| Popis: |
In this study, an inverse source problem of identifying an unknown spatial load f ( x , y ) in a system governed by the Kirchhoff plate equation u t t + D ▵ 2 u = g ( t ) f ( x , y ) , ( x , y , t ) ∈ Ω × ( 0 , T ) from available boundary observation (measured normal rotation) θ ( x , t ) : = u y ( x , 0 , t ) is studied. The inverse problem is reformulated as a minimization problem for the Tikhonov functional and existence of a minimizer is proved. The adjoint problem corresponding to the inverse problem is introduced to prove Frechet differentiability of the Tikhonov functional. An explicit gradient formula for the Tikhonov functional is derived by making use of the unique weak solution of the adjoint problem. Furthermore, new computational algorithm based on the conjugate gradient algorithm is developed for the numerical reconstruction of the spatial load. Numerical examples with random noisy measured output are presented to illustrate the validity and effectiveness of the proposed approach. |
