Geometrical Optics Solutions Research Articles

Stability of the determination of a coefficient for wave equations in an infinite waveguide

We consider the stability of the inverse problem consisting of the determination of a coefficient of order zero $q$, appearing in the Dirichlet initial-boundary value problem for a wave equation $\partial_t^2u-\Delta u+q(x)u=0$ in $(0,T)\times\Omega$, with $\Omega=\omega\times\mathbb{R}$ an unbounded cylindrical waveguide and $\omega$ a bounded smooth domain of $\mathbb{R}^2$, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to the wave equation. Using suitable geometric optics solutions, we prove a Holder stability estimate in the determination of $q$ from the Dirichlet to Neumann map. Moreover, provided that the coefficient $q$ is lying in a set of functions $\mathcal A$, where, for any $q_1,q_2\in\mathcal A$, $|q_1-q_2|$ attains its maximum in a fixed bounded subset of $\overline{\Omega}$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partial\Omega$.

Exceptional circles of radial potentials

A nonlinear scattering transform is studied for the two-dimensional Schrödinger equation at zero energy with a radial potential. Explicit examples are presented, both theoretically and computationally, of potentials with nontrivial singularities in the scattering transform. The singularities arise from non-uniqueness of the complex geometric optics solutions that define the scattering transform. The values of the complex spectral parameter at which the singularities appear are called exceptional points. The singularity formation is closely related to the fact that potentials of conductivity type are ‘critical’ in the sense of Murata.

Rotating Parabolic-Reflector Antenna Target in SAR Data: Model, Characteristics, and Parameter Estimation

Parabolic-reflector antennas (PRAs), usually possessing rotation, are a particular type of targets of potential interest to the synthetic aperture radar (SAR) community. This paper is aimed to investigate PRA’s scattering characteristics and then to extract PRA’s parameters from SAR returns, for supporting image interpretation and target recognition. We at first obtain both closed-form and numeric solutions to PRA’s backscattering by geometrical optics (GO), physical optics, and graphical electromagnetic computation, respectively. Based on the GO solution, a migratory scattering center model is at first presented for representing the movement of the specular point with aspect angle, and then a hybrid model, named the migratory/micromotion scattering center (MMSC) model, is proposed for characterizing a rotating PRA in the SAR geometry, which incorporates PRA’s rotation into its migratory scattering center model. Additionally, we in detail analyze PRA’s radar characteristics on radar cross-section, high-resolution range profiles, time-frequency distribution, and 2D images, which also confirm the models proposed. A maximal likelihood estimator is developed for jointly solving the MMSC model for PRA’s multiple parameters by optimization. By exploiting the aforementioned characteristics, the coarse parameter estimation guarantees convergency upon global minima. The signatures recovered can be favorably utilized for SAR image interpretation and target recognition.

Inverse diffusion from knowledge of power densities

This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography. We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.

On a Calderón Problem in Frequency Differential Electrical Impedance Tomography

Recent research in electrical impedance tomography produced images of biological tissue from frequency differential boundary voltages and corresponding currents. Physically one is to recover the electrical conductivity $\sigma$ and permittivity $\epsilon$ from the frequency differential boundary data. Let $\gamma=\sigma+i\omega\epsilon$ denote the complex admittivity, $\Lambda_{\gamma}$ be the corresponding Dirichlet-to-Neumann map, and $\frac{d\Lambda_{\gamma}}{d\omega}|_{\omega=0}$ be its frequency differential at $\omega=0$. If $\sigma\in C^{1,1}(\overline\Omega)$ is constant near the boundary and $\epsilon\in C^{1,1}_0(\Omega)$, we show that $\frac{d\Lambda_{\gamma}}{d\omega}|_{\omega=0}$ uniquely determines $\nabla\cdot(\nabla\epsilon-\epsilon\nabla\ln\sigma)/\sigma$ inside $\Omega$. In addition, if $\Lambda_{\gamma}|_{\omega=0}$ is also known, then $\epsilon$ and $\sigma$ can be reconstructed inside. The method of proof uses the complex geometrical optics solutions.

Quantitative photo-acoustic tomography with partial data

Photo-acoustic tomography is a newly developed hybrid imaging modality that combines a high-resolution modality with a high-contrast modality. We analyze the reconstruction of diffusion and absorption parameters in an elliptic equation and extend an earlier result of Bal and Uhlmann (2010 Inverse Problems 26 085010) to the partial data case. We show that the reconstruction can be uniquely determined by the knowledge of four internal data based on well-chosen partial boundary conditions. Stability of this reconstruction is ensured if a convexity condition is satisfied. A similar stability result is obtained without this geometric constraint if 4n well chosen partial boundary conditions are available, where n is the spatial dimension. The set of well chosen boundary measurements is characterized by some complex geometric optics solutions vanishing on a part of the boundary.

Transformation of Rice’s small perturbation solution for rough surface scattering into the high frequency reciprocal physical and geometric optics solutions

A step by step transformation of the low frequency small height and slope perturbation solution into the high frequency reciprocal and dual, physical and geometrical optic solutions is presented. The familiar Kirchhoff approximations for the fields impressed by the incident plane wave upon the rough surfaces results in nonreciprocal solutions. It is shown that the surface element scattering coefficients based on the Kirchhoff approximations agree with the corresponding reciprocal physical optics solutions only at the stationary phase, specular points on the rough surfaces. While the Kirchhoff approximations and physical optics approximations are based on the characterization of the surface fields by Fresnel reflection coefficients, the corresponding surface element scatter coefficient derived for the small perturbation solution and the full wave solutions are based on the imposition of boundary conditions for the tangential components of the electric and magnetic fields. A flow graph schematically depicting the relationships between these solutions for the scattered fields is also presented.

On the reconstruction of interfaces using complex geometrical optics solutions for the acoustic case

We deal with the problem of reconstruction of the coefficient discontinuities (or supports) of scalar divergence form equations with lower order terms from the Dirichlet-to-Neumann map using complex geometrical optics (CGO) solutions. We consider both penetrable and impenetrable obstacles. The usual proofs for justifying this method assume, in addition to the smoothness of the coefficients and the interfaces, the following two conditions.The finiteness of the touching points of the phase’s level curves (or surfaces) of the used CGO solutions with the interface.The positivity of the lower bound of the relative curvature of the interface.In this paper, we show how we can remove these two conditions and justify the reconstruction method considering L∞ coefficients and Lipschitz interfaces of discontinuity.

Electrical impedance tomography: 3D reconstructions using scattering transforms

In three dimensions the Calderón problem was addressed and solved in theory in the 1980s. The main ingredients in the solution of the problem are complex geometrical optics solutions to the conductivity equation and a (non-physical) scattering transform. The resulting reconstruction algorithm is in principle direct and addresses the full non-linear problem immediately. In this paper a new simplification of the algorithm is suggested. The method is based on solving a boundary integral equation for the complex geometrical optics solutions, and the method is implemented numerically using a Nyström method. Convergence estimates are obtained using hyperinterpolation operators. We compare the method numerically to two other approximations by evaluation on two numerical examples. In addition, a moment method for the numerical solution of the forward problem is given.

Reconstruction of penetrable inclusions in elastic waves by boundary measurements

We use Ikehataʼs enclosure method to reconstruct penetrable unknown inclusions in a plane elastic body in time-harmonic waves. Complex geometrical optics solutions with complex polynomial phases are adopted as the probing utility. In a situation similar to ours, due to the presence of a zeroth order term in the equation, some technical assumptions need to be assumed in early researches. In a recent work of Sini and Yoshida, they succeeded in abandoning these assumptions by using a different idea to obtain a crucial estimate. In particular the boundaries of the inclusions need only to be Lipschitz. In this work we apply the same idea to our model. Itʼs interesting that, with more careful treatment, we find the boundaries of the inclusions can in fact be assumed to be only continuous.

Inverse boundary value problem by measuring Dirichlet data and Neumann data on disjoint sets

We discuss the inverse boundary value problem of determining the conductivity in two dimensions from the pair of all input Dirichlet data supported on an open subset Γ+ and all the corresponding Neumann data measured on an open subset Γ−. We prove the global uniqueness under some additional geometric condition, in the case where , and we prove also the uniqueness for a similar inverse problem for the stationary Schrödinger equation. The key of the proof is the construction of appropriate complex geometrical optics solutions using Carleman estimates with a singular weight.

Quantitative thermo-acoustics and related problems

Thermo-acoustic tomography is a hybrid medical imaging modality that aims to combine the good optical contrast observed in tissues with the good resolution properties of ultrasound. Thermo-acoustic imaging may be decomposed into two steps. The first step aims at reconstructing an amount of electromagnetic radiation absorbed by tissues from boundary measurements of ultrasound generated by the heating caused by these radiations. We assume that this first step has been performed. Quantitative thermo-acoustics then consists of reconstructing the conductivity coefficient in the equation modeling radiation from the now known absorbed radiation. This second step is the problem of interest in this paper. Mathematically, quantitative thermo-acoustics consists of reconstructing the conductivity in Maxwell's equations from available internal data that are linear in the conductivity and quadratic in the electric field. We consider several inverse problems of this type with applications in thermo-acoustics as well as in acousto-optics. In this framework, we obtain uniqueness and stability results under a smallness constraint on the conductivity. This smallness constraint is removed in the specific case of a scalar model for electromagnetic wave propagation for appropriate illuminations constructed by the method of complex geometric optics solutions.

A direct numerical reconstruction algorithm for the 3D Calderón problem

In three dimensions Calderón's problem was addressed and solved in theory in the 1980s in a series of papers, but only recently the numerical implementation of the algorithm was initiated. The main ingredients in the solution of the problem are complex geometrical optics solutions to the conductivity equation and a (non-physical) scattering transform. The resulting reconstruction algorithm is in principle direct and addresses the full non-linear problem immediately. In this paper we will outline the theoretical reconstruction method and describe how the method can be implemented numerically. We will give three different implementations, and compare their performance on a numerical phantom.

Inverse problems for the anisotropic Maxwell equations

We prove that the electromagnetic material parameters are uniquely determined by boundary measurements for the time-harmonic Maxwell equations in certain anisotropic settings. We give a uniqueness result in the inverse problem for Maxwell equations on an admissible Riemannian manifold, and a uniqueness result for Maxwell equations in Euclidean space with admissible matrix coefficients. The proofs are based on a new Fourier analytic construction of complex geometrical optics solutions on admissible manifolds, and involve a proper notion of uniqueness for such solutions.

A Simple Numerical Method for Complex Geometrical Optics Solutions to the Conductivity Equation

This paper concerns numerical methods for computing complex geometrical optics (CGO) solutions to the conductivity equation $\nabla\cdot\sigma\nabla u(\cdot,k)=0$ in $\mathbb{R}^2$ for piecewise smooth conductivities $\sigma$, where k is a complex parameter. The key is to solve an $\mathbb R$-linear singular integral equation defined in the unit disk. Recently, Astala et al. [Appl. Comput. Harmon. Anal., 29 (2010), pp. 2–17] proposed a complicated method for numerical computation of CGO solutions by solving a periodic version of the $\mathbb{R}$-linear integral equation in a rectangle containing the unit disk. In this paper, based on the fast algorithms in [P. Daripa and D. Mashat, Numer. Algorithms, 18 (1998), pp. 133–157] for singular integral transforms, we propose a simpler numerical method which solves the $\mathbb{R}$-linear integral equation in the unit disk directly. For the resulting $\mathbb{R}$-linear operator equation, a minimal residual iterative method is proposed. Numerical examples illustrate the accuracy and efficiency of the new method.

Reconstruction of Penetrable Obstacles in Acoustic Scattering

We develop a reconstruction algorithm to determine penetrable obstacles inside a domain in the plane from acoustic measurements made on the boundary. This algorithm uses complex geometrical optics solutions to the Helmholtz equation with polynomial-type phase functions.

Reconstructions from boundary measurements on admissible manifolds

We prove that a potential $q$ can be reconstructed from the Dirichlet-to-Neumann map for the Schrodinger operator $-\Delta_g + q$ in a fixed admissible $3$-dimensional Riemannian manifold $(M,g)$. We also show that an admissible metric $g$ in a fixed conformal class can be constructed from the Dirichlet-to-Neumann map for $\Delta_g$. This is a constructive version of earlier uniqueness results by Dos Santos Ferreira et al. [10] on admissible manifolds, and extends the reconstruction procedure of Nachman [31] in Euclidean space. The main points are the derivation of a boundary integral equation characterizing the boundary values of complex geometrical optics solutions, and the development of associated layer potentials adapted to a cylindrical geometry.

Determination of second-order elliptic operators in two dimensions from partial Cauchy data

We consider the inverse boundary value problem in two dimensions of determining the coefficients of a general second-order elliptic operator from the Cauchy data measured on a nonempty arbitrary relatively open subset of the boundary. We give a complete characterization of the set of coefficients yielding the same partial Cauchy data. As a corollary we prove several uniqueness results in determining coefficients from partial Cauchy data for the isotropic conductivity equation, the Schrödinger equation, the convection-diffusion equation, the anisotropic conductivity equation modulo a group of diffeomorphisms that are the identity at the boundary, and the magnetic Schrödinger equations modulo gauge transformations. The key step is the construction of novel complex geometrical optics solutions using Carleman estimates.

Direct numerical reconstruction of conductivities in three dimensions using scattering transforms

A direct three-dimensional EIT reconstruction algorithm based on complex geometrical optics solutions and a nonlinear scattering transform is presented and implemented for spherically symmetric conductivity distributions. The scattering transform is computed both with a Born approximation and from the forward problem for purposes of comparison. Reconstructions are computed for several test problems. A connection to Calderón's linear reconstruction algorithm is established, and reconstructions using both methods are compared.

Increasing stability for the Schrödinger potential from the Dirichlet-to Neumann map

We derive some bounds which can be viewed as an evidence of increasing stability in the problem of recovery of the potential coefficient in the Schrodinger equation from the Dirichlet-to-Neumann map, when frequency (energy level) is growing. These bounds hold under certain a-priori bounds on the unknown coefficient. Proofs use complex- and real-valued geometrical optics solutions. We outline open problems and possible future developments.