Mathematical Sciences: Parameter Estimation Problems with Discontinuous Coefficients

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The subject of this project is parameter estimation problems for flow equations with discontinuous coefficients. These are the problems where parameters of the medium are required to be determined by indirect measurements. Specifically the objective of this project is developing of efficient algorithms for the identification of discontinuous parameters a(x) in 2-D and 3-D problems. The suggested algorithm uses finite elements approximations of the space variable u(x,t) and does not require any discretization of the parameter variable a(x). Special averaging procedure is introduced to eliminate numerical instabilities. The general theory is based on G-convergence of parabolic operators and principle od simultaneous minimization. Preliminary results for 2- d problems show examples of successful identification and significant possibilities for vectorization. We also study the problem of further improvements in the identification by considering additional data sets of observations.