A non-iterative method for recovering the space-dependent source and the initial value simultaneously in a parabolic equation

2020 ◽  
Vol 28 (4) ◽  
pp. 499-516
Author(s):  
Zewen Wang ◽  
Shuli Chen ◽  
Shufang Qiu ◽  
Bin Wu
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Source Term ◽  
Dimensional Space ◽  
Conditional Stability ◽  
Finite Dimensional Space ◽  
Initial Value ◽  
Equation Problem ◽  
Inverse Solutions ◽  
Direct Problems

AbstractThis paper is concerned with the inverse problem for determining the space-dependent source and the initial value simultaneously in a parabolic equation from two over-specified measurements. By means of transforming information of the initial value into the source term and obtaining a combined source term, the parabolic equation problem is converted into a parabolic problem with homogeneous conditions. Then the considered inverse problem is formulated into a regularized minimization problem, which is implemented by the finite element method based on solving a sequence of well-posed direct problems. The uniqueness of inverse solutions are proved by the solvability of the corresponding variational problem, and the conditional stability as well as the convergence rate of regularized solutions are also provided. Then the error estimate of approximate regularization solutions is presented in the finite-dimensional space. The proposed method is a very fast non-iterative algorithm, and it can successfully solve the multi-dimensional inverse problem for recovering the space-dependent source and the initial value simultaneously. Numerical results of five examples including one- and two-dimensional cases show that the proposed method is efficient and robust with respect to data noise.

scholarly journals Inverse parabolic problems of determining functions with one spatial-component independence by Carleman estimate

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Oleg Y. Imanuvilov ◽  
Yavar Kian ◽  
Masahiro Yamamoto
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Boundary Data ◽  
Stability Estimate ◽  
Carleman Estimate ◽  
Parabolic Problems ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Lateral Boundary ◽  
Spatial Component

Abstract For a parabolic equation in the spatial variable x = ( x 1 , … , x n ) {x=(x_{1},\ldots,x_{n})} and time t, we consider an inverse problem of determining a coefficient which is independent of one spatial component x n {x_{n}} by lateral boundary data. We apply a Carleman estimate to prove a conditional stability estimate for the inverse problem. Also, we prove similar results for the corresponding inverse source problem.

scholarly journals Determination of a nonlinear source term in a diffusion equation with final observations

2004 ◽  
Vol 2004 (14) ◽  
pp. 741-753 ◽  
Author(s):  
Gongsheng Li ◽  
Yichen Ma ◽  
Kaitai Li
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Integral Identity ◽  
Source Term ◽  
Conditional Stability ◽  
Inverse Source Problem ◽  
Source Terms ◽  
Nonlinear Source ◽  
Quasilinear Diffusion

This paper deals with an inverse problem of determining a nonlinear source term in a quasilinear diffusion equation with overposed final observations. Applying integral identity methods, data compatibilities are deduced by which the inverse source problem here is proved to be reasonable and solvable. Furthermore, with the aid of an integral identity that connects the unknown source terms with the known data, a conditional stability is established.

On the stability of recovering two sources and initial status in a stochastic hyperbolic-parabolic system

2021 ◽  
Author(s):  
Bin Wu ◽  
Jijun Liu
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Hyperbolic Equation ◽  
Wave Field ◽  
Time Derivative ◽  
Stability Index ◽  
Carleman Estimates ◽  
Conditional Stability ◽  
Coupling System ◽  
The Stability

Abstract Consider an inverse problem of determining two stochastic source functions and the initial status simultaneously in a stochastic thermoelastic system, which is constituted of two stochastic equations of different types, namely a parabolic equation and a hyperbolic equation. To establish the conditional stability for such a coupling system in terms of some suitable norms revealing the stochastic property of the governed system, we first establish two Carleman estimates with regular weight function and two large parameters for stochastic parabolic equation and stochastic hyperbolic equation, respectively. By means of these two Carleman estimates, we finally prove the conditional stability for our inverse problem, provided the source in the elastic equation be known near the boundary and the solution be in a prior bound set. Due to the lack of information about the time derivative of wave field at final moment, the stability index with respect to the wave field at final time is found to be halved, which reveals the special characteristic of our inverse problem for the coupling system.

scholarly journals A Note on the Inverse Problem for a Fractional Parabolic Equation

2012 ◽  
Vol 2012 ◽  
pp. 1-26 ◽  
Author(s):  
Abdullah Said Erdogan ◽  
Hulya Uygun
Keyword(s):  
Inverse Problem ◽  
Parabolic Equation ◽  
Operator Theory ◽  
Source Term ◽  
Approximate Solutions ◽  
Time Dependent ◽  
Stability Estimates ◽  
Theory Approach ◽  
One Dimensional ◽  
Term Stability

For a fractional inverse problem with an unknown time-dependent source term, stability estimates are obtained by using operator theory approach. For the approximate solutions of the problem, the stable difference schemes which have first and second orders of accuracy are presented. The algorithm is tested in a one-dimensional fractional inverse problem.

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.

Tikhonov Regularisation Method for Simultaneous Inversion of the Source Term and Initial Data in a Time-Fractional Diffusion Equation

2015 ◽  
Vol 5 (3) ◽  
pp. 273-300 ◽  
Author(s):  
Zhousheng Ruan ◽  
Jerry Zhijian Yang ◽  
Xiliang Lu
Keyword(s):  
Inverse Problem ◽  
Diffusion Equation ◽  
Source Term ◽  
Fractional Diffusion ◽  
Fractional Diffusion Equation ◽  
Conditional Stability ◽  
Smoothness Assumption ◽  
Tikhonov Regularisation ◽  
Time Fractional Diffusion Equation ◽  
Regularisation Method

AbstractThe inverse problem of identifying the time-independent source term and initial value simultaneously for a time-fractional diffusion equation is investigated. This inverse problem is reformulated into an operator equation based on the Fourier method. Under a certain smoothness assumption, conditional stability is established. A standard Tikhonov regularisation method is proposed to solve the inverse problem. Furthermore, the convergence rate is given for an a priori and a posteriori regularisation parameter choice rule, respectively. Several numerical examples, including one-dimensional and two-dimensional cases, show the efficiency of our proposed method.

Sign in / Sign up

Export Citation Format

Share Document