Inverse Prob Research Articles

Inverse problem for Hill equation with jump conditions

In the present study, the inverse nodal problem is discussed for the discontinuous periodic (or anti-periodic) Sturm–Liouville problem. Such types of problems are different from regular problems because of discontinuity in the boundary conditions. Firstly, results of the Sturm–Liouville problem including jump condition are present. Then, by deferring the zeros of eigenfunctions, the inverse problem is solved as we desired. The method is based on considering a translation so that the periodic (or anti-periodic) problem is reduced to a Dirichlet problem as in Cheng and Law (Inverse Prob 22: 891–901, 2006). But, our problem including also discontinuous conditions. In addition to all of these, although there are so many results on this subject, we combine both periodic conditions and discontinuity.

On optimal cloaking-by-mapping transformations

A central ingredient of cloaking-by-mapping is the diffeomorphism which transforms an annulus with a small hole into an annulus with a finite size hole, while being the identity on the outer boundary of the annulus. The resulting meta-material is anisotropic, which makes it difficult to manufacture. The problem of minimizing anisotropy among radial transformations has been studied in Griesmaier and Vogelius [Inverse Prob. 30 (2014) 17]. In this work, as in Griesmaier and Vogelius [Inverse Prob. 30 (2014) 17], we formulate the problem of minimizing anisotropy as an energy minimization problem. Our main goal is to provide strong evidence for the conjecture that for cloaks with circular boundaries, non-radial transformations do not lead to lower degree of anisotropy. In the final section, we consider cloaks with non-circular boundaries and show that in this case, non-radial cloaks may be advantageous, when it comes to minimizing anisotropy.

On numerical resolution of an inverse Cauchy problem modeling the airflow in the bronchial tree

This paper is devoted to the numerical resolution of an inverse Cauchy problem governed by Stokes equation modeling the airflow in the lungs. It consists in determining the air velocity and pressure on the artificial boundaries of the bronchial tree. This data completion problem is one of the highly ill-posed problems in the Hadamard sense (Hadamard in Lectures on Cauchy’s problem in linear partial differential equations. Dover, New York, 1953). This gives great importance to its numerical resolution and in particular to carry out stable numerical approaches, mostly in the case of noisy data. The main idea of this work is to extend some regularizing, stable and fast iterative algorithms for solving this problem based on the domain decomposition approach (Chakib et al. in Inverse Prob 35(1):015008, 2018). We discuss the efficiency and the feasibility of the proposed approach through some numerical tests performed using different domain decomposition algorithms. Finally, we opt for the Robin–Robin algorithm, which showed its performance, for the numerical simulation of the airflow in the bronchial tree configuration.

Inverse scattering for unbounded fractally corrugated surfaces.

The phaseless inverse scattering of optical or electromagnetic waves from unbounded fractally corrugated surfaces are taken into account. The main characteristics of this inverse scattering problem are that only the phaseless scattered field is measured and the incident field is a tapered wave. In the previous recent works of the authors about the inverse scattering of rough surfaces, the cases of the Dirichlet boundary condition were considered [Inverse Prob.32, 085002 (2016)INPEEY0266-561110.1088/0266-5611/32/8/085002; Inverse Probl. Sci. Eng.24, 1282 (2016)IPSECR1741-597710.1080/17415977.2016.1191071]. Here we consider the inverse rough surface problem with Neumann boundary conditions. The Fréchet derivatives of the scattered field with respect to the parameters of the surfaces are derived. Then a numerical method to identify the rough surfaces from the phaseless measurements of the scattered field at a fixed frequency is developed. Numerical examples are presented to show the validity and efficiency of the proposed method.

Passive Detection of Small Low-Emission Sources: Two-Dimensional Numerical Case Studies

In Allmaras et al. [Inverse Prob. Imaging, Vol. 7, No. 1, p. 47 (2013)], we present the mathematical foundations for the passive detection of small low-emission radioactive sources buried in background noise using Compton gamma cameras. Here, we apply these techniques to the detection of small amounts of special nuclear material in various cargo container geometries. Simulated two-dimensional data are used. We demonstrate the feasibility of such an approach using various realistic cargo container loadings and assess the effect of angular detection errors in Compton cameras.

A global minimization algorithm for Tikhonov functionals with convex () penalty terms in Banach spaces

We extend the globally convergent TIGRA method in Ramlau (2003 Inverse Prob. 19 433–65) for the computation of a minimizer of the Tikhonov-type functional with the convex () penalty terms Θ for nonlinear forward operators in Banach spaces. The Θ are allowed to be non-smooth to include or TV (total variation) functionals, which are significant in reconstructing special features of solutions such as sparsity and discontinuities. The proposed TIGRA-Θ method uses a dual gradient descent method in the inner iteration and linearly decreases the regularization parameter in the outer iteration. We present the global convergence analysis for the algorithm under suitable parameter selections, and the convergence rate results are provided under both a priori and a posteriori stopping rules. Two numerical examples—an auto-convolution problem and a parameter identification problem—are presented to illustrate the theoretic analysis and verify the effectiveness of the method.

Discretized Lavrent’ev regularization for the autoconvolution equation

Lavrent’ev regularization for the autoconvolution equation was considered by Janno J. in Lavrent’ev regularization of ill-posed problems containing nonlinear near-to-monotone operators with application to autoconvolution equation, Inverse Prob. 2000;16:333–348. Here this study is extended by considering discretization of the Lavrent’ev scheme by splines. It is shown how to maintain the known convergence rate by an appropriate choice of spline spaces and a proper choice of the discretization level. For piece-wise constant splines the discretized equation allows for an explicit solver, in contrast to using higher order splines. This is used to design a fast implementation by means of post-smoothing, which provides results, which are indistinguishable from results obtained by direct discretization using cubic splines.

A coupled complex boundary method for the Cauchy problem

Considered in this paper is a Cauchy problem governed by an elliptic partial differential equation. In the Cauchy problem, one wants to recover the unknown Neumann and Dirichlet data on a part of the boundary from the measured Neumann and Dirichlet data, usually contaminated with noise, on the remaining part of the boundary. The Cauchy problem is an inverse problem with severe ill-posedness. In this paper, a coupled complex boundary method (CCBM), originally proposed in [Cheng XL, Gong RF, Han W, et al. A novel coupled complex boundary method for solving inverse source problems. Inverse Prob. 2014;30:055002], is applied to solve the Cauchy problem stably. With the CCBM, all the data, including the known and unknown ones on the boundary are used in a complex Robin boundary on the whole boundary. As a result, the Cauchy problem is transferred into a complex Robin boundary problem of finding the unknown data such that the imaginary part of the solution equals zero in the domain. Then the Tikhonov regularization is applied to the resulting new formulation. Some theoretical analysis is performed on the CCBM-based Tikhonov regularization framework. Moreover, through the adjoint technique, a simple solver is proposed to compute the regularized solution. The finite-element method is used for the discretization. Numerical results are given to show the feasibility and effectiveness of the proposed method.

A conformal mapping method in inverse obstacle scattering

Akduman, Haddar and Kress [Akduman I, Kress R. Electrostatic imaging via conformal mapping. Inverse Prob. 2002;18:1659–1672; Haddar H, Kress R. Conformal mappings and inverse boundary value problems. Inverse Prob. 2005;21:935–953; Kress R. Inverse Dirichlet problem and conformal mapping. Math.Comput. Simul. 2004;66:255–265] have employed a conformal mapping technique for the inverse problem to reconstruct a perfectly conducting inclusion in a homogeneous background medium from Cauchy data for electrostatic imaging, that is, for solving an inverse boundary value problem for the Laplace equation. We propose an extension of this approach to inverse obstacle scattering for time-harmonic waves, that is, to the solution of an inverse boundary value problem for the Helmholtz equation. The main idea is to use the conformal mapping algorithm in an iterative procedure to obtain Cauchy data for a Laplace problem from the given Cauchy data for the Helmholtz problem. We present the foundations of the method together with a convergence result and exhibit the feasibility of the method via numerical examples.

Improved and extended results for enhanced convergence rates of Tikhonov regularization in Banach spaces

Even if the recent literature on enhanced convergence rates for Tikhonov regularization of ill-posed problems in Banach spaces shows substantial progress, not all factors influencing the best possible convergence rates under sufficiently strong smoothness assumptions were clearly determined. In particular, it was an open problem whether the structure of the residual term can limit the rates. For residual norms of power type in the functional to be minimized for obtaining regularized solutions, the latest rates results for nonlinear problems by Neubauer [On enhanced convergence rates for Tikhonov regularization of non-linear ill-posed problems in Banach spaces, Inverse Prob. 25 (2009), p. 065009] indicate an apparent qualification of the method caused by the residual norm exponent p. The new message of the present article is that optimal rates are shown to be independent of that exponent in the range 1 ≤ p < ∞. However, on the one hand, the smoothness of the image space influences the rates, and on the other hand best possible rates require specific choices of the regularization parameters α > 0. While for all p > 1 the regularization parameters have to decay to zero with some prescribed speed depending on p when the noise level tends to zero in order to obtain the best rates, the limiting case p = 1 shows some curious behaviour. For that case, the α-values must get asymptotically frozen at a fixed positive value characterized by the properties of the solution as the noise level decreases.

Electrical capacitance tomography for visualizing the flow of a mixture of high/low conducting components

This article deals with an inverse problem of electrical capacitance tomography (ECT) for visualizing cross-sectional images of the mixture of high/low conducting flow components in pipelines from boundary measurements using capacitive electrode sensors. The capacitive technique relies on changes in capacitances between electrodes due to the changes in permittivities and conductivities of flow components. In Woo et al. [A direct tracking method for a grounded conductor inside a pipeline from capacitance measurements, Inverse Prob. 22 (2006), pp. 481–494], the single-drive interleaved ECT system has been proposed for estimating the inclusion of high-contrast conductivity flow, and the corresponding inverse problem is based on a special 2D forward model which appears to be quite different from the simple 2D projection of the actual 3D forward model. In this system, we employ a single set of measured capacitance with a fixed applied voltage. In this article, we justify the validity of the 2D forward model proposed in Woo et al. as a good approximation of the actual 3D model. We also show that the solution of the simple 2D projection of the actual 3D model does not reflect the concrete relation between the conductive region and the measured capacitance data. Next, we provide a stability estimate for the corresponding inverse problem. Finally, we develop a very simple reconstruction algorithm called the sweeping plane reconstruction providing real-time estimate of a rough shape of the conductive region.

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

The objective of this paper is to reviewrecent developments in numerical re- construction methods for inverse transport problems in imaging applications, mainly optical tomography, fluorescence tomography and bioluminescence tomography. In those inverse problems, one aims at reconstructing physical parameters,suchas the ab- sorption coefficient, the scattering coefficient and the fluorescence light source, inside heterogeneous media, frompartialknowledge oftransport solutions onthe boundaries of the media. The physical parameters recovered can be used for diagnostic purpose. Numerical reconstruction techniques for those inverse transport problems can be roughly classified into two categories: linear reconstruction methods and nonlinear re- construction methods. In the first type of methods, the inverse problems are linearized around some known background to obtain linear inverse problems. Classical regular- ization techniques are then applied to solve those inverse problems. The second type of methods are either based on regularized nonlinear least-square techniques or based on gradient-driven iterative methods for nonlinear operator equations. In either case, the unknown parameters are iteratively updated until the solutions of the transport equations with the those parameters match the measurements to a certain extent. We reviewlinear and nonlinear reconstruction methods for inverse transport problems in medical imaging with stationary, frequency-domain and time-dependent data. The materials presented include both existing and new results. Meanwhile, we attempt to present similar algorithms for different problems in the same framework to make it more straightforward to generalize those algorithms to other inverse (transport) prob- lems.

Error estimation for Bregman iterations and inverse scale space methods in image restoration

In this paper, we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. Burger in Inverse Prob 20: 1411–1421, 2004; Resmerita in Inverse Prob 21: 1303–1314, 2005; Inverse Prob 22: 801–814, 2006), but so far it was not applied to image restoration in a systematic way. Due to the flexibility of the Bregman distances, this approach is particularly attractive for imaging tasks, where often singular energies (non-differentiable, not strictly convex) are used to achieve certain tasks such as preservation of edges. Besides the discussion of the variational image restoration schemes, our main goal in this paper is to extend the error estimation approach to iterative regularization schemes (and time-continuous flows) that have emerged recently as multiscale restoration techniques and could improve some shortcomings of the variational schemes. We derive error estimates between the iterates and the exact image both in the case of clean and noisy data, the latter also giving indications on the choice of termination criteria. The error estimates are applied to various image restoration approaches such as denoising and decomposition by total variation and wavelet methods. We shall see that interesting results for various restoration approaches can be deduced from our general results by just exploring the structure of subgradients.

Relationship of d-dimensional continuous multi-scale wavelet shrinkage with integro-differential equations

The goal of this paper is to extend the results of Didas and Weickert [Didas, S, Weickert, J. Integrodifferential equations for continuous multi-scale wavelet shrinkage. Inverse Prob Imag 2007;1:47–62.] to d-dimensional ( d ⩾ 1) case. Firstly, we relate a d-dimensional continuous mother wavelet ψ( x) with a fast decay and n vanishing moments to the sum of the order partial derivative of a group of functions θ k ( x)(∣ k∣ = n) with fast decay, which also makes wavelet transform equal to a sum of smoothed partial derivative operators. Moreover, d-dimensional continuous wavelet transform can be explained as a weighted average of pseudo-differential equations, too. For d = 1, our results are completely same as Didas and Weickert (2007), but for d > 1, it is different from the type of one variable. Finally, we exploit the reason with an example of 2-dimensional and 3-dimensional Mexican hat wavelet.

Geoacoustic inversion via acoustic‐seismic transfer function

Inversion methods for estimation of geoacoustic model parameters often use the scattered field data for obtaining the properties of viscoelastic layered media. This work presents a method to retrieve ground properties information using the outdoor acoustic–seismic transfer function (admittance). The resonance method developed in A. Nagle, H. Uberall, and K‐B. Yoo [Inverse Prob. 1, 99–110 (1985)] is extended and used for a preliminary estimation of a set of parameters for a three‐layered ground model. The least squares method is used afterward to choose the model with the best fit to experimental data. Results of the reconstruction show good agreement with the experimental data. A description of the resonant technique and the experimental setup are presented. The effect of a finite size of sound source on the acoustic–seismic transfer function is also discussed. [Work supported by ONR.]

Nonlinear model-following control application to airplane control

Nonlinear model-following control design is applied to the problem of control of the six degrees of freedom of an airplane that lacks direct control of lift and side force. The nonlinear expressions for the error dynamics of the model-following control are examined using Lyapunov stability analysis. The analysis results in nonlinear feedforward and feedback gains that are functions of the airplane and model states. As a consequence, gain scheduling requirements for the implementation of the model-following control are reduced to only those involving the estimation of stability and control derivatives of the airplane. The use of these gains is shown through an example application to the control of a nonlinear aerodynamic and engine model provided by NASA Ames-Dryden Flight Research Facility. The model being followed is based on a trajectory generation algorithm, and represents a form of dynamic inversion. HE design methodology to be used is based on the applica- tion of nonlinear model-following to the problem of the control of the six degrees of freedom of an airplane. This methodology is related to nonlinear inverse model theory. It is a more complete approach in that it provides a means for analysis of the dynamics of the errors involved in model-follow- ing. The particular approach has been successfully applied to the control of a nonlinear aerodynamic model of a high-angle- of-attack research vehicle (HARV) through large attitude and angle of attack changes. In general, model-following control attempts to make an actual airplane behave similarly to a prescribed mathematical model of an airplane with different force and moment character- istics than the actual airplane. The model behavior may be based on desirable flying qualities, and the matching of those flying qualities is taken to be the design objective. In this case the pilot controls are applied to the model (either conceptually or literally, to a simulation) and the airplane controls are deter- mined. Alternatively, the mathematical model may be a simplified representation of the actual airplane being controlled, in which case model-following control becomes a solution to the inverse problem. Here the state trajectory of the model is determined from a specification of a particular flight path or maneuver, and the airplane controls required to follow it are determined. Perfect, explicit model-following solutions to the inverse prob- lem provide more than the open-loop controls required to fly a maneuver, since this formulation allows control of the errors between the airplane and model during the maneuver. It is this application of model-following control that is used in this paper. To develop the nonlinear model-following controller, we will first review the model-following concepts used here. Initially a standard form of the airplane and model equations is presented with the conditions for perfect dynamic matching presented. Associated with the conditions for perfect dynamics matching are differential equations for the error. In many cases these error equations are linearized and standard linear control ideas applied to guarantee stability (i.e., they tend to go to zero in time). Hence one is led to a gain scheduling scheme. In the method presented, however, using an approach based on the stability theory of Lyapunov, a set of gains which insure stability of the nonlinear error dynamics can be found. These require no updates but are functions of the current state. The result of this analysis is illustrated through application to the nonlinear airplane simulation provided by NASA Ames- Dryden Flight Research Facility. In this application, the model being followed is a simplified description of the airplane being controlled. The model is not, however, directly flown by exter- nally applied (pilot) controls. Rather, it represents the states and state rates required to execute some prescribed maneuver.

New exactly solvable Hamiltonians: Shape invariance and self-similarity.

We discuss in some detail the self-similar potentials of Shabat and Spiridonov which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape invariant potentials of which the Shabat-Spiridonov ones are a special case. These new potentials can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions and transmission coefficients for these potentials are obtained. We show that these potentials can also be obtained numerically. Included as an intriguing case is a shape invariant double well potential whose supersymmetric partner potential is only a single well. Our class of exactly solvable Hamiltonians is further enlarged by examining two new directions: (i) changes of parameters which are different from the previously studied cases of translation and scaling; (ii) extending the usual concept of shape invariance in one step to a multi-step situation. These extensions can be viewed as q-deformations of the harmonic oscillator or multi-soliton solutions corresponding to the Rosen-Morse potential.