The factorization method for inverse problems.

*(English)*Zbl 1222.35001
Oxford Lecture Series in Mathematics and Its Applications 36 (ISBN 978-0-19-921353-5/hbk). xiv, 201 p. (2008).

The book is concerned with the identification of unknown open sets or domains of unknown functions related to stationary equations. The first problem dealt with arises in acoustics and is related to identifying the form of an unknown body \(D\) embedded in the whole of the space \({\mathbb R}^m\), where the model equation is Helmholtz’s equation, with given constant wavenumber \(k\) and a refraction index equal to \(1\), obtained under the assumption that the motion should be time periodic and subject to Sommmerfeld’s radiation condition. In this case it is assumed that a family of incident waves \(u^i\), depending on a direction parameter, is sent to the body \(D\) (having a connected complement set), which, in turn, spreads away a family of scattered waves \(u^s\).

On \(\partial D\) the field \(u^s\) satisfies \(u^s=-u^i\) or \(D_\nu u^s=-D_\nu u^i\), \(D_\nu\) denoting the outer normal derivative. The Dirichlet and Neumann conditions correspond to the sound-soft and sound-hard obstacles.

The information about the form of the body is collected in the so-called far field pattern \(u^\infty\) depending on two directions, the first related to the incident wave, the latter to the scattered one. Of course, it is assumed that \(u^\infty\) can be measured. By means of \(u^\infty\) the authors can define a far field (linear integral) operator \(F\), acting on \(L^2(S^2)\), \(S^2\) being the unit sphere in \({\mathbb R}^3\).

The basic idea in the book consists in factoring \(F=GTG^*\) as a product of three operators – whence the name of factorization method – and, by means of this factorization, recover all points \(z\in D\). This is done by using the fundamental fact that \(F\) is a normal operator and computing the infimum \(f(z)\) of the modulus of the quadratic form \(| (\psi,F\psi)_{L^2(S^2)}| \) under the constraint \((\psi,\phi_z)_{L^2(S^2)}=1\), where \(\phi_z(\widehat x)=e^{-ik\widehat x\cdot z}\). Under the assumption that \(k^2\) is not an eigenvalue of \(-\Delta\), a point \(z\) will belong to \(D\) if, and only if, \(f(z)\) turns out to be positive.

A similar technique is applied to the case where the scatterer \(D\) is characterized by an impedance boundary condition \(D_\nu u + \lambda u=0\) – i.e. a Robin boundary condition –, \(u\) and \(\lambda\) denoting the total field and an \(L^\infty(\partial D)\)-function such that \(\text{ Im}\lambda\geq 0\) a.e. on \(\partial D\). The reconstruction of \(D\) is done by using both far and near field operators. The latter denomination means that the field operator \(F\) is related to the radiating scattered wave \(v^s\) measured on the boundary of an open set \(\Omega\) containing \({\overline D}\).

We observe that in this case the operator \(F\) is not normal. Therefore the infimum procedure developed in the Dirichlet and Neumann cases – consuming a lot of time in explicit computations – has to be corrected suitably. The right operator is now \(F_\#=| \text{ Re}\,F| +| \text{ Im}\,F| \), where \(2\text{ Re}\,F=(F+F^*)\) and \(2\text{ Im}\,F=(F-F^*)\), \(F^*\) standing for the operator adjoint to \(F\). Since operator \(F_\#\) is (positive and) normal, it admits an eigensystem \((\lambda_j,\psi_j)\). In this case a point \(z\in D\) if, and only if, the condition \(\sum_{j=1}^{+\infty}\, \lambda_j^{-1}| (\psi_j,\phi_z)_{L^2(S^2)}| ^2<+\infty\) is satisfied, i.e. if the equation \(F_\#^{1/2}\phi=\phi_z\) is solvable.

Another important question is concerned with the so-called mixed boundary conditions. Explicitly this means that \(D\) consists of several disconnected open subsets on the boundaries of which either Dirichlet or Robin conditions are given. In the case when \(D=D_1\cup D_2\) it can be recovered by operator \(F_\#\) only if we assume to know some localization of \(D\), i.e. a pair of open sets \(\Omega_j\), \(j=1,2\), such that \({\overline D}_j\subset \Omega_j\), \(j=1,2\).

A chapter is then devoted to the case where the refraction index \(n\) is non-constant. However, the results proved concern only the determination of the domain of \(n-1\) and not the function \(n\) itself.

The latter part of the book is concerned with a problem similar to the acoustic one for time periodic solutions to the Maxwell equations subject to the Silver-Müller radiation condition – corresponding to the Sommerfeld condition in the acoustic case. The basic equation is now \[ \text{ curl}\,\big[\varepsilon^{-1}_r \text{ curl} H\big] - k^2\text{ curl} H=0, \] where the complex-valued function \(\varepsilon_r\in L^\infty(D)\), accounting for the electromagnetic properties of \(D\), with \(\text{ Re}\,\varepsilon_r\geq 0\) and \(\text{ Im}\,\varepsilon_r\geq 0\) a.e. in \(D\). In this case the incident magnetic field is \(H^i(x,\theta,p)=e^{ik\theta \cdot x}p\), where the polarization tensor \(p\in {\mathbb R}^3\) satisfies the orthogonality condition \(p\cdot \theta=0\).

After having factored the far field operator \(F\) and defined the somewhat involved (and new) concept of interior transmission eigenvalue, the authors can localize, under suitable assumptions on the function \(\varepsilon_r\), the support of the contrast \(q=1-(1/\varepsilon_r)\) when the function \(\phi_z\) is now defined by \(\phi_z(\widehat x)=ik(\widehat x\times p)e^{-ik\widehat x\cdot z}\). Finally, the authors exhibit sufficient conditions ensuring that the set of interior transmission eigenvalues is at most countable.

A further chapter is concerned with the factorization method for impedance tomography. The governing equation in this case reduces to the well-studied scalar elliptic one \(\text{ div}\,\big[ \gamma(x)\nabla u(x)\big]=0\), \(x\in B\), \(B\) being an open set, under the conormal boundary condition \(\gamma(x)D_\nu u(x)=0\), \(x\in \partial B\). Under the assumption that \(B\) contains an inclusion \({\overline D}\), \(D\) being an open set, by the knowledge of the classical Dirichlet-Neumann operator \(\Lambda\) and taking advantage of the factorization method, the authors can recover the support of the matrix-valued function \(q\), when the (matrix-valued) conductivity \(\gamma(x)\) is assumed to be of the particular form \(\gamma(x)=I+q(x)\), \(x\in D\) and \(\gamma(x)=I\), \(x\in B\setminus D\), \(I\) standing for the unit matrix.

Finally, the last chapter is devoted to illustrating alternative methods, aiming at recovering unknown bodies, such as the dual space, the linear sampling, the singular source and the probe ones.

On \(\partial D\) the field \(u^s\) satisfies \(u^s=-u^i\) or \(D_\nu u^s=-D_\nu u^i\), \(D_\nu\) denoting the outer normal derivative. The Dirichlet and Neumann conditions correspond to the sound-soft and sound-hard obstacles.

The information about the form of the body is collected in the so-called far field pattern \(u^\infty\) depending on two directions, the first related to the incident wave, the latter to the scattered one. Of course, it is assumed that \(u^\infty\) can be measured. By means of \(u^\infty\) the authors can define a far field (linear integral) operator \(F\), acting on \(L^2(S^2)\), \(S^2\) being the unit sphere in \({\mathbb R}^3\).

The basic idea in the book consists in factoring \(F=GTG^*\) as a product of three operators – whence the name of factorization method – and, by means of this factorization, recover all points \(z\in D\). This is done by using the fundamental fact that \(F\) is a normal operator and computing the infimum \(f(z)\) of the modulus of the quadratic form \(| (\psi,F\psi)_{L^2(S^2)}| \) under the constraint \((\psi,\phi_z)_{L^2(S^2)}=1\), where \(\phi_z(\widehat x)=e^{-ik\widehat x\cdot z}\). Under the assumption that \(k^2\) is not an eigenvalue of \(-\Delta\), a point \(z\) will belong to \(D\) if, and only if, \(f(z)\) turns out to be positive.

A similar technique is applied to the case where the scatterer \(D\) is characterized by an impedance boundary condition \(D_\nu u + \lambda u=0\) – i.e. a Robin boundary condition –, \(u\) and \(\lambda\) denoting the total field and an \(L^\infty(\partial D)\)-function such that \(\text{ Im}\lambda\geq 0\) a.e. on \(\partial D\). The reconstruction of \(D\) is done by using both far and near field operators. The latter denomination means that the field operator \(F\) is related to the radiating scattered wave \(v^s\) measured on the boundary of an open set \(\Omega\) containing \({\overline D}\).

We observe that in this case the operator \(F\) is not normal. Therefore the infimum procedure developed in the Dirichlet and Neumann cases – consuming a lot of time in explicit computations – has to be corrected suitably. The right operator is now \(F_\#=| \text{ Re}\,F| +| \text{ Im}\,F| \), where \(2\text{ Re}\,F=(F+F^*)\) and \(2\text{ Im}\,F=(F-F^*)\), \(F^*\) standing for the operator adjoint to \(F\). Since operator \(F_\#\) is (positive and) normal, it admits an eigensystem \((\lambda_j,\psi_j)\). In this case a point \(z\in D\) if, and only if, the condition \(\sum_{j=1}^{+\infty}\, \lambda_j^{-1}| (\psi_j,\phi_z)_{L^2(S^2)}| ^2<+\infty\) is satisfied, i.e. if the equation \(F_\#^{1/2}\phi=\phi_z\) is solvable.

Another important question is concerned with the so-called mixed boundary conditions. Explicitly this means that \(D\) consists of several disconnected open subsets on the boundaries of which either Dirichlet or Robin conditions are given. In the case when \(D=D_1\cup D_2\) it can be recovered by operator \(F_\#\) only if we assume to know some localization of \(D\), i.e. a pair of open sets \(\Omega_j\), \(j=1,2\), such that \({\overline D}_j\subset \Omega_j\), \(j=1,2\).

A chapter is then devoted to the case where the refraction index \(n\) is non-constant. However, the results proved concern only the determination of the domain of \(n-1\) and not the function \(n\) itself.

The latter part of the book is concerned with a problem similar to the acoustic one for time periodic solutions to the Maxwell equations subject to the Silver-Müller radiation condition – corresponding to the Sommerfeld condition in the acoustic case. The basic equation is now \[ \text{ curl}\,\big[\varepsilon^{-1}_r \text{ curl} H\big] - k^2\text{ curl} H=0, \] where the complex-valued function \(\varepsilon_r\in L^\infty(D)\), accounting for the electromagnetic properties of \(D\), with \(\text{ Re}\,\varepsilon_r\geq 0\) and \(\text{ Im}\,\varepsilon_r\geq 0\) a.e. in \(D\). In this case the incident magnetic field is \(H^i(x,\theta,p)=e^{ik\theta \cdot x}p\), where the polarization tensor \(p\in {\mathbb R}^3\) satisfies the orthogonality condition \(p\cdot \theta=0\).

After having factored the far field operator \(F\) and defined the somewhat involved (and new) concept of interior transmission eigenvalue, the authors can localize, under suitable assumptions on the function \(\varepsilon_r\), the support of the contrast \(q=1-(1/\varepsilon_r)\) when the function \(\phi_z\) is now defined by \(\phi_z(\widehat x)=ik(\widehat x\times p)e^{-ik\widehat x\cdot z}\). Finally, the authors exhibit sufficient conditions ensuring that the set of interior transmission eigenvalues is at most countable.

A further chapter is concerned with the factorization method for impedance tomography. The governing equation in this case reduces to the well-studied scalar elliptic one \(\text{ div}\,\big[ \gamma(x)\nabla u(x)\big]=0\), \(x\in B\), \(B\) being an open set, under the conormal boundary condition \(\gamma(x)D_\nu u(x)=0\), \(x\in \partial B\). Under the assumption that \(B\) contains an inclusion \({\overline D}\), \(D\) being an open set, by the knowledge of the classical Dirichlet-Neumann operator \(\Lambda\) and taking advantage of the factorization method, the authors can recover the support of the matrix-valued function \(q\), when the (matrix-valued) conductivity \(\gamma(x)\) is assumed to be of the particular form \(\gamma(x)=I+q(x)\), \(x\in D\) and \(\gamma(x)=I\), \(x\in B\setminus D\), \(I\) standing for the unit matrix.

Finally, the last chapter is devoted to illustrating alternative methods, aiming at recovering unknown bodies, such as the dual space, the linear sampling, the singular source and the probe ones.

Reviewer: Alfredo Lorenzi (Milano)

##### MSC:

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

35P25 | Scattering theory for PDEs |

35R30 | Inverse problems for PDEs |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35J25 | Boundary value problems for second-order elliptic equations |

47A68 | Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators |

78A45 | Diffraction, scattering |