The Inverse Problem Of Rectangular Resistor Networks

Every real positive functionγon the bounded open regionΩinduces the Dirichlet-to-Neumann mapΛ_γon functions onδΩ.The main inverse problems are to give a characterization of the mapsΛ_γand to find out whenΛ_γcan uniquely determineγ.There are similar Dirichlet-to-Neumann mapsΛ_γin discrete networks and have similar inverse problems.E. B.Curtis use Neumann-to-Dirichlet maps get all conductance of resistor network in[3],but to do this need to know full information about Neumann-to-Dirichlet maps.To do in this way not only need more time and expense,but also can't get a solution maybe,because the information have known is more than inversion parameters.We discuss the inverse problem of discrete resistor network in this paper.At first,we characterize the Neumann-to-Dirichlet map on the rectangular network of resistors.The existence and uniqueness of positive problem are verified by matrix theory and some properties of positive problem are given.Among these properties,the discrete version of Green's Formula,Extremum Principle and Strong Extremum Principle make a great use in verifying the existence and uniqueness of inverse problem with one unknown parameter.Using these properties,we give out the main conclusion of this paper:the solution of inverse problem with one unknown parameter exists and is unique under certain condition.An algorithm of such inverse problem is also given.If the condition is not satisfied,an adjusting method is given to make the inverse process work well.We discuss the inversion algorithms of the problem with more parameters in further. Using the Steepest Descent method and Newton method,we inverse the problem with more parameters.By examples,we explain why the speed of descent of Steepest Descent method is slowly and the inversion results of Newton method are not very precise when the initial value is too large.At last we give out a more efficient algorithm - revised Newton method.