摘要：A case of steady-case heat flow through a plane wall, which can be formulated as u t ( x , y , t ) − div ( k ( x , y ) ∇ u ( x , y , t ) ) = F ( x , y , t ) $u_{t}(x,y,t)- \operatorname{div} (k(x,y) \nabla u(x,y,t)) = F(x,y,t)$ with Robin boundary condition − k ( 1 , y ) u x ( 1 , y , t ) = ν 1 [ u ( 1 , y , t ) − T 0 ( t ) ] $-k(1,y)u_{x}(1,y,t)= \nu_) [u(1,y,t)-T_((t)]$ , − k ( x , 1 ) u y ( x , 1 , t ) = ν 2 [ u ( x , 1 , t ) − T 1 ( t ) ] $-k(x,1)u_{y}(x,1,t)= \nu_, [u(x,1,t)-T_)(t)]$ , where ω : = { F ( x , y , t ) ; T 0 ( t ) ; T 1 ( t ) } $\omega:=\{F(x,y,t);T_((t);T_)(t)\}$ is to be determined, from the measured final data μ T ( x , y ) = u ( x , y , T ) $\mu_{T}(x,y)=u(x,y,T)$ is investigated. It is proved that the Fréchet gradient of the cost functional J ( ω ) = ∥ μ T ( x , y ) − u ( x , y , T ; ω ) ∥ 2 $J(\omega )=\ \mu_{T}(x,y)-u(x,y,T;\omega)\ ^,$ can be found via the solution of the adjoint parabolic problem. Lipschitz continuity of the gradient is derived. The obtained results permit one to prove the existence of a quasi-solution of the inverse problem. A steepest descent method with line search, which produces a monotone iteration scheme based on the gradient, is formulated. Some convergence results are given.
