate of the solution of the dynamical systems equation is given,and the discrepancy principle of the posterior selection of regularized parameter is proposed to ensure the optimal rate of convergence of the solution of the dynamical systems equation.Compared with the traditional dynamical systems method,the simplified dynamical systems method reduces the computation amount of derivatives.

We propose a multiscale projection method for the numerical solution of the irtatively regularized Gauss-Newton method of nonlinear integral equations. An a posteriori rule is suggested to choose the stopping index of iteration and the rates of convergence are also derived under the Lipschitz condition. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

We propose a multilevel iteration method for the numerical solution of nonlinear ill-posed problems in the Hilbert space by using the Tikhonov regularization method. This leads to fast solutions of the discrete regularization methods for the nonlinear ill-posed equations. An adaptive choice of an a posteriori rule is suggested to choose the stopping index of iteration, and the rates of convergence are also derived. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

In this paper, we propose a projection method for the numerical solution of the simplified iteratively regularized Gauss-Newton method of nonlinear integral equations for which an a posteriori stopping rule is proposed to terminate the iteration. Such methods require only the computation of the Fréchet derivative at the initial approximation. Thus the computational work is considerably reduced. Under certain mild conditions, we give the convergence analysis and derive the order optimality. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method.

A simplified dynamical systems method for solving nonlinear ill-posed operator equation F（u）=f with monotone operator F is studied in this paper.The convergence of the simplified dynamical systems method is proved under certain conditions of the operator F and the exact solution y.In addition,the convergence of the discrete simplified dynamical systems method is studied.Finally,an example is given to verify the effectiveness of the new algorithm.

采用有效的自适应投影离散方法,用于求解第一类算子方程.该方法将分数阶渐近正则化与自适应投影方法相结合,与标准投影方法相比,所需的离散信息少得多,得到了近似解的最优收敛率.

本文采用多尺度配置法求解第一类弱扇形积分方程.将压缩配置法用于投影离散非定常迭代正则化方程,得到了近似解在Banach空间范数下误差估计,给出了迭代停止准则,确保近似解无穷范数下的最优收敛率.优点是确保了收敛率,减少了计算量.数值例子验证了算法的有效性.

考虑求解线性不适定问题的多尺度压缩投影算法,采用具有矩阵压缩策略的多尺度Galerkin方法,对Nesterov加速后的Landweber迭代正则化方程进行离散,给出近似解的先验误差估计,并提出后验参数选择策略,确保近似解的最优收敛率.数值实验表明将Nesterov加速方案应用到有限维空间求解线性不适定问题时,Land-weber迭代速度明显加快.

An adaptive multilevel iteration method via a coupled system is proposed, which leads to fast and effective algorithm for solving ill-posed integral equations. In this algorithm, the discrete layer and the iteration can be selected adaptively. By imposing projection conditions, convergence rates of an a priori parameter choice rule and two a posteriori parameter choice rule for this algorithm are established under certain source conditions. Numerical results are presented to illustrate the performance of adaptive multilevel iteration methods with the balance principle and the discrepancy principle, respectively.

研究了求解线性积分方程的多尺度压缩投影离散的Landweber算法,提出了Landweber迭代终止指标的停止准则,与传统的投影算法比较,减少了计算量,确保了近似解的最优收敛性.算例表明了该方法的有效性.

给出了求解第一类非线性积分方程的投影离散的改进Landweber迭代方法,给出了迭代停止准则,确保了近似解的收敛性与收敛率.数值例子说明了算法的有效性.

In this paper, Landweber iteration with a relaxation factor is proposed to solve nonlinear ill-posed integral equations. A compression multiscale Galerkin method that retains the properties of the Landweber iteration is used to discretize the Landweber iteration. This method leads to the optimal convergence rates under certain conditions. As a consequence, we propose a multiscale compression algorithm to solve nonlinear ill-posed integral equations. Finally, the theoretical analysis is verified by numerical results.

In this paper, we apply the multilevel augmentation method for solving ill-posed Fredholm integral equations of the first kind via iterated Tikhonov regularization method. The method leads to fast solutions of the discrete regularization methods for the equations. The convergence rates of iterated Tikhonov regularization are achieved by using a modified parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

In this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

In this paper, we consider a fast multiscale Galerkin method with compression technique for solving Fredholm integral equations of the first kind via the nonstationary iterated Tikhonov regularization. A modified a posteriori regularization parameter choice strategy is established, which leads to optimal convergence rates.

用多尺度快速配置法求解病态积分方程的隐式迭代方程.在积分算子是扇形紧算子时,该方法得到了离散隐式迭代方程的近似解.采用Morozov偏差原理作为停止准则,并证明了在该准则下隐式迭代正则化方法所得近似解的收敛率.最后,用数值实验证实理论结果和说明数值方法的有效性.

In this paper,we discuss the multiscale collocation method for solving Fredholm integral equation of the first kind.In the case that the integral operator is a sectorial compact operator,firstly the discrete iterative regularization equation is solved by using the multiscale collocation method with matrix compression strategy,then the error estimate of the discrete approximate solution is given and the selection method of iterative stopping criterion is proposed to ensure the optimal convergence rate of solution.Finally,numerical experiments are given to illustrate the efficiency of the proposed algorithm.

为了避免直接求解Tikhonov正则化方程,先将其分解成等价的方程组,再采用带有压缩策略的多尺度配置法离散方程组,然后,采用多层迭代法求解离散后的矩阵方程组,并提出先验和后验正则化参数选择策略,确保近似解的收敛率.