Iterative computational identification of a spacewise dependent the source in a parabolic equations
This work addresses the numerical solution of a linear inverse problem for parabolic PDEs, which is a known problem for applied mathematicians and engineers, but the methods appear incremental.
The paper proposes two iterative methods for identifying the spacewise-dependent source term in a parabolic equation from final or integral overdetermination, demonstrating their effectiveness on a 2D model problem with finite-element and implicit time discretization.
Coefficient inverse problems related to identifying the right-hand side of an equation with use of additional information is of interest among inverse problems for partial differential equations. When considering non-stationary problems, tasks of recovering the dependence of the right-hand side on time and spatial variables can be treated as independent. These tasks relate to a class of linear inverse problems, which sufficiently simplifies their study. This work is devoted to a finding the dependence of right-hand side of multidimensional parabolic equation on spatial variables using additional observations of the solution at the final point of time - the final overdetermination. More general problems are associated with some integral observation of the solution on time - the integral overdetermination. The first method of numerical solution of inverse problems is based on iterative solution of boundary value problem for time derivative with non-local acceleration. The second method is based on the known approach with iterative refinement of desired dependence of the right-hand side on spacial variables. Capabilities of proposed methods are illustrated by numerical examples for model two-dimensional problem of identifying the right-hand side of a parabolic equation. The standard finite-element approximation on space is used, the time discretization is based on fully implicit two-level schemes.