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In this paper, the identification problem of recovering the spatial sourceF  ∈    L  2   ⁢   (  0  ,  l  ){F\in L^{2}(0,l)}    in the wave equationu   t  ⁢  t    =    u   x  ⁢  x    +   F  ⁢   (  x  )   ⁢   cos  ⁡   (   ω  ⁢  t   ){u_{tt}=u_{xx}+F(x)\cos(\omega t)}   , with harmonically varying external sourceF  ⁢   (  x  )   ⁢   cos  ⁡   (   ω  ⁢  t   ){F(x)\cos(\omega t)}    and with the homogeneous boundaryu  ⁢   (  0  ,  t  )    =   u  ⁢   (  l  ,  t  )    =  0{u(0,t)=u(l,t)=0}   ,t  ∈   (  0  ,  T  ){t\in(0,T)}   , and initialu  ⁢   (  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 outputg  ⁢   (  t  ){g(t)}   , the Neumann-type boundary measurementg  ⁢   (  t  )    :=    u  x   ⁢   (  0  ,  t  ){g(t):=u_{x}(0,t)}   ,t  ∈   (  0  ,  T  ){t\in(0,T)}   , at the left boundaryx  =  0{x=0}    is used. It is assumed that the observationg  ∈    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 sourceF  ∈    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 operatorΦ  ⁢  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)}   , whereu  ⁢   (  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 functionalJ  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}}   , whereJ  ⁢   (  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 functionalJ  ⁢   (  F  ){J(F)}   . Use of the gradient formula in the conjugate gradient algorithm (CGA) allows to construct a fast algorithm for recovering the unknown sourceF  ⁢   (  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.

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