2016/04/15 by Petr N. Vabishchevich, Vabishchevich, Petr N.
Mathematics · #65M06 #65M32 #80A23 #Differential Equations and Boundary Problems #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Statistical and numerical algorithms
paper · pdf · doi:10.48550/arxiv.1604.04443
openalex publication_date 2016/04/15 · openalex created_date 2022/08/31 · openalex updated_date 2026/07/28
Coefficient inverse problems related to identifying the right-hand side of an\nequation with use of additional information is of interest among inverse\nproblems for partial differential equations. When considering non-stationary\nproblems, tasks of recovering the dependence of the right-hand side on time and\nspatial variables can be treated as independent. These tasks relate to a class\nof linear inverse problems, which sufficiently simplifies their study. This\nwork is devoted to a finding the dependence of right-hand side of\nmultidimensional parabolic equation on spatial variables using additional\nobservations of the solution at the final point of time - the final\noverdetermination. More general problems are associated with some integral\nobservation of the solution on time - the integral overdetermination. The first\nmethod of numerical solution of inverse problems is based on iterative solution\nof boundary value problem for time derivative with non-local acceleration. The\nsecond method is based on the known approach with iterative refinement of\ndesired dependence of the right-hand side on spacial variables. Capabilities of\nproposed methods are illustrated by numerical examples for model\ntwo-dimensional problem of identifying the right-hand side of a parabolic\nequation. The standard finite-element approximation on space is used, the time\ndiscretization is based on fully implicit two-level schemes.\n