Sergey Lapin

A kinematically coupled time-splitting scheme for fluid–structure interaction in blood flow

Abstract We present a new time-splitting scheme for the numerical simulation of fluid–structure interaction between blood flow and vascular walls. This scheme deals in a successful way with the problem of the added mass effect. The scheme is modular and it embodies the stability properties of implicit schemes at the low computational cost of loosely coupled ones.

Explicit algorithms to solve a class of state constrained parabolic optimal control problems

Abstract We consider an optimal control problem of a system governed by a linear parabolic equation with the following features: control is distributed, observation is either distributed or final, there are constraints on the state function and on its time derivative. Iterative solution methods are proposed and investigated for the finite difference approximations of these optimal control problems. Due to explicit in time approximation of the state equation and the appropriate choice of the preconditioners in the iterative methods, the implementation of all constructed methods is carried out by explicit formulae. Computational experiments confirm the theoretical results.

Easily implementable iterative methods for variational inequalities with nonlinear diffusion–convection operator and constraints to the gradient of solution

Abstract New iterative solution methods are proposed for the finite element approximation of a class of variational inequalities with nonlinear diffusion-convection operator and constraints to the gradient of solution. Implementation of every iteration of these methods reduces to the solution of a system of linear equations and a set of two-dimensional minimization problems. Convergence is proved by the application of a general result on the convergence of the iterative methods for a nonlinear constrained saddle point problem.

Computer-aided ophthalmic artery waveform analysis in healthy individuals and glaucoma patients

We thank Michael Brodsky for his interest in our study and for drawing our attention to a paper by Brodsky and Klaehn that examined optokinetic responses in humans with infantile esotropia using what the authors labeled an ‘‘optokinetic uncover test.’’ We commend Brodsky and Klaehn on the observations and conclusions that they draw from their study. However, we would like to point out some key differences between their study and our study. A major difference is in the construction and delivery of the stimulus. Brodsky and Klaehn presented the patients with an optokinetic stimulus, first monocularly to the fixating eye and then binocularly. Further, the stimulus itself was full field, that is, not localized to any part of the visual field of the deviated eye. In our study, we utilized a dichoptic presentation in which the optokinetic stimulus is presented only to the deviated eye (never binocularly) and the fixating eye sees only a stationary target. Moreover, the optokinetic stimulation is restricted to a 108 patch, and we observed optokinetic nystagmus (OKN) only when the patch occupied the central 108 of the deviated eye that included the fovea. We also made quantitative evaluations of the strength of the OKN response and its gradation with spatial location and contrast of the OKN stimulus, whereas the Brodsky and Klaehn study was essentially a qualitative evaluation of the OKN response. Therefore, in our view, the two studies are not directly comparable. However, our results do agree with one of the conclusions that they reached, which was that information is being processed via both eyes. In his letter, Brodsky does not question our results but perhaps suggests two additional discussion points with regard to our study: (1) The OKN responses that we observed are driven by a subcortical optokinetic circuit, that is, no cortical involvement, and (2) cortical suppression of the fovea of the deviating eye might still have been present because the OKN leaked through via the subcortical pathway. In response to his two points, we make the following arguments. (1) In strabismus, nasotemporal asymmetry is observed in motion detection, visually evoked potential (VEP) response, smooth pursuit, and OKN. Neurophysiological investigation in strabismic monkeys has shown loss of binocularity in cortical areas V1, MT, MST, and also in brainstem area nucleus of the optic tract (NOT). Therefore, as proposed by models in the literature, the loss of binocular connections in the pathway from V1 MT MST NOT could lead to asymmetric visual or oculomotor response to monocular motion stimuli. Subcortical projections (direct retina–NOT projections) may also play a role, but it is not clear that they play an exclusive or primary role in generating nasalward OKN in strabismus. (2) Our study was focused on identifying areas of retinal suppression in strabismus, and OKN was simply used as a readout to identify suppressed versus unsuppressed retina. Our data and conclusions fit in nicely with the previous work in the literature. For instance, Economides, Adams, and Horton used a visual psychophysical paradigm, and showed that the fovea of the deviated eye was not suppressed. In addition, other work from our lab in which we used a saccade paradigm to examine spatial patterns of fixation switch behavior (presumably driven by suppression) also revealed similar results. Taken together with these other studies, our current study does indeed support the idea of lack of suppression of the fovea of the deviated eye in exotropia. Finally, we would like to point out that the discussion points above are fundamentally speculative because neither Brodsky’s previous work nor our current study directly examined neural responses, and therefore cannot ascribe the optokinetic responses to cortical/subcortical pathways. As always in science, the best way to settle the issue would be to design and perform an appropriate experiment and quantitatively evaluate the data.

Non-overlapping domain decomposition method for a variational inequality with gradient constraints

Non-overlapping domain decomposition method is applied to a variational inequality with nonlinear diffusion-convection operator and gradient constraints. The method is based on the initial approximation of the problem and its subsequent splitting into subproblems. For the resulting constrained saddle point problem block relaxation-Uzawa iterative solution method is applied.

On the Iterative Solution Methods for Finite-Dimensional Inclusions with Applications to Optimal Control Problems

Abstract Iterative methods for finite-dimensional inclusions which arise in applying a finite-element or a finite-difference method to approximate state-constrained optimal control problems have been investigated. Specifically, problems of control on the right- hand side of linear elliptic boundary value problems and observation in the entire domain have been considered. The convergence and the rate of convergence for the iterative algorithms based on the finding of the control function or Lagrange multipliers are proved.

A penalty approach to the numerical simulation of a constrained wave motion

o The main goal of this article is to investigate the numerical solution of a vector-valued nonlinear wave equation, the nonlinearity being of the Ginzburg-Landau type, namely (j~ uj 2 1)~ u. This equation is obtained when treating by penalty a constrained wave-motion, where the displace- ment vector is of constant length (1 here, after rescaling). An important step of the approximation process is the construction of a time discretization scheme preserving - in some sense - the energy conservation property of the continuous model.The stability properties of the above scheme are dis- cussed. The authors discuss also the nite element approximation and the quasi-Newton solution of the nonlinear elliptic system obtained at each time step from the time discretization. The results of numerical experiments are presented; they show that for the constraint of the original wave problem to be accurately veried we need to use a small value of the penalty parameter.