A NUMERICAL METHOD FOR THE PROBLEM OF COEFFICIENT

IDENTIFICATION OF THE WAVE EQUATION BASED ON A LOCAL OBSERVATION ON THE BOUNDARY

KENJI SHIROTA IBARAKI UNIVERSITY BUNKYO 2-1-1, MITO, IBARAKI 310-8512, JAPAN. E-MAIL: SHIROTA@MITO.IPC.IBARAKI.AC.JP

In this study, we consider the problem of coefficient identification of the scalar wave equation based on a local observation on the boundary. This problem is to determine the space-dependent unknown coefficient by means of the knowledge of simultaneous Dirichlet and Neumann data on a part of boundary.

Let Ω ⊂ R^{n} (n = 2, 3) be a bounded domain with two smooth boundaries Γ_{i} and Γ_{d}, satisfying Γ_{i} ∩ Γ_{d} = ∅. A conventional problem is to find the function u[K, g] such that

\frac{∂^{2}u}{∂t^{2}} − ∇ · (K∇u) = 0 in Ω × (0, T],
u = \frac{∂u}{∂t} = 0  on Ω × \{0\},
u = \overline{u}  on Γ_{i} × (0, T),
K \frac{∂u}{∂n} = g  on Γ_{d} × (0, T).

Here we assume that the coefficient function K belongs to L^{∞}(Ω) and satisfies the condition K(x) ≥ C > 0 for all x ∈ Ω, where C is given positive constant. Our inverse problem is to determine the unknown coefficient function K(x) with the knowledge of the Dirichlet data \overline{u} and the Neumann data \overline{q} on the boundary Γ_{i}. This problem is not uniquely solvable.

To determine the unknown coefficient function, we adopt the direct variational method. The unknown coefficient function K and the Neumann data g are determined by minimizing the functional F, defined by

F (K, g) := ∫_{0}^{T} ∫_{Γ_{i}} |q[K, g] − \overline{q}|^{2} ds dt,

where q[K, g] := K \frac{∂u[K, g]}{∂n}|_{Γ_{i}×(0,T)}.

To find the minimum K and g, we make use of the gradient method: For k = 0, 1, 2, . . . . . .

K_{j+1} = K_{j} − α_{j}s_{j}, g_{j+1} = g_{j} − β_{j}F_{g}(K_{j}, g_{j}),

where s_{j} is the search direction determined by using the projected gradient method and J_{g} is the first variation with respect to the function g. To use this method, we must get concrete expressions of the search directions. However it is difficult to get the concrete expressions of the search directions. We make use of the approximate search directions instead of the exact ones.

We show a numerical example about our method. Let Ω be a unit disk and Γ_{i} := {(x_{1}, x_{2}) | x_{1}^{2}+x_{2}^{2} = 1, x_{2} ≥ 0}. Suppose that the exact coefficient function K(x_{1}, x_{2}) = 2.0 (0 ≤ \sqrt{x_{1}^{2} + x_{2}^{2}} < 0.25), = 1.0 (0.25 < \sqrt{x_{1}^{2} + x_{2}^{2}} < 1) (Fig.1). The initial estimates are set as K_{0} = 1.5 and g_{0} = 0.0. Figure 2 shows the distribution of the calculated coefficient function after 20 iteration. This calculated coefficient function is in good agreement with the exact one for the region x_{2} > 0.

Fig. 1 Exact coefficient function Fig. 2 Calculated coefficient function