Stability and Regularization Method for Inverse Initial Value Problem of Biparabolic Equation

Abstract

We consider an inverse initial value problem of the biparabolic equation; this problem is ill-posed and the regularization methods are needed to stabilize the numerical computations. This paper firstly establishes a conditional stability of Holder type, then uses a modified regularization method to overcome its ill-posedness and gives the convergence estimate under an a-priori assumption for the exact solution. Finally, a numerical example is presented to show that this method works well.

On terminal value problems for bi-parabolic equations driven by Wiener process and fractional Brownian motions

“…For the homogeneous case (when F = 0), Lakhdari [22] showed that the such problem is ill-posed and proposed a regularizing strategy based on the Kozlov-Mazýa iteration method to approximate the solution. Zhang, in [39], established a conditional stability of Holder type and used a modified regularization method to overcome the ill-posedness in this case. For the non-homogeneous problem, the very last paper [27] investigated the deterministic TVP with two cases of source function including linear and nonlinear sources.…”

Dynamical Model Updating Based on Gradient Regularization Method

“…This is a typical inverse problem of operator identification [1] . We can solve the problem by gradient regularization method(GR)which was first presented in reference [2] .The method proceed with the common of inverse problem,but not attached to any particular constraint,and the solution is not affected by spatial dimension limits.Thus,it is a kind of universal method to solve the inverse problem. Based on improved GR method [5,7] ,the general finite element program is compiled,the issues such as model error and the choice of measuring points frequencies are discussed as well through numerical examples.…”

“…This is an obvious ill-posed problem [1] . Applying regularization method to construct regular functionals [2] ，based on improved gradient regularization method [5,7] ,we can get the nonlinear inverse problem iterative formula as follows:…”