Geometrical Optics Solutions 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.

Hölder stability estimates in determining the time-dependent coefficients of the heat equation from the Cauchy data set

Abstract In this paper, we address stability results in determining the time-dependent scalar and vector potentials appearing in the convection-diffusion equation from the knowledge of the Cauchy data set. We prove Hölder-type stability estimates. The key tool used in this work is the geometric optics solution.

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.

SAR Image Simulations of Ocean Scenes Based on the Improved Facet TSM

The facet-based two scale model (FTSM) is widely applied in SAR image simulations of the anisotropic ocean surface. However, this model is sensitive to the cutoff parameter and facet size, and the choice of these two parameters is arbitrary. We propose to make an approximation of the cutoff invariant two scale model (CITSM) to improve the simulation efficiency while remaining the robustness to cutoff wavenumbers. Meanwhile, the robustness to facet sizes is obtained by correcting the geometrical optics (GO) solution, taking into account the slope probability density function (PDF) correction induced by the spectrum within an individual facet. The new FTSM, with less dependence on cutoff parameters and facet sizes, is proved to be reasonable in the comparisons with advanced analytical models and experimental data. Finally, SAR images of the ocean surface and ship wakes with various facet sizes are provided to prove the operability and applicability of our model.

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.

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.

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.

Linking Diffractive and Geometrical Optics Surface Scattering at a Fundamental Level

Optical surface scattering analyses based on diffractive optics (DO) are typically applied to one surface; however, there is a need for simulating surface scattering losses for devices having many surface interactions such as light pipes. Light pipes are often simulated with geometric optics (GO) using ray tracing, where surface scattering is driven by the surface slope distribution. In the DO case, surface scattering analyses depend on the spatial frequency distribution and amplitude as well as wavelength, with the sinusoidal grating as a fundamental basis. A better understanding of the link, or transition, between DO and GO scattering domains would be helpful for efficiently incorporating scattering loss analyses into ray trace simulations. A formula for the root-mean-square (rms) scattered angle width of a sinusoidal reflection grating that depends only on the surface rms slope is derived from the nonparaxial scalar diffraction theory, thereby linking it to GO. The scatter angle’s mean and rms width are evaluated over a range of grating amplitudes and periods using scalar theory and full vector simulations from the COMSOL® wave optic module for a sinusoidal reflection grating. The conditions under which the diffraction-based solution closely approximates the GO solution, as predicted by the rms slope, are identified. Close agreement is shown between the DO and GO solutions for the same surface rms slope scattering loss due to angular filtering near the critical angle of a total internal reflection (TIR) glass-to-air interface.

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>

Recovery of transversal metric tensor in the Schrödinger equation from the Dirichlet-to-Neumann map

<p style='text-indent:20px;'>In this paper, we deal with the inverse problem of determining simple metrics on a compact Riemannian manifold from boundary measurements. We take this information in the dynamical Dirichlet-to-Neumann map associated to the Schrödinger equation. We prove in dimension <inline-formula><tex-math id="M1">\begin{document}$ n\geq 2 $\end{document}</tex-math></inline-formula> that the knowledge of the Dirichlet-to-Neumann map for the Schrödinger equation uniquely determines the simple metric (up to an admissible set). We also prove a Hölder-type stability estimate by the construction of geometrical optics solutions of the Schrödinger equation and the direct use of the invertibility of the geodesical X-ray transform.</p>

Effect of ground-based environmental conditions on the level of dangerous ultraviolet solar radiation

An analytical model is suggested to determine the contribution of solar radiation scattered by dispersed layers of weakly absorbing highly-porous media on the surface of the earth to the local UV-index, which characterizes harmful ultraviolet irradiation. The transport approximation for the single-scattering phase function and the two-flux method for approximate radiative transfer calculations were used to calculate an additional UV irradiance due to multiple scattering in the layer of a substrate medium. Three important environment conditions are considered in the case problems: the snowpack in a mountain ski resort of Provo (Utah), the salt layer in the Great Salt Lake desert, and the layers of sand typical of sand resorts and beaches. The optical properties of the media were calculated using the data for spectral optical constants of substances in the near-ultraviolet and analytical geometrical optics solution for large spherical particles of a weakly absorbing material. The resulting spectral single-scattering transport albedo of the medium was used to determine an additional harmful UV radiation. It was found that corrected values of the UV-index in all three cases are substantially higher than those calculated without account for the light reflected from a scattering medium on the earth surface. The model of the present paper can be used for different dispersed media in a wide range of physical parameters.

Hybrid design of diffractive optical elements for optical beam shaping.

Hybrid methods combining the geometrical-optics and diffraction-theory methods enable designing diffractive optical elements (DOEs) with high performance due to the suppression of stray light and speckles and, at the same time, with a regular and fabrication-friendly microrelief. Here, we propose a geometrical-optics method for calculating the eikonal function of the light field providing the generation of a required irradiance distribution. In the method, the problem of calculating the eikonal function is formulated in a semi-discrete form as a problem of maximizing a concave function. For solving the maximization problem, a gradient method is used, with analytical expressions obtained for the gradient. In contrast to geometrical-optics approaches based on solving the Monge-Ampére equation using finite difference methods, the proposed method enables generating irradiance distributions defined on disconnected regions with non-smooth boundaries. As an example, we calculate an eikonal function, which provides the generation of a "discontinuous" irradiance distribution in the form of a hexagram. It is shown that the utilization of the hybrid approach, in which the obtained geometrical-optics solution is used as a starting point in iterative Fourier transform algorithms, enables designing DOEs with a quasi-regular or piecewise-smooth microrelief structure. The calculation results are confirmed by the results of experimental investigations of a DOE generating a hexagram-shaped irradiance distribution.

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.