Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint

We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi class="ltx_mathvariant_double-struck" mathvariant="double-struck">R</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> Rᵈ , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>d</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> d≥ 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.