In the last 30 days

• Open/close Left and right preconditioning for electrical impedance tomography with structural information

  Daniela Calvetti et al 2012 Inverse Problems 28 055015 Tag this article

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  A common problem in computational inverse problems is to find an efficient way of solving linear or nonlinear least-squares problems. For large-scale problems, iterative solvers are the method of choice for solving the associated linear systems, and for nonlinear problems, an additional effective local linearization method is required. In this paper, we discuss an efficient preconditioning scheme for Krylov subspace methods, based on the Bayesian analysis of the inverse problem. The model problem to which we apply this methodology is electrical impedance tomography (EIT) augmented with prior information coming from a complementary modality, such as x-ray imaging. The particular geometry considered here models the x-ray-guided EIT for breast imaging. The interest in applying EIT concurrently with x-ray breast imaging arises from the experimental observation that the impedivity spectra of certain types of malignant and benign tissues differ significantly from each other, thus offering a possibility of diagnosis without more invasive tissue sampling. After setting up the EIT inverse problem within a Bayesian framework, we present an inner and outer iteration scheme for computing a maximum a posteriori estimate. The prior covariance provides a right preconditioner and the modeling error covariance provides a left preconditioner for the iterative method used to solve the linear least-squares problem at each outer iteration of the optimization problem. Moreover, the stopping criterion for the inner iterations is coupled with the progress of the solution of the outer iteration. Besides the preconditioning scheme, the computational efficiency relies on a very efficient method to compute the Jacobian, obtained by carefully organizing the forward computation. Computed examples illustrate the robustness and computational efficiency of the proposed algorithm.

• Open/close A numerical differentiation method and its application to reconstruction of discontinuity

  Y B Wang et al 2002 Inverse Problems 18 1461 Tag this article

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  In this paper, we discuss a classical ill-posed problem—numerical differentiation by the Tikhonov regularization. Based on the conditional stability estimate for this ill-posed problem, a new simple method for choosing regularization parameters is proposed. We show that it has an almost optimal convergence rate when the exact solution is in H ². The advantages of our method are (1) we can get similar computational results with much less computation, in comparison with other methods, and (2) we can find the discontinuous points numerically.

• Open/close Analysis of the Hessian for inverse scattering problems: I. Inverse shape scattering of acoustic waves

  Tan Bui-Thanh and Omar Ghattas 2012 Inverse Problems 28 055001 Tag this article

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  We derive expressions for the shape Hessian operator of the data misfit functional corresponding to the inverse problem of inferring the shape of a scatterer from reflected acoustic waves, using a Banach space setting and the Lagrangian approach. The shape Hessian is then analyzed in both Hölder and Sobolev spaces. Using an integral equation approach and compact embeddings in Hölder and Sobolev spaces, we show that the shape Hessian can be decomposed into four components, of which the Gauss–Newton part is a compact operator, while the others are not. Based on the Hessian analysis, we are able to express the eigenvalues of the Gauss–Newton Hessian as a function of the smoothness of the shape space, which shows that the smoother the shape is, the faster the decay rate. Analytical and numerical examples are presented to validate our theoretical results. The implication of the compactness of the Gauss–Newton Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated shape.

• Open/close Regularization methods for ill-posed problems in multiple Hilbert scales

  Gisela L Mazzieri and Ruben D Spies 2012 Inverse Problems 28 055005 Tag this article

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  Several convergence results in Hilbert scales under different source conditions are proved and orders of convergence and optimal orders of convergence are derived. Also, relations between those source conditions are proved. The concept of a multiple Hilbert scale on a product space is introduced, and regularization methods on these scales are defined, both for the case of a single observation and for the case of multiple observations. In the latter case, it is shown how vector-valued regularization functions in these multiple Hilbert scales can be used. In all cases, convergence is proved and orders and optimal orders of convergence are shown. Finally, some potential applications and open problems are discussed.

• Open/close Analysis of the Hessian for inverse scattering problems: II. Inverse medium scattering of acoustic waves

  Tan Bui-Thanh and Omar Ghattas 2012 Inverse Problems 28 055002 Tag this article

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  We address the inverse problem for scattering of acoustic waves due to an inhomogeneous medium. We derive and analyze the Hessian in both Hölder and Sobolev spaces. Using an integral equation approach based on Newton potential theory and compact embeddings in Hölder and Sobolev spaces, we show that the Hessian can be decomposed into two components, both of which are shown to be compact operators. Numerical examples are presented to validate our theoretical results. The implication of the compactness of the Hessian is that for small data noise and model error, the discrete Hessian can be approximated by a low-rank matrix. This in turn enables fast solution of an appropriately regularized inverse problem, as well as Gaussian-based quantification of uncertainty in the estimated inhomogeneity.

• Open/close Strong robust generalized cross-validation for choosing the regularization parameter

  Mark A Lukas 2008 Inverse Problems 24 034006 Tag this article

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  Let f _λ be the Tikhonov regularized solution of a linear inverse or smoothing problem with discrete noisy data y _i, i = 1, ..., n. To choose λ we propose a new strong robust GCV method denoted by R ₁GCV that is part of a family of such methods, including the basic RGCV method. R ₁GCV chooses λ to be the minimizer of γ V(λ) + (1 − γ) F ₁(λ), where V(λ) is the GCV function, F ₁(λ) is a certain approximate total measure of the influence of each data point on f _λ with respect to the regularizing norm or seminorm, and γ (0, 1) is a robustness parameter. We show that R ₁GCV is less likely to choose a very small value of λ than both GCV and RGCV. RGCV and R ₁GCV also have good asymptotic properties for general problems with independent errors. Strengthening previous results for RGCV, it is shown that the (shifted) RGCV and R ₁GCV functions are consistent estimates of 'robust risk' functions, which place extra weight on the variance of f _λ. In addition RGCV is asymptotically equivalent to the modified GCV method. The results of numerical simulations for R ₁GCV are consistent with the asymptotic results, and, for suitable values of γ, R ₁GCV is more reliable and accurate than GCV.

• Open/close Determining the shape of defects in non-absorbing inhomogeneous media from far-field measurements

  Y Grisel et al 2012 Inverse Problems 28 055003 Tag this article

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  We consider non-absorbing inhomogeneous media represented by some refraction index. We have developed a method to reconstruct, from far-field measurements, the shape of the areas where the actual index differs from a reference index. Following the principle of the factorization method, we present a fast reconstruction algorithm relying on far-field measurements and near-field values, easily computed from the reference index. Our reconstruction result is illustrated by several numerical test cases.

• Open/close Reconstruction in the three-dimensional parallel scanning geometry with application in synchrotron-based x-ray tomography

  B Hahn and A K Louis 2012 Inverse Problems 28 045013 Tag this article

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  Common reconstruction approaches in the three-dimensional parallel scanning geometry, that occurs for example in synchrotron-based x-ray tomography, use layerwise 2D procedures. A finite number of layers of the investigated object are reconstructed by a 2D method, before a 3D visualization is obtained by interpolation between these layers. In this paper, a fully three-dimensional reconstruction algorithm for the parallel scanning geometry is presented. It allows us to calculate arbitrary slices or even the complete 3D density function directly without causing any interpolation errors. Furthermore, the 3D method enables the fast and stable combination of reconstruction and image analysis step to extract features of the investigated object, for example, derivatives used in canny edge detectors. The application to real data illustrates the stability and advantageous regularization properties of the new approach.

• Open/close Sparsity and incoherence in compressive sampling

  Emmanuel Candès and Justin Romberg 2007 Inverse Problems 23 969 Tag this article

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  We consider the problem of reconstructing a sparse signal from a limited number of linear measurements. Given m randomly selected samples of Ux ⁰, where U is an orthonormal matrix, we show that ℓ ₁ minimization recovers x ⁰ exactly when the number of measurements exceeds
  
  
  
  where S is the number of nonzero components in x ⁰ and μ is the largest entry in U properly normalized: . The smaller μ is, the fewer samples needed. The result holds for 'most' sparse signals x ⁰ supported on a fixed (but arbitrary) set T. Given T, if the sign of x ⁰ for each nonzero entry on T and the observed values of Ux ⁰ are drawn at random, the signal is recovered with overwhelming probability. Moreover, there is a sense in which this is nearly optimal since any method succeeding with the same probability would require just about as many samples.

• Open/close Estimation of position and intensity of a pollutant source in channel flow using transmittance functions

  T Maalej et al 2012 Inverse Problems 28 055010 Tag this article

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  This paper considers the one-dimension linear advection diffusion equation with constant coefficients for a transient pollutant point source. Theoretical transmittances deduced from the analytical solution of the transport equation (state-space), that is, transfer functions between concentrations, have been calculated. The convolution product between transmittances obtained by Laplace inversion and a reference concentration for channel flow application allows us to estimate the parameters of the model (velocity and diffusion coefficient). A nonlinear fixed point method based on least-squares estimation based on noisy concentration responses and using truncated singular value decomposition regularization is implemented to determinate gaseous contaminant source locations and strengths.