Identification of a harmonically varying external source in wave equation from Neumann-type boundary measurement

Abstract

Abstract In this paper, the identification problem of recovering the spatial source F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} in the wave equation u t ⁢ t = u x ⁢ x + F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {u_{tt}=u_{xx}+F(x)\cos(\omega t)} , with harmonically varying external source F ⁢ ( x ) ⁢ cos ⁡ ( ω ⁢ t ) {F(x)\cos(\omega t)} and with the homogeneous boundary u ⁢ ( 0 , t ) = u ⁢ ( l , t ) = 0 {u(0,t)=u(l,t)=0} , t ∈ ( 0 , T ) {t\in(0,T)} , and initial u ⁢ ( x , 0 ) = u t ⁢ ( x , 0 ) = 0 {u(x,0)=u_{t}(x,0)=0} , x ∈ ( 0 , l ) {x\in(0,l)} , conditions, is studied. As a measurement output g ⁢ ( t ) {g(t)} , the Neumann-type boundary measurement g ⁢ ( t ) := u x ⁢ ( 0 , t ) {g(t):=u_{x}(0,t)} , t ∈ ( 0 , T ) {t\in(0,T)} , at the left boundary x = 0 {x=0} is used. It is assumed that the observation g ∈ L 2 ⁢ ( 0 , T ) {g\in L^{2}(0,T)} may has a random noise. We propose combination of the boundary control for PDEs, adjoint method and Tikhonov regularization, for identification of the unknown source F ∈ L 2 ⁢ ( 0 , l ) {F\in L^{2}(0,l)} . Our approach based on weak solution theory of PDEs and, as a result, allows use of nonsmooth input/output data. Introducing the input-output operator Φ ⁢ F := u x ⁢ ( 0 , t ; F ) {\Phi F:=u_{x}(0,t;F)} , Φ : L 2 ⁢ ( 0 , l ) ↦ L 2 ⁢ ( 0 , T ) {\Phi:L^{2}(0,l)\mapsto L^{2}(0,T)} , where u ⁢ ( x , t ; F ) {u(x,t;F)} is the solution of the wave equation with above homogeneous boundary and initial conditions, we first prove the compactness of this operator. This allows to obtain the uniqueness of regularized solution of the identification problem, i.e. the minimum of the regularized cost functional J α ⁢ ( F ) := J ⁢ ( F ) + 1 2 ⁢ α ⁢ ∥ F ∥ L 2 ⁢ ( 0 , l ) 2 {J_{\alpha}(F):=J(F)+\frac{1}{2}\alpha\|F\|_{L^{2}(0,l)}^{2}} , where J ⁢ ( F ) = 1 2 ⁢ ∥ u x ⁢ ( 0 , ⋅ ; F ) - g ∥ L 2 ⁢ ( 0 , T ) 2 {J(F)=\frac{1}{2}\|u_{x}(0,\cdot\,;F)-g\|_{L^{2}(0,T)}^{2}} . Then the adjoint problem approach is used to derive a formula for the Fréchet gradient of the cost functional J ⁢ ( F ) {J(F)} . Use of the gradient formula in the conjugate gradient algorithm (CGA) allows to construct a fast algorithm for recovering the unknown source F ⁢ ( x ) {F(x)} . A comprehensive set of benchmark numerical examples, with up to 10 noise level random noisy data, illustrate the usefulness and effectiveness of the proposed approach.