Parameter Choice Rule Research Articles

Nesterov acceleration‐based iterative method for backward problem of distributed‐order time‐fractional diffusion equation

This paper is concerned with the inverse problem of determining the initial value of the distributed‐order time‐fractional diffusion equation from the final time observation data, which arises in some ultra‐slow diffusion phenomena in applied areas. Since the problem is ill‐posed, we propose an iterated regularization method based on the Nesterov acceleration strategy to deal with it. Convergence rates for the regularized approximation solution are given under both the a priori and a posteriori regularization parameter choice rules. It is shown that the proposed method can always yield the order optimal convergence rates as long as the iteration parameter which appears in the Nesterov acceleration strategy is chosen large enough. In numerical aspect, the main advantage of the proposed method lies in its simplicity. Specifically, due to the Nesterov acceleration strategy, only a few number of iteration steps are required to obtain the approximation solution, and at each iteration step, we only need to numerically solve the standard initial‐boundary value problem for the distributed‐order time‐fractional diffusion equation. Some numerical examples including one‐dimensional and two‐dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.

THE CONDITIONAL STABILITY AND AN ITERATIVE REGULARIZATION METHOD FOR A FRACTIONAL INVERSE ELLIPTIC PROBLEM OF TRICOMI-GELLERSTEDT-KELDYSH TYPE

The present paper is devoted to identifying an inaccessible boundary condition for a fractional elliptic problem of Tricomi-Gellerstedt-Keldysh-type. Using the expansion Fourier method, the considered problem can be reformulated as an operator equation of the first kind. To construct a stabilized approximate solution we employ a variant of the iterative method. We also present error estimates between the exact solution and the regularized solution by the a priori and the a posteriori parameter choice rules. Finally, some numerical verifications on the efficiency and accuracy of the proposed algorithm is presented.

Two Regularization Methods for Identifying the Spatial Source Term Problem for a Space-Time Fractional Diffusion-Wave Equation

In this paper, we delve into the challenge of identifying an unknown source in a space-time fractional diffusion-wave equation. Through an analysis of the exact solution, it becomes evident that the problem is ill-posed. To address this, we employ both the Tikhonov regularization method and the Quasi-boundary regularization method, aiming to restore the stability of the solution. By adhering to both a priori and a posteriori regularization parameter choice rules, we derive error estimates that quantify the discrepancies between the regularization solutions and the exact solution. Finally, we present numerical examples to illustrate the effectiveness and stability of the proposed methods.

A discretizing Tikhonov regularization method via modified parameter choice rules

Abstract 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.

Adaptive minimax optimality in statistical inverse problems via SOLIT—Sharp Optimal Lepskiĭ-Inspired Tuning

We consider statistical linear inverse problems in separable Hilbert spaces and filter-based reconstruction methods of the form fˆα=qαT∗TT∗Y , where Y is the available data, T the forward operator, qαα∈A an ordered filter, and α > 0 a regularization parameter. Whenever such a method is used in practice, α has to be appropriately chosen. Typically, the aim is to find or at least approximate the best possible α in the sense that mean squared error (MSE) E[∥fˆα−f†∥2] w.r.t. the true solution f† is minimized. In this paper, we introduce the Sharp Optimal Lepskiĭ-Inspired Tuning (SOLIT) method, which yields an a posteriori parameter choice rule ensuring adaptive minimax rates of convergence. It depends only on Y and the noise level σ as well as the operator T and the filter qαα∈A and does not require any problem-dependent tuning of further parameters. We prove an oracle inequality for the corresponding MSE in a general setting and derive the rates of convergence in different scenarios. By a careful analysis we show that no other a posteriori parameter choice rule can yield a better performance in terms of the order of the convergence rate of the MSE. In particular, our results reveal that the typical understanding of Lepskiĭ-type methods in inverse problems leading to a loss of a log factor is wrong. In addition, the empirical performance of SOLIT is examined in simulations.

Discrete Legendre spectral projection-based methods for Tikhonov regularization of first kind Fredholm integral equations

In this paper, we apply the discrete Legendre Galerkin and multi-Galerkin methods to find the approximate solution of the Tikhonov regularized equation of the Fredholm integral equations of the first kind. We evaluate the error bounds for the approximate solutions with the exact solution in the infinity norm. We provide an a priori parameter choice strategy to find the convergence rates under the infinity norm. Since smoothness of the solution is not known in applied problems, we discuss an adaptive parameter choice rule to choose the regularization parameter, and then using this regularization parameter, we obtain the order of convergence in infinity norm. We give test examples to justify the theoretical estimates.

Determine source of Cahn–Hilliard equation with time‐fractional derivative by iterative variational regularization method

In this paper, we consider the inverse problem for the time‐fractional two‐dimensional Cahn–Hilliard equation. The ill‐posedness and a conditional stability of the inverse problem are proved. We present a new iterative variational regularization method to solve it. The convergence rates of the regularized solutions under the a priori regularization parameter choice rule and a posteriori regularization parameter choice rule are obtained. Numerical examples illustrate the effectiveness and stability of our proposed method.

The iterative generalized quasi‐boundary value method for an inverse source problem of time fractional diffusion‐wave equation in a cylinder

This paper is devoted to solve an inverse source problem of time fractional diffusion‐wave equation in a cylinder. The ill‐posedness, existence, and uniqueness of the inverse source problem are proved. We propose an iterative generalized quasi‐boundary value regularization method to deal with the inverse source problem. The convergence rates of the regularized solution are obtained under the a priori regularization parameter choice rule and the a posteriori regularization parameter choice rule. Numerical examples are presented to show the validity and stability of the proposed method.

Filter Regularization Method for Inverse Source Problem of the Rayleigh–Stokes Equation

In this paper, we consider a problem of recovering a space-dependent source term for the Rayleigh–Stokes equation, where the additional data is the observation at a final moment $t = T$, which is ill-posed in the sense of Hadamard. Firstly, the uniqueness, ill-posedness and the conditional stability of inverse source problem is given. Next, we develop a filter regularization method to overcome the ill-posedness of the problem. Under reasonable a priori bound assumption about the source function, a Hölder-type error estimate of the regularized solution is proved for a priori regularization parameter choice rule. Furthermore, a logarithmic-type error estimate between the exact solution and the regularized solution is established based on a posteriori regularization parameter choice rule.

The quasi-reversibility regularization method for backward problems of the time-fractional diffusion-wave equation

This paper is devoted to two backward problems in a time-fractional diffusion-wave equation, with the aid of extra measurement data at a final time. Since these two problems are ill-posed, a quasi-reversibility method is obtained by employing eigenfunction expansion, and the existence, uniqueness and regularity of the regularized solutions are also proven. We can get order-optimal error estimates from the a-priori parameter choice rule. Finally, several numerical examples of both one- and two-dimension show that the proposed regularization method is effective and stable.

The Landweber Iterative Regularization Method for Identifying the Unknown Source of Caputo-Fabrizio Time Fractional Diffusion Equation on Spherically Symmetric Domain

In this article, the inverse problem for identifying the space-dependent source of time fractional diffusion equation on a spherically symmetric domain with Caputo–Fabrizio fractional derivative is discussed. This problem is a typical ill-posed problem and the Landweber iterative regularization method is used to obtain the approximation solution. The convergent error estimates under a priori regularization parameter choice rule and a posteriori regularization parameter choice rule are given, respectively. The numerical examples are given to show the effectiveness of the Landweber iterative regularization method.

Determining an unknown source in a time-fractional diffusion equation based on Jacobi polynomials expansion with a modified Tiknonov regularization

In this paper, we try to recover an unknown source in a time-fractional diffusion equation. In order to overcome the influence of boundary conditions on source conditions, we introduce the Jacobi polynomials to construct the approximation and a modified Tikhonov regularization method is proposed to deal with the illposedness. Error estimates are obtained under a discrepancy principle as the parameter choice rule. Numerical results are also presented to demonstrate the effectiveness of the proposed method.

Landweber Iterative Method for an Inverse Source Problem of Time-Space Fractional Diffusion-Wave Equation

Abstract In this paper, we apply a Landweber iterative regularization method to determine a space-dependent source for a time-space fractional diffusion-wave equation from the final measurement. In general, this problem is ill-posed, and a Landweber iterative regularization method is used to obtain the regularization solution. Under the a priori parameter choice rule and the a posteriori parameter choice rule, we give the error estimates between the regularization solution and the exact solution, respectively. Some numerical results in the one-dimensional and two-dimensional cases show the utility of the used regularization method.

An iterative generalized quasi-boundary value regularization method for the backward problem of time fractional diffusion-wave equation in a cylinder

In this paper, we consider the backward problem for a time fractional diffusion-wave equation in a cylinder. The ill-posedness and a conditional stability of the inverse problem are proved. Based on the generalized quasi-boundary value regularization method, we propose an iterative generalized quasi-boundary value regularization method to deal with the inverse problem, and this iterative method has a higher convergence rate. The convergence rates of the regularized solution under an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are obtained. Numerical examples illustrate the effectiveness and stability of our proposed method.

Regularization of the Final Value Problem for the Time-Fractional Diffusion Equation

We consider the backward problem of reconstructing the initial condition of a nonhomogeneous time-fractional diffusion equation from final measurements. The proposed method relies on the eigenfunction expansion of the forward solution and the Tikhonov regularization to control the instability of the underlying inverse problem. We establish stability results and we provide convergence rates under a priori and a posteriori parameter choice rules. The resulting algorithm is robust and computationally inexpensive. Two examples are included to illustrate the effectiveness and accuracy of the proposed method.

An iterative method based on Nesterov acceleration for identifying space-dependent source term in a time-fractional diffusion-wave equation

In this paper, we investigate an inverse problem of identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. Since the problem is ill-posed, we propose an iterative method based on the Nesterov acceleration strategy to deal with it. Two kinds of convergence rates for the regularized solution are given under both the a priori and a posteriori regularization parameter choice rules. Compared with the classical regularization methods, our method is easy to implement with a fewer number of iterations and can always yield the order optimal error estimates provided the iteration parameter is chosen large enough. Some numerical examples including one-dimensional and two-dimensional cases are presented to illustrate the validity and effectiveness of the proposed method.

FRACTIONAL TIKHONOV REGULARIZATION METHOD FOR SIMULTANEOUS INVERSION OF THE SOURCE TERM AND INITIAL DATA IN A TIME-FRACTIONAL DIFFUSION EQUATION

This paper is concerned with the problem of identifying the space-dependent source term and initial value simultaneously for a time-fractional diffusion equation. The inverse problem is ill-posed, and the idea of decoupling it into two operator equations is applied. In order to solve this inverse problem, a fractional Tikhonov regularization method is proposed. Furthermore, the corresponding convergence estimates are presented by using the a priori and a posteriori parameter choice rules. Several numerical examples compared with the classical Tikhonov regularization are constructed for verifying the accuracy and efficiency of the proposed method.

The backward problem for radially symmetric time‐fractional diffusion‐wave equation under Robin boundary condition

This paper is devoted to solve the backward problem for radially symmetric time‐fractional diffusion‐wave equation under Robin boundary condition. This problem is ill‐posed and we apply an iterative regularization method to solve it. The error estimates are obtained under the a priori and a posteriori parameter choice rules. Numerical results show that the proposed method is efficient and stable.

Simultaneous determination of source term and the initial value in the space-fractional diffusion problem by a novel modified quasi-reversibility regularization method

This paper deals with an inverse problem of determining source term and initial data simultaneously for a space-fractional diffusion equation in a strip domain, with the aid of extra measurement data at a fixed time. The uniqueness results are obtained by a simple trick based on the linear property of the proposed equation. Since this problem is ill-posed, a modified quasi-reversibility method is obtained by employing the Fourier transform. Error estimates for source term and initial value are obtained from a suitable parameter choice rule. Finally, several numerical examples show that the proposed regularization method is effective and stable.

Identifying a space-dependent source term in distributed order time-fractional diffusion equations

<p style='text-indent:20px;'>The aim of this paper is to investigate an inverse problem of recovering a space-dependent source term governed by distributed order time-fractional diffusion equations in Hilbert scales. Such a problem is ill-posed and has important practical applications. For this problem, we propose a general regularization method based on the idea of the filter method. With a suitable source condition, we prove that the method is of optimal order under various choices of regularization parameter. One is based on the a priori regularization parameter choice rule and another one is the discrepancy principle. Finally, the capabilities of our method are illustrated by both the Tikhonov and the Landweber method.</p>