An Explicit Method for Inverse Reconstruction of Equivalent Current Dipoles and Quadrupoles

Recovering multiple small inclusions in a three-dimensional domain using a single measurement

“…It is based on the methods introduced in [9,19,21] for the two-dimensional case. It generalizes the algorithm discussed in [23] in two ways: it is applicable for an arbitrary domains, and it is able to systematically handle multiple inclusions.…”

“…The requirement to solve such systems has arisen frequently in many recent works, (cf. [9,11,19,21,22,25,26]). A common and effective method to solve this problem, stated below, reduces the system (8) to finding the roots of a K -th degree polynomial.…”

“…Algorithm 1 combines the ideas of [9,19,21]. Given any harmonic function v(y) which satisfies v = 0 inside , multiply (5) by v and integrate by parts.…”

“…Following [19,21], we first consider the particular sequence of these identities corresponding to m = n. Note that |y| n Y n n (ŷ) = C(y 1 + iy 2 ) n for a known constant C independent of y and that ν · ∇ y |y| n Y n n (ŷ) = Cn(y 1 + iy 2 ) n−1 (ν 1 + iν 2 ).…”

“…The algorithm is an extension of the two-dimensional algorithm in [9] to three dimensions, using ideas developed in [19][20][21][22]. In two dimensions, we also show that the algorithm developed in [12] is a special case of the method presented in [9].…”