Parameter Choice Rule Research Articles

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.

Identification of the Initial Value for a Time-Fractional Diffusion Equation

In this manuscript, we study the inverse problem for identifying the initial value of a time-fractional diffusion equation in an axisymmetric region. This is an ill-posed problem, i.e., the solution does not depend continuously on the data. We choose the Landweber iterative regularization method to solve this problem. Under the a priori and the a posteriori regularization parameter choice rules, we present the error estimates between the regularization solutions and the exact solution. We present some examples to show this method’s effectiveness.

The backward problem of a stochastic PDE with bi-harmonic operator driven by fractional Brownian motion

ABSTRACT This paper is concerned with a backward problem of a stochastic partial differential equation with bi-harmonic operator. The source term is driven by fractional Brownian motion. Based on the Gevrey-type space, the regularity of the mild solution is studied. However, this problem is ill-posed since it is unstable. The instability is discussed in the sense of expectation and variance. Moreover, a regularization method is proposed. The error estimation between the regularization solution and the mild solution is given using an a prior parameter choice rule.

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.

Modeling a Spheroidal Particle Ensemble and Inversion by Generalized Runge–Kutta Regularizers from Limited Data

Extracting information about the shape or size of non-spherical aerosol particles from limited optical radar data is a well-known inverse ill-posed problem. The purpose of the study is to figure out a robust and stable regularization method including an appropriate parameter choice rule to address the latter problem. First, we briefly review common regularization methods and investigate a new iterative family of generalized Runge–Kutta filter regularizers. Next, we model a spheroidal particle ensemble and test with it different regularization methods experimenting with artificial data pertaining to several atmospheric scenarios. We found that one method of the newly introduced generalized family combined with the L-curve method performs better compared to traditional methods.

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.

Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method

This article is devoted to identifying a space-dependent source term in linear parabolic equations. Such a problem is ill posed, i.e., a small perturbation in the input data may cause a dramatically large error in the solution (if it exists). The conditional stability of the solution is analyzed. Based on a convoluting equation method, we can deal with the problem under the a priori parameter choice rule. Meanwhile, a modified version of Morozov’s discrepancy principle is provided to decide on an a posteriori regularization parameter choice strategy and a log-type error estimate is obtained. Two numerical results show that our proposed method works well.

A generalized regularization scheme for solving singularly perturbed parabolic PDEs

Many problems in science and engineering can be modeled as singularly perturbed partial differential equations. Solutions to such problems are not generally continuous with respect to the perturbation parameter(s) and hence developing stable and efficient numerical schemes to singularly perturbed problems are always more interesting and mathematically challenging to researchers. In this paper, we propose a generalized regularization scheme as an alternate numerical scheme for solving singularly perturbed parabolic PDEs with higher-order derivatives multiplied by a small parameter. Since the involved operators are unbounded, first, we propose a general theoretical framework for unbounded operators with an a posteriori parameter choice rule for selecting a regularization parameter, and then we apply it to singularly perturbed problems. We numerically implement the proposed scheme and compare it with other traditional schemes. The study illustrates that the proposed scheme can be considered as an alternate and competent method for solving singularly perturbed parabolic problems.

Inverse Source Problem for Sobolev Equation with Fractional Laplacian

In this paper, we are interested in the problem of determining the source function for the Sobolev equation with fractional Laplacian. This problem is ill-posed in the sense of Hadamard. In order to edit the instability instability of the solution, we applied the fractional Landweber method. In the theoretical analysis results, we show the error estimate between the exact solution and the regularized solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Finally, we investigate the convergence of the source function when fractional order β ⟶ 1 + .

Simultaneous determination of source term and initial value in the heat conduction problem by modified quasi-reversibility regularization method

In this article, we investigate the inverse problem of determining source term and initial data simultaneously in a parabolic equation where data are given at two fixed times. We can get the exact solution of the problem by the Fourier transform, and find that the problem is ill-posed, i.e., the solution does not depend continuously on the input data and small errors in the data can destroy the numerical solution. Hence, a modified quasi-reversibility regularization method is used to solve the problem and an order-optimal error estimate is obtained under a suitable parameter choice rule. Finally, some numerical examples including smooth and nonsmooth functions show that the proposed method is effective and stable.

Regularization Error Analysis for a Sideways Problem of the 2D Nonhomogeneous Time-Fractional Diffusion Equation

The inverse and ill-posed problem of determining a solute concentration for the two-dimensional nonhomogeneous fractional diffusion equation is investigated. This model is much worse than its homogeneous counterpart as the source term appears. We propose a modified kernel regularization technique for the stable numerical reconstruction of the solution. The convergence estimates under both a priori and a posteriori parameter choice rules are proven.

On regularization via frame decompositions with applications in tomography

In this paper, we consider linear ill-posed problems in Hilbert spaces and their regularization via frame decompositions, which are generalizations of the singular-value decomposition. In particular, we prove convergence for a general class of continuous regularization methods and derive convergence rates under both a priori and a posteriori parameter choice rules. Furthermore, we apply our derived results to a standard tomography problem based on the Radon transform.

A Tikhonov regularization method for solving a backward time–space fractional diffusion problem

In this paper, a backward problem for a time–space fractional diffusion equation is considered, which is to determine the initial data from a noisy final data. To deal with this ill-posed problem, combined the ideas of the Tikhonov regularization in Hilbert Schales proposed by Natterer in 1984 and the fractional Tikhonov method given by Hochstenbach–Reichel in 2011, a Tikhonov regularization method is constructed. Based on the Fourier transform, an a-priori and an a-posteriori regularization parameter choice rules are used to guarantee the order optimal convergence rates. By the finite difference methods and the Discrete Fourier transform, numerical examples in one-dimensional and two-dimensional cases are given for the a-posteriori regularization parameter choice rule. Theoretical and numerical results show that the proposed method works well for both the smooth and the non-smooth functions.

A generalized quasi-boundary value method for recovering a source in a fractional diffusion-wave equation

This paper is devoted to identifying a space-dependent source in a time-fractional diffusion-wave equation by using the final time data. By the series expression of the solution of the direct problem, the inverse source problem can be formulated by a first kind of Fredholm integral equation. The existence and uniqueness, ill-posedness and a conditional stability in Hilbert scale for the considered inverse problem are provided. We propose a generalized quasi-boundary value regularization method to solve the inverse source problem and also prove that the regularized problem is well-posed. Further, two kinds of convergence rates in Hilbert scale for the regularized solution can be obtained by using an a priori and an a posteriori regularization parameter choice rule, respectively. The numerical examples in one-dimensional case and two-dimensional case are given to confirm our theoretical results for the constant coefficients problem. We also propose a finite difference method based on a variant of L1 scheme to solve the regularized problem for the variable coefficients problem and give its convergence rate. One finite difference method based on a convolution quadrature is provided to solve the regularized problem for comparison. The numerical results for three examples by two algorithms are provided to show the effectiveness and stability of the proposed algorithms.

Stability results for backward heat equations with time-dependent coefficient in the Banach space [formula omitted

In this paper, we investigate the problem of backward heat equations with time-dependent coefficient in the Banach space Lp(R), (1<p<∞). For this problem, we first prove the stability estimates of Hölder type. After that the Tikhonov-type regularization is applied to solve the problem. A priori and a posteriori parameter choice rules are investigated, which yield error estimates of Hölder type. Numerical implementations are presented to show the validity of the proposed scheme.

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.

A numerical comparison of some heuristic stopping rules for nonlinear Landweber iteration

The choice of a suitable regularization parameter is an important part of most regularization methods for inverse problems. In the absence of reliable estimates of the noise level, heuristic parameter choice rules can be used to accomplish this task. While they are already fairly well understood and tested in the case of linear problems, not much is known about their behaviour for nonlinear problems and even less in the respective case of iterative regularization. Hence, in this paper, we numerically study the performance of some of these rules when used to determine a stopping index for Landweber iteration for various nonlinear inverse problems. These are chosen from different practically relevant fields such as integral equations, parameter estimation, and tomography.

Estimating Solution Smoothness and Data Noise with Tikhonov Regularization

A main drawback of classical Tikhonov regularization is that often the parameters required to apply theoretical results, e.g., the smoothness of the sought-after solution and the noise level, are unknown in practice. In this paper we investigate in new detail the residuals in Tikhonov regularization viewed as functions of the regularization parameter. We show that the residual carries, with some restrictions, the information on both the unknown solution and the noise level. By calculating approximate solutions for a large range of regularization parameters, we can extract both parameters from the residual given only one set of noisy data and the forward operator. The smoothness in the residual allows to revisit parameter choice rules and relate a-priori, a-posteriori, and heuristic rules in a novel way that blurs the lines between the classical division of the parameter choice rules. All results are accompanied by numerical experiments.

Landweber Iterative Regularization Method for Identifying the Initial Value Problem of the Rayleigh–Stokes Equation

In this paper, we study an inverse problem to identify the initial value problem of the homogeneous Rayleigh–Stokes equation for a generalized second-grade fluid with the Riemann–Liouville fractional derivative model. This problem is ill posed; that is, the solution (if it exists) does not depend continuously on the data. We use the Landweber iterative regularization method to solve the inverse problem. Based on a conditional stability result, the convergent error estimates between the exact solution and the regularization solution by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule are given. Some numerical experiments are performed to illustrate the effectiveness and stability of this method.