Research On Numerical Solution Methods For Some Inverse Problems

Since the 1960s, inverse problems for"tracing the reason or preimage (in-put) from the e?ect or representation (output)"have appeared in many fieldsin science and technology, such as geophysics, life sciences, material science,remote sensing technique, signal and image processing, industrial control, andeven economic decision. We call such inverse problems mathematical physicsinverse problems. Since most inverse problems cannot be solved analytically,numerical solution methods play a fundamental role. Many scholars have beenattracted to the field of inverse problems because the theories and algorithmsof such problems are of great challenging, and the applications are widespread.In the last few decades, the field of researches in mathematical physics inverseproblems has been one of the fastest developing field.The contents of the field in inverse problems are widespread. In this thesiswe will mainly discuss three inverse problems. 1) The numerical inversion ofLaplace transform. In this case, we are concerned with smooth solutions. 2) Nu-merical solution methods for piecewise constant solutions of Fredholm integralequations of the first kind. This is an ill-posed linear problem. 3) Numerical so-lution methods for the Robin inverse problem. We are concerned with piecewiseconstant Robin coe?cients. This is an ill-posed nonlinear problem.This thesis consists of four chapters. The main contents are as follow:In Chapter one, we give some introductory materials and some back groundabout inverse problems, including the well-posedness and ill-posedness. We alsointroduce some numerical methods for the Fredholm integral equation of the firstkind, including the idea and construction of regularization operator, the errorestimation of regularized solution and the determination of regularization pa-rameter. In the last section of this chapter, we will introduce some backgroundabout the Robin inverse problem.In Chapter two, we propose numerical algorithms for inversion of Laplace transform based on high order numerical quadratures. It is shown that bychoosing suitable quadrature rules and sample points, small discretization er-rors can be guaranteed. Thus, by applying a suitable regularization to thelinear system, a numerical solution of high accuracy can be found. Numericalresults show that the approximate inverse Laplace transform obtained by thealgorithms proposed in this paper can be very accurate.In Chapter three, we consider numerical solutions for Fredholm integralequations of the first kind with piecewise constant solutions. We assume thatthe true solution has k di?erent function values. We consider the cases whichthe function values are known or unknown respectively and propose a modifiedTikhonov-TV regularization method. The modified Tikhonov-TV functionalis then approximated by a positive quadratic functional and thus we derivean approximate modified Newton method. Numerical results show that if thenumber of function values and the number of discontinuous points are small, wecan obtain quite good numerical results. That is, we can recover the functionvalues and the discontinuous points accurately.In Chapter four, we consider the Robin inverse problem where the Robin co-e?cient is piecewise constant. This is a nonlinear inverse problem. Our methodsare based on [F. Lin and W. Fang, Inverse Problems, 21 (2005), pp. 1757–1772]:transform the problem into a boundary value integral equation first and thenintroduce a new function to transform the nonlinear problem into a linear one.As in Chapter three, we also consider two cases. In the first case, we assumethat the Robin coe?cient takes k di?erent values {c1,c2,...,ck} with knownci (i = 1,2,...,k). In this case, we insert this information into the Tikhonovfunctional to build up a variant Modica-Mortola-functional (or variant MM-functional for short), and approximate the functional by a positive quadraticfunctional. In this way, we obtain a modified approximate Newton method. Inthe second case, we only know that the Robin coe?cient is piecewise constantand the bound of the coe?cient. For this case, we consider using Tikhonov-TVagain. Similarly, we approximate the functional a positive quadratic functional.Then we derive a sequence quadratic programming algorithm for the problem.Numerical results show that we can estimate the piecewise constant Robin co-e?cient quite well by using the algorithms proposed in this chapter.