fully implicit methodology for the dynamics of biomembranes and capillary interfaces by combining the level set and newton methods abstract this framework is concerned with the numerical modeling of the dynamics of individual biomembranes and capillary interfaces in a surrounding newtonian fluid a level set approach helps to follow the interface motion our method features the use of high order fully implicit time integration schemes that enable to overcome stability issues related to the explicit discretization of the highly non linear bending force or capillary force at each time step the tangent systems are derived and the resulting nonlinear problems are solved by a newton raphson method based on the signed distance assumption several inexact newton strategies are employed to solve the capillary and vesicle problems and guarantee the second order convergence behavior we address in detail the main features of the proposed method and we report several experiments in the two dimensional case with the aim of illustrating its accuracy and efficiency comparative investigations with respect to the fully explicit scheme depict the stabilizing effect of the new method which allows to use significantly larger time step sizes

computing the dynamics of biomembranes by combining conservative level set and adaptive finite element methods the numerical simulation of the deformation of vesicle membranes under simple shear external fluid flow is considered in this paper a saddle point approach is proposed for the imposition of the fluid incompressibility and the membrane inextensibility constraints through lagrange multipliers defined in the fluid and on the membrane respectively using a level set formulation the problem is approximated by mixed finite elements combined with an automatic adaptive mesh procedure at the vicinity of the membrane boundary numerical experiments show that this combination of the saddle point and adaptive mesh method enhances the robustness of the method the effect of inertia on the stability of the vesicle in a shear flow is also investigated

simulating the dynamics of inextensible vesicles by the penalty immersed boundary method in this paper we develop an immersed boundary ib method to simulate the dynamics of inextensible vesicles interacting with an incompressible fluid in order to take into account the inextensibility constraint of the vesicle the penalty immersed boundary pib method is used to virtually decouple the fluid and vesicle dynamics as numerical tests of our current pib method the dynamics of single and multiple inextensible vesicles under shear flows have been extensively explored and compared with the previous literature the method is also validated by a series of convergence study which confirms its consistent first order accuracy on the velocity field the vesicle configuration the vesicle area and the perimeter errors in addition the method is also applied to study a binary component vesicle problem

an optimal one way multigrid algorithm for discrete time stochastic control the numerical solution of discrete time stationary infinite horizon discounted stochastic control problems is considered for the case where the state space is continuous and the problem is to be solved approximately within a desired accuracy after a discussion of problem discretization the authors introduce a multigrid version of the successive approximation algorithm that proceeds one way from coarse to fine grids and analyze its computational requirements as a function of the desired accuracy and of the discount factor they also study the effects of a certain mixing ergodicity condition on the algorithm s performance it is shown that the one way multigrid algorithm improves upon the complexity of its single grid variant and is in a certain sense optimal

optimal approximation schedules for a class of iterative algorithms with an application to multigrid value iteration many iterative algorithms employ operators which are difficult to evaluate exactly but for which a graduated range of approximations exist in such cases coarse to fine algorithms are often used in which a crude initial operator approximation is gradually refined with new iterations in such algorithms because the computational complexity increases over iterations the algorithm s convergence rate is properly calculated with respect to cumulative computation time this suggests the problem of determining an optimal rate of refinement for the operator approximation this paper shows that for linearly convergent algorithm the optimal rate of refinement approaches the rate of convergence of the exact algorithm itself regardless of the tolerance complexity relationship we illustrate this result with an analysis of coarse to fine grid algorithms for markov decision processes with continuous state spaces using the methods proposed here we deduce an algorithm that presents optimal complexity results and consists solely of a non adaptive schedule for the gradual decrease of grid size

accelerating the convergence of value iteration by using partial transition functions abstract this work proposes an algorithm that makes use of partial information to improve the convergence properties of the value iteration algorithm in terms of the overall computational complexity the algorithm iterates on a series of increasingly refined approximate models that converges to the true model according to an optimal linear rate which coincides with the convergence rate of the original value iteration algorithm the paper investigates the properties of the proposed algorithm and features a series of switchover queue examples which illustrates the efficacy of the method

stability and optimality of a multi product production and storage system under demand uncertainty this work develops a discrete event model for a multi product multi stage production and storage p s problem subject to random demand the intervention problem consists of three types of possible decisions made at the end of one stage which depend on the observed demand or lack of for each item i to proceed further with the production of the same product ii to proceed with the production of another product or iii to halt the production the intervention problem is formulated in terms of dynamic programming dp operators and each possible solution induces an homogeneous markov chain that characterizes the dynamics however solving directly the dp problem is not a viable task in situations involving a moderately large number of products with many production stages and the idea of the paper is to detach from strict optimality with monitored precision and rely on stability the notion of stochastic stability brought to bear requires a finite set of positive recurrent states and the paper derives necessary and sufficient conditions for a policy to induce such a set in the studied p s problem an approximate value iteration algorithm is proposed which applies to the broader class of control problems described by homogeneous markov chains that satisfy a structural condition pointed out in the paper this procedure iterates in a finite subset of the state space circumventing the computational burden of standard dynamic programming to benchmark the approach the proposed algorithm is applied to a simple two product p s system

dynamic simulation of locally inextensible vesicles suspended in an arbitrary two dimensional domain a boundary integral method we consider numerical algorithms for the simulation of hydrodynamics of two dimensional vesicles suspended in a viscous stokesian fluid the motion of vesicles is governed by the interplay between hydrodynamic and elastic forces continuum models of vesicles use a two phase fluid system with interfacial forces that include tension to maintain local surface inextensibility and bending vesicle flows are challenging to simulate on the one hand explicit time stepping schemes suffer from a severe stability constraint due to the stiffness related to high order spatial derivatives in the bending term on the other hand implicit time stepping schemes can be expensive because they require the solution of a set of nonlinear equations at each time step our method is an extension of the work of veerapaneni et al s k veerapaneni d gueyffier d zorin g biros a boundary integral method for simulating the dynamics of inextensible vesicles suspended in a viscous fluid in 2d journal of computational physics 228 7 2009 2334 2353 in which a semi implicit time marching scheme based on a boundary integral formulation of the stokes problem for vesicles in an unbounded medium was proposed in this paper we consider two important generalizations i confined flows within arbitrary shaped stationary moving geometries and ii flows in which the interior to the vesicle and exterior fluids have different viscosity in the rest of the paper we will refer to this as the viscosity contrast these two problems require solving additional integral equations and cause nontrivial modifications to the previous numerical scheme our method does not have severe time step stability constraints and its computational cost per time step is comparable to that of an explicit scheme the discretization is pseudo spectral in space and multistep bdf in time we conduct numerical experiments to investigate the stability accuracy and the computational cost of the algorithm overall our method achieves several orders of magnitude speed up compared to standard explicit schemes as a preliminary validation of our scheme we study the dependence of the inclination angle of a single vesicle in shear flow on the viscosity contrast and the reduced area of the vesicle the lateral migration of vesicles in shear flow the dispersion of two vesicles and the effective viscosity of a dilute suspension of vesicles

differential equation based constrained reinitialization for level set methods a partial differential equation based reinitialization method is presented in the framework of a localized level set method two formulations of the new reinitialization scheme are derived these formulations are modifications of the partial differential equation introduced by sussman et al m sussman p smereka s osher a level set approach for computing solutions to incompressible two phase flow j comput phys 114 1994 146 159 and in particular improvements of the second order accurate modification proposed by russo and smereka g russo p smereka a remark on computing distance functions j comput phys 163 2000 51 67 the first formulation uses the least squares method to explicitly minimize the displacement of the zero level set within the reinitialization the overdetermined problem which is solved in the first formulation of the new reinitialization scheme is reduced to a determined problem in another formulation such that the location of the interface is locally preserved within the reinitialization the second formulation is derived by systematically minimizing the number of constraints imposed on the reinitialization scheme for both systems the resulting algorithms are formulated in a three dimensional frame of reference and are remarkably simple and efficient the new formulations are second order accurate at the interface when the reinitialization equation is solved with a first order upwind scheme and do not diminish the accuracy of high order discretizations of the level set equation the computational work required for all components of the localized level set method scales with o n detailed analyses of numerical solutions obtained with different discretization schemes evidence the enhanced accuracy and the stability of the proposed method which can be used for localized and global level set methods

approximate fixed point iteration with an application to infinite horizon markov decision processes many approximate iterative algorithms can be represented by the form v n tv n 1 u n n geq1 where v n 1 u n n geq1 are elements of a seminormed linear space mathcal v cdot and t is a contractive operator the objective is usually to calculate a fixed point v of t here the quantities u n may be interpreted as the errors obtained when the operator t can be evaluated only approximately as is well known in the absence of such an approximation error the algorithm converges to a fixed point with a geometric rate under general conditions in this article the convergence properties of such algorithms in the presence of an approximation error u n are studied it is shown that the error v n v is dominated by u n so that convergence of u n to zero implies the convergence of v n v to zero at a rate determined by u n the results are naturally extended to a relative error of the form u n v n 1 as well as to j stage contractions the utility of this general theory is then demonstrated by an extended application to the problem of model based approximate and adaptive control of markov decision processes the theory is shown to permit a sharpening of known convergence rates under more general conditions additionally bounds on regret for adaptive controls with forced exploration are calculated in terms of a stagewise exploration rate this permits the determination of an optimal choice of exploration rate within the class of certainty equivalence adaptive control policies

approximate dynamic programming via direct search in the space of value function approximations this paper deals with approximate value iteration avi algorithms applied to discounted dynamic programming dp problems for a fixed control policy the span semi norm of the so called bellman residual is shown to be convex in the banach space of candidate solutions to the dp problem this fact motivates the introduction of an avi algorithm with local search that seeks to minimize the span semi norm of the bellman residual in a convex value function approximation space the novelty here is that the optimality of a point in the approximation architecture is characterized by means of convex optimization concepts and necessary and sufficient conditions to local optimality are derived the procedure employs the classical avi algorithm direction bellman residual combined with a set of independent search directions to improve the convergence rate it has guaranteed convergence and satisfies at least the necessary optimality conditions over a prescribed set of directions to illustrate the method examples are presented that deal with a class of problems from the literature and a large state space queueing problem setting

a balanced force refined level set grid method for two phase flows on unstructured flow solver grids a balanced force refined level set grid method for two phase flows on structured and unstructured flow solver grids is presented to accurately track the phase interface location an auxiliary high resolution equidistant cartesian grid is introduced in conjunction with a dual layer narrow band approach this refined level set grid method allows for parallel efficient grid convergence and error estimation studies of the interface tracking method the navier stokes equations are solved on an unstructured flow solver grid with a novel balanced force algorithm for level set methods based on the recently proposed method by francois et al m m francois s j cummins e d dendy d b kothe j m sicilian m w williams a balanced force algorithm for continuous and sharp interfacial surface tension models within a volume tracking framework j comput phys 213 2006 141 173 for volume of fluid methods on structured grids to minimize spurious currents a second order converging curvature evaluation technique for level set methods is presented the results of several different test cases demonstrate the effectiveness of the proposed method showing good mass conservation properties and second order converging spurious current magnitudes

technical communique policy set iteration for markov decision processes this communique presents an algorithm called policy set iteration psi for solving infinite horizon discounted markov decision processes with finite state and action spaces as a simple generalization of policy iteration pi psi generates a monotonically improving sequence of stationary markovian policies p k based on a set manipulation as opposed to pi s single policy manipulation at each iteration k when the set involved with psi at k contains n independently generated sample policies from a given distribution d the probability that the expected value of any sampled policy from d with respect to an initial state distribution is greater than that of p k converges to zero with o n k rate moreover psi converges to an optimal policy no slower than pi in terms of the number of iterations for any d

an immersed boundary method for simulating the dynamics of three dimensional axisymmetric vesicles in navier stokes flows in this paper we develop a simple immersed boundary method to simulate the dynamics of three dimensional axisymmetric inextensible vesicles in navier stokes flows instead of introducing a lagrange s multiplier to enforce the vesicle inextensibility constraint we modify the model by adopting a spring like tension to make the vesicle boundary nearly inextensible so that solving for the unknown tension can be avoided we also derive a new elastic force from the modified vesicle energy and obtain exactly the same form as the originally unmodified one in order to represent the vesicle boundary we use fourier spectral approximation so we can compute the geometrical quantities on the interface more accurately a series of numerical tests on the present scheme have been conducted to illustrate the applicability and reliability of the method we first perform the accuracy check of the geometrical quantities of the interface and the convergence check for different stiffness numbers as well as fluid variables then we study the vesicle dynamics in quiescent flow and in gravity finally the shapes of vesicles in poiseuille flow are investigated in detail to study the effects of the reduced volume the confinement and the mean flow velocity the numerical results are shown to be in good agreement with those obtained in literature

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

an adaptive level set approach for incompressible two phase flows we present a numerical method using the level set approach for solving incompressible two phase flow with surface tension in the level set approach the free surface is represented as the zero level set of a smooth function this has the effect of replacing the advection of density which has steep gradients at the free surface with the advection of the level set function which is smooth in addition the free surface can merge or break up with no special treatment we maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects in this work we couple the level set scheme to an adaptive projection method for the incompressible navier stokes equations in order to achieve higher resolution of the free surface with a minimum of additional expense we present two dimensional axisymmetric and fully three dimensional results of air bubble and water drop computations

simulating the deformation of vesicle membranes under elastic bending energy in three dimensions in this paper we study the three dimensional deformation of a vesicle membrane under the elastic bending energy with prescribed bulk volume and surface area both static and dynamic deformations are considered a newly developed energetic variational formulation is employed to give an effective eulerian description efficient time and spatial discretizations are considered and implemented numerical experiments illustrate some fascinating phenomena that are of interests in real applications

optimization of a special case of continuous time markov decision processes with compact action set performance optimization is considered for average cost multichain markov decision processes mdps with compact action set since for a general compact multichain model the optimality equation system may have no solution and also a policy iteration algorithm may yield a suboptimal policy rather than an optimal one we concentrate only on a special case of multichain models in this paper where we assume that the classifications of states are fixed identically rather than varying with policies by using the concept of performance potentials the existence of solutions to the optimality equation system is established and then a potential based policy iteration algorithm is supposed to solve this system in addition the optimality convergence for recurrent classes of the algorithm has been proved finally a numerical example is provided

solving average cost markov decision processes by means of a two phase time aggregation algorithm abstract this paper introduces a two phase approach to solve average cost markov decision processes which is based on state space embedding or time aggregation in the first phase time aggregation is applied for policy optimization in a prescribed subset of the state space and a novel result is applied to expand the evaluation to the whole state space this evaluation is then used in the second phase in a policy improvement step and the two phases are then alternated until convergence is attained some numerical experiments illustrate the results

the constrained reinitialization equation for level set methods based on the constrained reinitialization scheme d hartmann m meinke w schroder differential equation based constrained reinitialization for level set methods j comput phys 227 2008 6821 6845 a new constrained reinitialization equation incorporating a forcing term is introduced two formulations for high order constrained reinitialization hcr are presented combining the simplicity and generality of the original reinitialization equation m sussman p smereka s osher a level set approach for computing solutions to incompressible two phase flow j comput phys 114 1994 146 159 in terms of high order standard discretization and the accuracy of the constrained reinitialization scheme in terms of interface displacement the novel hcr schemes represent simple extensions of standard implementations of the original reinitialization equation the results evidence the significantly increased accuracy and robustness of the novel schemes

computational parametric willmore flow we propose a new algorithm for the computation of willmore flow this is the l 2 gradient flow for the willmore functional which is the classical bending energy of a surface willmore flow is described by a highly nonlinear system of pdes of fourth order for the parametrization of the surface the spatially discrete numerical scheme is stable and consistent the discretization relies on an adequate calculation of the first variation of the willmore functional together with a derivation of the second variation of the area functional which is well adapted to discretization techniques with finite elements the algorithm uses finite elements on surfaces we give numerical examples and tests for piecewise linear finite elements a convergence proof for the full algorithm remains an open question

lao a heuristic search algorithm that finds solutions with loops classic heuristic search algorithms can find solutions that take the form of a simple path a a tree or an acyclic graph ao in this paper we describe a novel generalization of heuristic search called lao that can find solutions with loops we show that lao can be used to solve markov decision problems and that it shares the advantage heuristic search has over dynamic programming for other classes of problems given a start state it can find an optimal solution without evaluating the entire state space 2001 elsevier science b v all rights reserved

eulerian finite element method for the numerical modeling of fluid dynamics of natural and pathological aortic valves we present a finite element methodology tailored for the simulation of pulsatile flow in the full aorta and sinus of valsalva interacting with highly deformable thin leaflets we describe an extension of the so called resistive immersed surface method to circumvent stability issues resulting from the bad conditioning of the linear system especially when flow and geometry become complex after the inclusion of the aorta we use a lagrange multiplier technique that couples the dynamics of valve and flow a banded level set variant allows to address the singularity of the resulting linear system while featuring in addition to the parallel implementation higher accuracy and an affordable computational burden high fidelity computational geometries are built and simulations are performed under physiological conditions several numerical experiments illustrate the ability of the model to capture the basic fluidic phenomena in both healthy and pathological configurations we finally examine numerically the flow dynamics in the sinus of valsalva after transcatheter aortic valve implantation we show numerically that flow may be subject to stagnation in the lower part of the sinuses we highlight the far reaching implications of this phenomenon and we hope inciting adequate studies to further investigate its potential clinical consequences display omitted hemodynamics in full aorta and aortic sinuses coupled with highly deformable valve an exact lagrange multiplier technique couples the dynamics of valve and flow a damped newton strategy allows more stability for relatively large reynolds numbers numerical examples in 2d and 3d with healthy and pathological valves numerical investigations pinpoint a risk of blood stagnation after tavi

analysis on an ode arisen from studying the shape of a red blood cell we study the existence problem of a special solution to the helfrich functional such that it corresponds to a surface of the shape of a red blood cell the helfrich functional is also a perturbation of the willmore functional involving some parameters with physical meanings with the expected symmetry of the surface it reduces to an analysis on an ode with certain shape requirements we discover a sufficient condition on the parameters which ensures the existence of such a special solution to the ode

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

on the reinitialization procedure in a narrow band locally refined level set method for interfacial flows a local level set algorithm for simulating interfacial flows described by the two dimensional incompressible navier stokes equations is presented the governing equations are solved using a finite difference discretization on a cartesian grid and a second order approximate projection method the level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid which is reconstructed every time step and refined using a simple uniform cell splitting operation within the band instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band the influence of different pde based reinitialization strategies on the accuracy of the results is investigated the ability of the proposed method to accurately compute interfacial flows is discussed using different tests namely the advection of a circle of fluid in two different time reversed vortex flows the advection of zalesak s rotating disk the propagation of small amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water and a classical test of rayleigh taylor instability with and without surface tension effects the interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method copyright 2005 john wiley sons ltd

a level set projection model of lipid vesicles in general flows a new numerical method to model the dynamic behavior of lipid vesicles under general flows is presented a gradient augmented level set method is used to model the membrane motion to enforce the volume and surface incompressibility constraints a four step projection method is developed to integrate the full navier stokes equations this scheme is implemented on an adaptive non graded cartesian grid convergence results are presented along with sample two dimensional results of vesicles under various flow conditions

square root bryson frazier smoothing algorithms some new square root algorithms for bryson frazier smoothing formulas are suggested square root algorithms and a fast square root or so called chandrasekhar type algorithm the new square root algorithms use square root arrays composed of smoothed estimates and their error covariances these algorithms provide many advantages over the conventional algorithms with respect to systolic array and parallel implementations as well as numerical stability and conditioning for the case of constant parameter systems a fast square root algorithm is suggested which requires less computation than others

a phase field approach in the numerical study of the elastic bending energy for vesicle membranes in this paper we compute the equilibrium configurations of a vesicle membrane under elastic bending energy with prescribed volume and surface area a variational phase field method is developed for such a problem discrete finite difference approximations and numerical simulations are carried out in the axial symmetrical cases different energetic bifurcation phenomena are discussed

simulation of two fluid flows using a finite element level set method application to bubbles and vesicle dynamics a new framework for two fluid flows using a finite element level set method is presented and verified through the simulation of the rising of a bubble in a viscous fluid this model is then enriched to deal with vesicles which mimic red blood cells mechanical behavior by introducing a lagrange multiplier to constrain the inextensibility of the membrane moreover high order polynomial approximation is used to increase the accuracy of the simulations a validation of this model is finally presented on known behaviors of vesicles under flow such as tank treading and tumbling motions