Semidefnite Relaxation For Parameter Identifcation Problems

A large variety of natural and social phenomena can be modeled by (systems of)partialdiferentialequations(PDEs）. WhendoingsimulationsbasedonaPDEmodel,it is assumed that all involved parameters—such as coefcients or source terms—areknown, so people often need to solve a Parameter Identifcation Problem, which is aninverse problem. In our paper, we consider an elliptic partial diferential equation anddetermine the parameter q from the observation of solution. Since the inverse prob-lem is ill-posed, we use the output-least-square methods with certain regularizationtechniques, and we transform the Parameter Identifcation Problem into a constrainedregularization problem. Then we discretize the Parameter Identifcation Problem andget a Quadratic Constraints Quadratic Programming(QCQP) problem. After that, wediscusshowto formulatea SDP relaxationforQCQP problem. Atlast,wedetailaprac-tical implementation based on this approach, both one dimension and two dimension,and numerical results illustrate the feasibility of our method.