Posteriori Regularization Parameter Choice 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.

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.

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.

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 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.

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.

Source term identification of time‐fractional diffusion equation under Robin boundary condition

The source term identification of the time‐fractional diffusion equation under Robin boundary condition is studied. This problem is ill‐posed. Therefore, we apply Landweber iterative regularization method, fractional Landweber iterative regularization method, TSVD method, combining TSVD method, and fractional Landweber iterative regularization method, respectively. The comparisons of these four methods are given, which can help us select the most effective method. The error estimates between the regularized approximate solutions and the exact solution are given under the a priori and a posteriori regularization parameter choice rules. Finally, numerical examples verify the effectiveness of the methods.

The quasi-reversibility method for an inverse source problem for time-space fractional parabolic equations

In this paper, we apply the quasi-reversibility method to solve an inverse source problem for a time-space fractional parabolic equation. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. The theoretical error estimates are confirmed with numerical tests for one and two dimensional equations.

Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative

This research determines an unknown source term in the fractional diffusion equation with the Riemann–Liouville derivative. This problem is ill-posed. Conditional stability for the inverse source problem can be given. Further, a fractional Tikhonov regularization method was applied to regularize the inverse source problem. In the theoretical results, we propose a priori and a posteriori regularization parameter choice rules and obtain the convergence estimates.

A fast iterative method for identifying the radiogenic source for the helium production-diffusion equation

ABSTRACT In this paper, we consider an inverse source problem for the helium production-diffusion equation. That is to determine a space-dependent source term in the helium production-diffusion equation from a noisy final data. Since the problem is ill-posed, we propose an effective iterative scheme to deal with the problem. More specifically, we use an acceleration technique to speed up a simple iterative scheme. We also give two kinds of convergence rates by using an a priori and a posteriori regularization parameter choice rule, respectively. Numerical examples are presented to illustrate the validity and effectiveness of the proposed method.

Source Identification for the Time-Fractional Diffusion Equation With Robin Boundary Conditions

The source term identification of the time-fractional diffusion equation with Robin boundary conditions is studied. This problem is ill-posed. Therefore, an iterative regularization method is constructed to calculate the regularized approximate solution of the source term. The error estimates between the regularized approximate solution and the exact solution are given under the priori and the posteriori regularization parameter choice rules. Finally, numerical examples verify the effectiveness of the method.

Identifying initial value problem for time‐fractional diffusion equation with Caputo‐like counterpart hyper‐Bessel operator: Optimal error bound analysis and regularization method

In this paper, we consider the inverse problem for identifying the initial value problem for time‐fractional diffusion equation with hyper‐Bessel operator. Firstly, we show that this problem is ill‐posed and analyze the optimal error bound. Next, we use the fractional Landweber iterative regularization method to solve this problem and give the error estimates between the regularization solution and the exact solution under the a priori regularization parameter choice rule and a posteriori regularization parameter choice rule. Finally, we use numerical examples to verify the effectiveness of this method.

A fractional Landweber iterative regularization method for stable analytic continuation

In this paper, we consider the problem of analytic continuation of the analytic function $g(z) = g(x+iy)$ on a strip domain Ω = $\{z = x+iy\in \mathbb{C}|\, x\in\mathbb{R}, 0 < y < y_0\}$, where the data is given only on the line $y = 0$. This problem is a severely ill-posed problem. We propose the fraction Landweber iterative regularization method to deal with this problem. Under the a priori and a posteriori regularization parameter choice rule, we all obtain the error estimates between the regularization solution and the exact solution. Some numerical examples are given to verify the efficiency and accuracy of the proposed methods.

Regularization of backward time-fractional parabolic equations by Sobolev-type equations

Abstract The problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

Identifying an unknown source term of a parabolic equation in Banach spaces

A new regularization method for an inverse source problem for a parabolic equation in a Banach space is proposed. Hölder-type error estimates for the regularized solutions are proved for both a priori and a posteriori regularization parameter choice rules. Some numerical examples are presented for illustrating the efficiency of the method.

An extension of the landweber regularization for a backward time fractional wave problem

In this paper, we investigate numerical methods for a backward problem of the time-fractional wave equation in bounded domains. We propose two fractional filter regularization methods, which can be regarded as an extension of the classical Landweber iteration for the time-fractional wave backward problem. The idea is first to transform the ill-posed backward problem into a weighted normal operator equation, then construct the regularization methods for the operator equation by introducing suitable fractional filters. Both a priori and a posteriori regularization parameter choice rules are investigated, together with an estimate for the smallest regularization parameter according to a discrepancy principle. Furthermore, an error analysis is carried out to derive the convergence rates of the regularized solutions generated by the proposed methods. The theoretical estimate shows that the proposed fractional regularizations efficiently overcome the well-known over-smoothing drawback caused by the classical regularizations. Some numerical examples are provided to confirm the theoretical results. In particular, our numerical tests demonstrate that the fractional regularization is actually more efficient than the classical methods for problems having low regularity.

Convergence results and low-order rates for nonlinear Tikhonov regularization with oversmoothing penalty term

For Tikhonov regularization of ill-posed nonlinear operator equations, convergence is studied in a Hilbert scale setting. We include the case of oversmoothing penalty terms, which means that the exact solution does not belong to the domain of definition of the considered penalty functional. In this case, we try to close a gap in the present theory, where Holder-type convergence rates results have been proven under corresponding source conditions, but assertions on norm convergence for regularized solutions without source conditions are completely missing. A result of the present work is to provide sufficient conditions for convergence under a priori and a posteriori regularization parameter choice strategies without any additional smoothness assumption on the solution. The obtained error estimates moreover allow us to prove low-order convergence rates under associated (for example logarithmic) source conditions. Some numerical illustrations are also given.

The Truncation Regularization Method for Identifying the Initial Value on Non-Homogeneous Time-Fractional Diffusion-Wave Equations

In the essay, we consider an initial value question for a mixed initial-boundary value of time-fractional diffusion-wave equations. This matter is an ill-posed problem; the solution relies discontinuously on the measured information. The truncation regularization technique is used for restoring the initial value functions. The convergence estimations are given in a priori regularization parameter choice regulations and a posteriori regularization parameter choice regulations. Numerical examples are given to demonstrate this is effective and practicable.

Optimal regularization for an unknown source of space-fractional diffusion equation

We study an inverse problem of finding the source Q(x, t) for diffusion equation with space-fractional Laplacian ut(x,t)=−μ(−Δ)γu(x,t)+Q(x,t)(0<γ≤1,μ>0,x∈R) from the initial and the final data. Determining the unknown source is to identify the nature of diffusion process and that is one of the most important and well-studied problems in many branches of engineering sciences. This problem must be solved since its result will be used in scientific and technical applications. We limit the severely ill-posed problem to the one of finding Q(x,t)=Φ(t)S(x) with Φ being known. The cases that the variable x is in bounded domains of Rn or the differential operator has the divergence form (−∇·(μ∇u))γ with μ=μ(x,t) or μ=μ(x,t,u) are still not investigated in this paper. The uniqueness and the conditional stability for S are obtained. We apply the Fourier truncation method to construct stable approximation problem and we show the asymptotically optimal estimates for the worst case error of the method. This optimality cannot deduce directly from Tautenhaln’s results. We also establish a general method of a posteriori regularization parameter choice rule which has an error estimate independent of the smoothing parameter of S and, moreover, we prove that the choice rule gives in some cases the best error order ϵ. The μ-depended results show the incremental influent of the diffusivity μ on the optimal choice rules when μ is close to zero.

A quasi-boundary regularization method for identifying the initial value of time-fractional diffusion equation on spherically symmetric domain

Abstract In this paper, an inverse problem to identify the initial value for high dimension time fractional diffusion equation on spherically symmetric domain is considered. This problem is ill-posed in the sense of Hadamard, so the quasi-boundary regularization method is proposed to solve the problem. The convergence estimates between the regularization solution and the exact solution are presented under the a priori and a posteriori regularization parameter choice rules. Numerical examples are provided to show the effectiveness and stability of the proposed method.