Model-based elastography is fraught with problems owing to the ill-posed nature of the inverse elasticity problem. To overcome this limitation, we have recently developed a novel inversion scheme that incorporates a priori information concerning the mechanical properties of the underlying tissue structures, and the variance incurred during displacement estimation in the modulus image reconstruc...

The generalized singular value decomposition (GSVD) often is used to solve Tikhonov regularization problems with a regularization matrix without exploitable structure. This paper describes how the standard methods for the computation of the GSVD of a matrix pair can be simplified in the context of Tikhonov regularization. Also, other regularization methods, including truncated GSVD, are conside...

We propose a regularization method for solving ill-posed problems, under the assumption that the solutions are piecewise constant functions with unknown level sets and unknown level values. A level set framework is established for the inverse problem and a Tikhonov regularization approach is proposed. Existence of generalized minimizers for the Tikhonov functional is proven. Moreover, we establ...

We analyze a multiple level-set method for solving inverse problems with piecewise constant solutions. This method corresponds to an iterated Tikhonov method for a particular Tikhonov functional Gα based on TV–H 1 penalization. We define generalized minimizers for our Tikhonov functional and establish an existence result. Moreover, we prove convergence and stability results of the proposed Tikh...

The problem of convergence of a special form of the generalized pulsespectrum technique (GPST) for solving inverse problems of one-dimensional diffusion equations in space-time domain is considered. Under the assumptions that a Tikhonov regularized solution exists and the derivative operator of the regularized forward problem at the regularized solution is invertible, the iterative solutions of...

The intention of this paper is to show the applicability of a generalized conditional gradient method for the minimization of Tikhonov-type functionals, which occur in the regularization of nonlinear inverse problems with sparsity constraints. We consider functionals of Tikhonov type where the usual quadratic penalty term is replaced by the pth power of a weighted p-norm. First of all, we analy...

We consider Tikhonov regularization of large linear discrete ill-posed problems with a regularization operator of general form and present an iterative scheme based on a generalized Krylov subspace method. This method simultaneously reduces both the matrix of the linear discrete ill-posed problem and the regularization operator. The reduced problem so obtained may be solved, e.g., with the aid ...

This paper introduces two new algorithms, belonging to the class of Arnoldi-Tikhonov regularization methods, which are particularly appropriate for sparse reconstruction. The main idea is to consider suitable adaptively-defined regularization matrices that allow the usual 2-norm regularization term to approximate a more general regularization term expressed in the p-norm, p ≥ 1. The regularizat...