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In this paper, we consider a Levenbergâ€"Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gâteaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (2016 Numer. Math. 133 655â€"684) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the non-existing Gâteaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example. [ABSTRACT FROM AUTHOR]

In this paper we study the inverse boundary value problem for the Schrödinger equation with a potential and the conductivity equation using partial Cauchy data. We derive stability estimates for these inverse problems. [ABSTRACT FROM AUTHOR]

In this paper, the problem of uniqueness concerning the inverse scattering problem in two-dimensional acoustics for one incident plane wave and one wavenumber is considered. Using the fact that the optimal lower estimate for the eigenvalues of the Laplacian for a domain is given by the Faber–Krahn inequality, which relates the area of the domain to the first eigenvalue of a disc of equal area, it is proved that the uniqueness holds under the restriction that the possible scatterers do not deviate ‘too much’ in area. Also an improvement of the results due to Colton and Sleeman (1983 IMA J. Appl. Math. 31 253–9) is presented, based on the a priori information that the unknown scatterers lie inside a given ball and that the far field is known for a finite number of incident plane waves. The main advantage of this work is that it provides uniqueness for the half number of the needed incoming waves in Colton and Sleeman (1983 IMA J. Appl. Math. 31 253–9). For the case of one incoming plane wave uniqueness is satisfied if the scatterers are contained in a ball of radius R such that kR < t10 = 4.4939, where t10 is the first root of the spherical Bessel function of first order j1(x). The result of local uniqueness is applied to a class of star-shaped scatterers which are smooth perturbations of discs with common centre in [inline image] for one incident plane-wave direction. Numerical implementations are presented for smooth perturbations of discs. [ABSTRACT FROM AUTHOR]

In applications of imaging science, such as emission tomography, fluorescence microscopy and optical/infrared astronomy, image intensity is measured via the counting of incident particles (photons, g-rays, etc). Fluctuations in the emission-counting process can be described by modeling the data as realizations of Poisson random variables (Poisson data). A maximum-likelihood approach for image reconstruction from Poisson data was proposed in the mid-1980s. Since the consequent maximization problem is, in general, ill-conditioned, various kinds of regularizations were introduced in the framework of the so-called Bayesian paradigm. A modification of the well-known Tikhonov regularization strategy results in the data-fidelity function being a generalized Kullback-Leibler divergence. Then a relevant issue is to find rules for selecting a proper value of the regularization parameter. In this paper we propose a criterion, nicknamed discrepancy principle for Poisson data, that applies to both denoising and deblurring problems and fits quite naturally the statistical properties of the data. The main purpose of the paper is to establish conditions, on the data and the imaging matrix, ensuring that the proposed criterion does actually provide a unique value of the regularization parameter for various classes of regularization functions. A few numerical experiments are performed to demonstrate its effectiveness. More extensive numerical analysis and comparison with other proposed criteria will be the object of future work. [ABSTRACT FROM AUTHOR]

This paper is concerned with the inverse problem of scattering of time-harmonic electromagnetic waves by an impenetrable obstacle buried in the lower half-space of a two-layered background medium. We prove that the buried obstacle and its physical property can be uniquely determined from a knowledge of the tangential component of the electric field on some two-dimensional device in the upper half-space corresponding to all incident electric dipoles located on another two-dimensional device in the upper half-space with two different polarizations. The key ingredient of our proof is a novel reciprocity relation established in this paper for the solution of the scattering problem of the electric dipole located at two different source points. [ABSTRACT FROM AUTHOR]

This paper gives a note on an application of the enclosure method to an inverse obstacle scattering problem governed by the Helmholtz equation in two dimensions. It is shown that one can uniquely determine the convex hull of an unknown sound-hard polygonal obstacle from the trace of the total wave that was exerted by a single point source onto a known circle surrounding the obstacle provided the source is sufficiently far from the obstacle. The result contains a formula that extracts the value of the support function of the obstacle at a generic direction. Some other applications to thin obstacles, obstacles in a layered medium and the far-field equation in the linear sampling method are also included. [ABSTRACT FROM AUTHOR]

In this paper, we prove Lipschitz stability results for inverse source problems relative to parabolic equations. We use the method introduced by Imanuvilov and Yamamoto in 1998 based on Carleman estimates. What is new here is that we study a class of one-dimensional degenerate parabolic equations. In our model, the diffusion coefficient vanishes at one extreme point of the domain. Instead of the classical Carleman estimates obtained by Fursikov and Imanuvilov for non degenerate equations, we use and extend some recent Carleman estimates for degenerate equations obtained by Cannarsa, Martinez and Vancostenoble. Finally, we obtain Lipschitz stability results in inverse source problems for our class of degenerate parabolic equations both in the case of a boundary observation and in the case of a locally distributed observation. [ABSTRACT FROM AUTHOR]

In this paper, we describe two stable methods for the inversion of the Abel integral operator of the first kind. The first method is based on the use of appropriate families of orthonormal polynomials of Jacobi type that constitute orthonormal bases of the L2([0, 1])-space. We prove that this first method for solving the ill-posed problem described by the first kind of Abel integral equation is stable. This stability is proved by using some properties of our orthogonal polynomials as well as using a convenient weighted subspace of L2([0, 1]). The second method is based on expanding the solution with respect to the basis of Legendre polynomials. More precisely, by using a special technique for the computation of the Legendre expansion coefficients of the solution, we prove that our second method is stable under weak conditions on the smoothness of the solution as well as on the data function. Also, we give a brief review of a modified Thikhonov method for solving the ill-posed Abel problem and we show how our second proposed method fits in the modified Tikhonov regularization scheme. Finally, we provide the reader with some numerical tests that illustrate the results of this work. [ABSTRACT FROM AUTHOR]

Let fλ be the regularized solution for the problem of estimating a function or vector ƒ0 from noisy data yi = Liƒ0 + εi, i = l, ...,n, where Li are linear functionals. A prominent method for the selection of the crucial regularization parameter λ is generalized cross-validation (GCV). It is known that GCV has good asymptotic properties as n→∞but it may not be reliable for small or medium sized n, sometimes giving an estimate that is far too small. We propose a new robust GCV method (RGCV) which chooses λ to be the minimizer of γV (λ) + (1 − γ )F(λ), where V (λ) is the GCV function, F(λ) is an approximate average measure of the influence of each data point on fλ and γ ∈ (0, 1) is a robustness parameter. We show that for any n, RGCV is less likely than GCV to choose a very small value of λ, resulting in a more robust method. We also show that RGCV has good asymptotic properties as n→∞ for general linear operator equations with uncorrelated errors. The function EF(λ) approximates the risk ER(λ) for values of λ that are asymptotically a bit smaller than the minimizer of ER(λ) (where V (λ) may not approximate well). The ‘expected’ RGCV estimate is asymptotically optimal as n → ∞ with respect to the ‘robust risk’ γ ER(λ) + (1 − γ )v(λ), where v(λ) is the variance component of the risk, and it has the optimal decay rate with respect to ER(λ) and stronger error criteria. The GCV and RGCV methods are compared in numerical simulations for the problem of estimating the second derivative from noisy data. The results for RGCV with n = 51 are consistent with the asymptotic results, and, for a large range of γ values, RGCV is more reliable and accurate than GCV. [ABSTRACT FROM AUTHOR]

A new procedure is firstly proposed to construct soliton equations with self-consistent sources (SESCSs) in bilinear forms, starting from the Gram-type determinant solution or Gram-type Pfaffian solution of soliton equations without sources; the soliton solutions of SESCSs can then be given. This procedure is applied to the 2D Toda lattice equation, the discrete KP equation and the BKP equation. [ABSTRACT FROM AUTHOR]

The factorization method for inverse elliptic problems finds inclusions in a domain, given the Neumann-to-Dirichlet operator associated with the elliptic equation. In practice, one only knows an approximation of this data set in some finite-dimensional space. This work deals with the question of in which sense the series criterion of the factorization method can be extended to such situations. We show that regularization by spectral cut-off can be used to construct finite-dimensional factorization methods. [ABSTRACT FROM AUTHOR]

For a 2 × 2 reaction-diffusion system on a bounded domain, we give a simultaneous stability result for one coefficient and for the initial conditions. The key ingredient is a global Carleman-type estimate with a single observation acting on a subdomain. [ABSTRACT FROM AUTHOR]

We introduce the concept of generalized projection operators, i.e., projection integrals over a body in that generalize the usual result of projected area in a given direction by taking into account shadowing and scattering effects as well as additional convolution functions in the integral. Such operators arise naturally in connection with various observation instruments and data types. We review and discuss some properties of these operators and the related inverse problems, particularly in the cases pertaining to photometric and radar data. We also prove an ambiguity theorem for a special observing geometry common in astrophysics, and uniqueness theorems for radar inverse problems of a spherical target. These theorems are obtained by employing the intrinsic rotational properties of the observing geometries and function representations. We then present examples of the mathematical modelling of the shape and rotation state of a body by simultaneously using complementary data sources corresponding to different generalized projection operators. We show that generalized projection operators unify a number of mathematical considerations and physical observation types under the same concept. [ABSTRACT FROM AUTHOR]

We study one of the multidimensional inverse scattering problems for quantum systems governed by the Stark Hamiltonians. By applying the time-dependent method developed by Enss and Weder (1995 J. Math. Phys. 36 3902–21), we prove that the high-velocity limit of the scattering operator uniquely determines the short-range interaction potentials. Moreover, we prove that, when a long-range interaction potential is given, the high-velocity limit of the Dollard-type modified scattering operator uniquely determines the short-range part of the interactions. We allow the potential functions to belong to very broad classes. These results are improvements on the previous results obtained by Adachi and Maehara (2007 J. Math. Phys. 48 042101; Adachi et al 2013 Inverse Problems29 085012). [ABSTRACT FROM AUTHOR]

A demanding challenge in Bayesian inversion is to efficiently characterize the posterior distribution. This task is problematic especially in high-dimensional non-Gaussian problems, where the structure of the posterior can be very chaotic and difficult to analyse. Current inverse problem literature often approaches the problem by considering suitable point estimators for the task. Typically the choice is made between the maximum a posteriori (MAP) or the conditional mean (CM) estimate. The benefits of either choice are not well-understood from the perspective of infinite-dimensional theory. Most importantly, there exists no general scheme regarding how to connect the topological description of a MAP estimate to a variational problem. The recent results by Dashti and others (Dashti et al 2013 Inverse Problems29 095017) resolve this issue for nonlinear inverse problems in Gaussian framework. In this work we improve the current understanding by introducing a novel concept called the weak MAP (wMAP) estimate. We show that any MAP estimate in the sense of Dashti et al (2013 Inverse Problems29 095017) is a wMAP estimate and, moreover, how the wMAP estimate connects to a variational formulation in general infinite-dimensional non-Gaussian problems. The variational formulation enables to study many properties of the infinite-dimensional MAP estimate that were earlier impossible to study. In a recent work by the authors (Burger and Lucka 2014 Maximum a posteriori estimates in linear inverse problems with logconcave priors are proper bayes estimators preprint) the MAP estimator was studied in the context of the Bayes cost method. Using Bregman distances, proper convex Bayes cost functions were introduced for which the MAP estimator is the Bayes estimator. Here, we generalize these results to the infinite-dimensional setting. Moreover, we discuss the implications of our results for some examples of prior models such as the Besov prior and hierarchical prior. [ABSTRACT FROM AUTHOR]