In the last 30 days

• Open/close An algorithm for total variation regularization in high-dimensional linear problems

  Michel Defrise et al 2011 Inverse Problems 27 065002 Tag this article

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  This paper describes an iterative algorithm for high-dimensional linear inverse problems, which is regularized by a differentiable discrete approximation of the total variation (TV) penalty. The algorithm is an interlaced iterative method based on optimization transfer with a separable quadratic surrogate for the TV penalty. The surrogate cost function is optimized using the block iterative regularized algebraic reconstruction technique (RSART). A proof of convergence is given and convergence is illustrated by numerical experiments with simulated parallel-beam computerized tomography (CT) data. The proposed method provides a block-iterative and convergent, hence efficient and reliable, algorithm to investigate the effects of TV regularization in applications such as CT.

• Open/close Comparison of optimal design methods in inverse problems

  H T Banks et al 2011 Inverse Problems 27 075002 Tag this article

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  Typical optimal design methods for inverse or parameter estimation problems are designed to choose optimal sampling distributions through minimization of a specific cost function related to the resulting error in parameter estimates. It is hoped that the inverse problem will produce parameter estimates with increased accuracy using data collected according to the optimal sampling distribution. Here we formulate the classical optimal design problem in the context of general optimization problems over distributions of sampling times. We present a new Prohorov metric-based theoretical framework that permits one to treat succinctly and rigorously any optimal design criteria based on the Fisher information matrix. A fundamental approximation theory is also included in this framework. A new optimal design, SE-optimal design ( standard error optimal design), is then introduced in the context of this framework. We compare this new design criterion with the more traditional D-optimal and E-optimal designs. The optimal sampling distributions from each design are used to compute and compare standard errors; the standard errors for parameters are computed using asymptotic theory or bootstrapping and the optimal mesh. We use three examples to illustrate ideas: the Verhulst–Pearl logistic population model (Banks H T and Tran H T 2009 Mathematical and Experimental Modeling of Physical and Biological Processes (Boca Raton, FL: Chapman and Hall/CRC)), the standard harmonic oscillator model (Banks H T and Tran H T 2009) and a popular glucose regulation model (Bergman R N, Ider Y Z, Bowden C R and Cobelli C 1979 Am. J. Physiol. 236 E667–77; De Gaetano A and Arino O 2000 J. Math. Biol. 40 136–68; Toffolo G, Bergman R N, Finegood D T, Bowden C R and Cobelli C 1980 Diabetes 29 979–90).

• Open/close Reconstruction of the interface between two-layered media using far field measurements

  Lahcène Chorfi and Patricia Gaitan 2011 Inverse Problems 27 075001 Tag this article

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  We consider an inverse transmission scattering problem. This problem consists in determining an interface between two-layered media by far field measurements. We prove that the interface is uniquely determined by the measurements of the far field pattern associated with incoming plane waves at a fixed frequency. For the reconstruction of the interface we solve a nonlinear integral equation using a truncated Newton conjugate gradient algorithm.

• Open/close Linear convergence rates for Tikhonov regularization with positively homogeneous functionals

  Markus Grasmair 2011 Inverse Problems 27 075014 Tag this article

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  The goal of this paper is the formulation of an abstract setting that can be used for the derivation of linear convergence rates for a large class of sparsity promoting regularization functionals for the solution of ill-posed linear operator equations. Examples where the proposed setting applies include joint sparsity and group sparsity, but also (possibly higher order) discrete total variation regularization. In all of these cases, a range condition together with some kind of restricted injectivity imply linear convergence rates. That is, the error in the approximate solution, measured in terms of the regularization functional, is of the same order as the noise level.

• Open/close Convergence rates for total variation regularization of coefficient identification problems in elliptic equations I

  Dinh Nho Hào and Tran Nhan Tam Quyen 2011 Inverse Problems 27 075008 Tag this article

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  We investigate the convergence rates for total variation regularization of the problem of identifying (i) the coefficient q in the Neumann problem for the elliptic equation , and (ii) the coefficient a in the Neumann problem for the elliptic equation , when u is imprecisely given by z ^δ in . We regularize these problems by correspondingly minimizing the convex functionals
  
  
  
  

  and
  
  
  
  

  over the admissible sets, where U( q) ( U( a)) is the solution of the first (second) Neumann boundary value problem; ρ > 0 is the regularization parameter. Taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems, we obtain the convergence rates of them to a total variation-minimizing solution in the sense of the Bregman distance under relatively simple source conditions without the smallness requirement on the source functions.

• Open/close An adaptive finite element reconstruction of distributed fluxes

  Jingzhi Li et al 2011 Inverse Problems 27 075009 Tag this article

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  Based on a posteriori error estimates, we propose an adaptive finite element method for a distributed flux reconstruction in a diffusion system, recovering the unknown distributed flux on some inaccessible boundary using partial measurement data on part of the accessible boundary. A posteriori error estimates are first derived, and then the efficiency of the error estimator is addressed by showing that it provides upper and lower bounds on the discretization errors of the quantities of interest. Numerical experiments are presented to show the applicability and efficiency of the proposed adaptive method based on the derived error estimator, which provides a robust guidance for the adaptive refining of meshes to locate the singularities of the fluxes.

• Open/close Inverse problems in ion channel modelling

  Martin Burger 2011 Inverse Problems 27 083001 Tag this article

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  This paper reviews several inverse problems related to biological ion channels and polymer nanopores. Different problems arise at different scales and thus lead to different inversion tasks, which can be modelled as parameter identification in systems of partial differential equations, ordinary differential equations, or as the solution of integral equations. Due to the high potential for future developments, we will not only discuss the previous results and approaches, but in particular highlight a variety of open issues and unsolved inverse problems, as well as biological questions that could be modeled as inverse problems and might lead to interesting future developments.

• Open/close Dynamical inverse problem on a metric tree

  S A Avdonin et al 2011 Inverse Problems 27 075011 Tag this article

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  We consider the problem of reconstruction of the potential for the wave equation on a specific star graph using the dynamical Dirichlet-to-Neumann map. Our algorithm is based on the boundary control method. We reduce the problem of reconstruction to a second kind Fredholm integral equation, the kernel and the right-hand side of which arise from an auxiliary second kind Volterra integral equation. A second-order accurate numerical method for the equations is described and implemented. Then several numerical examples demonstrate that the algorithm works and can be used to reconstruct an unknown potential accurately.

• Open/close Contrast source inversion method applied to relatively high contrast objects

  Paul-André Barrière et al 2011 Inverse Problems 27 075012 Tag this article

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  This paper presents a contrast source inversion method for microwave tomography with improved performance with respect to the popular multiplicative regularized contrast source inversion (MR-CSI) method when the contrast of the object under study is relatively high. More precisely, some shortcomings of MR-CSI which may prevent it from performing adequately on higher contrast objects are outlined. The proposed method does not suffer from such shortcomings, but necessitates the tuning of parameters. However, a unique set of parameters can be used in a wide variety of experimental conditions related to a given application. Simple guidelines to determine their values are provided.

• Open/close Singular value decomposition for the truncated Hilbert transform: part II

  A Katsevich 2011 Inverse Problems 27 075006 Tag this article

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  Hilbert transform is a very important tool in computed tomography. Image reconstruction from truncated tomographic data can be reduced to the problem of inverting the Hilbert transform knowing ψ on the interval [ a ₂, a ₃], where a ₁ < a ₂ < a ₃ < a ₄. In this paper, we obtain a singular value decomposition for the operator .