Regularization Of Three Inverse Boundary Value Problems

In this paper we consider three inverse boundary value problems: the high-dimensional Helmholtz equation and backward heat conduction problems on the bounded region and on the cylindrical region. To Helmholtz equation we want to get the value u(y, z) on the region 0≤z < 1 from the given Cauchy data u(y, 1) and to the last two problems we want to get the value u(x, t) on the region 0≤t≤T from the given data u(x, T). They are all severely ill-posed problems, and the ill-posedness becomes sharper as the unknown solutions are closer to the boundary point. So given some effective regularization methods to restore the stability of the solutions is very important not only for practice applications but also theoretical research. In the paper we use the simplified Tikhonov Regularization method, discrete regularization method and filtering regularization method to solve these problems and obtain very good convergence estimates. Especially,all the methods but the filtering regularization method are convergent at the boudary point.