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

a semi implicit finite element method for viscous lipid membranes a finite element formulation to approximate the behavior of lipid membranes is proposed the mathematical model incorporates tangential viscous stresses and bending elastic forces together with the inextensibility constraint and the enclosed volume constraint the membrane is discretized by a surface mesh made up of planar triangles over which a mixed formulation velocity curvature is built based on the viscous bilinear form boussinesq scriven operator and the laplace beltrami identity relating position and curvature a semi implicit approach is then used to discretize in time with piecewise linear interpolants for all variables two stabilization terms are needed the first one stabilizes the inextensibility constraint by a pressure gradient projection scheme codina and blasco 1997 33 the second couples curvature and velocity to improve temporal stability as proposed by bansch 2001 36 the volume constraint is handled by a lagrange multiplier which turns out to be the internal pressure and an analogous strategy is used to filter out rigid body motions the nodal positions are updated in a lagrangian manner according to the velocity solution at each time step an automatic remeshing strategy maintains suitable refinement and mesh quality throughout the simulation numerical experiments show the convergent and robust behavior of the proposed method stability limits are obtained from numerous relaxation tests and convergence with mesh refinement is confirmed both in the relaxation transient and in the final equilibrium shape virtual tweezing experiments are also reported computing the dependence of the deformed membrane shape with the tweezing velocity a purely dynamical effect for sufficiently high velocities a tether develops which shows good agreement both in its final radius and in its transient behavior with available analytical solutions finally simulation results of a membrane subject to the simultaneous action of six tweezers illustrate the robustness of the method

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

local likelihood density estimation based on smooth truncation two existing density estimators based on local likelihood have properties that are comparable to those of local likelihood regression but they are much less used than their counterparts in regression we consider truncation as a natural way of localising parametric density estimation based on this idea a third local likelihood density estimator is introduced our main result establishes that the three estimators coincide when a free multiplicative constant is used as an extra local parameter

geometrically consistent mesh modification a new paradigm of adaptivity is to execute refinement coarsening and smoothing of meshes on manifolds with incomplete information about their geometry and yet preserve position and curvature accuracy we refer to this collectively as geometrically consistent gc mesh modification we discuss the concept of discrete gc show the failure of naive approaches and propose and analyze a simple algorithm that is gc and accuracy preserving

semiparametric density estimation by local l2 fitting this article examines density estimation by combining a parametric approach with a nonparametric factor the plug in parametric estimator is seen as a crude estimator of the true density and is adjusted by a nonparametric factor the nonparametric factor is derived by a criterion called local l 2 fitting a class of estimators that have multiplicative adjustment is provided including estimators proposed by several authors as special cases and the asymptotic theories are developed theoretical comparison reveals that the estimators in this class are better than or at least competitive with the traditional kernel estimator in a broad class of densities the asymptotically best estimator in this class can be obtained from the elegant feature of the bias function

weighted maximum likelihood for dynamic factor analysis and forecasting with mixed frequency data for the purpose of forecasting key macroeconomic or financial variables from a panel of time series variables we adopt the dynamic factor model and propose a weighted likelihood based method for parameter estimation the loglikelihood function is split into two parts that are weighted differently the first part is associated with the key variables while the second part is associated with the related variables which may contribute to the forecasting of key variables we derive asymptotic properties including consistency and asymptotic normality of the weighted maximum likelihood estimator we show that this estimator outperforms the standard likelihood based estimator in approximating the true unknown distribution of the data as well as in out of sample forecasting accuracy we verify the new estimation method in a monte carlo study and investigate the role of different weights in different settings in the context of forecasting gross domestic product growth this key variable is typically observed at a low quarterly frequency while the supporting variables are observed at a high monthly frequency we adopt a low frequency representation of the mixed frequency dynamic factor model and discuss the computational efficiencies of this approach in our empirical study for the u s economy we present improvements in nowcasting and forecasting accuracy when the weighted likelihood based estimation procedure is adopted

free boundary regularity on the focusing problem for the gauss curvature flow with flat sides we consider the motion of a compact weakly convex two dimensional surface of revolution sigma under the gauss curvature flow we assume that the initial surface has a flat side and as a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate at points where the curvature is zero expressing the strictly convex part of the surface near the interface as the graph of a function z f r t we show that if at ti me t 0 g sqrt f vanishes linearly at the flat side then g r t will become smooth up to the interface for t 0 and it will remain smooth up to the focusing time t of the flat side we also show that at the focusing time of the flat side the function g is of class c 1 beta for all beta and no better than c 1 2 over 5 this implies that at the focusing time the surface sigma is of class c 2 beta for all beta and no better than c 2 2 over 5 in the case of the evolution monge ampere equation we find the exact self similar profile of the function g at its focusing time

change point estimation using nonparametric regression we consider a regression model in which the mean function may have a discontinuity at an unknown point we propose an estimate of the location of the discontinuity based on one side nonparametric regression estimates of the mean function the change point estimate is shown to converge in probability at rate 0 n 1 and to have the same asymptotic distribution as maximum likelihood estimates considered by other authors under parametric regression models confidence regions for the location and size of the change are also discussed

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

viscous regularization and r adaptive remeshing for finite element analysis of lipid membrane mechanics as two dimensional fluid shells lipid bilayer membranes resist bending and stretching but are unable to sustain shear stresses this property gives membranes the ability to adopt dramatic shape changes in this paper a finite element model is developed to study static equilibrium mechanics of membranes in particular a viscous regularization method is proposed to stabilize tangential mesh deformations and improve the convergence rate of nonlinear solvers the augmented lagrangian method is used to enforce global constraints on area and volume during membrane deformations as a validation of the method equilibrium shapes for a shape phase diagram of lipid bilayer vesicle are calculated these numerical techniques are also shown to be useful for simulations of three dimensional large deformation problems the formation of tethers long tube like extensions and ginzburg landau phase separation of a two lipid component vesicle to deal with the large mesh distortions of the two phase model modification of viscous regularization is explored to achieve r adaptive mesh optimization

triangular surface mesh fairing via gaussian curvature flow surface mesh fairing by the mean curvature flow and its various modifications have become a popular topic however very few researches have been attempted on using the gaussian curvature flow in surface fairing the aim of this paper is to investigate such a problem we find that gaussian curvature flow can only be used to smooth convex meshes hence it cannot be used to smooth noisy surface meshes because a noisy surface mesh is not convex to overcome this difficulty we design a new diffusion equation whose evolution direction depends on the mean curvature normal and the magnitude is a properly defined function of the gaussian curvature experimental results show that the designed fairing scheme can effectively remove the noise and simultaneously preserve the sharp features such as corners and edges

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

parametric fem for geometric biomembranes we consider geometric biomembranes governed by an l 2 gradient flow for bending energy subject to area and volume constraints helfrich model we give a concise derivation of a novel vector formulation based on shape differential calculus and corresponding discretization via parametric fem using quadratic isoparametric elements and a semi implicit euler method we document the performance of the new parametric fem with a number of simulations leading to dumbbell red blood cell and toroidal equilibrium shapes while exhibiting large deformations

a class of local likelihood methods and near parametric asymptotics observations in the neighbourhood of t the paper studies the sense in which f t t is closer to the true distribution g t than the usual estimate f t is asymptotic results are presented for the case in which the model misspecification becomes vanishingly small as the sample size tends to1 i n this setting the relative entropy risk of the local method is better than that of maximum likelihood the form of optimum weights for the local likelihood is obtained and illustrated for the normal distribution

variational discretization of axisymmetric curvature flows we present natural axisymmetric variants of schemes for curvature flows introduced earlier by the present authors and analyze them in detail although numerical methods for geometric flows have been used frequently in axisymmetric settings numerical analysis results so far are rare in this paper we present stability equidistribution existence and uniqueness results for the introduced approximations numerical computations show that these schemes are very efficient in computing numerical solutions of geometric flows as only a spatially one dimensional problem has to be solved the good mesh properties of the schemes also allow them to compute in very complex axisymmetric geometries

a multi level adaptive mesh refinement for an integrated finite element level set formulation to simulate multiphase flows with surface tension abstract a multi level adaptive mesh refinement technique for an integrated finite element level set formulation was implemented to simulate the incompressible interfacial flows with surface tension by using unstructured p2p1 triangular and tetrahedral meshes linear quadratic elements near the interface were dynamically refined as an interface evolves in order to produce an accurate solution with a lower computational cost an adaptive mesh refinement algorithm for a linear element has been extended to a quadratic element by using a node reordering the proposed algorithm was successfully validated by solving several benchmark problems simulation of a static bubble problem was found to give an accurate solution with negligible spurious currents compared with existing results simulation results of rising bubble problems in 2d and 3d were also compared against existing numerical and experimental results in terms of terminal rising speed and terminal bubble shape we have found that the present algorithm provided stable and accurate solutions even for a high density ratio and that the solution accuracy strongly depended on the mesh resolution near the interface further the present algorithm was shown to be very efficient because the computational overhead by mesh refinement algorithm was very small compared to the elapsed times consumed for solving the incompressible navier stokes equations and the advection re initialization equations of level set method lastly the coalescence scenario of two oblique bubbles was well reproduced by the present 3d simulation

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

adaptive time stepping for vesicle suspensions we present an adaptive high order accurate time stepping numerical scheme for the flow of vesicles suspended in stokesian fluids our scheme can be summarized as an approximate implicit spectral deferred correction sdc method applying a fully implicit sdc scheme to vesicle flows is prohibitively expensive for this reason we introduce several approximations our scheme is based on a semi implicit linearized low order time stepping method our discretization is spectrally accurate in space we expect that the accuracy can be arbitrary order but our examples suffer from order reduction which limits the observed accuracy to third order we also use invariant properties of vesicle flows constant area and boundary length in two dimensions to reduce the computational cost of error estimation for adaptive time stepping we present results in two dimensions for single vesicle flows constricted geometry flows converging flows and flows in a couette apparatus we experimentally demonstrate that the proposed scheme enables automatic selection of the time step size and high order accuracy

an immersed boundary method for simulating vesicle dynamics in three dimensions we extend our previous immersed boundary ib method for 3d axisymmetric inextensible vesicle in navier stokes flows hu et al 2014 17 to general three dimensions despite a similar spirit in numerical algorithms to the axisymmetric case the fully 3d numerical implementation is much more complicated and is far from straightforward a vesicle membrane surface is known to be incompressible and exhibits bending resistance as in 3d axisymmetric case instead of keeping the vesicle locally incompressible we adopt a modified elastic tension energy to make the vesicle surface patch nearly incompressible so that solving the unknown tension lagrange multiplier for the incompressible constraint can be avoided nevertheless the new elastic force derived from the modified tension energy has exactly the same mathematical form as the original one except the different definitions of tension the vesicle surface is discretized on a triangular mesh where the elastic tension and bending force are calculated on each vertex lagrangian marker in the ib method of the triangulation a series of numerical tests on the present scheme are conducted to illustrate the robustness and applicability of the method we perform the convergence study for the immersed boundary forces and the fluid velocity field we then study the vesicle dynamics in various flows such as quiescent simple shear and gravitational flows our numerical results show good agreements with those obtained in previous theoretical experimental and numerical studies

an a posteriori error estimator for anisotropic refinement besides an algorithm for local refinement an a posteriori error estimator is the basic tool of every adaptive finite element method using information generated by such an error estimator the refinement of the grid is controlled for 2nd order elliptic problems we present an error estimator for anisotropically refined grids like n d cuboidal and 3 d prismatic grids that gives correct information about the size of the error additionally it generates information about the direction into which some element has to be refined to reduce the error in a proper way numerical examples are presented for 2 d rectangular and 3 d prismatic grids

adaptive finite element method for a phase field bending elasticity model of vesicle membrane deformations in this paper a three dimensional adaptive finite element method is developed for a variational phase field bending elasticity model of vesicle membrane deformations using a mixed finite element formulation residual type a posteriori error estimates are derived for the associated nonlinear system of equations and they are used to introduce the mesh refinement and coarsening the resulting mesh adaptivity significantly improves the efficiency of the phase field simulation of vesicle membranes and enhances its capability in handling complex shape and topological changes the effectiveness of the adaptive method is further demonstrated through numerical examples

a convergent evolving finite element algorithm for mean curvature flow of closed surfaces a proof of convergence is given for semi and full discretizations of mean curvature flow of closed two dimensional surfaces the numerical method proposed and studied here combines evolving finite elements whose nodes determine the discrete surface like in dziuk s method and linearly implicit backward difference formulae for time integration the proposed method differs from dziuk s approach in that it discretizes huisken s evolution equations for the normal vector and mean curvature and uses these evolving geometric quantities in the velocity law projected to the finite element space this numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five the error analysis combines stability estimates and consistency estimates to yield optimal order h 1 norm error bounds for the computed surface position velocity normal vector and mean curvature the stability analysis is based on the matrix vector formulation of the finite element method and does not use geometric arguments the geometry enters only into the consistency estimates numerical experiments illustrate and complement the theoretical results

biomembrane modeling with isogeometric analysis abstract we consider the numerical approximation of lipid biomembranes at equilibrium described by the canham helfrich model according to which the bending energy is minimized under area and volume constraints energy minimization is performed via l 2 gradient flow of the canham helfrich energy using two lagrange multipliers to weakly enforce the constraints this yields a highly nonlinear high order time dependent geometric partial differential equation pde we represent the biomembranes as single patch nurbs closed surfaces we discretize the geometric pdes in space with nurbs based isogeometric analysis and in time with backward differentiation formulas we tackle the nonlinearity in our formulation through a semi implicit approach by extrapolating at each time level the geometric quantities of interest from previous time steps we report the numerical results of the approximation of the canham helfrich problem on ellipsoids of different aspect ratio which leads to the classical biconcave shape of lipid vesicles at equilibrium we show that this framework permits an accurate approximation of the canham helfrich problem while being computationally efficient

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

a parametric finite element method for fourth order geometric evolution equations we present a finite element approximation of motion by minus the laplacian of curvature and related flows the proposed scheme covers both the closed curve case and the case of curves that are connected via triple junctions on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches it turns out that the new scheme has very good properties with respect to area conservation and the equidistribution of mesh points we state also an extension of our scheme to willmore flow of curves and discuss possible further generalizations

isogeometric analysis of geometric partial differential equations abstract we consider the numerical approximation of geometric partial differential equations pdes defined on surfaces in the 3d space in particular we focus on the geometric pdes deriving from the minimization of an energy functional by l 2 gradient flow we analyze two energy functionals the area which leads to the mean curvature flow a nonlinear second order pde and the willmore energy leading to the willmore flow a nonlinear fourth order pde we consider surfaces represented by single patch tensor product nurbs and discretize the pdes by means of nurbs based isogeometric analysis in the framework of the galerkin method to approximate the high order geometric pdes we use high order continuous nurbs basis functions for the time discretization of the nonlinear geometric pdes we use backward differentiation formulas bdf with extrapolation of the geometric quantities involved in the weak formulation of the problem in this manner we solve a linear problem at each time step we report numerical results concerning the mean curvature and willmore flows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme

constrained local likelihood estimators for semiparametric skew normal distributions summary a local likelihood estimator for a nonparametric nuisance function is proposed in the context of semiparametric skew normal distributions constraints imposed on such functions result in a nonparametric estimator with a different target function for maximization from classical local likelihood estimators the optimal asymptotic semiparametric efficiency bound on parameters of interest is achieved by using this estimator in conjunction with an estimating equation formed by summing efficient scores a generalized profile likelihood approach is also proposed this method has the advantage of providing a unique estimate in cases where an estimating equation has multiple solutions our nonparametric estimator of the nuisance function leads to an estimator of the semiparametric skew normal density both the estimating equation and profile likelihood approaches are applicable to more general skew symmetric distributions

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

preconditioning a class of fourth order problems by operator splitting we develop preconditioners for systems arising from finite element discretizations of parabolic problems which are fourth order in space we consider boundary conditions which yield a natural splitting of the discretized fourth order operator into two discrete linear second order elliptic operators and exploit this property in designing the preconditioners the underlying idea is that efficient methods and software to solve second order problems with optimal computational effort are widely available we propose symmetric and non symmetric preconditioners along with theory and numerical experiments they both document crucial properties of the preconditioners as well as their practical performance it is important to note that we neither need h s regularity s 1 of the continuous problem nor quasi uniform grids

a finite element method to compute three dimensional equilibrium configurations of fluid membranes we introduce a finite element method to compute equilibrium configurations of fluid membranes identified as stationary points of a curvature dependent bending energy functional under certain geometric constraints the reparameterization symmetries in the problem pose a challenge in designing parametric finite element methods and existing methods commonly resort to lagrange multipliers or penalty parameters in contrast we exploit these symmetries by representing solution surfaces as normal offsets of given reference surfaces and entirely bypass the need for artificial constraints we then resort to a galerkin finite element method to compute discrete c 1 approximations of the normal offset coordinate the variational framework presented is suitable for computing deformations of three dimensional membranes subject to a broad range of external interactions we provide a systematic algorithm for computing large deformations wherein solutions at subsequent load steps are identified as perturbations of previously computed ones we discuss the numerical implementation of the method in detail and demonstrate its optimal convergence properties using examples we discuss applications of the method to studying adhesive interactions of fluid membranes with rigid substrates and to investigate the influence of membrane tension in tether formation

a finite difference algorithm for multiple moving boundary problems using real and virtual grid networks abstract most numerical methods developed for moving boundary or stefan problems deal with the case of a single moving boundary mb separating two different media although this is applicable to a large number of engineering problems there are many problems where more than one mb exists simultaneously during the process a heat transfer process involving heating of a solid melting and partial vaporisation of liquid can be considered as a three phase stefan or two mb problem where the time of appearance and disappearance of phases are to be determined as a part of the solution an explicit unconditionally stable numerical scheme for such problems is presented and tested herein the approach originates from the explicit variable time step evts method developed by the same authors for single mb problems during the vaporisation stage where two mbs exist simultaneously the method uses a virtual distorted grid network moving in parallel to the vapour liquid interface in order to determine its position vis a vis the real grid network the method has been tested by solving both the collapse of an adiabatic wall and a normalised two mb problem whose exact solution is known

a finite element method for surface diffusion the parametric case surface diffusion is a fourth order highly nonlinear geometric driven motion of a surface with normal velocity proportional to the surface laplacian of mean curvature we present a novel variational formulation for parametric surfaces with or without boundaries the method is semi implicit requires no explicit parametrization and yields a linear system of elliptic pde to solve at each time step we next develop a finite element method propose a schur complement approach to solve the resulting linear systems and show several significant simulations some with pinch off in finite time we introduce a mesh regularization algorithm which helps prevent mesh distortion and discuss the use of time and space adaptivity to increase accuracy while reducing complexity

local gaussian correlation a new measure of dependence abstract it is a common view among finance analysts and econometricians that the correlation between financial objects becomes stronger as the market is going down and that it approaches one when the market crashes having the effect of destroying the benefit of diversification the purpose of this paper is to introduce a local dependence measure that gives a precise mathematical description and interpretation of such phenomena we propose a new local dependence measure a local correlation function based on approximating a bivariate density locally by a family of bivariate gaussian densities using local likelihood at each point the correlation coefficient of the approximating gaussian distribution is taken as the local correlation existence uniqueness and limit results are established a number of properties of the local gaussian correlation and its estimate are given along with examples from both simulated and real data this new method of modelling carries with it the prospect of being able to do locally for a general density what can be done globally for the gaussian density in a sense it extends gaussian analysis from a linear to a non linear environment

metric tensors for anisotropic mesh generation it has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features in practice an anisotropic mesh is commonly generated as a quasi uniform mesh in the metric determined by a tensor specifying the shape size orientation of elements thus it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation in this paper we develop a general formula for the metric tensor for use in any spatial dimension the formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements numerical results in two dimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality the procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates

contact aware simulations of particulate stokesian suspensions we present an efficient accurate and robust method for simulation of dense suspensions of deformable and rigid particles immersed in stokesian fluid in two dimensions we use a well established boundary integral formulation for the problem as the foundation of our approach this type of formulation with a high order spatial discretization and an implicit and adaptive time discretization have been shown to be able to handle complex interactions between particles with high accuracy yet for dense suspensions very small time steps or expensive implicit solves as well as a large number of discretization points are required to avoid non physical contact and intersections between particles leading to infinite forces and numerical instability our method maintains the accuracy of previous methods at a significantly lower cost for dense suspensions the key idea is to ensure interference free configuration by introducing explicit contact constraints into the system while such constraints are unnecessary in the formulation in the discrete form of the problem they make it possible to eliminate catastrophic loss of accuracy by preventing contact explicitly introducing contact constraints results in a significant increase in stable time step size for explicit time stepping and a reduction in the number of points adequate for stability

on the parametric finite element approximation of evolving hypersurfaces in r 3 we present a variational formulation of motion by minus the laplacian of curvature and mean curvature flow as well as related second and fourth order flows of a closed hypersurface in r 3 on introducing a parametric finite element approximation we prove stability bounds and compare our scheme with existing approaches the presented scheme has very good properties with respect to the distribution of mesh points and if applicable volume conservation

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

3d vesicle dynamics simulations with a linearly triangulated surface simulations of biomembranes have gained an increasing interest in the past years specificities of these membranes propose new challenges for the numerics in particular vesicle dynamics are governed by bending forces as well as a surface incompressibility constraint a method to compute the bending force density resultant onto piecewise linearly triangulated surface meshes is described this method is coupled with a boundary element method solver for inner and outer fluids to compute vesicle dynamics under external flows the surface incompressibility constraint is satisfied by the construction of a projection operator

a boundary element method for multiple moving boundary problems a boundary element formulation for the solution of multiple moving boundary problems is presented and tested herein a heat transfer problem involving heating of solid melting of solid and partial vaporisation of liquid is considered numerical results show that the boundary element method is more suitable and more accurate than both finite difference and finite element methods for this kind of problem

a moving mesh finite element method for the two dimensional stefan problems an r adaptive moving mesh method is developed for the numerical solution of an enthalpy formulation of two dimensional heat conduction problems with a phase change the grid is obtained from a global mapping of the physical to the computational domain which is designed to cluster mesh points around the interface between the two phases of the material the enthalpy equation is discretised using a semiimplicit galerkin finite element method using linear basis functions the moving finite element method is applied to problems where the phase front is cusp shaped and where the interface changes topology

removability of point singularities of willmore surfaces we investigate point singularities of willmore surfaces which for example appear as blowups of the willmore flow near singularities and prove that closed willmore surfaces with one unit density point singularity are smooth in codimension one as applications we get in codimension one that the willmore flow of spheres with energy less than 8 exists for all time and converges to a round sphere and further that the set of willmore tori with energy less than

finite element approximation of large bending isometries a finite element scheme for the approximation of large isometric deformations with minimal bending energy is devised and analyzed the convergence to a stationary point and energy decreasing property of an iterative algorithm for the numerical solution of the scheme is proved numerical experiments illustrate the performance of the iteration and show that the discretization leads to accurate approximations for large vertical loads and compressive boundary conditions

relative efficiencies of kernel and local likelihood density estimators summary local likelihood methods enjoy advantageous properties such as good performance in the presence of edge effects that are rarely found in other approaches to nonparametric density estimation however as we argue in this paper standard kernel methods can have distinct advantages when edge effects are not present we show that whereas the integrated variances of the two methods are virtually identical the integrated squared bias of a conventional kernel estimator is less than that of a local log linear estimator by as much as a factor of 4 moreover the greatest bias improvements offered by kernel methods occur when they are needed most i e when the effect of bias is particularly high similar comparisons can also be made when high degree local log polynomial fits are assessed against high order kernel methods for example although as is well known high degree local polynomial fits offer potentially infinite efficiency gains relative to their kernel competitors the converse is also true indeed the asymptotic value of the integrated squared bias of a local log quadratic estimator can exceed any given constant multiple of that for the competing kernel method in all cases the densities that suffer problems in the context of local log likelihood methods can be chosen to be symmetric either unimodal or bimodal either infinitely or compactly supported and to have arbitrarily many derivatives as functions on the real line they are not pathological however our results reveal quantitative differences between global performances of local log polynomial estimators applied to unimodal or multimodal distributions

on local likelihood density estimation this paper considers a class of local likelihood methods introduced by eguchi and copas unified asymptotic results are presented in the usual smoothing context of the bandwidth h tending to zero as the sample size tends to infinity we present our results pointwise in the univariate case but then go on to extend them to global properties and to indicate how to cope with the multivariate case specific members of the class due to copas and hjort and jones are seen to be members of a subset of the whole class with the same and best small h behavior further comparisons between members of the class are alluded to based on the complementary large h asymptotic results of eguchi and copas

interface evolution in three dimensions with curvature dependent energy and surface diffusion interface controlled evolution phase transitions epitaxial growth of elastic films when the interfacial energy is a nonconvex function of orientation the anisotropic curvature flow equation becomes backward parabolic to overcome the instability thus generated a regularization of the equation that governs the evolution of the interface is needed in this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk surface interactions the theory is based on a superficial mass balance configurational forces and couples consistent with superficial force and moment balances a mechanical version of the second law that includes via the configurational moments work that accompanies changes in the curvature of the interface a constitutive theory whose main ingredient is a positive definite isotropic quadratic dependence of the interfacial energy on the curvature tensor two special cases are investigated i the interface is a boundary between bulk phases or grains and ii the interface separates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface

parametric approximation of isotropic and anisotropic elastic flow for closed and open curves deckelnick and dziuk math comput 78 266 645 671 2009 proved a stability bound for a continuous in time semidiscrete parametric finite element approximation of the elastic flow of closed curves in mathbb r d d geq2 we extend these ideas in considering an alternative finite element approximation of the same flow that retains some of the features of the formulations in barrett et al j comput phys 222 1 441 462 2007 siam j sci comput 31 1 225 253 2008 ima j numer anal 30 1 4 60 2010 in particular an equidistribution mesh property for this new approximation we obtain also a stability bound for a continuous in time semidiscrete scheme apart from the isotropic situation we also consider the case of an anisotropic elastic energy in addition to the evolution of closed curves we also consider the isotropic and anisotropic elastic flow of a single open curve in the plane and in higher codimension that satisfies various boundary conditions

local likelihood based on kernel censoring by drawing an analogy with likelihood for censored data a local likelihood function is proposed which gives more weight to observations near a region of interest in the sample space resulting methods can be used for assessing local departures from a parametric model and for semiparametric density estimation some theory and three examples is given

locally parametric nonparametric density estimation this paper develops a nonparametric density estimator with parametric overtones suppose f x is some family of densities indexed by a vector of parameters we define a local kernel smoothed likelihood function which for each x can be used to estimate the best local parametric approximant to the true density this leads to a new density estimator of the form f x o x thus inserting the best local parameter estimate for each new value of x when the bandwidth used is large this amounts to ordinary full likelihood parametric density estimation while for moderate and small bandwidths the method is essentially nonparametric using only local properties of data and the model alternative ways more general than via the local likelihood are also described the methods can be seen as ways of nonparametrically smoothing the parameter within a parametric class properties of this new semiparametric estimator are investigated our preferred version has approximately the same variance as the ordinary kernel method but potentially a smaller bias the new method is seen to perform better than the traditional kernel method in a broad nonparametric vicinity of the parametric model employed while at the same time being capable of not losing much in precision to full likelihood methods when the model is correct other versions of the method are approximately equivalent to using particular higher order kernels in a semiparametric framework the methodology we develop can be seen as the density estimation parallel to local likelihood and local weighted least squares theory in nonparametric regression

an algorithm for evolutionary surfaces an algorithm is presented which allows to split the calculation of the mean curvature flow of surfaces with or without boundary into a series of poisson problems on a series of surfaces this gives a new method to solve plateau s problem forh surfaces

parametrically assisted nonparametric estimation of a density in the deconvolution problem nonparametric estimation of a density from contaminated data is a difficult problem for which convergence rates are notoriously slow we introduce parametrically assisted nonparametric estimators which can dramatically improve on the performance of standard nonparametric estimators when the assumed model is close to the true density without degrading much the quality of purely nonparametric estimators in other cases we establish optimal convergence rates for our problem and discuss estimators that attain these rates the very good numerical properties of the methods are illustrated via a simulation study supplementary materials for this article are available online

efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice boltzmann finite element method the deformation of an initially spherical capsule freely suspended in simple shear flow can be computed analytically in the limit of small deformations d barthes biesel j m rallison the time dependent deformation of a capsule freely suspended in a linear shear flow j fluid mech 113 1981 251 267 those analytic approximations are used to study the influence of the mesh tessellation method the spatial resolution and the discrete delta function of the immersed boundary method on the numerical results obtained by a coupled immersed boundary lattice boltzmann finite element method for the description of the capsule membrane a finite element method and the skalak constitutive model r skalak a tozeren r p zarda s chien strain energy function of red blood cell membranes biophys j 13 1973 245 264 have been employed our primary goal is the investigation of the presented model for small resolutions to provide a sound basis for efficient but accurate simulations of multiple deformable particles immersed in a fluid we come to the conclusion that details of the membrane mesh as tessellation method and resolution play only a minor role the hydrodynamic resolution i e the width of the discrete delta function can significantly influence the accuracy of the simulations the discretization of the delta function introduces an artificial length scale which effectively changes the radius and the deformability of the capsule we discuss possibilities of reducing the computing time of simulations of deformable objects immersed in a fluid while maintaining high accuracy

probit transformation for kernel density estimation on the unit interval kernel estimation of a probability density function supported on the unit interval has proved difficult because of the well known boundary bias issues a conventional kernel density estimator would necessarily face in this situation transforming the variable of interest into a variable whose density has unconstrained support estimating that density and obtaining an estimate of the density of the original variable through back transformation seems a natural idea to easily get rid of the boundary problems in practice however a simple and efficient implementation of this methodology is far from immediate and the few attempts found in the literature have been reported not to perform well in this article the main reasons for this failure are identified and an easy way to correct them is suggested it turns out that combining the transformation idea with local likelihood density estimation produces viable density estimators mostly free from boundary issues their asymptotic properties are derived an

minimum complexity density estimation the authors introduce an index of resolvability that is proved to bound the rate of convergence of minimum complexity density estimators as well as the information theoretic redundancy of the corresponding total description length the results on the index of resolvability demonstrate the statistical effectiveness of the minimum description length principle as a method of inference the minimum complexity estimator converges to true density nearly as fast as an estimator based on prior knowledge of the true subclass of densities interpretations and basic properties of minimum complexity estimators are discussed some regression and classification problems that can be examined from the minimum description length framework are considered

phase field approximations of the willmore functional and flow we discuss in this paper phase field approximations of the willmore functional and the associated mathrm l 2 l2 flow after recollecting known results on the approximation of the willmore energy and its mathrm l 1 l1 relaxation we derive the expression of the flows associated with various approximations and we show their behavior by formal arguments based on matched asymptotic expansions we introduce an accurate numerical scheme whose local convergence is proved to describe with more details the behavior of two flows the classical and the flow associated with an approximation model due to mugnai we propose a series of numerical simulations in 2d and 3d to illustrate their behavior in both smooth and singular situations

parametric approximation of willmore flow and related geometric evolution equations we present various variational approximations of willmore flow in mathbb r d d 2 3 as well as the classic willmore flow we also consider variants that are a volume preserving and b volume and area preserving the latter evolution law is the so called helfrich flow in addition we consider motion by gaus curvature the presented fully discrete schemes are easy to solve as they are linear at each time level and they have good properties with respect to the distribution of mesh points finally we present numerous numerical experiments including simulations for energies appearing in the modeling of biological cell membranes

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

moving grid scheme for multiple moving boundaries a numerical method which employs moving grid has been presented for computing multiple moving boundaries numerical solution has been obtained for a solidification problem in which freezing takes place over a temperature range it has been shown that implicit boundary conditions at the moving boundaries can be tackled by considering an appropriate explicit boundary condition two models have been considered for the solid fraction in the mushy region and the validity of these models has been discussed in the light of integral heat balance

boosting method for local learning in statistical pattern recognition we propose a local boosting method in classification problems borrowing from an idea of the local likelihood method our proposal local boosting includes a simple device for localization for computational feasibility we proved the bayes risk consistency of the local boosting in the framework of probably approximately correct learning inspection of the proof provides a useful viewpoint for comparing ordinary boosting and local boosting with respect to the estimation error and the approximation error both boosting methods have the bayes risk consistency if their approximation errors decrease to zero compared to ordinary boosting local boosting may perform better by controlling the trade off between the estimation error and the approximation error ordinary boosting with complicated base classifiers or other strong classification methods including kernel machines may have classification performance comparable to local boosting with simple base classifiers for example decision stumps local boosting however has an advantage with respect to interpretability local boosting with simple base classifiers offers a simple way to specify which features are informative and how their values contribute to a classification rule even though locally several numerical studies on real data sets confirm these advantages of local boosting

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

electrohydrodynamics of three dimensional vesicles a numerical approach a three dimensional numerical model of vesicle electrohydrodynamics in the presence of dc electric fields is presented the vesicle membrane is modeled as a thin capacitive interface through the use of a semi implicit gradient augmented level set jet scheme the enclosed volume and surface area are conserved both locally and globally by a new navier stokes projection method the electric field calculations explicitly take into account the capacitive interface by an implicit immersed interface method formulation which calculates the electric potential field and the transmembrane potential simultaneously the results match well with previously published experimental analytic and two dimensional computational works

a stable numerical method for the dynamics of fluidic membranes we develop a finite element scheme to approximate the dynamics of two and three dimensional fluidic membranes in navier stokes flow local inextensibility of the membrane is ensured by solving a tangential navier stokes equation taking surface viscosity effects of boussinesq scriven type into account in our approach the bulk and surface degrees of freedom are discretized independently which leads to an unfitted finite element approximation of the underlying free boundary problem bending elastic forces resulting from an elastic membrane energy are discretized using an approximation introduced by dziuk numer math 111 55 80 2008 the obtained numerical scheme can be shown to be stable and to have good mesh properties finally the evolution of membrane shapes is studied numerically in different flow situations in two and three space dimensions the numerical results demonstrate the robustness of the method and it is observed that the conservation properties are fulfilled to a high precision

droplet footprint control controlling droplet shape via surface tension has numerous technological applications such as droplet lenses and lab on a chip this motivates a partial differential equation constrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate we use shape differential calculus to derive an l 2 gradient flow approach to compute equilibrium shapes for sessile droplets on substrates we then develop a gradient based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint i e the liquid solid interface has a desired shape moreover we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint numerical results are also presented to showcase the method

computation of three dimensional dendrites with finite elements starting from an initial seed crystal inside an undercooled liquid the solid phase begins to grow rapidly and develops unstable growth patterns some growth directions are preferred because of anisotropic parameters in the physical model this results in the development of dendrites the physical model includes the heat equation for both the liquid and solid phases the gibbs thomson law couples velocity and curvature of the interface and the temperature we describe a numerical method that enables us to compute dendritic growth of crystals in two and three space dimensions the method consists of two coupled finite element algorithms the first one solves the heat equation the other operates on a discretization of the free boundary and computes the evolution of this moving interface the two methods work with totally independent grids by using timedependent locally refined and coarsened adaptive meshes in both methods we are able to reach a spatial resolution necessary to compute dendritic growth in two and three space dimensions

new anisotropic a priori error estimates we prove a priori anisotropic estimates for the l 2 and h 1 interpolation error on linear finite elements the full information about the mapping from a reference element is employed to separate the contribution to the elemental error coming from different directions this new h 1 error estimate does not require the maximal angle condition the analysis has been carried out for the 2d case but may be extended to three dimensions numerical experiments have been carried out to test our theoretical results

shape optimization of self avoiding curves this paper presents a softened notion of proximity or self avoidance for curves we then derive a sensitivity result based on shape differential calculus for the proximity this is combined with a gradient based optimization approach to compute three dimensional parameterized curves that minimize the sum of an elastic bending energy and a proximity energy that maintains self avoidance by a penalization technique minimizers are computed by a sequential quadratic programming sqp method where the bending energy and proximity energy are approximated by a finite element method we then apply this method to two problems first we simulate adsorbed polymer strands that are constrained to be bound to a surface and be locally inextensible this is a basic model of semi flexible polymers adsorbed onto a surface a current topic in material science several examples of minimizing curve shapes on a variety of surfaces are shown an advantage of the method is that it can be much faster than using molecular dynamics for simulating polymer strands on surfaces second we apply our proximity penalization to the computation of ideal knots we present a heuristic scheme utilizing the sqp method above for minimizing rope length and apply it in the case of the trefoil knot applications of this method could be for generating good initial guesses to a more accurate but expensive knot tightening algorithm a new softened version of proximity or self contact for 1 d curves is proposed sensitivity of the proximity with respect to shape differentiation is given a method for finding minimizing curve shapes that respect self contact is given the method is applied to simulating polymer strands adsorbed onto surfaces a heuristic method is given for approximating ideal knots

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

a numerical method for simulation dynamics of incompressible lipid membranes in viscous fluid a numerical scheme is presented for modeling dynamics of incompressible elastic lipid membranes imbedded into viscous fluid a new elliptic equation for tension in the membrane implying its local incompressibility is derived the membranes dynamics is approximated in a semi implicit way the lattice boltzmann method is used to approximate the fluid flow forces acting on the fluid from the lipid membrane are implemented using the immersed boundary method by peskin the method is illustrated by examples of axisymmetric membranes with deformations and flows typical for experiments with lipid vesicles and nanotubes where deformations and flows can be modulated by external forces applied to the membrane

eliminating spurious velocities with a stable approximation of viscous incompressible two phase stokes flow abstract we present a parametric finite element approximation of two phase flow this free boundary problem is given by the stokes equations in the two phases which are coupled via jump conditions across the interface using a novel variational formulation for the interface evolution gives rise to a natural discretization of the mean curvature of the interface in addition the mesh quality of the parametric approximation of the interface does not deteriorate in general over time and an equidistribution property can be shown for a semidiscrete continuous in time variant of our scheme in two space dimensions moreover on using a simple xfem pressure space enrichment we obtain exact volume conservation for the two phase regions furthermore our fully discrete finite element approximation can be shown to be unconditionally stable we demonstrate the applicability of our method with some numerical results which in particular demonstrate that spurious velocities can be avoided in the classical test cases