Using Perturbation Method To Reduce Dimensions Of The Inverse Problem Of The Wear-line Of An Axi-symmetric Melting Container

In this paper,we advance a new method which is a combination of perturbation method and expansion with eigenfunction to deal with a high-dimension inverse problem of corrosion and reduce its dimension.The corrosion will destroy the original approximate axis-symmetry and complicate the shape,of the container.Thus the original one-dimension or two-dimension axis-symmetric problem probably become a two-dimension or three-dimension problem.Since the increase of complexity of geometry leads to the increase of complexity of computation,when we solve this kind of inverse problem numerically we often need to solve a lot of direct problems with flexible boundary.Especially,when we deal with a high-dimension problem the sick increase of complexity and amount of computation leads to great difficulty in solving the problem.In this paper,we use perturbation method to transfer the problem into a problem with fixed boundary and axis-symmetric domain. Then the boundary condition is not axis-symmetric and we use expansion with eigenfunction to transfer it into a group of lower-dimension problems. Using this method,we can transfer a two-dimension problem with unknown boundary into a group of one-dimension axis-symmetric problems and we also can transfer a three-dimension problem with unknown boundary into a group of two-dimension axis-symmetric problems with fixed boundary,sometimes even into a group of one-dimension problems.This method overcome the difficulty brought by the complexity of geometry and high-dimension.Large numbers of computational results show that the methods are effective and accurate.