Numerical Methods For Three Kinds Of Ill-posed Problems

In this paper, we consider three kinds of inverse problems for partial differen-tial equations, i.e., boundary identification problems, non-point source identifica-tion problems and Cauchy problem of heat equation.Boundary identification problem is to estimate the shape of corrosion bound-ary from measurements. We study the moving boundary identification problem for heat equation, which contain one-dimensional single layer and multilayer do-main and two-dimensional cases, and Laplace equation, respectively. We use the quasi-reversibility regularization method to transform the original problem into well-posed problem, and then apply the method of lines to estimate the shape of the moving boundary. Numerical experiments show that our proposed method is effective and feasible.Inverse source identification problem is to consider Poisson equation in two-dimensional and three-dimensional space, aim is to detect the number, the location and size, and the shape of hidden inclusions within a body from measured data. We transform the source identification problem into an optimization problem for find-ing the minimum of an objective function. We apply Nelder-Mead simplex algo-rithm(NMA)、Gradient descent algorithm(GDA)、Leverberg-Marquardt optimiza-tion algorithm (LM A) and Trust-region-reflective optimization algorithm(TRA) to solve the optimization problem to get the minimum. From the results of numerical experiments, we can see that our proposed iterative algorithms are effective.Cauchy problem of heat equation is to determine temperature and flux of the entire region from the measured Cauchy data on one side. We mainly study the Cauchy problem of heat equation without initial values, and we apply the quasi-reversibility regularization method to transform the ill-posed problem into well-posed problem. If we impose some a priori assumption on the exact solution and choose the appropriate regularization parameter, then we can get the convergence estimates of temperature and flux in the entire region.