CTSVD Method And Numerical Differentiation

A new method—cononical Truncation Singular Value Decomposition(cTSVD) method for solving ill-posed problems will be present in this paper,which is a modification method of classical TSVD method.Both theorical and numerical results show that limitations of TSVD method have been overcome by new method and no additive computation is needed.In general,calculating the singular system of a operator takes a lot of time.So the methods which depend on singular system are always not be valued.But in this paper we will find that a better result will be obtained by TSVD-like methods for the genuine solution with a high smooth scale.So a further research to TSVD-like method is necessary.In practical,the singular systems of some operators can be obtained easily.For these problems, the dominant of TSVD-like method will be more remarkable and numerical differentiation just is this kind of problem.Numerical differentiation is a classical ill-posed problem.The small errors in the measurement may lead to huge errors in the numerical results.This problem has been treated by several methods.In this paper,We will point out the limitations of solving operator in some classical method and corresponding modification will be present.The convergence property of solution changed significant only by a petty change for solving operator,which shows that it is important to choose a reasonable approach for ill-posed problems.Next,we will give a further disscussion to numerical differentiation in one dimension case with the L-generalized cTSVD mehtod.we will introduce mollification method at a new angle.First we introduce the auxiliary equation,which will build a bridge between mollification method and L—genaralized solution regularization method. Theories of L—genaralized solution regularization methods have been well developed,so we can obtain the theorical results easily and which construct a framework for solving numerical differentiation for one-dimensional case.In the following,we will deal with the numerical differentiation problems for two-dimensional case.With the mollification idea and a sound choice of operator L,the theoretical results which have been obtained for one-dimensional case can be generalized to two-dimensional case and arbitrary finite-dimensional cases.Morever,considering the singular systems of solving operators for regular domain can be obtained easily,so our method can be realized easily and fast.In term of irregular domain,on the one hand we can obtain singular systems by numerical method,on the other hand due to the theoretical results which we have obtained are fit for mostly regularization methods,so the mollification idea can be realized by others regularization methods which don't need singular system.In general,we can obtain the convergence results corresponding to smooth scale of accurate functions in Sobolev space for numerical differentiation problems,which is of pith and moment for development of ill-posed problem theory.