Identification of a space- and time-dependent source in a variable coefficient advection-diffusion equation from Dirichlet and Neumann boundary measured outputs

Abstract

Abstract This paper considers the inverse problem of identifying an unknown space- and time-dependent source function F ⁢ ( x , t ) F(x,t) in the variable coefficient advection-diffusion equation u t = ( D ⁢ ( x ) ⁢ u x ) x - ( V ⁢ ( x ) ⁢ u ) x + F ⁢ ( x , t ) u_{t}=(D(x)u_{x})_{x}-(V(x)u)_{x}+F(x,t) from the Dirichlet ν ⁢ ( t ) := u ⁢ ( ℓ , t ) \nu(t):=u(\ell,t) and Neumann f ⁢ ( t ) := - D ⁢ ( 0 ) ⁢ u x ⁢ ( 0 , t ) f(t):=-D(0)u_{x}(0,t) , t ∈ ( 0 , T ] t\in(0,T] , boundary measured outputs. This problem was motivated by several important real-world applications in the field of contaminant hydrogeology, and the novel analysis presented here is highly relevant to problems of practical interest. The input-output operators corresponding to the Dirichlet and Neumann measured boundary data are introduced. The inverse problem is then formulated as a system of operator equations consisting of these operators and the measured outputs. The compactness and Lipschitz continuity of the input-output operators are proved in the relevant classes of admissible source functions ℱ and F r \mathcal{F}_{r} . These results together with the derived trace estimates allow us to show the existence of a quasi-solution of the inverse source problem as a minimum of the Tikhonov functional, under minimal regularity assumptions with respect to the source function and other inputs. An explicit gradient formula for the Fréchet gradient of the Tikhonov functional is also derived by means of an appropriate adjoint problem.