Volume 26, Issue 2

In this paper we apply the multilevel augmentation method to solve an ill-posed integral equation via the iterated Lavrentiev regularization. This method leads to fast solutions of discrete iterated Lavrentiev regularization. The convergence rates of the iterated Lavrentiev regularization are achieved by using a certain parameter choice strategy. Finally, numerical experiments are given to illustrate the efficiency of the method.

In this paper, the iterative reweighted least squares (IRLS) algorithm for sparse signal recovery with partially known support is studied. We establish a theoretical analysis of the IRLS algorithm by incorporating some known part of support information as a prior, and obtain the error estimate and convergence result of this algorithm. Our results show that the error bound depends on the best (s+k){(s+k)}-term approximation and the regularization parameter λ, and convergence result depends only on the regularization parameter λ. Finally, a series of numerical experiments are carried out to demonstrate the effectiveness of the algorithm for sparse signal recovery with partially known support, which shows that an appropriate q (0<q<1{0<q<1}) can lead to a better recovery performance than that of the case q=1{q=1}.

The aim of this paper is to discuss potential advances in PET kinetic models and direct reconstruction of kinetic parameters. As a prominent example we focus on a typical task in perfusion imaging and derive a system of transport-reaction-diffusion equations, which is able to include macroscopic flow properties in addition to the usual exchange between arteries, veins, and tissues. For this system we propose an inverse problem of estimating all relevant parameters from PET data. We interpret the parameter identification as a nonlinear inverse problem, for which we formulate and analyze variational regularization approaches. For the numerical solution we employ gradient-based methods and appropriate splitting methods, which are used to investigate some test cases.

We study the inverse problem of determining the magnetic field and the electric potential appearing in the magnetic Schrödinger equation from the knowledge of a finite number of lateral observations of the solution. We prove a Lipschitz stability estimate for both coefficients simultaneously by choosing the “initial” conditions suitably.

We consider the inverse Stefan type free boundary problem, where information on the boundary heat flux and the density of the sources are missing and must be found along with the temperature and the free boundary. We pursue the optimal control framework analyzed in [1, 2], where the boundary heat flux, the density of the sources, and the free boundary are components of the control vector. We prove the Frechet differentiability in Besov spaces, and derive the formula for the Frechet differential under minimal regularity assumptions on the data. The result implies a necessary condition for optimal control and opens the way to the application of projective gradient methods in Besov spaces for the numerical solution of the inverse Stefan problem.

Acoustic impedance (AI) inversion is a desirable tool to extract rock-physical properties from recorded seismic data. It plays an important role in seismic interpretation and reservoir characterization. When one of recursive inversion schemes is employed to obtain the AI, the spatial coherency of the estimated reflectivity section may be damaged through the trace-by-trace processing. Meanwhile, the results are sensitive to noise in the data or inaccuracies in the generated reflectivity function. To overcome the above disadvantages, in this paper, we propose a data-driven inversion scheme to directly invert the AI from seismic reflection data. We first explain in principle that the anisotropic total variation (ATV) regularization is more suitable for inverting the impedance with sharp interfaces than the total variation (TV) regularization, and then establish the nonlinear objective function of the AI model by using anisotropic total variation (ATV) regularization. Next, we solve the nonlinear impedance inversion problem via the alternating split Bregman iterative algorithm. Finally, we illustrate the performance of the proposed method and its robustness to noise with synthetic and real seismic data examples by comparing with the conventional methods.

Full waveform inversion is a large-scale nonlinear and ill-posed problem. We consider applying the regularization technique for full waveform inversion with structure constraints. The structure information was extracted with difference operators with respect to model parameters. And then we establish an lp{l_{p}}-lq{l_{q}}-norm constrained minimization model for different choices of parameters p and q . To solve this large-scale optimization problem, a fast gradient method with projection onto convex set and a multiscale inversion strategy are addressed. The regularization parameter is estimated adaptively with respect to the frequency range of the data. Numerical experiments on a layered model and a benchmark SEG/EAGE overthrust model are performed to testify the validity of this proposed regularization scheme.

This paper intends to develop practical marching schemes for Cauchy problems of the Helmholtz equation in laterally varying waveguides. We arrive at a stable representation of the marching solutions in waveguides. Based on the representation, a second-order marching scheme is then constructed to eliminate the ill-conditioning and compute the wave propagation in waveguides with laterally variable mediums. In the end, extensive experiments are implemented to verify the efficiency and accuracy of the marching scheme in various waveguides, and we also point out the application scope of the scheme.

Variational source conditions proved to be useful for deriving convergence rates for Tikhonov’s regularization method and also for other methods. Up to now, such conditions have been verified only for few examples or for situations which can be also handled by classical range-type source conditions. Here we show that for almost every ill-posed inverse problem variational source conditions are satisfied. Whether linear or nonlinear, whether Hilbert or Banach spaces, whether one or multiple solutions, variational source conditions are a universal tool for proving convergence rates.

We consider different concepts of well-posedness and ill-posedness and their relations for solving nonlinear and linear operator equations in Hilbert spaces. First, the concepts of Hadamard and Nashed are recalled which are appropriate for linear operator equations. For nonlinear operator equations, stable respective unstable solvability is considered, and the properties of local well-posedness and ill-posedness are investigated. Those two concepts consider stability in image space and solution space, respectively, and both seem to be appropriate concepts for nonlinear operators which are not onto and/or not, locally or globally, injective. Several example situations for nonlinear problems are considered, including the prominent autoconvolution problems and other quadratic equations in Hilbert spaces. It turns out that for linear operator equations, well-posedness and ill-posedness are global properties valid for all possible solutions, respectively. The special role of the nullspace is pointed out in this case. Finally, non-injectivity also causes differences in the saturation behavior of Tikhonov and Lavrentiev regularization of linear ill-posed equations. This is examined at the end of this study.

We consider a distributed system of a type which is encountered in the study of diffusion processes with memory and in viscoelasticity. The key feature of such a system is the persistence in the future of the past actions due the memory described via a certain relaxation kernel; see below. The parameters of the kernel have to be inferred from experimental measurements. Our main result in this paper is that by using two boundary measurements, the identification of a relaxation kernel which is a linear combination of Abel kernels (as often assumed in applications) can be reduced to the solution of a (linear) deconvolution problem.