Stability and regularization of a backward parabolic PDE with variable coefficients
نویسندگان
چکیده
We consider a backward parabolic partial differential equation with variable coefficient a(x, t) in time. A new modification is used on the logarithmic convexity method to obtain a conditional stability estimate. Based on a formal solution, we reveal the essence of the ill-posedness and propose a simple regularization method. Moreover, we apply the regularization method to two representative cases. The results of both theoretical and numerical performance show the validity of our method.
منابع مشابه
A numerical scheme for solving nonlinear backward parabolic problems
In this paper a nonlinear backward parabolic problem in one dimensional space is considered. Using a suitable iterative algorithm, the problem is converted to a linear backward parabolic problem. For the corresponding problem, the backward finite differences method with suitable grid size is applied. It is shown that if the coefficients satisfy some special conditions, th...
متن کاملMultigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints
We present a multigrid algorithm for the solution of distributed parameter inverse problems constrained by variable-coefficient linear parabolic partial differential equations. We consider problems in which the inversion variable is a function of space only; for stability we use an L2 Tikhonov regularization. The main feature of our algorithm is that its convergence rate is mesh-independent—eve...
متن کاملA regularization method for solving a nonlinear backward inverse heat conduction problem using discrete mollification method
The present essay scrutinizes the application of discrete mollification as a filtering procedure to solve a nonlinear backward inverse heat conduction problem in one dimensional space. These problems are seriously ill-posed. So, we combine discrete mollification and space marching method to address the ill-posedness of the proposed problem. Moreover, a proof of stability and<b...
متن کاملA probabilistic approach to large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions
This paper is devoted to the study of the large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions. This work is the sequel of [13] in which a probabilistic method was developed to show that the solution of a parabolic semilinear PDE behaves like a linear term λT shifted with a function v, where (v, λ) is the solution of the ergodic PDE associated to t...
متن کاملA Numerical Approach to a Nonlinear and Degenerate Parabolic Problem by Regularization Scheme
In this work we propose a numerical scheme for a nonlinear and degenerate parabolic problem having application in petroleum reservoir and groundwater aquifer simulation. The degeneracy of the equation includes both locally fast and slow diffusion (i.e. the diffusion coefficients may explode or vanish in some point). The main difficulty is that the true solution is typically lacking in regularit...
متن کامل