Complex Geometrical Optics Research Articles

Applications of the Stone–Weierstrass theorem in the Calderón problem

We give examples on the use of the Stone–Weierstrass theorem in inverse problems. We show uniqueness in the linearized Calderón problem on holomorphically separable Kähler manifolds and in the Calderón problem for nonlinear equations on conformally transversally anisotropic manifolds. We also study the holomorphic separability condition in terms of plurisubharmonic functions. The Stone–Weierstrass theorem allows us to generalize and simplify earlier results. It also makes it possible to circumvent the use of complex geometrical optics solutions and inversion of explicit transforms in certain cases.

Inverse problem of recovering a time-dependent nonlinearity appearing in third-order nonlinear acoustic equations *

This paper is devoted to some inverse problems of recovering the nonlinearity for the Jordan–Moore–Gibson–Thompson equation, which is a third order nonlinear acoustic equation. This equation arises, for example, from the wave propagation in viscous thermally relaxing fluids. The well-posedness of the nonlinear equation is obtained with the small initial and boundary data. By the second order linearization to the nonlinear equation, and construction of complex geometric optics solutions for the linearized equation, the uniqueness of recovering the nonlinearity is derived.

Increasing stability of a linearized inverse boundary value problem for a nonlinear Schrödinger equation on transversally anisotropic manifolds

We consider the problem of recovering a nonlinear potential function in a nonlinear Schrödinger equation on transversally anisotropic manifolds from the linearized Dirichlet-to-Neumann map at a large wavenumber. By calibrating the complex geometric optics solutions according to the wavenumber, we prove the increasing stability of recovering the coefficient of a cubic term as the wavenumber becomes large.

Reconstructing unknown inclusions for the biharmonic equation

Herein, we study an inverse problem for detecting unknown obstacles by the enclosure method using the Dirichlet–to–Neumann map for measurements. We justify the method for an penetrable obstacle case involving a biharmonic equation. We use complex geometrical optics solutions with a logarithmic phase to reconstruct some non–convex parts of the obstacle.

Lipschitz stability of recovering the conductivity from internal current densities

In recent years, coupled physics imaging techniques have been developed to produce clearer images than those produced by electrical impedance tomography. This paper focuses on the inverse problem arising in current density impedance imaging and magneto-acousto-electric tomography. We consider the electrostatic equation in a bounded domain with either the Dirichlet or Neumann boundary condition b, where σ is a scalar conductivity function. The inverse problem is formulated as recovering σ from vector fields with different boundary conditions b. We provide a local Lipschitz stability, stating that near some known σ 0 and under some regularity assumptions, we can find b 1 and b 2 by constructing complex geometrical optics (CGO) solutions such that . Furthermore, we modify the CGO solutions using the reflection method to make b 1 and b 2 vanish on a portion of a plane, and prove a local Lipschitz stability with partial data.

Unveiling the Role of the Beam Shape in Photothermal Beam Deflection Measurements: A 1D and 2D Complex Geometrical Optics Model Approach

The preponderance of laser beam shapes cannot be ruled out during the implementation of an optical experiment nor during the formulation of its theoretical background. The present work elucidates the role of Gaussian and top-hat beam shapes in generating and analysing the photothermal beam deflection (PBD) signals. The complex geometrical optics models encompassing the perturbations in the phase and amplitude of the probe beam with one-dimensional (1D) and two-dimensional (2D) approaches is employed to curve fit the PBD signal and are compared. From the fitted curve, the thermal diffusivity and conductivity of the sample are calculated with the 1D and 2D models. A uniform intensity distribution over the sample, like a top-hat beam, is achieved using an optical lens system and verified using a beam profiler. When the phase and amplitude of the PBD signal are fitted at different positions of the lens, i.e., in focussed and defocussed conditions, it is observed that difference in the measured thermal characteristics is about 30% for the Gaussian pump beam profile, whereas it is only <4% for top-hat beam. Even though the fitting accuracy and sum of residues estimated for the 2D model are better than 1D, the ease of computation with the 1D model employing top-hat excitation suggests the application of the top-hat profile in photothermal experiments.

Stability of inverse scattering problem for the damped biharmonic plate equation

This paper is concerned with the stability of the inverse scattering problem for the damped biharmonic plate equation in a bounded domain with the Cauchy data. A sharp estimate for the mass density is established using a priori information concerning Sobolev norms and a priori information about the support of the inhomogeneity. Our results improve previous estimates and explicitly depend on the damping coefficient. The proof mainly relies on the complex geometric optics solution method and the Fourier analysis.

Quasi‐Optical Ray Tracing of Gaussian Beams in the Magnetosphere

AbstractA method for tracing rays in tokamak plasmas is applied to tracing rays in the Earth's magnetosphere. The problem of computing power flux from pencil rays using classical geometric optics that occurs when constituent rays exhibit coplanarity is addressed by appealing to complex geometric optics. The construction of Gaussian beams is discussed and the quasi‐optical method of tracing such beams is given. Examples of tracing Gaussian beams from a ground transmitter, a partial beam, and from a satellite location are presented. A median method for dealing with nonviable contributions to the power flux from sections of the beam encountering caustics or otherwise experiencing coplanarity is described. The effect of beam parameter choices on the resulting power flux computation is provided for each of the examples.

Simultaneous recoveries for semilinear parabolic systems

In this paper, we study inverse boundary problems associated with semilinear parabolic systems in several scenarios where both the nonlinearities and the initial data can be unknown. We establish several simultaneous recovery results showing that the passive or active boundary Dirichlet-to-Neumann operators can uniquely recover both of the unknowns, even stably in a certain case. It turns out that the nonlinearities play a critical role in deriving these recovery results. If the nonlinear term belongs to a general C 1 class but fulfilling a certain growth condition, the recovery results are established by the control approach via Carleman estimates. If the nonlinear term belongs to an analytic class, the recovery results are established through successive linearization in combination with special complex geometrical optics solutions for the parabolic system.

A surface deformation measurement algorithm for reflector antennas based on complex geometrical optics

AbstractThis paper presents a new method to reveal the relation between the surface deformation and near-field amplitude of a reflector antenna based on complex geometrical optics, which could be used as an efficient way to estimate the antenna surface verified by simulation results. The measurement process based on this method is envisaged to be realized by a single scanning of the near-field amplitude which would overcome many limitations of radio holography and phase retrieval methods such as the frequency and elevation. The largest source of error in the original deformation-amplitude equation (DAE) has been corrected by considering the Gaussian feed as a complex point source. To track the ray trajectory so that the improved DAE could be solved, an iteration method including a golden section search algorithm is designed to make the solution converge. By solving the modified DAE, simulation result shows that a more accurate solution could be obtained, and the antenna surface could be recovered to a root mean square error of under 30 microns.

Large ∣k∣ Behavior of Complex Geometric Optics Solutions to d‐bar Problems

AbstractComplex geometric optics solutions to a system of d‐bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey‐Stewartson II equations are studied for large values of the spectral parameter k. For potentials for some , it is shown that the solution converges as the geometric series in . For potentials q being the characteristic function of a strictly convex open set with smooth boundary, this still holds with s = 3/2, i.e., with instead of . The leading‐order contributions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk. © 2022 Courant Institute of Mathematics and Wiley Periodicals LLC.

A new type of CGO solutions and its applications in corner scattering

We consider corner scattering for the operator ∇ ⋅ γ(x)∇ + k 2 ρ(x) in , with γ a positive definite symmetric matrix and ρ a positive scalar function. A corner is referred to one that is on the boundary of the (compact) support of γ(x) − I or ρ(x) − 1, where I stands for the identity matrix. We assume that γ is a scalar function in a small neighborhood of the corner. We show that any admissible incident field will be scattered by such corners, which are allowed to be concave. Moreover, we provide a brief discussion on the existence of non-scattering waves when γ − I has a jump across the corner. In order to prove the results, we construct a new type of complex geometric optics solutions.

Direct regularized reconstruction for the three-dimensional Calderón problem

<p style='text-indent:20px;'>Electrical Impedance Tomography gives rise to the severely ill-posed Calderón problem of determining the electrical conductivity distribution in a bounded domain from knowledge of the associated Dirichlet-to-Neumann map for the governing equation. The uniqueness and stability questions for the three-dimensional problem were largely answered in the affirmative in the 1980's using complex geometrical optics solutions, and this led further to a direct reconstruction method relying on a non-physical scattering transform. In this paper, the reconstruction problem is taken one step further towards practical applications by considering data contaminated by noise. Indeed, a regularization strategy for the three-dimensional Calderón problem is presented based on a suitable and explicit truncation of the scattering transform. This gives a certified, stable and direct reconstruction method that is robust to small perturbations of the data. Numerical tests on simulated noisy data illustrate the feasibility and regularizing effect of the method, and suggest that the numerical implementation performs better than predicted by theory.</p>

Self-focusing of a spatially modulated beam within the paraxial complex geometrical optics framework in low-density plasmas

Accurate modelling of ponderomotive laser self-focusing may represent a key for the success of inertial confinement fusion, especially within the shock ignition approach. From a numerical point of view, implementation of a paraxial complex geometrical optics (PCGO) method has improved the performance of the hydrodynamics code CHIC, but (1) overestimating ponderomotive speckle self-focusing in reduced two-dimensional geometry, and (2) not accounting for speckle intensity statistics. The first issue was addressed in our previous work (Ruocco et al 2019 Plasma Phys. Control. Fusion 61 115009). Based on those results, here we propose a novel PCGO scheme for modelling spatially modulated laser beams by (1) creating Gaussian speckles, and (2) emulating the realistic speckle intensity statistics. Self-focusing of spatially modulated beams in a homogeneous stationary plasma with this method is studied. This investigation evidences that plasma smoothing does not reduce the speckle intensity enhancement at long time scales when the average beam intensity is twice above the speckle critical intensity. Comparison against electromagnetic simulations confirms that this approach improves the description of self-focusing of high-intensity speckles within the PCGO model.

Calderón's Method with a Spatial Prior for 2-D EIT Imaging of Ventilation and Perfusion.

Bedside imaging of ventilation and perfusion is a leading application of 2-D medical electrical impedance tomography (EIT), in which dynamic cross-sectional images of the torso are created by numerically solving the inverse problem of computing the conductivity from voltage measurements arising on electrodes due to currents applied on electrodes on the surface. Methods of reconstruction may be direct or iterative. Calderón’s method is a direct reconstruction method based on complex geometrical optics solutions to Laplace’s equation capable of providing real-time reconstructions in a region of interest. In this paper, the importance of accurate modeling of the electrode location on the body is demonstrated on simulated and experimental data, and a method of including a priori spatial information in dynamic human subject data is presented. The results of accurate electrode modeling and a spatial prior are shown to improve detection of inhomogeneities not included in the prior and to improve the resolution of ventilation and perfusion images in a human subject.

The Calderón inverse problem for isotropic quasilinear conductivities

We prove a global uniqueness result for the Calderón inverse problem for a general quasilinear isotropic conductivity equation on a bounded open set with smooth boundary in dimension n≥3. Performing higher order linearizations of the nonlinear Dirichlet–to–Neumann map, we reduce the problem of the recovery of the differentials of the quasilinear conductivity, which are symmetric tensors, to a completeness property for certain anisotropic products of solutions to the linearized equation. The completeness property is established using complex geometric optics solutions to the linearized conductivity equation, whose amplitudes concentrate near suitable two dimensional planes.

Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds

We show that a continuous potential q can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the Schrödinger operator −Δg+q on a conformally transversally anisotropic manifold of dimension ≥3, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [12]. A crucial role in our reconstruction procedure is played by a constructive determination of the boundary traces of suitable complex geometric optics solutions based on Gaussian beams quasimodes concentrated along non-tangential geodesics on the transversal manifold, which enjoy uniqueness properties. This is achieved by applying the simplified version of the approach of [33] to our setting. We also identify the main space introduced in [33] with a standard Sobolev space on the boundary of the manifold. Another ingredient in the proof of our result is a reconstruction formula for the boundary trace of a continuous potential from the knowledge of the Dirichlet–to–Neumann map.

Stable recovery of noncompactly supported electromagnetic potentials in unbounded domain

We consider the inverse problem of determining an electromagnetic potential appearing in an infinite cylindrical domain from boundary measurements. More precisely, we prove the stable recovery of a general class of magnetic field and electric potential from boundary measurements. Assuming the knowledge of the unknown coefficients close to the boundary, we obtain other results of stable recovery with measurements restricted to some portion of the boundary. Our approach combines construction of complex geometric optics solutions and Carleman estimates suitably designed for our stability results stated in an unbounded domain.

The enclosure method for a generalized anisotropic complex conductivity equation

We study how to apply the enclosure method to reconstruct an unknown inclusion within a medium in a domain in which satisfies the conductivity equation ∇ ⋅ ((σ 0 + iɛ 0)∇u) = 0 with σ 0 and ɛ 0 being real matrix functions. Motivated by some real world applications, we assume the unknown inclusion satisfies an equation of the more general form , where σ, ɛ, ζ are also real matrix functions. Due to the anisotropy, it is in general difficult to find complex geometric optics solutions. Therefore, we construct the oscillating decaying solutions, which is used to test whether a given half-space intersects the unknown inclusion or not.

Inverse problems for elliptic equations with power type nonlinearities

We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for certain nonlinear equations in cases where the solution for a corresponding linear equation is not known. Assuming the knowledge of a nonlinear Dirichlet-to-Neumann map, we determine both a potential and a conformal manifold simultaneously in dimension 2, and a potential on transversally anisotropic manifolds in dimensions n≥3. In the Euclidean case, we show that one can solve the Calderón problem for certain semilinear equations in a surprisingly simple way without using complex geometrical optics solutions.