BACKUS-GILBERT ALGORITHM FOR A CAUCHY PROBLEM OF LAPLACE EQUATION

Y.C. HON AND T. WEI

Consider a bounded and connected domain Ω ∈ ℜ^{d} with Lipschitz boundary ∂Ω. Assume that an inaccessible part of the boundary has been damaged by corrosion. The non-destructive evaluation technique was to apply an alternating current to an accessible boundary from which the corroded boundary could be determined by observing the output current fields. This non-destructive evaluation technique is very important and in high demand by the engineering industry. The mathematical formulation of the problem as an inverse problem was given as follows:

(0.1)  Δu = 0,  x ∈ Ω
(0.2)  u(x) = f(x),  x ∈ Γ
(0.3)  \frac{∂u(x)}{∂ν} = g(x),  x ∈ Γ

where Δ is the Laplacian operator, Γ is the accessible part of the boundary ∂Ω with given Cauchy data f and g, and ν is the outward normal unit vector with respect to Γ. Without loss of generality, we may assume that Γ is connected. The inverse problem is then to determine an unknown boundary Υ in Ω which satisfies u(x) = 0 on Υ in a weak sense. It is not required that Υ ∪ Γ = ∂Ω.

The purpose of this talk is to devise a numerical approximation of the solution of (0.1) – (0.3). The main key of the proposed method is to transform the Cauchy problem to a moment problem whose numerical algorithms are available. The unknown solution can then be obtained by solving a mixed boundary value problem for Laplace equation based on the approximation of the solution on the boundary ∂Ω \ Γ.