Optimization Method For Identifying Unknown Coefficient In Parabolic Equation

In many kinds of theoretical and practical areas, one often encounters the problem of recovering the unkown coefficients in differential equations which belongs to the typical catageory of inverse problems mathematically and physically. It is well-known that these kinds of problems are usually ill-posed, therefore some regularization techniques are needed to obtain the stable approximate solution.The purpose of this thesis is to investigate optimization method for recovering the coefficient q(x) in the following parabolic initial-boundary value problem:from final temperature u(x, T) = z(x) as inversion input data, where initial temperature u₀(x) is given. The output least square method, one of the typical methods to treat ill-posed problems, is used to identify the unknown coefficient in this thesis. The solution of the above system depends on the unknown coefficient q(x), so this problem is a nonlinear and ill-posed inverse problem. On the other hand, the information of q(x) contained in u(x,T) is very weak due to the exponentially decay of u(x, t) with respect to t, and there is no uniqueness for this inverse problem with general initial data u₀ and single measurement z(x), the optimization technique should be applied to get some general solution.The recovering process applied in this thesis is to seek coefficient q{x) in a way that u(q)(x, t) matches its terminal observation data z(x) optimally around final time T with small length σ in L²-norm sense. In the numerical implementation, σ is taken to be one discrete time step size.The main work of this thesis is as follows. Fistly, the identifying problem is reformulated as a constrained minimization problem using the output least square approach. The existence of minimizer in orginal infinite demension space is proven. Secondly, a finite element method is used to approximate the constrained minimization problem, both existence of minimizer for cost functional in finite demensional space and the convergence of minimizer are proven. Finally, the discrete constrained problem is reduced to a sequence of unconstrained minimization problems and numerical experiments are obtained by using the Armijo-type algorithm. The numerical examples carried out in this thesis show the validity of the proposed inversion method and the stability of the algorithm.