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The non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 ( α ⩾ β ⩾ 0 ) regularization is a new approach for sparse recovery. A minimizer of the α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 regularized function can be computed by applying the ST-(αℓ 1 − βℓ 2) algorithm which is similar to the classical iterative soft thresholding algorithm (ISTA). It is known that ISTA converges quite slowly, and a faster alternative to ISTA is the projected gradient (PG) method. However, the conventional PG method is limited to solve problems with the classical ℓ 1 sparsity regularization. In this paper, we present two accelerated alternatives to the ST-(αℓ 1 − βℓ 2) algorithm by extending the PG method to the non-convex α ‖ ⋅ ‖ ℓ 1 − β ‖ ⋅ ‖ ℓ 2 sparsity regularization. Moreover, we discuss a strategy to determine the radius R of the ℓ 1-ball constraint by Morozov’s discrepancy principle. Numerical results are reported to illustrate the efficiency of the proposed approach.

This paper presents a novel regularization with a non-convex, non-smooth term of the form with parameters to solve ill-posed linear problems with sparse solutions. We investigate the existence, stability and convergence of the regularized solution. It is shown that this type of regularization is well-posed and yields sparse solutions. Under an appropriate source condition, we get the convergence rate in the -norm for a priori and a posteriori parameter choice rules, respectively. A numerical algorithm is proposed and analyzed based on an iterative threshold strategy with the generalized conditional gradient method. We prove the convergence even though the regularization term is non-smooth and non-convex. The algorithm can easily be implemented because of its simple structure. Some numerical experiments are performed to test the efficiency of the proposed approach. The experiments show that regularization with performs better in comparison with the classical sparsity regularization and can be used as an alternative to the regularizer.

The Cauchy problem for an elliptic partial differential equation is ill-posed. In this paper, we study a numerical method for solving the Cauchy problem. The numerical method is based on a reformulation of the Cauchy problem through an optimal control approach coupled with a regularization term which is included to treat the severe ill-conditioning of the corresponding discretized formulation. We prove convergence of the numerical method and present theoretical results for the limiting behaviors of the numerical solution as the regularization parameter approaches zero. Results from some numerical examples are reported.

Bioluminescence tomography (BLT) is a rapidly developing new area of molecular imaging. The goal of BLT is to produce a quantitative reconstruction of a bioluminescent source distribution within a living mouse from bioluminescent signals measured on the body surface of the mouse. While in most BLT studies so far the optical parameters of the key anatomical regions are assumed known from the literature or diffuse optical tomography (DOT), these parameters cannot be very accurate in general. In this paper, we propose and study a new BLT approach that optimizes optical parameters when an underlying bioluminescent source distribution is reconstructed to match the measured data. We prove the solution existence and the convergence of numerical methods. Also, we present numerical results to illustrate the utility of our approach and evaluate its performance.

Molecular imaging is widely recognized as the main stream in the next generation of biomedical imaging. Bioluminescence tomography (BLT) is a rapidly developing new area of molecular imaging. The goal of BLT is to provide quantitative three-dimensional reconstruction of a bioluminescent source distribution within a small animal from optical signals on the surface of the animal body. In this paper, a mathematical framework is established for BLT. Solution existence and uniqueness are established. Continuous dependence of the solution is demonstrated with respect to data. Stable BLT schemes are studied, leading to error estimates and convergence of the methods. A numerical example is presented to illustrate the algorithmic performance.

In reflection seismology, knowing the seismic wavelet is important both for processing seismic data and for modeling the seismic response. There are two approaches to obtain the seismic wavelet. One approach is deterministic and the other is statistic. This work belongs to the second category. A seismic wavelet is determined by the product of amplitude spectrum and phase spectrum. A conventional method uses a two-step procedure to estimate the seismic wavelet. The first step is for the amplitude spectrum, and the second step is for the phase spectrum. So extracting the amplitude spectrum of a seismic wavelet (ASSW) from the amplitude spectrum of a seismic trace is a key step. The commonly used methods are correlation-based method, the log-spectrum-averaging method and spectrum-shaping method. All these methods assume the reflection coefficient sequence is white, which may not be valid under some conditions. In this paper, we propose a new approach to obtain ASSW without the whiteness assumption about reflectivity. We define an operator on a properly chosen function space and prove that the operator is contractive in this space. Then, we convert the problem of estimating the ASSW into one of finding the fixed point of the operator. We give the algorithm in detail based on the contraction operator mapping (COM method). We compare our method with the widely used methods by synthetic signals in which the reflectivity is not white. The results show that our method performs satisfactorily for nonwhite reflection series, on which other methods do not work well. Moreover, our method is robust for the noise and the frequency interval on which COM works.

In this paper, we consider an inverse source problem for elliptic partial differential equations with Dirichlet and Neumann boundary data. The unknown source term is to be determined from additional boundary conditions. Unlike the existing methods found in the literature, which usually use some of the boundary conditions to form a boundary value problem for the elliptic partial differential equation and the remaining boundary conditions in the objective functional for optimization to determine the source term, the novel method that we propose here has coupled complex boundary conditions. We use a complex elliptic partial differential equation with a Robin boundary condition coupling the Dirichlet and Neumann boundary data, and optimize with respect to the imaginary part of the solution in the domain to determine the source term. Then, on the basis of the complex boundary value problem, Tikhonov regularization is used to obtain a stable approximate source function and the finite element method is used for discretization. Theoretical analysis is given for both the continuous model and the discrete model. Several numerical examples are provided to show the usefulness of the proposed coupled complex boundary method.

This paper provides a theoretical study of reconstructing absorption and scattering coefficients based on the radiative transport equation (RTE) by using the total variation regularization method. The function space for solutions of the RTE is a natural one from the form of the boundary value problem of the RTE. We analyze the continuity and differentiability of the forward operator. We then show that the total variation regularization method can be applied for a stable solution. Convergence of the total variation-minimizing solution in the sense of the Bregman distance is also obtained.

A regularization method is proposed for the function reconstruction from noisy local averages in any dimension. Error bounds for the approximate solution in L2-norm are derived. A number of numerical examples are provided to show computational performance of the method, with the regularization parameters selected by different strategies.