This paper is concerned with the numerical approximation of the minimizer of the continuous Rudin-Osher-Fatemi (ROF) model for image denoising. A new discrete total variation is proposed and the associated Hilbertian total variation denoising model is used to construct continuous piecewise linear functions that approximate the minimizer of the ROF model in the strong topology of L 2 (Ω), provided that the data function is bounded and weakly regular in the sense of Lip(α, L 2 (Ω)).

A detailed biophysical model for a neuron/astrocyte network is developed to explore mechanisms responsible for the initiation and propagation of cortical spreading depolarizations and the role of astrocytes in maintaining ion homeostasis, thereby preventing these pathological waves. Simulations of the model illustrate how properties of spreading depolarizations, such as wave speed and duration of depolarization, depend on several factors, including the neuron and astrocyte Na(+)-K(+) ATPase pump strengths. In particular, we consider the neuroprotective role of astrocyte gap junction coupling. The model demonstrates that a syncytium of electrically coupled astrocytes can maintain a physiological membrane potential in the presence of an elevated extracellular K(+) concentration and efficiently distribute the excess K(+) across the syncytium. This provides an effective neuroprotective mechanism for delaying or preventing the initiation of spreading depolarizations.

In this paper we discuss what is known about the classification of symmetry groups and patterns of phase-shift synchrony for periodic solutions of coupled cell networks. Specifically, we compare the lists of spatial and spatiotemporal symmetries of periodic solutions of admissible vector fields to those of equivariant vector fields in the three cases of R n (coupled equations), T n (coupled oscillators), and (R k) n where k ≥ 2 (coupled systems). To do this we use the H/K Theorem of Buono and Golubitsky applied to coupled equations and coupled systems and prove the H/K theorem in the case of coupled oscillators. Josić and Török [17] prove that the H/K lists for equivariant vector fields and admissible vector fields are the same for transitive coupled systems. We show that the corresponding theorem is false for coupled equations. We also prove that the pairs of subgroups H ⊃ K for coupled equations are contained in the pairs for coupled oscillators which are contained in the pairs for coupled systems. Finally, we prove that patterns of rigid phase-shift synchrony for coupled equations are contained in those of coupled oscillators and those of coupled systems.

Total variation smoothing methods have been proven to be very efficient at discriminating between structures (edges and textures) and noise in images. Recently, it was shown that such methods do not create new discontinuities and preserve the modulus of continuity of functions. In this paper, we propose a Galerkin-Ritz method to solve the Rudin-Osher-Fatemi image denoising model where smooth bivariate spline functions on triangulations are used as approximating spaces. Using the extension property of functions of bounded variation on Lipschitz domains, we construct a minimizing sequence of continuous bivariate spline functions of arbitrary degree, d, for the TV-L 2 energy functional and prove the convergence of the finite element solutions to the solution of the Rudin, Osher, and Fatemi model. Moreover, an iterative algorithm for computing spline minimizers is developed and the convergence of the algorithm is proved.

We present a numerical method for solving the Monge-Ampère equation based on the characterization of the solution of the Dirichlet problem as the minimizer of a convex functional of the gradient and under convexity and nonlinear constraints. When the equation is discretized with a certain monotone scheme, we prove that the unique minimizer of the discrete problem solves the finite difference equation. For the numerical results we use both the standard finite difference discretization and the monotone scheme. Results with standard tests confirm that the numerical approximations converge to the Aleksandrov solution.