Reconstruction of small inhomogeneities from boundary measurements.

*(English)*Zbl 1113.35148
Lecture Notes in Mathematics 1846. Berlin: Springer (ISBN 3-540-22483-1/pbk). ix, 238 p. (2004).

The book, according to the authors, describes a “fresh and promising techiques for the reconstruction of small inclusions from boundary measurements” and the presentation is “intended to be self-contained”.

The book consists of three parts.The first part deals with conductivity inclusions, the second part deals with the elastic ones, and the third part deals with electromagnetic inclusions. Besides the smallness of the inclusions, the authors require the inclusions to be weak. The weakness of the inclusions means that \(|a-1|\) is sufficiently small, where \(a\) is defined below.

In the first part the governing equation is the static equation \(\Delta u + \text{div} (a\operatorname{grad} u)=0 \) in \(\Omega\), \(u_N|_S=g\), where \(a= \text{const}>0, a\neq 1,\) in \(z+\varepsilon D\subset \Omega\) and \(a=0\) in the complement of the region \(z+\varepsilon D\), \(\varepsilon>0\) is a small parameter, \(D\) is a ball (in many cases discussed in the book) or, more generally, \(D\) is a Lipschitz star-shaped domain, \(S\) is the boundary of \(\Omega\), \(N\) is the unit outer normal to \(S\), and \(\int_Sgds=0\). The solution \(u\) has to satisfy the requirement \(\int_Suds=0\), and the transmission boundary conditions on \(\partial D\).

The method allows one to treat the Dirichlet condition also.

Given \(g\) and \(a\), one finds uniquely this solution \(u\) and, therefore, \(f:=u|_S\).

The problems in parts 2 and 3 are modelled by the problems for scalar equations with scalar boundary conditions.

The authors derive asymptotics of \(u\) as \(\varepsilon \to 0\). The asymptotics obtained depend on the parameter \(\varepsilon\), which is not dimensionless, while in electromagnetic problems the usual asymptotics are obtained with respect to a dimensionless parameter \(k\varepsilon\gg1\), where \(k\) is the wavenumber and \(\varepsilon\) is the charactersistic dimension of a small body.

Given \(g\) and \(f:=u|_S\), the authors want to recover the location of \(z\), the size of \(D\) and its orientation, from these asymptotics, which involve the polarizability tensors and their analogs. The same scheme is used in Parts 2 and 3.

The reader may raise some questions concerning the author’s presentation.

First, the influence of noise in the boundary measurements is not discussed. However, this issue is important: suppose that the noise level is one percent: \(10^{-2}\), and the parameter \(\varepsilon<10^{-1}\), while the size of the body is of order one. In the asymptotics formulas, derived by the authors, the main informative term is of order \(\varepsilon^d\), where \(d\) is the dimension of the space. If \(d=3\), then this term is of order \(10^{-3}\), which is much less than the noise level \(10^{-2}\). This leaves the reader with some doubts about the practical usefulness of the suggested approach, because the measured quantities are much less in absolute value than the noise level. Of course, one may argue that the measurements should be done more accurately, so that the noise level be decreased to, say, \(10^{-4}\). The question is: can one do this practically?

Theoretically speaking, the asymptotics of the solution \(u\) corresponding to a scattering by a small inclusion with the Dirichlet boundary condition \(u|_{\partial D}=0\) (which corresponds to a perfect conductor in electromagnetic-type problems) is not covered. In this case the scattered field is of order \(\varepsilon\), rather than \(\varepsilon^d\).

The asymptotics in this case is derived in the monograph (*) [A. G. Ramm, Iterative methods for calculating static fields and wave scattering by small bodies, Springer Verlag, New York (1982; Zbl 0479.45007)], which is not mentioned by the authors. In the same monograph one can find analytical formulas for the polarizability tensors for bodies of arbitrary shapes, which allow one to compute these tensors with arbitrary accuracy.

In the monographs [A. G.Ramm, Scattering by obstacles, D. Reidel Dordrecht (1986; Zbl 0607.35006) and Inverse problems, Springer, New York (2005; Zbl 1083.35002)], the basic approach to a study of wave scattering by small bodies consists in a reduction of this problem to finding finitely many numbers (parameters, such as polarizabilty tensors and capacitances), which define the scattering amplitude. While for the scattering by bodies that are not small one has to solve some integral equations, and their solutions allow one to compute the scattering amplitude, in the case when the bodies are small one has to find not functions, but some numbers, and these numbers determine the scattering amplitude (see (*)). To find these numbers, it is not necessary to solve some integral equations, as shown in the abovementioned monographs.

Occasionally, it is not clear what the authors claim: for example, if \(d=2\) then formula (2.17) on p.17 (referenced to the author’s paper [170]) implies that the values (2.16) vanish for every density \(\phi\in L^2(\partial D)\), while obviosly this is not true for \(\phi=1\).

The references in the book are not always adequate. For example, estimate (2.21) on p.20, is a well-known consequnce of the old results by Mikhlin and Vishik [see e.g., the book by S. Mikhlin, Multidimensional singular integrals and integral equations, Nauka, Moscow (1962; Zbl 0105.30301), p. 214.]

The problems discussed in the book are of interest both theoretically and practically. This book hopefully will stimulate the research in the area of finding small inhomogeneities from experimental data.

The book consists of three parts.The first part deals with conductivity inclusions, the second part deals with the elastic ones, and the third part deals with electromagnetic inclusions. Besides the smallness of the inclusions, the authors require the inclusions to be weak. The weakness of the inclusions means that \(|a-1|\) is sufficiently small, where \(a\) is defined below.

In the first part the governing equation is the static equation \(\Delta u + \text{div} (a\operatorname{grad} u)=0 \) in \(\Omega\), \(u_N|_S=g\), where \(a= \text{const}>0, a\neq 1,\) in \(z+\varepsilon D\subset \Omega\) and \(a=0\) in the complement of the region \(z+\varepsilon D\), \(\varepsilon>0\) is a small parameter, \(D\) is a ball (in many cases discussed in the book) or, more generally, \(D\) is a Lipschitz star-shaped domain, \(S\) is the boundary of \(\Omega\), \(N\) is the unit outer normal to \(S\), and \(\int_Sgds=0\). The solution \(u\) has to satisfy the requirement \(\int_Suds=0\), and the transmission boundary conditions on \(\partial D\).

The method allows one to treat the Dirichlet condition also.

Given \(g\) and \(a\), one finds uniquely this solution \(u\) and, therefore, \(f:=u|_S\).

The problems in parts 2 and 3 are modelled by the problems for scalar equations with scalar boundary conditions.

The authors derive asymptotics of \(u\) as \(\varepsilon \to 0\). The asymptotics obtained depend on the parameter \(\varepsilon\), which is not dimensionless, while in electromagnetic problems the usual asymptotics are obtained with respect to a dimensionless parameter \(k\varepsilon\gg1\), where \(k\) is the wavenumber and \(\varepsilon\) is the charactersistic dimension of a small body.

Given \(g\) and \(f:=u|_S\), the authors want to recover the location of \(z\), the size of \(D\) and its orientation, from these asymptotics, which involve the polarizability tensors and their analogs. The same scheme is used in Parts 2 and 3.

The reader may raise some questions concerning the author’s presentation.

First, the influence of noise in the boundary measurements is not discussed. However, this issue is important: suppose that the noise level is one percent: \(10^{-2}\), and the parameter \(\varepsilon<10^{-1}\), while the size of the body is of order one. In the asymptotics formulas, derived by the authors, the main informative term is of order \(\varepsilon^d\), where \(d\) is the dimension of the space. If \(d=3\), then this term is of order \(10^{-3}\), which is much less than the noise level \(10^{-2}\). This leaves the reader with some doubts about the practical usefulness of the suggested approach, because the measured quantities are much less in absolute value than the noise level. Of course, one may argue that the measurements should be done more accurately, so that the noise level be decreased to, say, \(10^{-4}\). The question is: can one do this practically?

Theoretically speaking, the asymptotics of the solution \(u\) corresponding to a scattering by a small inclusion with the Dirichlet boundary condition \(u|_{\partial D}=0\) (which corresponds to a perfect conductor in electromagnetic-type problems) is not covered. In this case the scattered field is of order \(\varepsilon\), rather than \(\varepsilon^d\).

The asymptotics in this case is derived in the monograph (*) [A. G. Ramm, Iterative methods for calculating static fields and wave scattering by small bodies, Springer Verlag, New York (1982; Zbl 0479.45007)], which is not mentioned by the authors. In the same monograph one can find analytical formulas for the polarizability tensors for bodies of arbitrary shapes, which allow one to compute these tensors with arbitrary accuracy.

In the monographs [A. G.Ramm, Scattering by obstacles, D. Reidel Dordrecht (1986; Zbl 0607.35006) and Inverse problems, Springer, New York (2005; Zbl 1083.35002)], the basic approach to a study of wave scattering by small bodies consists in a reduction of this problem to finding finitely many numbers (parameters, such as polarizabilty tensors and capacitances), which define the scattering amplitude. While for the scattering by bodies that are not small one has to solve some integral equations, and their solutions allow one to compute the scattering amplitude, in the case when the bodies are small one has to find not functions, but some numbers, and these numbers determine the scattering amplitude (see (*)). To find these numbers, it is not necessary to solve some integral equations, as shown in the abovementioned monographs.

Occasionally, it is not clear what the authors claim: for example, if \(d=2\) then formula (2.17) on p.17 (referenced to the author’s paper [170]) implies that the values (2.16) vanish for every density \(\phi\in L^2(\partial D)\), while obviosly this is not true for \(\phi=1\).

The references in the book are not always adequate. For example, estimate (2.21) on p.20, is a well-known consequnce of the old results by Mikhlin and Vishik [see e.g., the book by S. Mikhlin, Multidimensional singular integrals and integral equations, Nauka, Moscow (1962; Zbl 0105.30301), p. 214.]

The problems discussed in the book are of interest both theoretically and practically. This book hopefully will stimulate the research in the area of finding small inhomogeneities from experimental data.

Reviewer: Alexander G. Ramm (Manhattan)

##### MSC:

35R30 | Inverse problems for PDEs |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

31B20 | Boundary value and inverse problems for harmonic functions in higher dimensions |

74G75 | Inverse problems in equilibrium solid mechanics |

78A70 | Biological applications of optics and electromagnetic theory |

92C55 | Biomedical imaging and signal processing |