On a Nonlinear Model for Tumor Growth in a Cellular Medium

We investigate the dynamics of a nonlinear model for tumor growth within a cellular medium. In this setting the “tumor” is viewed as a multiphase flow consisting of cancerous cells in either proliferating phase or quiescent phase and a collection of cells accounting for the “waste” and/or dead cells in the presence of a nutrient. Here, the tumor is thought of as a growing continuum \(\Omega \) with boundary \(\partial \Omega \) both of which evolve in time. In particular, the evolution of the boundary \(\partial \Omega \) is prescibed by a given velocity \({{{\varvec{V}}}.}\) The key characteristic of the present model is that the total density of cancerous cells is allowed to vary, which is often the case within cellular media. We refer the reader to the articles (Enault in Mathematical study of models of tumor growth, 2010; Li and Lowengrub in J Theor Biol, 343:79–91, 2014) where compressible type tumor growth models are investigated. Global-in-time weak solutions are obtained using an approach based on penalization of the boundary behavior, diffusion, viscosity and pressure in the weak formulation, as well as convergence and compactness arguments in the spirit of Lions (Mathematical topics in fluid dynamics. Compressible models, 1998) [see also Donatelli and Trivisa (J Math Fluid Mech 16: 787–803, 2004), Feireisl (Dynamics of viscous compressible fluids, 2014)].     

Iterated Tikhonov regularization with a general penalty term

Tikhonov regularization is one of the most popular approaches to solving linear discrete ill‐posed problems. The choice of the regularization matrix may significantly affect the quality of the computed solution. When the regularization matrix is the identity, iterated Tikhonov regularization can yield computed approximate solutions of higher quality than (standard) Tikhonov regularization. This paper provides an analysis of iterated Tikhonov regularization with a regularization matrix different from the identity. Computed examples illustrate the performance of this method.     

Fast deconvolution with approximated PSF by RSTLS with antireflective boundary conditions

The problem of reconstructing signals and images from degraded ones is considered in this paper. The latter problem is formulated as a linear system whose coefficient matrix models the unknown point spread function and the right hand side represents the observed image. Moreover, the coefficient matrix is very ill-conditioned, requiring an additional regularization term. Different boundary conditions can be proposed. In this paper antireflective boundary conditions are considered. Since both sides of the linear system have uncertainties and the coefficient matrix is highly structured, the Regularized Structured Total Least Squares approach seems to be the more appropriate one to compute an approximation of the true signal/image. With the latter approach the original problem is formulated as an highly nonconvex one, and seldom can the global minimum be computed. It is shown that Regularized Structured Total Least Squares problems for antireflective boundary conditions can be decomposed into single variable subproblems by a discrete sine transform. Such subproblems are then transformed into one-dimensional unimodal real-valued minimization problems which can be solved globally. Some numerical examples show the effectiveness of the proposed approach.     

Image deblurring by sparsity constraint on the Fourier coefficients

This paper is concerned with the image deconvolution problem. For the basic model, where the convolution matrix can be diagonalized by discrete Fourier transform, the Tikhonov regularization method is computationally attractive since the associated linear system can be easily solved by fast Fourier transforms. On the other hand, the provided solutions are usually oversmoothed and other regularization terms are often employed to improve the quality of the restoration. Of course, this weighs down on the computational cost of the regularization method. Starting from the fact that images have sparse representations in the Fourier and wavelet domains, many deconvolution methods have been recently proposed with the aim of minimizing the ?₁-norm of these transformed coefficients. This paper uses the iteratively reweighted least squares strategy to introduce a diagonal weighting matrix in the Fourier domain. The resulting linear system is diagonal and hence the regularization parameter can be easily estimated, for instance by the generalized cross validation. The method benefits from a proper initial approximation that can be the observed image or the Tikhonov approximation. Therefore, embedding this method in an outer iteration may yield further improvement of the solution. Finally, since some properties of the observed image, like continuity or sparsity, are obviously changed when working in the Fourier domain, we introduce a filtering factor which keeps unchanged the large singular values and preserves the jumps in the Fourier coefficients related to the low frequencies. Numerical examples are given in order to show the effectiveness of the proposed method.     

An Iterative Multigrid Regularization Method for Toeplitz Discrete Ill-Posed Problems

Iterative regularization multigrid methods have been successfully applied to signal/image deblurring problems. When zero-Dirichlet boundary conditions are imposed the deblurring matrix has a Toeplitz structure and it is potentially full. A crucial task of a multilevel strategy is to preserve the Toeplitz structure at the coarse levels which can be exploited to obtain fast computations. The smoother has to be an iterative regularization method. The grid transfer operator should preserve the regularization property of the smoother. This paper improves the iterative multigrid method proposed in [11] introducing a wavelet soft-thresholding denoising post-smoother. Such post-smoother avoids the noise amplification that is the cause of the semi-convergence of iterative regularization methods and reduces ringing effects. The resulting iterative multigrid regularization method stabilizes the iterations so that the imprecise (over) estimate of the stopping iteration does not have a deleterious effect on the computed solution. Numerical examples of signal and image deblurring problems confirm the effectiveness of the proposed method.     

Magnetic Cobalt Ferrite Nanocrystals For an Energy Storage Concentration Cell

Energy‐storage concentration cells are based on the concentration gradient of redox‐active reactants; the increased entropy is transformed into electric energy as the concentration gradient reaches equilibrium between two half cells. A recyclable and flow‐controlled magnetic electrolyte concentration cell is now presented. The hybrid inorganic–organic nanocrystal‐based electrolyte, consisting of molecular redox‐active ligands adsorbed on the surface of magnetic nanocrystals, leads to a magnetic‐field‐driven concentration gradient of redox molecules. The energy storage performance of concentration cells is dictated by magnetic characteristics of cobalt ferrite nanocrystal carriers. The enhanced conductivity and kinetics of redox‐active electrolytes could further induce a sharp concentration gradient to improve the energy density and voltage switching of magnetic electrolyte concentration cells.     

On the condition number of the antireflective transform

Deconvolution problems with a finite observation window require appropriate models of the unknown signal in order to guarantee uniqueness of the solution. For this purpose it has recently been suggested to impose some kind of antireflectivity of the signal. With this constraint, the deconvolution problem can be solved with an appropriate modification of the fast sine transform, provided that the convolution kernel is symmetric. The corresponding transformation is called the antireflective transform. In this work we determine the condition number of the antireflective transform to first order, and use this to show that the so-called reblurring variant of Tikhonov regularization for deconvolution problems is a regularization method. Moreover, we establish upper bounds for the regularization error of the reblurring strategy that hold uniformly with respect to the size n of the algebraic system, even though the condition number of the antireflective transform grows with n. We briefly sketch how our results extend to higher space dimensions.     

Fast transforms for high order boundary conditions in deconvolution problems

We study strategies to increase the precision in deconvolution models, while maintaining the complexity of the related numerical linear algebra procedures (matrix-vector product, linear system solution, computation of eigenvalues, etc.) of the same order of the celebrated fast Fourier transform. The key idea is the choice of a suitable functional basis to represent signals and images. Starting from an analysis of the spectral decomposition of blurring matrices associated to the antireflective boundary conditions introduced in Serra Capizzano (SIAM J. Sci. Comput. 25(3):1307–1325, 2003), we extend the model for preserving polynomials of higher degree and fast computations also in the nonsymmetric case.     

Synthesis of 5-methyl-1,3,2-benzodithiazoles

The novel helerocycles 5-methyl-1,3,2-benzodithiazoles (7) were prepared in 30–50% yields from toluene-3,4-dithiol (6) and appropriate primary amines in the presence of 2 equivalents of triethylamine under high-dilution conditions. These compounds, which exhibit one reversible oxidation potential around +0.90 V vs. SCE, serve as a model study in the quest of synthesizing a new donor, the “boron-nitrogen” analog of tetrathiafulvalene (BNTTF).