Limited memory restarted l(p)-l(q) minimization methods using generalized Krylov subspaces

Regularization of certain linear discrete ill-posed problems, as well as of certain regression problems, can be formulated as large-scale, possibly nonconvex, minimization problems, whose objective function is the sum of the p(th) power of the l(p)-norm of a fidelity term and the qth power of the lq-norm of a regularization term, with 0 < p,q = 2. We describe new restarted iterative solution methods that require less computer storage and execution time than the methods described by Huang et al. (BIT Numer. Math. 57,351-378, 14). The reduction in computer storage and execution time is achieved by periodic restarts of the method. Computed examples illustrate that restarting does not reduce the quality of the computed solutions

Generalized cross validation for ℓ p-ℓ q minimization

Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence of noise in the data often makes it difficult to compute an accurate approximate solution. To reduce the sensitivity of the computed solution to the noise, one replaces the original problem by a nearby well-posed minimization problem, whose solution is less sensitive to the noise in the data than the solution of the original problem. This replacement is known as regularization. We consider the situation when the minimization problem consists of a fidelity term, that is defined in terms of a p-norm, and a regularization term, that is defined in terms of a q-norm. We allow 0 < p,q ≤ 2. The relative importance of the fidelity and regularization terms is determined by a regularization parameter. This paper develops an automatic strategy for determining the regularization parameter for these minimization problems. The proposed approach is based on a new application of generalized cross validation. Computed examples illustrate the performance of the method proposed

An Arnoldi-based preconditioner for iterated Tikhonov regularization

Many problems in science and engineering give rise to linear systems of equations that are commonly referred to as large-scale linear discrete ill-posed problems. These problems arise, for instance, from the discretization of Fredholm integral equations of the first kind. The matrices that define these problems are typically severely ill-conditioned and may be rank-deficient. Because of this, the solution of linear discrete ill-posed problems may not exist or be very sensitive to perturbations caused by errors in the available data. These difficulties can be reduced by applying Tikhonov regularization. We describe a novel "approximate Tikhonov regularization method" based on constructing a low-rank approximation of the matrix in the linear discrete ill-posed problem by carrying out a few steps of the Arnoldi process. The iterative method so defined is transpose-free. Our work is inspired by a scheme by Donatelli and Hanke, whose approximate Tikhonov regularization method seeks to approximate a severely ill-conditioned block-Toeplitz matrix with Toeplitz-blocks by a block-circulant matrix with circulant-blocks. Computed examples illustrate the performance of our proposed iterative regularization method

Range restricted iterative methods for linear discrete ill-posed problems

Linear systems of equations with a matrix whose singular values decay to zero with increasing index number, and without a significant gap, are commonly referred to as linear discrete ill-posed problems. Such systems arise, e.g., when discretizing a Fredholm integral equation of the first kind. The right-hand side vectors of linear discrete ill-posed problems that arise in science and engineering often represent an experimental measurement that is contaminated by measurement error. The solution to these problems typically is very sensitive to this error. Previous works have shown that error propagation into the computed solution may be reduced by using specially designed iterative methods that allow the user to select the subspace in which the approximate solution is computed. Since the dimension of this subspace often is quite small, its choice is important for the quality of the computed solution. This work describes algorithms for three iterative methods that modify the GMRES, block GMRES, and global GMRES methods for the solution of appropriate linear systems of equations. We contribute to the work already available on this topic by introducing two block variants for the solution of linear systems of equations with multiple right-hand side vectors. The dominant computational aspects are discussed, and software for each method is provided. Additionally, we illustrate the utility of these iterative subspace methods through numerical examples focusing on image reconstruction. This paper is accompanied by software

Theoretical and numerical aspects of a non-stationary preconditioned iterative method for linear discrete ill-posed problems

This work considers some theoretical and computational aspects of the recent paper (Buccini et al., 2021), whose aim was to relax the convergence conditions in a previous work by Donatelli and Hanke, and thereby make the iterative method discussed in the latter work applicable to a larger class of problems. This aim was achieved in the sense that the iterative method presented convergences for a larger class of problems. However, while the analysis presented is correct, it does not establish the superior behavior of the iterative method described. The present note describes a slight modification of the analysis that establishes the superiority of the iterative method. The new analysis allows to discuss the behavior of the algorithm when varying the involved parameters, which is also useful for their empirical estimation

SARS-CoV-2 and extracellular vesicles: An intricate interplay in pathogenesis, diagnosis and treatment

Extracellular vesicles (EVs) are widely recognized as intercellular communication mediators. Among the different biological processes, EVs play a role in viral infections, supporting virus entrance and spread into host cells and immune response evasion. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection became an urgent public health issue with significant morbidity and mortality worldwide, being responsible for the current COVID-19 pandemic. Since EVs are implicated in SARS-CoV-2 infection in a morphological and functional level, they have gained growing interest for a better understanding of SARS-CoV-2 pathogenesis and represent possible diagnostic tools to track the disease progression. Furthermore, thanks to their biocompatibility and efficient immune activation, the use of EVs may also represent a promising strategy for the development of new therapeutic strategies against COVID-19. In this review, we explore the role of EVs in viral infections with a focus on SARS-CoV-2 biology and pathogenesis, considering recent morphometric studies. The common biogenesis aspects and structural similarities between EVs and SARS-CoV-2 will be examined, offering a panoramic of their multifaceted interplay and presenting EVs as a machinery supporting the viral cycle. On the other hand, EVs may be exploited as early diagnostic biomarkers and efficient carriers for drug delivery and vaccination, and ongoing studies will be reviewed to highlight EVs as potential alternative therapeutic strategies against SARS-CoV-2 infection

A comparison of parameter choice rules for ℓp - ℓq minimization

Images that have been contaminated by various kinds of blur and noise can be restored by the minimization of an ℓp-ℓq functional. The quality of the reconstruction depends on the choice of a regularization parameter. Several approaches to determine this parameter have been described in the literature. This work presents a numerical comparison of known approaches as well as of a new one

An Efficient Implementation of the Gauss-Newton Method Via Generalized Krylov Subspaces

The solution of nonlinear inverse problems is a challenging task in numerical analysis. In most cases, this kind of problems is solved by iterative procedures that, at each iteration, linearize the problem in a neighborhood of the currently available approximation of the solution. The linearized problem is then solved by a direct or iterative method. Among this class of solution methods, the Gauss-Newton method is one of the most popular ones. We propose an efficient implementation of this method for large-scale problems. Our implementation is based on projecting the nonlinear problem into a sequence of nested subspaces, referred to as Generalized Krylov Subspaces, whose dimension increases with the number of iterations, except for when restarts are carried out. When the computation of the Jacobian matrix is expensive, we combine our iterative method with secant (Broyden) updates to further reduce the computational cost. We show convergence of the proposed solution methods and provide a few numerical examples that illustrate their performance

Large-scale regression with non-convex loss and penalty

We describe a computational method for parameter estimation in linear regression, that is capable of simultaneously producing sparse estimates and dealing with outliers and heavy-tailed error distributions. The method used is based on the image restoration method proposed in Huang et al. (2017) [13]. It can be applied to problems of arbitrary size. The choice of certain parameters is discussed. Results obtained for simulated and real data are presented