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1.	Carlsson, Marcus, et al. (författare) On phase retrieval via matrix completion and the estimation of low rank PSD matrices 2020 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 36:1 Tidskriftsartikel (refereegranskat)abstract Given underdetermined measurements of a positive semi-definite (PSD) matrix X of known low rank K, we present a new algorithm to estimate X based on recent advances in non-convex optimization schemes. We apply this in particular to the phase retrieval problem for Fourier data, which can be formulated as a rank 1 PSD matrix recovery problem. Moreover, we provide a theory for how oversampling affects the stability of the lifted inverse problem.
2.	Shestopalov, Yury, 1953-, et al. (författare) Determination of permittivity of an inhomogeneous dielectric body in a waveguide 2011 Ingår i: Inverse Problems. - Philadelphia : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 27, s. 095010-1-095010-12 Tidskriftsartikel (refereegranskat)abstract The determination of permittivity of an inhomogeneous dielectric body located in a rectangular waveguide is considered. An iteration method for the numerical solution of the problem is proposed. Convergence of the method is proved. Numerical results for the determination of permittivity of a dielectric body are presented
3.	Andersson, Fredrik (författare) The Doppler moment transform in Doppler tomography 2005 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 21:4, s. 1249-1274 Tidskriftsartikel (refereegranskat)abstract The problem of reconstructing two-dimensional vector fields from tomographic measurements based on Doppler techniques is addressed. Measurements are made along lines, from which distributions of the field component parallel to the line are obtained. A new transform, the Doppler moment transform, which extracts local information from such data, is introduced. In contrast to most previous approaches in vector tomography that are only able to treat solenoidal fields, the Doppler moment transform provides information also of the irrotational part of the field. Specifically, it gives rise to partial differential equations in the field components. Properties of the new transform and the corresponding equations are discussed, shedding light on questions of uniqueness. It is also shown that the differential equations can be combined to form algebraic equations in the field components. Finally, a reconstruction scheme is constructed and some numerical experiments are provided.
4.	Carlsson, Marcus, et al. (författare) An unbiased approach to compressed sensing 2020 Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 36:11 Tidskriftsartikel (refereegranskat)abstract In compressed sensing a sparse vector is approximately retrieved from an under-determined equation system Ax = b. Exact retrieval would mean solving a large combinatorial problem which is well known to be NP-hard. For b of the form Ax(0) + epsilon, where x(0) is the ground truth and epsilon is noise, the 'oracle solution' is the one you get if you a priori know the support of x(0), and is the best solution one could hope for. We provide a non-convex functional whose global minimum is the oracle solution, with the property that any other local minimizer necessarily has high cardinality. We provide estimates of the type parallel to(x) over cap - x(0)parallel to(2) <= C parallel to epsilon parallel to(2) that are significantly lower than for competing methods or theorems, and our theory relies on soft assumptions on the matrix A, in comparison with standard results in the field. The framework also allows to incorporate a priori information on the cardinality of the sought vector. In this case we show that despite being non-convex, our cost functional has no spurious local minima and the global minima is again the oracle solution, thereby providing the first method which is guaranteed to find this point for reasonable levels of noise, without resorting to combinatorial methods.
5.	Kurasov, Pavel (författare) Scattering matrices with finite phase shift and the inverse scattering problem 1996 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 12:3, s. 295-307 Tidskriftsartikel (refereegranskat)abstract The inverse scattering problem for the Schrödinger operator on the half-axis is studied. it is shown that this problem can be solved for the scattering matrices with arbitrary finite phase shift on the real axis if one admits potentials with long-range oscillating tails at infinity. The solution of the problem is constructed with the help of the Gelfand-Levitan-Marchenko procedure. The inverse problem has no unique solution for the standared set of scattering data which includes the scattering matrix, energies of the bound states and corresponding normalizing constants. This fact is related to zeros of the spectral density on the real axis. It is proven that the inverse problem has a unique solution in the defined class of potentials if the zeros of the spectral density are added to the set of scattering data.
6.	Kohlenberg, Jennifer, et al. (författare) The inverse spectral problem for the discrete cubic string 2007 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 23:1, s. 99-121 Tidskriftsartikel (refereegranskat)abstract Given a measure m on the real line or a finite interval, the cubic string is the third-order ODE - ′′′ ≤ zm where z is a spectral parameter. If equipped with Dirichlet-like boundary conditions this is a non-self-adjoint boundary value problem which has recently been shown to have a connection to the Degasperis-Procesi nonlinear water wave equation. In this paper, we study the spectral and inverse spectral problem for the case of Neumann-like boundary conditions which appear in a high-frequency limit of the Degasperis-Procesi equation. We solve the spectral and inverse spectral problem for the case of m being a finite positive discrete measure. In particular, explicit determinantal formulae for the measure m are given. These formulae generalize Stieltjes' formulae used by Krein in his study of the corresponding second-order ODE -″ ≤ zm. © 2007 IOP Publishing Ltd.
7.	Shestopalov, Youri, 1953-, et al. (författare) Existence and uniqueness of solution to the inverse problem of complex permittivity reconstruction of a dielectric body in a waveguide 2010 Ingår i: Inverse Problems. - Bristol : IoP. - 0266-5611 .- 1361-6420. ; 26:10 Tidskriftsartikel (refereegranskat)abstract This paper presents a statement, a proof of uniqueness, and a method of solution to the inverse problem of the determination of permittivity of a lossy dielectric inclusion in a three-dimensional waveguide of rectangular cross-section fromthe transmission characteristics. The approach is based on the solution to a volume singular integral equation (VSIE). The examination of this equationis based on the analysis of the corresponding boundary value problem (BVP) for the system of Maxwell equations and the equivalence of this BVP andVSIE. The existence and uniqueness for VSIE in the space of square-integrable functions are proved. The permittivity reconstruction employs a method ofiterations applied to the solution of VSIE
8.	Andersson, Fredrik, et al. (författare) A variational formulation for interpolation of seismic traces with derivative information 2015 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 31:5 Tidskriftsartikel (refereegranskat)abstract We construct a variational formulation for the problem of interpolating seismic data in the case of missing traces. We assume that we have derivative information available at the traces. The variational problem is essentially the minimization of the integral over the smallest eigenvalue of the structure tensor associated with the interpolated data. This has the physical meaning of penalizing the local presence of more than one direction in the interpolation. The variational problem is used to justify the solutions of a non-standard anisotropic diffusion problem as reasonable interpolated images. We show existence and uniqueness for this type of anisotropic diffusion. In particular, the uniqueness property is important as it guarantees that the solution can be obtained by the numerical schemes we propose.
9.	Asadzadeh, Mohammad, 1952, et al. (författare) A posteriori error estimates in a globally convergent FEM for a hyperbolic coefficient inverse problem 2010 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:11 Tidskriftsartikel (refereegranskat)abstract This study concerns a posteriori error estimates in a globally convergent numerical method for a hyperbolic coefficient inverse problem. Using the Laplace transform the model problem is reduced to a nonlinear elliptic equation with a gradient dependent nonlinearity. We investigate the behavior of the nonlinear term in both a priori and a posteriori settings and derive optimal a posteriori error estimates for a finite-element approximation of this problem. Numerical experiments justify the efficiency of a posteriori estimates in the globally convergent approach.
10.	Beilina, Larisa, 1970, et al. (författare) A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem 2010 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:4 Tidskriftsartikel (refereegranskat)abstract A synthesis of a globally convergent numerical method for a coefficient inverse problem and the adaptivity technique is presented. First, the globally convergent method provides a good approximation for the unknown coefficient. Next, this approximation is refined via the adaptivity technique. The analytical effort is focused on a posteriori error estimates for the adaptivity. A numerical test is presented.
11.	Beilina, Larisa, 1970, et al. (författare) Reconstruction from blind experimental data for an inverse problem for a hyperbolic equation 2014 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 30:2 Tidskriftsartikel (refereegranskat)abstract We consider the problem of the reconstruction of dielectrics from blind backscattered experimental data. The reconstruction is done from time domain data, as opposed to a more conventional case of frequency domain data. Experimental data were collected using a microwave scattering facility which was built at the University of North Carolina at Charlotte. This system sends electromagnetic pulses into the medium and collects the time-resolved backscattered data on a part of a plane. The spatially distributed dielectric constant epsilon(r)(x), x is an element of R-3 is the unknown coefficient of a wave-like PDE. This coefficient is reconstructed from those data in blind cases. To do this, a globally convergent numerical method is used.
12.	Beilina, Larisa, 1970, et al. (författare) Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm 2010 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:12 Tidskriftsartikel (refereegranskat)abstract The validity of the synthesis of a globally convergent numerical method with the adaptive FEM technique for a coefficient inverse problem is verified on time-resolved experimental data. The refractive indices, locations and shapes of dielectric abnormalities are accurately imaged.
13.	Beilina, Larisa, 1970, et al. (författare) Reconstruction of shapes and refractive indices from backscattering experimental data using the adaptivity 2014 Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 30:10, s. Art. no. 105007- Tidskriftsartikel (refereegranskat)abstract We consider the inverse problem of the reconstruction of the spatially distributed dielectric constant epsilon(r)(x), x is an element of R-3, which is an unknown coefficient in the Maxwell's equations, from time-dependent backscattering experimental radar data associated with a single source of electric pulses. The refractive index is n(x) = root epsilon(r)(x). The coefficient epsilon(r)(x) is reconstructed using a two-stage reconstruction procedure. In the first stage an approximately globally convergent method proposed is applied to get a good first approximation of the exact solution. In the second stage a locally convergent adaptive finite element method is applied, taking the solution of the first stage as the starting point of the minimization of the Tikhonov functional. This functional is minimized on a sequence of locally refined meshes. It is shown here that all three components of interest of targets can be simultaneously accurately imaged: refractive indices, shapes and locations.
14.	Burger, Raimund, et al. (författare) Convexity-preserving flux identification for scalar conservation laws modelling sedimentation 2013 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 29:4 Tidskriftsartikel (refereegranskat)abstract Sedimentation of a suspension of small particles dispersed in a viscous fluid can be described by a scalar, nonlinear conservation law, whose flux function usually has one inflection point. The identification of the flux function is of theoretical interest and practical importance for plant-scale simulators of continuous sedimentation. For a real suspension, the Kynch test and the Diehl test, which are based on an initially homogenous suspension either filling the whole settling column or being initially located above clear liquid, respectively, provide data points that represent curved (convex or concave, respectively) suspension-supernate interfaces from which it is possible to reconstruct portions of the flux function to either side of the inflection point. Several functional forms can be employed to generate a provably convex or concave, twice differentiable accurate approximation of these data points via the solution of a constrained least-squares minimization problem. The resulting spline-like estimated trajectory can be converted into an explicit formula for the flux function. It is proved that the inverse problem of flux identification solved this way has a unique solution. The problem of gluing together the portions of the flux function from the Kynch and Diehl tests is addressed. Examples involving synthetic data are presented.
15.	Cheney, Margaret, et al. (författare) Three-dimensional inverse scattering: layer-stripping formulas and ill-posedness results 1988 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 4:3, s. 625-642 Tidskriftsartikel (refereegranskat)abstract The authors consider the three-dimensional direct and inverse scattering problems for the Schrodinger equation and for the reduced wave equation with variable velocity. The scatterer is probed with either point sources or plane waves of fixed frequency. They ask the question, 'How does the wave field change when the scatterer is truncated?' Simple formulae for the derivative of the wave field with respect to the truncation parameter are obtained. Similar formulae are obtained for the scattering amplitudes. These formulae are used to derive ill-posedness results for various inverse scattering problems. The ill-posedness results apply when data are collected over a range of frequencies.
16.	Elden, Lars, 1944-, et al. (författare) A stability estimate for a Cauchy problem for an elliptic partial differential equation 2005 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 21:5, s. 1643-1653 Tidskriftsartikel (refereegranskat)abstract A two-dimensional inverse steady state heat conduction problem in the unit square is considered. Cauchy data are given for y ≤ 0, and boundary data are for x ≤ 0 and x ≤ 1. The elliptic operator is self-adjoint with non-constant, smooth coefficients. The solution for y ≤ 1 is sought. This Cauchy problem is ill-posed in an L2-setting. A stability functional is defined, for which a differential inequality is derived. Using this inequality a stability result of Hölder type is proved. It is demonstrated explicitly how the stability depends on the smoothness of the coefficients. The results can also be used for rectangle-like regions that can be mapped conformally onto a rectangle. © 2005 IOP Publishing Ltd.
17.	Folkow, Peter, 1968 (författare) Time domain inversion of a viscoelastically restrained Timoshenko beam 1999 Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 15, s. 551-562 Tidskriftsartikel (refereegranskat)abstract This paper deals with the inverse scattering problem for a homogeneous Timoshenko beam suspended on a semi-infinite viscoelastic layer. The purpose is to derive the properties of the suspension from knowledge of the reflection data. The method used is the imbedding technique, which render solutions that are independent of the excitation. From a numerical point of view, the reflection equation in its usual form is inappropriate when solving the inverse problem. In order to bypass such complications, the reflection equation is modified and solved. In the numerical results, the properties of the layer are calculated using noisy reflection data.
18.	Fridén, Jonas, et al. (författare) Transient external 3D excitation of a dispersive and anisotropic slab 1997 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 13:3, s. 691-709 Tidskriftsartikel (refereegranskat)abstract Propagation of a transient electromagnetic field in a stratified, dispersive and anisotropic slab, and related direct and inverse problems, are investigated. The field is generated by a transient external three-dimensional (3D) source. The analysis relies on the wave splitting concept and a two-dimensional Fourier transformation in the transverse spatial coordinates. An investigation of the physical properties of the split fields is made. To solve the direct and inverse scattering problems, wave propagators are used. This method is a generalization and a unification of the previously used imbedding and Green functions methods. The wave propagator approach provides an exact solution of the transmission operator. From this solution it is possible to extract the first precursor (the Sommerfeld forerunner). These results also hold for a bi-anisotropic slab. An inverse problem is outlined using reflection and transmission data corresponding to four, two-dimensional Fourier parameters. Due to the stratification of the medium, the inverse Fourier transformation is not needed in the inverse problem.
19.	Fuks, Peter, et al. (författare) Direct and inverse scattering from dispersive media 1994 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 10:3, s. 555-571 Tidskriftsartikel (refereegranskat)abstract Direct and inverse scattering from dispersive media are studied in the time domain. The media are one-dimensional and homogeneous but can be impedance mismatched to the surrounding media. A time-domain method is introduced where the reflection and transmission kernels are obtained from Volterra equations of the second kind. Numerical examples are given based upon both synthetic and measured data for the inverse problem.
20.	Gustafsson, Mats, et al. (författare) Cramer-Rao Lower Bounds for Inverse Scattering Problems of Multilayer Structures 2006 Ingår i: Inverse Problems. - : IOP Publishing. ; 22:4, s. 1359-1380, s. 1-1 Tidskriftsartikel (refereegranskat)abstract In this paper, the inverse scattering problem of amultilayer structure is analysed with the Fisher information matrix and the Cramer–Rao lower bound (CRLB). The CRLB quantifies the ill-posedness of the inverse scattering problem in terms of resolution versus estimation accuracy based on the observation of noisy data. The limit for feasible inversion is identified by an asymptotic eigenvalue analysis of the Toeplitz Fisher information matrix and an application of the sampling theorem. It is shown that the resolution is inversely proportional to the bandwidth of the reflection data and that the CRLB increases linearly with the number of slabs. The transmission data give a rank-1 Fisher information matrix which can approximately reduce the CRLB by a factor of 4. Moreover, the effect of dispersive material parameters and simultaneous estimation of two material parameters are analysed. The results are illustrated with numerical examples.
21.	Hellberg, Rasmus, et al. (författare) Design of reflectionless media for transient electromagnetic waves 1995 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 11:1, s. 147-164 Tidskriftsartikel (refereegranskat)abstract A method for modelling reflectionless conductive dispersive media is presented. The media are either half-spaces or finite dispersive slabs with a spatially varying impedance. The problem of finding reflectionless media for plane waves at normal incidence is formulated as an inverse problem where the constitutive relation is to be determined as a function of depth given a reflection kernel which is zero. The inverse problem is solved by a time-domain Green function technique. It is seen that non-reflecting half-spaces can be constructed in a number of different ways, whereas non-reflecting slabs can only be found if the back-wall is non-conducting.
22.	Klibanov, M. V., et al. (författare) Picosecond scale experimental verification of a globally convergent algorithm for a coefficient inverse problem 2010 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 26:4 Tidskriftsartikel (refereegranskat)abstract A globally convergent algorithm by the first and third authors for a 3D hyperbolic coefficient inverse problem is verified on experimental data measured in the picosecond scale regime. Quantifiable images of dielectric abnormalities are obtained. The total measurement timing of a 100 ps pulse for one detector location was 1.2 ns with 20 ps (= 0.02 ns) time step between two consecutive readings. Blind tests have consistently demonstrated an accurate imaging of refractive indexes of dielectric abnormalities. At the same time, it is shown that a modified gradient method is inapplicable to this kind of experimental data. This inverse algorithm is also applicable to other types of imaging modalities, e. g. acoustics. Potential applications are in airport security, imaging of land mines, imaging of defects in non-distractive testing, etc.
23.	Kristensson, Gerhard, et al. (författare) Direct and inverse scattering for transient electromagnetic waves in nonlinear media 1998 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; II, s. 587-589 Tidskriftsartikel (refereegranskat)abstract Nonlinear propagation of electromagnetic waves is an important problem in optics. Often the properties of the nonlinear media are not fully understood. The solution of an inverse problem can provide an aid to that understanding. An inverse transmission problem is posed; it is one of reconstructing the medium parameters, by measurement of a wave that has been propagated through the nonlinear medium. The nonlinear medium is assumed to be homogeneous and isotropic. The methods have application to nonlinear optics, and the numerical results for both the direct and inverse problems presented are based on the nonlinear Kerr effect, which is observed in the optical wavelength band. However, the mathematical techniques that are developed are applicable to any set of nonlinear first-order equations. The method is therefore model independent.
24.	Kuzhuget, A. V., et al. (författare) Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm 2012 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 28:9 Tidskriftsartikel (refereegranskat)abstract An approximately globally convergent numerical method for a 1D coefficient inverse problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A posteriori analysis has revealed that computed and tabulated values of dielectric constants are in good agreement. Convergence analysis is presented.
25.	Lundmark, Hans, 1970-, et al. (författare) Multi-peakon solutions of the Degasperis-Procesi equation 2003 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 19:6, s. 1241-1245 Tidskriftsartikel (refereegranskat)abstract We present an inverse scattering approach for computing n-peakon solutions of the Degasperis-Procesi equation (a modification of the Camassa-Holm (CH) shallow water equation). The associated non-self-adjoint spectral problem is shown to be amenable to analysis using the isospectral deformations induced from the n-peakon solution, and the inverse problem is solved by a method generalizing the continued fraction solution of the peakon sector of the CH equation.
26.	Marchesini, Stefano, et al. (författare) Augmented projections for ptychographic imaging 2013 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 29:11, s. 115009- Tidskriftsartikel (refereegranskat)abstract Ptychography is a popular technique to achieve diffraction limited resolution images of a two- or three-dimensional sample using high frame rate detectors. We introduce a relaxation of common projection algorithms to account for instabilities given by intensity and background fluctuations, position errors, or poor calibration using multiplexing illumination. This relaxation introduces an additional phasing optimization at every step that enhances the convergence rate of common projection algorithms. Numerical tests exhibit the exact recovery of the object and the perturbations when there is high redundancy in the data.
27.	Nordebo, Sven, et al. (författare) A Systematic Approach to Robust Preconditioning for Gradient Based Inverse Scattering Algorithms 2008 Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 24:2, s. 025027- Tidskriftsartikel (refereegranskat)abstract This paper presents a systematic approach to robust preconditioning for gradient-based nonlinear inverse scattering algorithms. In particular, one- and two-dimensional inverse problems are considered where the permittivity and conductivity profiles are unknown and the input data consist of the scattered field over a certain bandwidth. A time-domain least-squares formulation is employed and the inversion algorithm is based on a conjugate gradient or quasi-Newton algorithm together with an FDTD-electromagnetic solver. A Fisher information analysis is used to estimate the Hessian of the error functional. A robust preconditioner is then obtained by incorporating a parameter scaling such that the scaled Fisher information has a unit diagonal. By improving the conditioning of the Hessian, the convergence rate of the conjugate gradient or quasi-Newton methods are improved. The preconditioner is robust in the sense that the scaling, i.e. the diagonal Fisher information, is virtually invariant to the numerical resolution and the discretization model that is employed. Numerical examples of image reconstruction are included to illustrate the efficiency of the proposed technique. © 2008 IOP Publishing Ltd.
28.	Nordebo, Sven, et al. (författare) An adjoint field approach to Fisher information-based sensitivity analysis in electrical impedance tomography 2010 Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 26:12 Tidskriftsartikel (refereegranskat)abstract An adjoint field approach is used to formulate a general numerical framework for Fisher information-based sensitivity analysis in electrical impedance tomography. General expressions are given for the gradients used in standard least-squares optimization, i.e. the Jacobian related to the forward problem, and it is shown that these gradient expressions are compatible with commonly used electrode models such as the shunt model and the complete electrode model. By using the adjoint field formulations together with a variational analysis, it is also shown how the computation of the Fisher information can be integrated with the gradient calculations used for optimization. It is furthermore described how the Fisher information analysis and the related sensitivity map can be used in a preconditioning strategy to obtain a well-balanced parameter sensitivity and improved performance for gradient-based quasi-Newton optimization algorithms in electrical impedance tomography. Numerical simulations as well as reconstructions based on experimental data are used to illustrate the sensitivity analysis and the performance of the improved inversion algorithm in a four-electrode measurement set-up.
29.	Nordebo, Sven, et al. (författare) Fisher information analysis for two-dimensional microwave tomography 2007 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 23:3, s. 859-877 Tidskriftsartikel (refereegranskat)abstract In this paper, a Fisher information analysis is employed to establish some important physical performance bounds in microwave tomography. As a canonical problem, the two-dimensional electromagnetic inverse problem of imaging a cylinder with isotropic dielectric losses is considered. A fixed resolution is analysed by introducing a finite basis, i.e., pixels representing the material properties. The corresponding Cramer-Rao bound for estimating the pixel values is computed based on a calculation of the sensitivity field which is obtained by differentiating the observed field with respect to the estimated parameter. An optimum trade-off between the accuracy and the resolution is defined based on the Cramer-Rao bound, and its application to assess a practical resolution limit in the inverse problem is discussed. Numerical examples are included to illustrate how the Fisher information analysis can be used to investigate the significance of measurement distance, operating frequency and losses in the canonical tomography set-up.
30.	Olsson, Peter, 1956 (författare) Directional decomposition of the acoustic wave equation for fluids and metafluids in spherical geometries, with application to transformational acoustics 2016 Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 32:3 Tidskriftsartikel (refereegranskat)abstract A new directional decomposition of the acoustic 3D wave equation is derived for spherically symmetric geometries, where the wave fields do not need to possess such a symmetry. This provides an alternative basis for various applications of techniques like invariant embedding and time domain Green functions in spherically symmetric geometries. Contrary to previous results on spherical wave splittings, the new decomposition is given in a very explicit form. The wave equation considered incorporates effects from radially varying compressibility and density, but also from anisotropic density, a property of certain so called metafluids. By applying the new spherical wave splitting, we show that all spherically symmetric acoustic metafluid cloaks are diffeomorphic images of a homogeneous and isotropic spherical ball of perfect fluid.
31.	Olsson, Peter, 1956, et al. (författare) Partial elastodynamic cloaking by means of fiber-reinforced composites 2011 Ingår i: Inverse Problems. - : IOP Publishing. - 1361-6420 .- 0266-5611. ; 27:4 Tidskriftsartikel (refereegranskat)abstract In this paper, we show that if inextensible fibers are embedded in an elastic material (or the material is by some other means made considerably stiffer in a particular but possibly variable direction), one may obtain equations of motion which are form invariant under certain diffeomorphism, allowing for partial cloaking (or 'illusion optics') schemes in both 2D and 3D. The schemes are valid at all frequencies without requiring any active material properties, and will thus work in the time domain without requiring active materials. While being mathematically exact at all frequencies, the applicability is of course limited to where the continuum approximation holds. Additional limits, set by idealization in the modeling of fibers and core material, are also present. However, contrary to some other approaches, the solution does not require metamaterials with non-scalar mass densities, nor does it require the breaking of the supersymmetry of the elasticity tensor.
32.	Rikte, Sten (författare) Time-domain direct and inverse scattering for bi-anisotropic slabs at oblique incidence 2002 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 18:2, s. 467-493 Tidskriftsartikel (refereegranskat)abstract Using the Cayley–Hamilton theorem and unique solubility of scalar Volterra convolution equations of the second kind, the inverse problem of determining the four time-dependent susceptibility dyadics of a linear, homogeneous, bi-anisotropic slab from generic scattering data at oblique incidence is shown to be well posed. An explicit formula for the crucial step is given.
33.	Rullgård, Hans (författare) Stability of the inverse problem for the attenuated Radon transform with 180 degrees data 2004 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 20:3, s. 781-797 Tidskriftsartikel (refereegranskat)abstract We consider the inverse problem for the attenuated Radon transform in two dimensions. It has recently been shown that this problem has a unique solution when the attenuation function is Hölder continuous, even with limited angle data. In this paper, we study stability of the inverse problem, with data from a 180° range of angles. We show that when the attenuation function is Hölder continuous and the support of the original function f is restricted to a given compact set, the Hölder norm of f with exponent -1/2 can be estimated in terms of the L2 norm of the attenuated Radon transform of f. We also obtain new stability results for the forward problem, that is, estimates of the norm of the Radon transform in terms of norms of the original function and the attenuation function.
34.	Sjöberg, Daniel (författare) Reconstruction of nonlinear material properties for homogeneous, isotropic slabs using electromagnetic waves 1999 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 15:2, s. 44-431 Tidskriftsartikel (refereegranskat)abstract This paper addresses the inverse problem of reconstructing a medium’s instantaneous, nonlinear response to electromagnetic excitation. Using reflection and transmission data for an almost arbitrary incident field on a homogeneous slab, we are able to obtain the nonlinear constitutive relations for both electric and magnetic fields, with virtually no assumptions made on the specific form of the relations. It is shown that for a nonmagnetic material, reflection data suffices to obtain the electrical nonlinear response. We also show that the algorithms are well posed. Numerical examples illustrate the analysis presented in this paper.
35.	Tenorio, L., et al. (författare) Data analysis tools for uncertainty quantification of inverse problems 2011 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 27:4 Tidskriftsartikel (refereegranskat)abstract We present exploratory data analysis methods to assess inversion estimates using examples based on l(2)- and l(1)-regularization. These methods can be used to reveal the presence of systematic errors such as bias and discretization effects, or to validate assumptions made on the statistical model used in the analysis. The methods include bounds on the performance of randomized estimators of a large matrix, confidence intervals and bounds for the bias, resampling methods for model validation and construction of training sets of functions with controlled local regularity.
36.	Åberg, Ingegerd, et al. (författare) Propagation of transient electromagnetic waves in time-varying media-direct and inverse scattering problems 1995 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 11:1, s. 29-49 Tidskriftsartikel (refereegranskat)abstract Wave propagation of transient electromagnetic waves in time-varying media is considered. The medium, which is assumed to be inhomogeneous and dispersive, lacks invariance under time translations. The spatial variation of the medium is assumed to be in the depth coordinate, i.e. it is stratified. The constitutive relations of the medium are a time integral of a generalized susceptibility kernel and the field. The generalized susceptibility kernel depends on one spatial and two time coordinates. The concept of wave splitting is introduced. The direct and inverse scattering problems are solved by the use of an imbedding or a Green function approach. The direct and the inverse scattering problems are solved for a homogeneous semi-infinite medium. Explicit algorithms are developed. In this inverse scattering problem, a function depending on two time coordinates is reconstructed. Several numerical computations illustrate the performance of the algorithms.
37.	Öktem, Ozan, 1969-, et al. (författare) A component-wise iterated relative entropy regularization method with updated prior and regularization parameter 2007 Ingår i: Inverse Problems. - : Institute of Physics (IOP). - 0266-5611 .- 1361-6420. ; 23:5, s. 2121-2139 Tidskriftsartikel (refereegranskat)abstract We present a componentwise iterated relative entropy regularization method (COMET) where the prior and regularization parameter could be updated in the iterates. Such a reconstruction method could be useful for multicomponent inverse problems, such as the one occurring in electron tomography. The paper also contains a brief introduction to regularization theory with emphasis on variational based regularization methods, and we rigorously prove that the tolerance-based entropy reconstruction method that occurs in the COMET iterates is a regularization method. We conclude by showing examples of COMET applied to electron tomography data.
38.	Adler, Jonas, et al. (författare) Solving ill-posed inverse problems using iterative deep neural networks 2017 Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 33:12 Tidskriftsartikel (refereegranskat)abstract We propose a partially learned approach for the solution of ill-posed inverse problems with not necessarily linear forward operators. The method builds on ideas from classical regularisation theory and recent advances in deep learning to perform learning while making use of prior information about the inverse problem encoded in the forward operator, noise model and a regularising functional. The method results in a gradient-like iterative scheme, where the 'gradient' component is learned using a convolutional network that includes the gradients of the data discrepancy and regulariser as input in each iteration. We present results of such a partially learned gradient scheme on a non-linear tomographic inversion problem with simulated data from both the Sheep-Logan phantom as well as a head CT. The outcome is compared against filtered backprojection and total variation reconstruction and the proposed method provides a 5.4 dB PSNR improvement over the total variation reconstruction while being significantly faster, giving reconstructions of 512 x 512 pixel images in about 0.4 s using a single graphics processing unit (GPU).
39.	Adler, Jonas, et al. (författare) Task adapted reconstruction for inverse problems 2022 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 38:7 Tidskriftsartikel (refereegranskat)abstract The paper considers the problem of performing a post-processing task defined on a model parameter that is only observed indirectly through noisy data in an ill-posed inverse problem. A key aspect is to formalize the steps of reconstruction and post-processing as appropriate estimators (non-randomized decision rules) in statistical estimation problems. The implementation makes use of (deep) neural networks to provide a differentiable parametrization of the family of estimators for both steps. These networks are combined and jointly trained against suitable supervised training data in order to minimize a joint differentiable loss function, resulting in an end-to-end task adapted reconstruction method. The suggested framework is generic, yet adaptable, with a plug-and-play structure for adjusting both the inverse problem and the post-processing task at hand. More precisely, the data model (forward operator and statistical model of the noise) associated with the inverse problem is exchangeable, e.g., by using neural network architecture given by a learned iterative method. Furthermore, any post-processing that can be encoded as a trainable neural network can be used. The approach is demonstrated on joint tomographic image reconstruction, classification and joint tomographic image reconstruction segmentation.
40.	Andersson, Joel, et al. (författare) Stability estimates for the local Radon transform 2018 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 34:3 Tidskriftsartikel (refereegranskat)abstract We consider the inverse problem for the 2-dimensional local Radon transform R[f], where f is supported in y >= x(2) and R[f](xi, eta) = integral f (x, xi x + eta) dx is defined near (xi, eta) = (0, 0). We give logarithmic estimates of f in terms of R[f] for functions f that satisfy an a priori bound. For a certain class of smooth, positive weight functions m we give similar estimates for the weighted Radon transform R-m vertical bar f](xi, eta) = integral f (x, xi x + eta) m(xi, eta, x) dx.
41.	Behrndt, Jussi, 1975-, et al. (författare) Inverse problems with partial data for elliptic operators on unbounded Lipschitz domains 2020 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 36:3 Tidskriftsartikel (refereegranskat)abstract For a second order formally symmetric elliptic differential expression we show that the knowledge of the Dirichlet-to-Neumann map or Robin-to-Dirichlet map for suitably many energies on an arbitrarily small open subset of the boundary determines the self-adjoint operator with a Dirichlet boundary condition or with a (possibly non-self-adjoint) Robin boundary condition uniquely up to unitary equivalence. These results hold for general Lipschitz domains, which can be unbounded and may have a non-compact boundary, and under weak regularity assumptions on the coefficients of the differential expression.
42.	Beilina, Larisa, 1970, et al. (författare) Lipschitz stability for an inverse hyperbolic problem of determining two coefficients by a finite number of observations 2018 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 34:1 Tidskriftsartikel (refereegranskat)abstract We consider an inverse problem of reconstructing two spatially varying coefficients in an acoustic equation of hyperbolic type using interior data of solutions with suitable choices of initial condition. Using a Carleman estimate, we prove Lipschitz stability estimates which ensure unique reconstruction of both coefficients. Our theoretical results are justified by numerical studies on the reconstruction of two unknown coefficients using noisy backscattered data.
43.	Burman, Erik, et al. (författare) Solving ill-posed control problems by stabilized finite element methods : an alternative to Tikhonov regularization 2018 Ingår i: Inverse Problems. - : IOP PUBLISHING LTD. - 0266-5611 .- 1361-6420. ; 34:3 Tidskriftsartikel (refereegranskat)abstract Tikhonov regularization is one of the most commonly used methods for the regularization of ill-posed problems. In the setting of finite element solutions of elliptic partial differential control problems, Tikhonov regularization amounts to adding suitably weighted least squares terms of the control variable, or derivatives thereof, to the Lagrangian determining the optimality system. In this note we show that the stabilization methods for discretely illposed problems developed in the setting of convection-dominated convection-diffusion problems, can be highly suitable for stabilizing optimal control problems, and that Tikhonov regularization will lead to less accurate discrete solutions. We consider some inverse problems for Poisson's equation as an illustration and derive new error estimates both for the reconstruction of the solution from the measured data and reconstruction of the source term from the measured data. These estimates include both the effect of the discretization error and error in the measurements.
44.	De Hoop, Maarten V, et al. (författare) Inverse problem for Love waves in a layered, elastic half-space 2024 Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; :045013, s. 1-44 Tidskriftsartikel (refereegranskat)abstract In this paper we study Love waves in a layered, elastic half-space. We first address the direct problem and we characterize the existence of Love waves through the dispersion relation. We then address the inverse problem and we show how to recover the parameters of the elastic medium from the empirical knowledge of the frequency–wavenumber couples of the Love waves.
45.	de Hoop, Maarten, V, et al. (författare) Semiclassical inverse spectral problem for seismic surface waves in isotropic media : part I. Love waves 2020 Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 36:7 Tidskriftsartikel (refereegranskat)abstract We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we focus on Love waves. Under certain generic conditions, we establish uniqueness and present a reconstruction scheme for theS-wavespeed with multiple wells from the semiclassical spectrum of these waves.
46.	de Hoop, Maarten, V, et al. (författare) Semiclassical inverse spectral problem for seismic surface waves in isotropic media : part II. Rayleigh waves 2020 Ingår i: Inverse Problems. - : Institute of Physics Publishing (IOPP). - 0266-5611 .- 1361-6420. ; 36:7 Tidskriftsartikel (refereegranskat)abstract We analyze the inverse spectral problem on the half line associated with elastic surface waves. Here, we extend the treatment of Love waves [5] to Rayleigh waves. Under certain conditions, and assuming that the Poisson ratio is constant, we establish uniqueness and present a reconstruction scheme for the S-wave speed with multiple wells from the semiclassical spectrum of these waves.
47.	Esteve-Yague, Carlos, et al. (författare) Spectral decomposition of atomic structures in heterogeneous cryo-EM 2023 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 39:3, s. 034003- Tidskriftsartikel (refereegranskat)abstract We consider the problem of recovering the three-dimensional atomic structure of a flexible macromolecule from a heterogeneous cryogenic electron microscopy (cryo-EM) dataset. The dataset contains noisy tomographic projections of the electrostatic potential of the macromolecule, taken from different viewing directions, and in the heterogeneous case, each cryo-EM image corresponds to a different conformation of the macromolecule. Under the assumption that the macromolecule can be modelled as a chain, or discrete curve (as it is for instance the case for a protein backbone with a single chain of amino-acids), we introduce a method to estimate the deformation of the atomic model with respect to a given conformation, which is assumed to be known a priori. Our method consists on estimating the torsion and bond angles of the atomic model in each conformation as a linear combination of the eigenfunctions of the Laplace operator in the manifold of conformations. These eigenfunctions can be approximated by means of a well-known technique in manifold learning, based on the construction of a graph Laplacian using the cryo-EM dataset. Finally, we test our approach with synthetic datasets, for which we recover the atomic model of two-dimensional and three-dimensional flexible structures from simulated cryo-EM images.
48.	Gris, Barbara, et al. (författare) Image reconstruction through metamorphosis 2020 Ingår i: Inverse Problems. - : IOP PUBLISHING LTD. - 0266-5611 .- 1361-6420. ; 36:2 Tidskriftsartikel (refereegranskat)abstract The paper describes a method for reconstructing an image from noisy and indirect observations by registering, via metamorphosis, a template. The image registration part consists of two components, one is a geometric deformation that moves intensities without changing them and the other that changes intensity values. Unlike a registration with only geometrical deformation, this framework gives good results also when intensities of the template are poorly chosen. It also allows for appearance of a new structure. The approach is applicable to general inverse problems in imaging and we prove existence, stability and convergence, which implies that the method is a well-defined regularisation method. We also present several numerical examples from tomography.
49.	Pouchol, Camille, et al. (författare) The ML-EM algorithm in continuum : sparse measure solutions 2020 Ingår i: Inverse Problems. - : Institute of Physics (IOP). - 0266-5611 .- 1361-6420. ; 36:3 Tidskriftsartikel (refereegranskat)abstract Linear inverse problems A mu = y with Poisson noise and non-negative unknown mu > 0 are ubiquitous in applications, for instance in positron emission tomography (PET) in medical imaging. The associated maximum likelihood problem is routinely solved using an expectation-maximisation algorithm (ML-EM). This typically results in images which look spiky, even with early stopping. We give an explanation for this phenomenon. We first regard the image mu as a measure. We prove that if the measurements y are not in the cone {A mu, mu > 0}, which is typical of low injected dose, likelihood maximisers must be sparse, i.e., typically a sum of point masses. We also show a weak sparsity result for cluster points of ML-EM. On the other hand, in the low noise regime, we prove that cluster points of ML-EM are optimal measures with full support. Finally, we provide concentration bounds for the probability to be in the sparse case, and a set of numerical experiments supporting our claims.
50.	Purisha, Zenith, et al. (författare) Probabilistic approach to limited-data computed tomography reconstruction 2019 Ingår i: Inverse Problems. - : IOP Publishing. - 0266-5611 .- 1361-6420. ; 35:10 Tidskriftsartikel (refereegranskat)

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