ON THE DUALITY OF THE POTENTIAL METHODAND THE POINT SOURCE METHOD ININVERSE SCATTERING PROBLEMS

摘要

The reconstruction of scattered wave fromits far-field pattern is of great importance in inverse scatter-ing problems. The classic potential method due to Kirsch andKress is a well known scheme by solving an integral equation ofthe first kind with respect to a density function, which relatesthe scattered wave to its far-field pattern. In recent years,a filtering scheme known as point source method, is also welldeveloped, which is based on the point source decompositionand the reciprocity principle. This paper aims to consider thequantitative relation between these two regularizing methods.We prove that these two schemes will generate exactly thesame approximate solution when used with identical geomet-ric setup and if their own regularizing parameters are takenas a constant multiple (a golden rule). Our key step is toemploy an adjoint relation between the Herglotz wave opera-tor and the far-field operator. Further we provide estimatesof the solutions with regularization parameters different fromthe golden rule. As illustration and for practical testing ofthese results numerical examples are presented to show thenumerical equivalence of these two methods.