Mathematical Aspects and Numerical Computations of an Inverse Boundary   Value Identification Using the Adjoint Method
  The purpose of this study is to show some mathematical aspects of the adjoint method that is a numerical method for the Cauchy problem, an inverse boundary value problem. The adjoint method is an iterative method based on the variational formulation, and the steepest descent method minimizes an objective functional derived from our original problem. The conventional adjoint method is time-consuming in numerical computations because of the Armijo criterion, which is used to numerically determine the step size of the steepest descent method. It is important to find explicit conditions for the convergence and the optimal step size. Some theoretical results about the convergence for the numerical method are obtained. Through numerical experiments, it is concluded that our theories are effective. 
Takemi Shigeta