An iterative regularizing method for an incomplete boundary data problem for the biharmonic equation

Abstract

An incomplete boundary data problem for the biharmonic equation is considered, where the displacement is known throughout the boundary of the solution domain whilst the normal derivative and bending moment are specified on only a portion of the boundary. For this inverse ill‐posed problem an iterative regularizing method is proposed for the stable data reconstruction on the underspecified boundary part. Convergence is proven by showing that the method can be written as a Landweber‐type procedure for an operator formulation of the incomplete data problem. This reformulation renders a stopping rule, the discrepancy principle, for terminating the iterations in the case of noisy data. Uniqueness of a solution to the considered problem is also shown.

Integral equations for biharmonic data completion

“…We assume that data is given such that there exists a classical or weak solution to (1)- (2). In [10], it is shown that (1)-(2) is ill-posed and that there exists at most one solution. Further in [10], an iterative method is proposed and analysed for the biharmonic data completion.…”

“…In [10], it is shown that (1)-(2) is ill-posed and that there exists at most one solution. Further in [10], an iterative method is proposed and analysed for the biharmonic data completion. At each iteration step, mixed boundary value problems are solved for (1) updating functions on the boundary curves.…”

“…At each iteration step, mixed boundary value problems are solved for (1) updating functions on the boundary curves. It is mentioned at the end of [10] that a single-layer approach can be used to solve the mixed problems numerically.…”

“…This in total constitute the novelty of the presented work. Related inverse problems for the biharmonic equation are mentioned in the introduction to [10]; for history and applications of the biharmonic equation, see [21].…”

“…For the outline of the work, in Section 2, the iterative method of [10] is briefly surveyed. In Section 3, the potential representation is given of the solution to the mixed biharmonic problems needed in the iterative procedure.…”