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

2021 ◽  
Vol 5 (4) ◽  
pp. 193
Author(s):  
Dun-Gang Li ◽  
Jun-Liang Fu ◽  
Fan Yang ◽  
Xiao-Xiao Li
Keyword(s):  
Inverse Problem ◽  
Stokes Equation ◽  
Initial Value Problem ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Choice Rule ◽  
Conditional Stability ◽  
Parameter Choice ◽  
Initial Value ◽  
Iterative Regularization

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.

scholarly journals A Posteriori Regularization Parameter Choice Rule for Truncation Method for Identifying the Unknown Source of the Poisson Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Xiao-Xiao Li ◽  
Dun-Gang Li
Keyword(s):  
Poisson Equation ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Choice Rule ◽  
Conditional Stability ◽  
A Posteriori ◽  
Parameter Choice ◽  
Convergence Estimate ◽  
The Poisson Equation ◽  
Unknown Source

We consider the problem of determining an unknown source which depends only on one variable in two-dimensional Poisson equation. We prove a conditional stability for this problem. Moreover, we propose a truncation regularization method combined with an a posteriori regularization parameter choice rule to deal with this problem and give the corresponding convergence estimate. Numerical results are presented to illustrate the accuracy and efficiency of this method.

scholarly journals SOLUTION OF THE PROBLEM OF COMPUTATIONAL STABILITY AND CONSISTENCY OF SAMPLE ESTIMATES OF THE CORRELATION MATRIX OF OBSERVATIONS BY THE METHOD OF DYNAMIC REGULARIZATION

2019 ◽  
Vol 26 ◽  
pp. 47-60
Author(s):  
V. SKACHKOV ◽  
Keyword(s):  
Inverse Problem ◽  
Regularization Method ◽  
Correlation Matrix ◽  
Regularization Parameter ◽  
A Priori ◽  
Computational Cost ◽  
Spectral Composition ◽  
Adaptive Antenna ◽  
Computational Stability ◽  
A Priori Uncertainty

The problem of forming sample estimates of the correlation matrix of observations that satisfy the criterion "computational stability – consistency" is considered. The variants in which the direct and inverse asymptotic forms of the correlation matrix of observations are approximated by various types of estimates formed from a sample of a fixed volume are investigated. The consistency of computationally stable estimates of the correlation matrix for their static regularization was analyzed. The contradiction inherent in the problem of regularization of the estimates with a fixed parameter is revealed. The dynamic regularization method as an alternative approach is proposed, which is based on the uniqueness theorem for solving the inverse problem with perturbed initial data. An optimal mean-square approximation algorithm has been developed for dynamic regularization of sample estimates of the correlation matrix of observations, using the law of monotonic decrease in the regularizing parameter with increasing sample size. An optimal dynamic regularization function was obtained for sample estimates of the correlation matrix under conditions of a priori uncertainty with respect to their spectral composition. The preference of this approach to the regularization of sample estimates of the correlation matrix under conditions of a priori uncertainty is proved, which allows to exclude the domain of computational instability from solving the inverse problem and obtain its solution in real time without involving prediction data and additional computational cost for finding the optimal value of the regularization parameter. The application of the dynamic regularization method is shown for solving the problem of detecting a signal at the output of an adaptive antenna array in a nondeterministic clutter and jamming environment. The results of a computational experiment that confirm the main conclusions are presented.

On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation

2021 ◽  
Vol 23 (4) ◽  
Author(s):  
Nguyen Hoang Luc ◽  
Do Lan ◽  
Donal O’Regan ◽  
Nguyen Anh Tuan ◽  
Yong Zhou
Keyword(s):  
Stokes Equation ◽  
Initial Value Problem ◽  
Initial Value

A wavelet method for numerical fractional derivative with noisy data

2016 ◽  
Vol 14 (05) ◽  
pp. 1650038 ◽  
Author(s):  
Xiangtuan Xiong ◽  
Qiang Cheng ◽  
Yanfeng Kong ◽  
Jin Wen
Keyword(s):  
Fractional Derivative ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Large Change ◽  
Stability Estimates ◽  
Parameter Choice ◽  
Wavelet Regularization ◽  
Ill Posed ◽  
Meyer Wavelet ◽  
Parameter Choice Rules

Numerical fractional differentiation is a classical ill-posed problem in the sense that a small perturbation in the data can cause a large change in the fractional derivative. In this paper, we consider a wavelet regularization method for solving a reconstruction problem for numerical fractional derivative with noise. A Meyer wavelet projection regularization method is given, and the Hölder-type stability estimates under both apriori and aposteriori regularization parameter choice rules are obtained. Some numerical examples show that the method works well.

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

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1007 ◽  
Author(s):  
Fan Yang ◽  
Qu Pu ◽  
Xiao-Xiao Li ◽  
Dun-Gang Li
Keyword(s):  
Wave Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Initial Boundary ◽  
Fractional Diffusion ◽  
Value Functions ◽  
Parameter Choice ◽  
Initial Value ◽  
Diffusion Wave ◽  
Initial Boundary Value

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.

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