the primal dual active set strategy as a semismooth newton method this paper addresses complementarity problems motivated by constrained optimal control problems it is shown that the primal dual active set strategy which is known to be extremely efficient for this class of problems and a specific semismooth newton method lead to identical algorithms the notion of slant differentiability is recalled and it is argued that the max function is slantly differentiable in lp spaces when appropriately combined with a two norm concept this leads to new local convergence results of the primal dual active set strategy global unconditional convergence results are obtained by means of appropriate merit functions

finite element approximations of a ladyzhenskaya model for stationary incompressible viscous flow some finite element approximation procedures are presented for a model proposed by ladyzhenskaya for stationary incompressible viscous flow the approximate problems are proved to be well posed and stable under standard assumptions on the finite element families the solutions of the approximate problems converge to the solution of the original problem under minimum regularity assumptions some error estimates are derived the optimal order of accuracy is assured with or even without using exact integration rules in the approximation procedure iterative methods for solving the discrete nonlinear problems and comments on some computational experiments are provided special attention is also paid to the common properties as well as differences between the approximation procedure presented here and the approximation for the stationary navier stokes equations

adaptivity in space and time for reaction diffusion systems in electrocardiology the paper introduces and studies numerical methods that are fully adaptive in both three dimensional 3d space and time to challenging multiscale cardiac reaction diffusion models in these methods temporal adaptivity comes via stepsize control in function space oriented linearly implicit time integration while spatial adaptivity is realized within multilevel finite element methods controlled by a posteriori local error estimators in contrast to other recent adaptivity approaches to cardiac modeling that discretize first in space and then in time so called method of lines our method discretizes first in time and then in space so called rothe method an approach that has already proven to be highly efficient in a number of challenging multiscale problems in science and technology kardos code library with this method the evolution of a complete heartbeat from the excitation to the recovery phase is simulated both in the frame of the anisotropic monodomain models and in the more realistic anisotropic bidomain models coupled with either a variant of the simple fitzhugh nagumo model or the more complex phase i luo rudy ionic model the numerical results exhibit a rather satisfactory performance of our adaptive method for complex cardiac reaction diffusion models on 3d domains up to moderate sizes in particular the method accurately resolves the evolution of the intra and extracellular potentials gating variables and ion concentrations during the excitation plateau and recovery phases

primal dual strategy for constrained optimal control problems an algorithm for efficient solution of control constrained optimal control problems is proposed and analyzed it is based on an active set strategy involving primal as well as dual variables for discretized problems sufficient conditions for convergence in finitely many iterations are given numerical examples are given and the role of the strict complementarity condition is discussed

variational bound on energy dissipation in plane couette flow we present numerical solutions to the extended doering constantin variational principle for upper bounds on the energy dissipation rate in turbulent plane couette flow using the compound matrix technique in order to reformulate this principle s spectral constraint we derive a system of equations that is amenable to numerical treatment in the entire range from low to asymptotically high reynolds numbers our variational bound exhibits a minimum at intermediate reynolds numbers and reproduces the busse bound in the asymptotic regime as a consequence of a bifurcation of the minimizing wavenumbers there exist two length scales that determine the optimal upper bound the effective width of the variational profile s boundary segments and the extension of their flat interior part

on the determination of the grad div criterion abstract grad div stabilization adding a term grad div u to penalize violation of incompressibility has proven to be a useful tool in the simulation of incompressible flows such a term requires a choice of the coefficient and studies have begun appearing with various suggestions for its value we give an analysis herein that provides a restricted range of possible values for the coefficient in 3d turbulent flows away from walls if u l denote the large scale velocity and length respectively and is the signal to noise ratio of the body force estimates suggest that should be restricted to the range 2 24 l u 2 4 r e l u mesh independent case 2 24 l u 2 4 h l 4 3 l u mesh dependent case

semi implicit time discretization schemes for the bidomain model the bidomain model is a system of partial differential equations used to model the propagation of electrical potential waves in the myocardium it is composed of coupled parabolic and elliptic partial differential equations as well as at least one ordinary differential equation to model the ion activity through the cardiac cell membranes the purpose of this paper is to propose and analyze several implicit semi implicit and explicit time stepping methods to solve that model in particular to avoid the expensive resolution of a nonlinear system through the newton raphson method we identify necessary stability conditions on the time step delta t for the proposed methods through a theoretical analysis based on energy estimates we next compare the methods for one and two dimensional test cases in terms of both stability and accuracy of the numerical solutions the theoretical stability conditions are seen to be consistent with those observed in practice our analysis allows us to recommend using either the crank nicolson adams bashforth method or the second order semi implicit backward differention method these semi implicit methods produce a good numerical solution unlike the explicit methods their stability does not depend on the spatial grid size and unlike the implicit methods they do not require the resolution of a system of nonlinear equations

on the finite element solution of the pure neumann problem this paper considers the finite element approximation and algebraic solution of the pure neumann problem our goal is to present a concise variational framework for the finite element solution of the neumann problem that focuses on the interplay between the algebraic and variational problems while many of the results that stem from our analysis are known by some experts they are seldom derived in a rigorous fashion and remain part of numerical folklore as a result this knowledge is not accessible or appreciated by many practitioners both novices and experts in one source our paper contributes a simple yet insightful link between the continuous and algebraic variational forms that will prove useful

variational bounds on shear driven turbulence and turbulent boussinesq convection abstract based on earlier studies by hopf 1941 doering and constantin 1992 1994 1995 have recently formulated a new background technique for obtaining upper bounds on turbulent fluid flow quantities this method produces upper bounds on the limit supremum of long time averages making no statistical assumptions about the flow in contrast to the well known howard busse approach the full optimisation problems posed by this method for the momentum transport in turbulent couette flow and the heat transport with zero background flow in turbulent boussinesq convection are solved here for the first time at asymptotically large reynolds number and rayleigh number within busse s multiple boundary layer approximation to extract the best lowest possible upper bounds available intriguingly the original bounds isolated by busse 1969 1970 within the confines of statistical stationarity are recovered exactly using this new formalism the optimal background velocity profile for turbulent couette flow is found to be shearless in the interior thus differing from busse s 1 4 mean shear result in the convective case an interesting degeneracy in the formulation of the background variational problem leads to an indeterminacy in the optimal background temperature profile only for one special choice is the isothermal core feature of busse s mean profile recovered

second fundamental measure of geometric sets and local approximation of curvatures using the theory of normal cycles we associate with each geometric subset of a riemannian manifold a tensor valued curvature measure which we call its second fundamental measure this measure provides a finer description of the geometry of singular sets than the standard curvature measures moreover we deal with approximation of curvature measures we get a local quantitative estimate of the difference between curvature measures of two geometric subsets when one of them is a smooth hypersurface

a 3d boundary optimal control for the bidomain bath system modeling the thoracic shock therapy for cardiac defibrillation this work is dedicated to study the cardiac defibrillation problem by using an optimal thoracic electroshock treatment the problem is formulated as an optimal control problem in a 3d domain surrounded by the bath and including the heart the control corresponds to the thoracic electroshock and the model describing the electrical activity in the heart is the bidomain model the bidomain model is coupled with the quasi static maxwell s equation to consider the effect of an external bathing medium the existence and uniqueness of a weak solution for the direct problem is assessed as well as the existence of a weak solution for the adjoint problem the numerical discretization is realized using a finite element method for the spatial discretization and linearly implicit runge kutta methods for the temporal discretization of the partial differential equations the numerical results are demonstrated for the termination of re entry waves

energy transfer and dissipation in forced isotropic turbulence a model for the reynolds number dependence of the dimensionless dissipation rate ce was derived from the dimensionless karman howarth equation resulting in ce ce c rl o 1 rl where rl is the integral scale reynolds number the coefficients c and ce arise from asymptotic expansions of the dimensionless second and third order structure functions this theoretical work was supplemented by direct numerical simulations dns of forced isotropic turbulence for integral scale reynolds numbers up to rl 5875 r 435 which were used to establish that the decay of dimensionless dissipation with increasing reynolds number took the form of a power law rln with exponent value n 1 000 0 009 and that this decay of ce was actually due to the increase in the taylor surrogate u3 l the model equation was fitted to data from the dns which resulted in the value c 18 9 1 3 and in an asymptotic value for ce in the infinite reynolds number limit of ce 0 468 0 006

primal dual active set strategy for large scale optimization of cardiac defibrillation a feasibility study of optimal control techniques for cardiac defibrillation on anatomical three spatial dimensional rabbit ventricle geometry in the presence of bilateral control constraints is presented the work addresses the numerical treatment of multi scale and multi domain simulations of the bidomain equations and is based on the primal dual active set method to solve the optimality system for this large scale optimization problem numerical results are presented for a successful defibrillation study robustness of the optimization algorithm w r t to variations in the model parameters is demonstrated a feasibility study for multiple small boundary control support is included as well finally the numerical convergence of the optimization algorithm and the parallel efficiency is demonstrated

numerical analysis for an optimal control of bidomain bath model this work is concerned with the study of the convergence analysis for an optimal control of bidomain bath model by using the finite element scheme the bidomain bath model equations describe the cardiac bioelectric activity at the tissue and bath volumes where the control acts at the boundary of the tissue domain we establish the existence of the finite element scheme and convergence of the unique weak solution of the direct bidomain bath model the convergence proof is based on deriving a series of a priori estimates and using a general l 2 compactness criterion moreover the well posedness of the adjoint problem and the first order necessary optimality conditions are shown comparing to the direct problem the convergence proof of the adjoint problem is based on using a general l 1 compactness criterion the numerical tests are demonstrated which achieve the successful cardiac defibrillation by utilizing less total current finally the robustness of the newton optimization algorithm is presented for different finer mesh geometries 1 introduction the electrical behavior of the cardiac tissue surrounded by a nonconductive bath is described by a coupled partial and ordinary differential equations which are so called bidomain model equations 17 22 24 the bidomain model equations consist of two parabolic partial differential equations pdes which describe the dynamics of the intra and the extracellular potentials the pdes coupled with an ordinary differential equations which model the ionic currents associated with the reaction terms furthermore an additional poisson problem has to be solved when the cardiac tissue is immersed in a conductive fluid e g tissue bath in an experimental context or a surrounding torso to model in vivo scenarios

approximation of stationary statistical properties of dissipative dynamical systems time discretization we consider temporal approximation of stationary statistical properties of dissipative infinite dimensional dynamical systems we demonstrate that stationary statistical properties of the time discrete approximations i e numerical scheme converge to those of the underlying continuous dissipative infinite dimensional dynamical system under three very natural assumptions as the time step approaches zero the three conditions that are sufficient for the convergence of the stationary statistical properties are 1 uniform dissipativity of the scheme in the sense that the union of the global attractors for the numerical approximations is pre compact in the phase space 2 convergence of the solutions of the numerical scheme to the solution of the continuous system on the unit time interval 0 1 uniformly with respect to initial data from the union of the global attractors 3 uniform continuity of the solutions to the continuous dynamical system on the unit time interval 0 1 uniformly for initial data from the union of the global attractors the convergence of the global attractors is established under weaker assumptions an application to the infinite prandtl number model for convection is discussed

kolmogorov theory via finite time averages abstract several relations from the kolmogorov theory of fully developed three dimensional turbulence are rigorously established for finite time averages over leray hopf weak solutions of the navier stokes equations the navier stokes equations are considered with periodic boundary conditions and an external forcing term the main parameter is the grashof number associated with the forcing term the relations rigorously proved in this article include estimates for the energy dissipation rate the kolmogorov wavenumber the taylor wavenumber the reynolds number and the energy cascade process for some estimates the averaging time depends on the macroscale wavenumber and the kinematic viscosity alone while for others such as the kolmogorov energy dissipation law and the energy cascade the estimates depend also on the grashof number as compared with earlier works by some of the authors the more physical concept of finite time average is replacing the concept of infinite time average used before

convergence of time averaged statistics of finite element approximations of the navier stokes equations when discussing numerical solutions of the navier stokes equations especially when turbulent flows are concerned there are at least two questions that can be raised what is meaningful to compute how to determine the fidelity of the computed solution with respect to the true solution this paper takes a step towards the answer of these questions for turbulent flows we consider long time averages of weak solutions of the navier stokes equations rather than strong solutions we present error estimates for the time averaged energy dissipation rate drag and lift most of them under the assumption of small reynolds generalized grashof number for shear flows we address the question of fidelity of the computed solution with respect to the true solution in view of kolmogorov s energy cascade theory

analysis of mesh effects on turbulent flow statistics abstract turbulence models such as the smagorinsky model herein are used to represent the energy lost from resolved to under resolved scales due to the energy cascade i e non linearity analytic estimates of the energy dissipation rates of a few turbulence models have recently appeared but none yet study the energy dissipation restricted to resolved scales i e after spatial discretization with h micro scale we do so herein for the smagorinsky model upper bounds are derived on the computed time averaged energy dissipation rate e u h for an under resolved mesh h for turbulent shear flow for coarse mesh size o r e 1 h l it is proven e u h c s h 2 l 5 c s 4 h l 5 2 c s 4 h 3 2 u 3 l where u and l are global velocity and length scale and c s and are model parameters this upper bound being independent of the viscosity at high reynolds number is in accord with the equilibrium dissipation law kolmogrov s conventional turbulence theory this estimate suggests over dissipation for any of c s 0 and 0 consistent with numerical evidence on the effects of model viscosity without wall damping function moreover the analysis indicates that the turbulent boundary layer is a more important length scale for shear flow than the kolmogorov microscale

a varifold approach to surface approximation we show that the theory of varifolds can be suitably enriched to open the way to applications in the field of discrete and computational geometry using appropriate regularizations of the mass and of the first variation of a varifold we introduce the notion of approximate mean curvature and show various convergence results that hold in particular for sequences of discrete varifolds associated with point clouds or pixel voxel type discretizations of d surfaces in the euclidean n space without restrictions on dimension and codimension the variational nature of the approach also allows us to consider surfaces with singularities and in that case the approximate mean curvature is consistent with the generalized mean curvature of the limit surface a series of numerical tests are provided in order to illustrate the effectiveness and generality of the method

the primal dual active set method for nonlinear optimal control problems with bilateral constraints the primal dual active set method has proved to be an efficient numerical tool in the context of diverse applications so far it has been investigated mainly for linear problems this paper is devoted to the study of global convergence of the primal dual active set method for nonlinear problems with bilateral constraints utilizing the close relationship between the primal dual active set method and semismooth newton methods local superlinear convergence of the method is investigated as well