Fractional Tikhonov Regularization Research Articles

Iterated fractional Tikhonov method for recovering the source term and initial data simultaneously in a two-dimensional diffusion equation

In this paper, an inverse problem of a two-dimensional diffusion equation is considered. The purpose here includes recovering not only the source term but also the initial value from given observations at two fixed times t=T1 and t=T2. The uniqueness of the solutions for the inverse problem is given. Instead of the classical Tikhonov regularization strategy, we propose an iterated fractional Tikhonov regularization method to solve this problem, an a priori and an a posterior parameter selection rules and corresponding convergence rates are derived. For verification of the theoretical estimates, several numerical examples are constructed and compared with the standard iterated Tikhonov regularization method.

A new improved fractional Tikhonov regularization method for moving force identification

To address the problem of poor recognition accuracy of integer-order Tikhonov regularization method (Tik) in identifying bridge moving loads, a bridge moving load recognition method based on Improved fractional Tikhonov (IF-Tik) regularization method is proposed in this paper. The bridge moving load identification model is established according to the theory of time domain method, and the process of driving a two-axle vehicle on the bridge is simulated. The moving load is represented as the differential form of the kernel function of bending moment response and acceleration response. The differential equation is transformed into a linear system of equations by the discretization method, and solved by the improved fractional-order Tikhonov regularization method. The results show that the IF-Tik method has certain advantages over the F-Tik and Tik methods in terms of recognition accuracy and noise immunity, and the IF-Tik method has a higher accuracy of load recognition, and the recognition error is only 15% of that of the F-Tik method; the recognition results are less affected by the moment response, and the proposed method has good robustness, and is more suitable for the bridge moving load identification.

Fractional Tikhonov Regularization Method for Higher-Order Numerical Derivatives

Fractional Tikhonov Regularization Method for Higher-Order Numerical Derivatives

Two regularization methods for identifying the source term of Caputo–Hadamard time‐fractional diffusion equation

In this paper, the inverse problem for identifying the unknown source on time‐fractional diffusion equation with Caputo–Hadamard derivative is considered. This problem is ill‐posed, and two regularization methods are used to solve it. Firstly, we prove that the problem is ill‐posed and give the conditional stability and the optimal error bound. Secondly, the error estimates of the fractional Landweber iterative regularization method and the fractional Tikhonov regularization method under a priori and a posteriori parameter selection rules are given, respectively. Finally, numerical examples are given to illustrate the effectiveness of two regularization methods.

Finite dimensional realization of a parameter choice strategy for fractional Tikhonov regularization method in Hilbert scales

One of the most crucial parts of applying a regularization method to solve an ill-posed problem is choosing a regularization parameter to obtain an optimal order error estimate. In this paper, we consider the finite dimensional realization of the parameter choice strategy proposed in [C. Mekoth, S. George and P. Jidesh, Appl. Math. Comput. 392, 125701, 2021] for Fractional Tikhonov regularization method for linear ill-posed operator equations in the setting of Hilbert scales.

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.

Optimization of beam arrangement for tunable diode laser absorption tomography reconstruction based on fractional Tikhonov regularization

Beam arrangement with limited projections based on tunable diode laser absorption spectroscopy is the key to achieving a more accurate two-dimensional reconstruction of the combustion. Using fractional calculus theory, a beam optimization method based on fractional Tikhonov regularization is proposed. The beam arrangement function based on fractional Tikhonov regularization is established by extending the standard Tikhonov regularization to fractional modes. The genetic algorithm is used to analyze the calculation results of different orders in a range of (0, 1), and the beam arrangement is obtained. Using 20 laser beams to scan the characteristic absorption spectrum of H<sub>2</sub>O in the near-infrared band 7185.6 cm<sup>–1</sup>, modeling the calculations in a 10×10 element discrete tomography domain, and comparing the reconstruction results of the five beam arrangements for different Gaussian distribution models, the beam arrangement based on fractional Tikhonov regularization shows more obvious advantages. This design method proposed in this work is valuable for the theoretical study of the optimal design of two-dimensional measurement beams based on the tunable diode laser absorption spectroscopy technique, which can promote the application of this technique in the two-dimensional reconstruction of complex engine combustion and combustion efficiency improvement.

On Fractional Tikhonov Regularization: Application to the Adaptive Network-Based Fuzzy Inference System for Regression Problems

In this article, we introduce a variant of the adaptive network-based fuzzy inference system (ANFIS). The proposed variant does not use backpropagation and grid partitioning, but the least-squares method with fractional Tikhonov regularization. The fractional regularization is a generalization of the standard regularization and is applied here to the learning process of the ANFIS scheme for the first time. This results in a simpler rule base, with a low number of rules, allowing to handle problems with many input variables with relatively low computational time while keeping high accuracy. We present new theoretical results on the fractional Tikhonov regularization. Such results are the basis for a formal discussion on how much the choice of a different architecture, resulting in a different matrix in the least-squares minimization, could affect the accuracy. We perform several numerical experiments on benchmark examples, first to assess the impact of the fractional regularization on the accuracy and then to compare our results against the most recent ones reported in the literature by other ANFIS-like or neuro-fuzzy systems. The numerical results show the good performance of the proposed approach.

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.

Finite dimensional realization of fractional Tikhonov regularization method in Hilbert scales

One of the intuitive restrictions of infinite dimensional Fractional Tikhonov Regularization Method (FTRM) for ill-posed operator equations is its numerical realization. This paper addresses the issue to a considerable extent by using its finite dimensional realization in the setting of Hilbert scales. Using adaptive parameter choice strategy, we choose the regularization parameter and obtain an optimal order error estimate. Also, the proposed method is applied to the well known examples in the setting of Hilbert scales.

A Posteriori Fractional Tikhonov Regularization Method for the Problem of Analytic Continuation

In this paper, the numerical analytic continuation problem is addressed and a fractional Tikhonov regularization method is proposed. The fractional Tikhonov regularization not only overcomes the difficulty of analyzing the ill-posedness of the continuation problem but also obtains a more accurate numerical result for the discontinuity of solution. This article mainly discusses the a posteriori parameter selection rules of the fractional Tikhonov regularization method, and an error estimate is given. Furthermore, numerical results show that the proposed method works effectively.

A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations

• Two types of inverse problems for diffusion equations involving Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space are given. • Two types of inverse problems are proved to be ill-posed in the sense of Hadamard whenever an additional condition at a final time is given. • A new fractional Tikhonov regularization method is used for the reconstruction of the stable solutions. • An error estimates between the exact and it regularize solutions are obtained. In this research, we deal with two types of inverse problems for diffusion equations involving Caputo fractional derivatives in time and fractional Sturm-Liouville operator for space. The first one is to identify the source term and the second one is to identify the initial value along with the solution in both cases. These inverse problems are proved to be ill-posed in the sense of Hadamard whenever an additional condition at the final time is given. A new fractional Tikhonov regularization method is used for the reconstruction of the stable solutions. Under the a-priori and the a-posteriori parameter choice rules, the error estimates between the exact and its regularized solutions are obtained. To illustrate the validity of our study, we give numerical examples. A final note is utilized in the ultimate section.

Iterated fractional Tikhonov regularization method for solving the spherically symmetric backward time-fractional diffusion equation

In this paper, we focus on a backward problem for an inhomogeneous time-fractional diffusion equation on a spherical symmetric domain. This problem is ill-posed in the sense of Hadamard. We propose an iterated fractional Tikhonov regularization method in both cases: the deterministic case and random noise case. The a-priori and the a-posteriori choice rules for regularization parameters are discussed and both rules yield the corresponding convergence rates. The iterated fractional Tikhonov regularization method goes beyond the saturation results of classical Tikhonov method, and iterative fractional Tikhonov regularization method is superior to classical iterative Tikhonov regularization method under the a-priori parameter choice rule. A numerical example is conducted for showing the validity of the proposed method.

Fractional Tikhonov regularization method in Hilbert scales

Fractional Tikhonov regularization method (FTRM) for linear ill-posed operator equations in the setting of Hilbert scales is being studied in this paper. Using a general Holder type source condition, we obtain an error estimate. A new parameter choice strategy is being proposed for choosing the regularization parameter in FTRM in the setting of Hilbert scales. Also, the proposed method is applied to the well known examples in the setting of Hilbert scales.

The fractional Tikhonov regularization methods for identifying the initial value problem for a time-fractional diffusion equation

In this paper, we identify the initial value for a time-fractional diffusion equation on a columnar axis-symmetric domain. This problem is ill-posed, i.e., the solution (if it exists) does not depend continuously on the domain. Two different kinds of fractional Tikhonov methods are used to solve this problem. Under the a priori and the a posteriori regularization parameter choice rules, the error estimates between the regularization solutions and the exact solution are obtained, respectively. Different numerical examples are presented to illustrate the validity and effectiveness of our methods.

Fractional Tikhonov regularization to improve the performance of extreme learning machines

Extreme learning machine (ELM) is a single-hidden-layer feed-forward neural network in which the input weights linking the input layer to the hidden layer are randomly chosen. The output weights which link the hidden layer to the output layer are analytically determined by solving a linear system of equations and hence is one of the fastest learning algorithms. The Moore–Penrose (MP) generalized inverse is normally employed to obtain the output weights of the neural network. Although the random weight parameters between input and hidden layers are need not be tuned, ELM provides good generalization performance with fast learning speed. In general, the data sets from real-world problems tend to make the linear system of ELM ill-conditioned due to the presence of inconsistent noise levels in the input data which leads to unreliable solutions and over-fitting problems. The regularization techniques are developed to address such issues in ELM and it involves estimation of additional variables termed as a regularization parameter. In this context, the proper selection of the regularization parameter is a crucial task as it is going to decide the quality of the solution obtained from the linear system. Further, the popular choice is the Tikhonov regularization technique which penalizes the ℓ2-norm of the model parameters. In ELM, such inclusion results are giving equal weight to singular values of the matrix irrespective of the noise level present in the data. In the presented work, a fractional framework is introduced in the Tikhonov regularized ELM to weigh the singular values with respect to a fractional parameter to reduce the effect of different noise levels. Moreover, an automated golden-section method is applied to choose the optimal fractional parameter. Finally, the generalized cross-validation method is applied for obtaining the suitable value of the regularization parameter. The proposed strategy of applying fractional Tikhonov regularization to ELM results in improvement of performance when compared with the conventional methods with respect to the performance measures. Finally, the results obtained from the proposed fractional regularization is also shown to be statistically significant.

Discrepancy principles for fractional Tikhonov regularization method leading to optimal convergence rates

Fractional Tikhonov regularization (FTR) method was studied in the last few years for approximately solving ill-posed problems. In this study we consider the Schock-type discrepancy principle for choosing the regularization parameter in FTR and obtained the order optimal convergence rate. Numerical examples are provided in this study.

On a terminal value problem for a generalization of the fractional diffusion equation with hyper‐Bessel operator

In this paper, we consider an inverse problem of recovering the initial value for a generalization of time‐fractional diffusion equation, where the time derivative is replaced by a regularized hyper‐Bessel operator. First, we investigate the existence and regularity of our terminal value problem. Then we show that the backward problem is ill‐posed, and we propose a regularizing scheme using a fractional Tikhonov regularization method. We also present error estimates between the regularized solution and the exact solution using two parameter choice rules.

A fractional Tikhonov regularization method for identifying a space-dependent source in the time-fractional diffusion equation

In this paper, we are concerned with an inverse source problem for the time-fractional diffusion equation with variable coefficients in a general bounded domain. The problem is mildly ill-posed. A new fractional Tikhonov method is proposed. We discuss the a-priori regularization parameter choice rule and the a-posteriori regularization parameter choice rule, and prove the corresponding convergence estimates. Numerical experiments are conducted for illustrating effectiveness of the proposed method.

Fractional Tikhonov Method for an Inverse Time-Fractional Diffusion Problem in 2-Dimensional Space

In this paper, we present a new fractional Tikhonov regularization method for solving an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting. Fractional Tikhonov regularization method not only retains the advantage of classical Tikhonov method, but also overcomes the effect of over-smoothing of classical Tikhonov method. We give the selection of regularization parameters of the new method and the corresponding error estimation. Furthermore, numerical results show that the fractional Tikhonov method outperforms the classical one.