This article presents a mathematical and numerical analysis of the adjoint problem approach for inverse coefficient problems related to linear parabolic equations. Based on maximum principle a structure of the coefficient-to-data mapping is derived. The obtained integral identities permit one to prove the monotonicity and invertibility of the input-output mappings, as well as formulate the gradient of the cost functional via the solutions of the direct and adjoint problems. In the second part of the paper a numerical algorithm for determining the diffusion coefficient k = k(x) in the linear parabolic equation u_t = (k(x)u _x)_x from the measured output data is presented. The main distinguished feature of the proposed algorithm is the use of a fine mesh for the numerical solution of the well-posed forward and backward parabolic problems, and a coarse mesh for the interpolation of unknown coefficient k = k(x). The nodal values of the unknown coefficient on the coarse mesh are recovered sequentially, solving on each step the well-posed forward and the sequence of backward initial value problems. This guarantees a compromise between the accuracy and stability of the solution of the considered inverse problem. An efficiency and applicability of the method is demonstrated on various numerical examples with noisy free and noisy data.