A subgradient method with constant step-size for ℓ1\ell_1-composite optimization

Subgradient methods are the natural extension to the non-smooth case of the classical gradient descent for regular convex optimization problems. However, in general, they are characterized by slow convergence rates, and they require decreasing step-sizes to converge. In this paper we propose a subgradient method with constant step-size for composite convex objectives with $\ell_1$-regularization. If the smooth term is strongly convex, we can establish a linear convergence result for the function values. This fact relies on an accurate choice of the element of the subdifferential used for the update, and on proper actions adopted when non-differentiability regions are crossed. Then, we propose an accelerated version of the algorithm, based on conservative inertial dynamics and on an adaptive restart strategy, that is guaranteed to achieve a linear convergence rate in the strongly convex case. Finally, we test the performances of our algorithms on some strongly and non-strongly convex examples.Comment: 18 pages, 3 figures. Minor changes, extended bibliographical references, new example in Section

Modeling Ventricular Excitation : axial and orthotropic anisotropy effects on wavefronts and potentials.

By applying the eikonal approximation to the bidomain model of the cardiac tissue we investigate the influence of the axially isotropic and orthotropic conductivity tensors on the propagation of the excitation wave fronts and on the associated potential distribution and electrograms

Computational modeling of the electromechanical response of a ventricular fiber affected by eccentric hypertrophy

AbstractThe aim of this work is to study the effects of eccentric hypertrophy on the electromechanics of a single myocardial ventricular fiber by means of a one-dimensional finite-element strongly-coupled model. The electrical current ow model is written in the reference configuration and it is characterized by two geometric feedbacks, i.e. the conduction and convection ones, and by the mechanoelectric feedback due to stretchactivated channels. First, the influence of such feedbacks is investigated for both a healthy and a hypertrophic fiber in case of isometric simulations. No relevant discrepancies are found when disregarding one or more feedbacks for both fibers. Then, all feedbacks are taken into account while studying the electromechanical responses of fibers. The results from isometric tests do not point out any notable difference between the healthy and hypertrophic fibers as regards the action potential duration and conduction velocity. The length-tension relationships show increased stretches and reduced peak values for tension instead. The tension-velocity relationships derived from afterloaded isotonic and quick- release tests depict higher values of contraction velocity at smaller afterloads. Moreover, higher maximum shortenings are achieved during the isotonic contraction. In conclusion, our simulation results are innovative in predicting the electromechanical behavior of eccentric hypertrophic fibers

Cardiac kinematic parameters computed from video of in situ beating heart

Mechanical function of the heart during open-chest cardiac surgery is exclusively monitored by echocardiographic techniques. However, little is known about local kinematics, particularly for the reperfused regions after ischemic events. We report a novel imaging modality, which extracts local and global kinematic parameters from videos of in situ beating hearts, displaying live video cardiograms of the contraction events. A custom algorithm tracked the movement of a video marker positioned ad hoc onto a selected area and analyzed, during the entire recording, the contraction trajectory, displacement, velocity, acceleration, kinetic energy and force. Moreover, global epicardial velocity and vorticity were analyzed by means of Particle Image Velocimetry tool. We validated our new technique by i) computational modeling of cardiac ischemia, ii) video recordings of ischemic/reperfused rat hearts, iii) videos of beating human hearts before and after coronary artery bypass graft, and iv) local Frank-Starling effect. In rats, we observed a decrement of kinematic parameters during acute ischemia and a significant increment in the same region after reperfusion. We detected similar behavior in operated patients. This modality adds important functional values on cardiac outcomes and supports the intervention in a contact-free and non-invasive mode. Moreover, it does not require particular operator-dependent skills

On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model

We study the Γ -convergence of a family of vectorial integral functionals, which are the sum of a vanishing anisotropic quadratic form in the gradients and a penalizing double-well potential depending only on a linear combination of the components of their argument. This particular feature arises from the study of the so-called 'bidomain model' for the cardiac electric field; one of its consequences is that the L1-norm of a minimizing sequence can be unbounded and therefore a lack of coercivity occurs. We characterize the Γ -limit as a surface integral functional, whose integrand is a convex function of the normal and can be computed by solving a localized minimization problem

A parallel solver for reaction-diffusion systems in computational electrocardiology

In this work, a parallel three-dimensional solver for numerical simulations in com- putational electrocardiology is introduced and studied. The solver is based on the anisotropic Bidomain cardiac model, consisting of a system of two degenerate parabolic reaction–diffusion equations describing the intra and extracellular potentials of the myocardial tissue. This model includes intramural fiber rotation and anisotropic con- ductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. The solver also includes the simpler anisotropic Monodomain model, consisting of only one reaction–diffusion equation. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations that can vary from the simple FitzHugh–Nagumo (FHN) model to the more complex phase-I Luo–Rudy model (LR1). The solver employs structured isoparametric Q1 finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library. Large-scale computations with up to O(107 ) unknowns have been run on parallel computers, simulating excitation and repolarization phenomena in three-dimensional domains

Parallel solution of cardiac reaction-diffusion models

Summary. We present and study a parallel iterative solver for reaction-diffusion systems in three dimensions arising in computational electrocardiology, such as the Bidomain and Monodomain models. The models include intramural fiber rotation and anisotropic conductivity coefficients that can be fully orthotropic or axially symmetric around the fiber direction. These cardiac models are coupled with a membrane model for the ionic currents, consisting of a system of ordinary differential equations. The solver employs structured isoparametric Q1 finite elements in space and a semi-implicit adaptive method in time. Parallelization and portability are based on the PETSc parallel library and large-scale computations with up to O(10 7) unknowns have been run on parallel computers. These simulation of the full Bidomain model (without operator or variable splitting) for a full cardiac cycle are, to our knowledge, among the most complete in the available literature. 1 The cardiac Bidomain and Monodomain models Cardiac tissue is traditionally modeled as an arrangement of cardiac fibers that rotate counterclockwise from the epicardium to the endocardium, (se