a general method for simulation of fluid flows with moving and compliant boundaries on unstructured grids abstract in this paper a method for simulating fluid structure interaction is described the numerical model is based on a 2d navier stokes incompressible flow solver on unstructured moving grid using the arbitrary lagrangian eulerian ale approach a high order upwind characteristics based finite volume scheme and an implicit dual time stepping method are employed for the simulation of unsteady flows and the fluid structure interaction furthermore a new dynamic mesh method is adopted to handle large deformation of the flow field with the dynamic mesh method the efficiency of unsteady flow simulation is significantly improved and more complex fluid structure interaction problems can be easily tackled the dynamic mesh method and the solver are validated for an unsteady channel flow with a moving indentation it is observed that the dynamic mesh method is very robust and able to handle large deformation without excessive distortion of the dense mesh near the wall in order to study fluid flow in a channel with an elastic membrane wall and their interaction a new membrane model has been adopted which can be applied to calculate arbitrary wall movement and variable tension along the membrane the membrane equation is also solved by the same implicit dual time stepping scheme for improved stability and efficiency it is shown that the proposed method is able to simulate unsteady flow with self excited oscillations that develop in a collapsible channel high frequency flutter is found in all the cases studied which is in the form of travelling waves along the membrane and the flutter is superimposed on lower frequency large amplitude oscillation due to the wall inertia and its tension the travelling waves tend to have larger amplitude and higher frequency with increased inertia and reduced tension

on the rate of convergence for the approximation of nonlinear problems in this paper we present an abstract theory to obtain error estimates in various norms for the approximation of solution branches of nonlinear equations by the aid of known estimates for corresponding linear problems

a stabilized multi level method for non singular finite volume solutions of the stationary 3d navier stokes equations this paper proposes and analyzes a stabilized multi level finite volume method fvm for solving the stationary 3d navier stokes equations by using the lowest equal order finite element pair without relying on any solution uniqueness condition this multi level stabilized fvm consists of solving the nonlinear problem on the coarsest mesh and then performing one newton correction step on each subsequent mesh thus only solving a large linear system an optimal convergence rate for the finite volume approximations of nonsingular solutions is first obtained with the same order as that for the usual finite element solution by using a relationship between the stabilized fvm and a stabilized finite element method then the multi level finite volume approximate solution is shown to have a convergence rate of the same order as that of the stabilized finite volume solution of the stationary navier stokes equations on a fine mesh with an appropriate choice of the mesh size h j h j 1 2 j 1 ldots j finally numerical results presented validate our theoretical findings

a c 0 c0 interior penalty method for a von karman plate we investigate a quadratic c 0 c0 interior penalty method for the approximation of isolated solutions of a von karman plate we prove that the discrete problem is uniquely solvable near an isolated solution and establish optimal order error estimates numerical results that illustrate the theoretical estimates are also presented

a conforming finite element method for the time dependent navier stokes equations we study a spatial discretization of the stokes problem in a domain of mathbb r 2 or mathbb r 3 by a finite element method the unknowns being the velocity and the pressure optimal error

numerical solution of the three dimensional ginzburg landau models using artificial boundary for three dimensional simulations of the vortex phenomena in superconductors based on the ginzburg landau gl theory the physical variables must be solved in the whole space in general exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain approximations of these boundary conditions with high accuracy are given and their convergence properties are examined error estimates are derived for the finite element approximations with the approximate boundary conditions

a matrix free implicit unstructured multigrid finite volume method for simulating structural dynamics and fluid structure interaction a new three dimensional 3d matrix free implicit unstructured multigrid finite volume fv solver for structural dynamics is presented in this paper the solver is first validated using classical 2d and 3d cantilever problems it is shown that very accurate predictions of the fundamental natural frequencies of the problems can be obtained by the solver with fast convergence rates this method has been integrated into our existing fv compressible solver x lv y zhao et al an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects journal of computational physics 215 2 2006 661 690 based on the immersed membrane method imm x lv y zhao et al as mentioned above results for the interaction between the fluid and an immersed fixed free cantilever are also presented to demonstrate the potential of this integrated fluid structure interaction approach

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

a new algorithm for two dimensional numerical continuation abstract consider a smooth mapping f from r n 2 into r n such a mapping can be interpreted as a system of n functions in n 2 unknowns the zero set of the mapping is typically a smooth two dimensional manifold surface embedded in the ambient space of dimension n 2 our paper describes a numerical algorithm for generating a representation of such a surface which can be displayed graphically the defining equations are given in implicit form not only is such a formulation convenient but it allows surfaces with folds in which there is not a functional relationship between the n state space variables and the two distinguished continuation parameters the algorithm is built as a layer on top of an existing one dimensional continuation solver and has been successfully applied to problems with a high dimensional state space using efficient sparse matrix techniques applications in electronic circuit simulation are described

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

numerical continuation methods a perspective in this historical perspective the principal numerical approaches to continuation methods are outlined in the framework of the mathematical sources that contributed to their development notably homotopy and degree theory simplicial complexes and mappings submanifolds defined by submersions and singularity and foldpoint theory

a fictitious domain method for dirichlet problem and applications abstract in this article we discuss the solution of the dirichlet problem for a class of elliptic operators by a lagrange multiplier fictitious domain method this approach allows the use of regular grids and therefore of fast specialized solvers for problems on complicated geometries the resulting saddle point system can be solved by an uzawa conjugate gradient algorithm in the case of two dimensional problems a quasi optimal preconditioner has been found by fourier analysis and numerical experiments confirm its nice scaling properties the resulting methodology is applied to a nonlinear time dependent problem namely the flow of a viscous plastic medium in a cylindrical pipe showing the potential of this methodology for some classes of nonlinear problems

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

preconditioned multigrid methods for unsteady incompressible flows a highly efficient numerical approach based on multigrid and preconditioning methods is developed for modeling 3d steady and time dependent incompressible flows thek turbulence model is used to estimate the effects of turbulence the model equations are solved together with the n s equations in a strongly coupled way and acceleration techniques like the multigrid method are also used for the turbulence model equations for unsteady problems a dual time stepping procedure is adopted to satisfy the divergence free constraint and to obtain a time accurate solution to improve the performance of this approach for small physical time steps a modification to residual smoothing parameters is proposed the numerical algorithm and the turbulence model are validated first by calculating unsteady inviscid flow around an oscillating cylinder unsteady laminar flow past a circular cylinder and steady high reynolds number turbulent flow over a 6 1 prolate spheroid then the three dimensional time dependent turbulent flow over a spheroid when it is undergoing a pitch up maneuver is calculated and compared with experimental data

numerical solution to a mixed navier stokes darcy model by the two grid approach we study numerical methods for a coupled navier stokes darcy model which describes a fluid flow filtrating through porous media a decoupled and linearized two grid algorithm is proposed numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm

a hybrid cartesian immersed boundary method for simulating flows with 3d geometrically complex moving bodies a numerical method is developed for solving the 3d unsteady incompressible navier stokes equations in cartesian domains containing immersed boundaries of arbitrary geometrical complexity moving with prescribed kinematics the governing equations are discretized on a hybrid staggered non staggered grid layout using second order accurate finite difference formulas the discrete equations are integrated in time via a second order accurate dual time stepping artificial compressibility iteration scheme unstructured triangular meshes are employed to discretize complex immersed boundaries the nodes of the surface mesh constitute a set of lagrangian control points used to track the motion of the flexible body at every instant in time the influence of the body on the flow is accounted for by applying boundary conditions at cartesian grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface grid convergence tests are carried out for the flow induced by an oscillating sphere in a cubic cavity which show that the method is second order accurate the method is validated by applying it to calculate flow in a cartesian domain containing a rigid sphere rotating at constant angular velocity as well as flow induced by a flapping wing the ability of the method to simulate flows in domains with arbitrarily complex moving bodies is demonstrated by applying to simulate flow past an undulating fish like body and flow past an anatomically realistic planktonic copepod performing an escape like maneuver euver

finite element approximation of incompressible navier stokes equations with slip boundary condition we consider mixed finite element approximations of the stationary incompressible navier stokes equations with slip boundary condition simultaneously approximating the velocity pressure and normal stress component the stability of the schemes is achieved by adding suitable consistent penalty terms corresponding to the normal stress component and to the pressure a new method of proving the stability of the discretizations allows us to obtain optimal error estimates for the velocity pressure and normal stress component in natural norms without using duality arguments and without imposing uniformity conditions on the finite element partition the schemes can easily be implemented into existing finite element codes for the navier stokes equations with standard dirichlet boundary conditions

how to get around a simple quadratic fold we analyse from a theoretical point of view a computational technique previously introduced 2 5 for tracing branches of solutions of nonlinear equations near simple quadratic folds

on the computation of multi dimensional solution manifolds of parametrized equations a new algorithm is presented for computing vertices of a simplicial triangulation of thep dimensional solution manifold of a parametrized equationf x 0 wheref is a nonlinear mapping fromr n tor m p n m 1 an essential part of the method is a constructive algorithm for computing moving frames on the manifold that is of orthonormal bases of the tangent spaces that vary smoothly with their points of contact the triangulation algorithm uses these bases together with a chord form of the gauss newton process as corrector to compute the desired vertices the jacobian matrix of the mapping is not required at all the vertices but only at the centers of certain local triangulation patches several numerical examples show that the method is very efficient in computing triangulations even around singularities such as limit points and bifurcation points this opens up new possibilities for determining the form and special features of such solution manifolds

on a computational method for the second fundamental tensor and its application to bifurcation problems an algorithm is presented for the computation of the second fundamental tensorv of a riemannian submanifoldm ofr n fromv the riemann curvature tensor ofm is easily obtained moreover v has a close relation to the second derivative of certain functionals onm which in turn provides a powerful new tool for the computational determination of multiple bifurcation directions frequently in applications thed dimensional manifoldm is defined implicitly as the zero set of a submersionf onr n in this case the principal cost of the algorithm for computingv p at a given pointp m involves only the decomposition of the jacobiandf p off atp and the projection ofd d 1 neighboring points ontom by means of a local iterative process usingdf p several numerical examples are given which show the efficiency and dependability of the method

a dual time method for two dimensional unsteady incompressible flow calculations a method for computing unsteady incompressible viscous flows on moving or deforming meshes is described it uses a well established time marching finite volume flow solver developed for steady compressible flows past rigid bodies time marching methods cannot be applied directly to incompressible flows because the governing equations are not hyperbolic such methods can be extended to steady incompressible flows using an artificial compressibility scheme a time accurate scheme for unsteady incompressible flows is achieved by using an implicit real time discretization and a dual time approach which uses a technique similar to the artificial compressibility scheme results are presented for test cases on both fixed and deforming meshes experimental numerical and theoretical data have been included for comparison where available and reasonable agreement has been achieved 1998 john wiley sons ltd

optimal order error estimates for finite element approximations of a bifurcation function the bifurcation function associated with an elliptic boundary value problem au g u 0 is a vector field b on rd with the property that the solutions of the boundary value problem are in a one to one correspondence with the zeros of b a finite element approximation bh of b is formulated and optimal order error estimates are derived implementation issues are also discussed

an optimization based domain decomposition method for pdes with random inputs an optimization based domain decomposition method for stochastic elliptic partial differential equations is presented the main idea of the method is a constrained optimization problem for which the minimization of an appropriate functional forces the solutions on the two subdomains to agree on the interface the constraints are the stochastic partial differential equations the existence of optimal solutions for the stochastic optimal control problem is shown as is the convergence to the exact solution of the given problem we prove the existence of a lagrange multiplier and derive an optimality system from which solutions of the domain decomposition problem may be determined finite element approximations to the solutions of the optimality system are defined and analyzed with respect to both spatial and random parameter spaces then the results of some numerical experiments are given to confirm theoretical error estimate results

a high order characteristics upwind fv method for incompressible flow and heat transfer simulation on unstructured grids abstract this paper presents a new unstructured grid upwind finite volume algorithm for accurate numerical simulation of incompressible flows and convection heat transfer on unstructured grids it is an upwind method at both the differential equation level and discretized equation level based on the method of characteristics this is made possible with the introduction of chorin s j comput phys 2 1967 12 26 artificial compressibility formulation to the incompressible flow equations flow variables are calculated along characteristics and their initial values are interpolated based on the signs of the corresponding characteristic speed in addition an upwind biased interpolation method of third order accuracy is used for interpolating flow variables on unstructured grids with these inherent upwinding techniques for evaluating convection fluxes at control volume surfaces no artificial viscosity is required and the method is accurate and less sensitive to the grid orientation thus suitable for unstructured grid application the use of the finite volume method combined with unstructured grids makes the scheme very flexible in dealing with very complex boundary geometries while remaining to be fully conservative at the cell as well as global level the discretized equations are solved by an explicit multistage runge kutta time stepping scheme which is found to be efficient in terms of cpu and memory overheads a computer code has been developed which uses the numerical methods presented and a highly efficient edge based data structure for flux calculation and data storage a number of well documented test cases including 2d laminar flow in a channel with a backward facing step forced and natural convection flows have been calculated in order to validate the code and evaluate the performance of the numerical algorithm good agreement between the computed and measured results have been obtained for all the cases to demonstrate the accuracy of the third order scheme flow in another channel with a backward facing step studied by gartling int j numer meth fluids 1990 11 is simulated with both the third order and first order schemes it is shown that the third order scheme can produce a velocity profile across the channel that has perfect agreement with that of gartling local citation whilst the first order scheme cannot even predict the separation near the upper wall the proposed numerical methods are also used to calculate 3d incompressible inviscid flows and results show that the method is accurate and robust for general 3d simulation the 3d navier stokes code is later used to study a lid driven cavity flow at a reynolds number of 400 the unstructured grid used has 97 336 nodes and 455 625 tetrahedral elements with strong grid clustering near the walls the convergence rate is found to be satisfactory which shows that the method is also robust for 3d navier stokes simulation the results obtained agree well with those reported by jiang et al comput meth appl mech eng 114 109 1994 results obtained using the second order central scheme with artificial viscosity are also presented for comparison it is observed that the third order characteristics scheme is more accurate than the latter thus confirming its third order accuracy

on the computation of manifolds of foldpoints for parameter dependent problems for parameter dependent nonlinear equations in r n a new augmentation method for computing submanifolds of foldpoints is presented the underlying theory is based on a characterization developed earlier of foldpoints on solution manifolds of parameterized equations the method allows for a locally convergent iterative computation of foldpoints of a certain class for a continuation along paths of such foldpoints and for a computation of simplicial approximations of higher dimensional submanifolds of foldpoints of the same type

computational error estimates and adaptive processes for some nonlinear structural problems abstract in earlier papers the authors introduced and analyzed the calculation of reliable a posteriori error estimates for finite element solutions and discussed the design and effectivity of adaptive procedures based upon them while most of this work concerned linear problems this paper is intended to show that the same approaches remain also highly effective in the nonlinear case as a model problem a planar elastic rod is considered involving both material as well as geometrical nonlinearities but as far as possible the discussion is kept general for nonlinear problems of this type interest centers on an analysis of the shape and features of the equilibrium surface after some general remarks about such surfaces and the related computational problems a general continuation process is summarized and some of its extensions for determining limit points and tracing stability paths are discussed then estimators are introduced for the error along the solution paths and of the computed critical values experimental results for the model problem show the effectivity of these estimators and of the adaptive procedures based upon them then some aspects relating to the occurrence and identification of spurious solutions are discussed in order to illustrate desirable generalizations of these results the paper ends with an outline of some results for linear problems of the type that should be achievable also in the nonlinear case

on the discretization error of parametrized nonlinear equations many applications lead to nonlinear parameter dependent equations h y t y 0 where h y times t to y y 0 in operatorname rge h and the state space y is infinite dimensional while the parameter space t has finite dimension the case dim t 1 is of special interest in connection with continuation methods for this case a general theory is developed which provides for the existence of solution paths of a rather general class of such equations and of their finite dimensional approximations and which allows for an assessment of the error between these paths a principal tool in this analysis is the theory of nonlinear fredholm operators the results cover a more general class of operators than the mildly nonlinear mappings to which other approaches appear to be restricted

mixtures of probabilistic principal component analyzers principal component analysis pca is one of the most popular techniques for processing compressing and visualizing data although its effectiveness is limited by its global linearity while nonlinear variants of pca have been proposed an alternative paradigm is to capture data complexity by a combination of local linear pca projections however conventional pca does not correspond to a probability density and so there is no unique way to combine pca models therefore previous attempts to formulate mixture models for pca have been ad hoc to some extent in this article pca is formulated within a maximum likelihood framework based on a specific form of gaussian latent variable model this leads to a well defined mixture model for probabilistic principal component analyzers whose parameters can be determined using an expectationmaximization algorithm we discuss the advantages of this model in the context of clustering density modeling and local dimensionality reduction and we demonstrate its application to image compression and handwritten digit recognition

a posteriori existence in numerical computations a typical a posteriori error analysis for numerical methods for differential equations consists of two elements the stability of the continuous problem and an a posteriori computed residual estimate if the problem is nonlinear then this is possible only if it is strongly monotone in this paper we will estimate the stability constant a posteriori as well in order to apply the inverse function theorem to derive the existence of solutions from the numerical computation and thus obtain rigorous a posteriori error estimates for nonmonotone equations

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

semidiscrete approximations of optimal robin boundary control problems constrained by semilinear parabolic pde an optimal robin boundary control problem associated with semilinear parabolic partial differential equations is considered existence of an optimal solution is proved and an optimality system of equations is derived semidiscrete finite element approximations of the optimality system are defined and error estimates are obtained

least squares methods for incompressible newtonian fluid flow linear stationary problems this paper develops and analyzes two least squares methods for the numerical so lution of linear stationary incompressible newtonian fluid flow in two and three dimensions both approaches use the l 2 norm to define least squares functionals one is based on the stress velocity formulation see section 3 2 and it applies to general boundary conditions the other is based on an equivalent formulation for the pseudostress and velocity see section 4 2 and it applies to pure velocity dirichlet boundary conditions the velocity gradient and vorticity can be obtained alge braically from this new tensor variable it is shown that the homogeneous least squares functionals are elliptic and continuous in the h div d h 1 d norm this immediately implies optimal error estimates for conforming finite element approximations as well it admits optimal multigrid solution methods if raviart thomas finite element spaces are used to approximate the stress or the pseudostress tensor

large sparse continuation problems we study and develop efficient and versatile predictor corrector continuation methods for large sparse problems the first object is to show how special solving methods for a large sparse linear systems can be incorporated into the basic steps of a continuation method next we describe how to use a special nonlinear conjugate gradient method to perform the corrector phase it is shown how such methods can be used to detect bifurcation points and how to trace bifurcating solution branches by using local perturbations finally a numerical example involving bifurcating branches of a nonlinear eigenvalue problem is given

parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids a novel immersed object method is developed for simulating two dimensional unsteady incompressible viscous flows around arbitrarily moving rigid bodies it has been implemented in a parallel unstructured finite volume incompressible navier stokes solver based on the artificial compressibility ac approach using a higher order characteristics based upwind scheme and matrix free implicit dual time stepping in the immersed object method an object is immersed in the flow field and it is supposed to contain frozen fluid which moves like a solid body this is realized by introducing source terms in the momentum equations during the ac sub iterations an internal mesh within the object is employed to search and locate all the eulerian nodes within the object in every time step for imposing the source terms unlike many existing methods this method does not require complex searching extrapolation and interpolation to find the intersections of the object boundary with the unstructured background mesh and assign flow condition onto the object boundary if it is necessary to capture the boundary layer accurately then a dense overlapping grid can then be constructed around the object for further refined calculation the immersed object method has been used to simulate steady and unsteady incompressible viscous flows over a stationary circular cylinder rotating square cylinder and moving disk in cavity the results agree well with published numerical solutions and experimental measurements

equation free analysis of agent based models and systematic parameter determination abstract agent based models abm s are increasingly used in social science economics mathematics biology and computer science to describe time dependent systems in circumstances where a description in terms of equations is difficult yet few tools are currently available for the systematic analysis of abm behaviour numerical continuation and bifurcation analysis is a well established tool for the study of deterministic systems recently equation free ef methods have been developed to extend numerical continuation techniques to systems where the dynamics are described at a microscopic scale and continuation of a macroscopic property of the system is considered to date the practical use of ef methods has been limited by 1 the over head of application specific implementation 2 the laborious configuration of problem specific parameters and 3 large ensemble sizes potentially leading to computationally restrictive run times in this paper we address these issues with our tool for the ef continuation of stochastic systems which includes algorithms to systematically configuration problem specific parameters and enhance robustness to noise our tool is generic and can be applied to any black box simulator and determines the essential ef parameters prior to ef analysis robustness is significantly improved using our convergence constraint with a corrector repeat c3r method this algorithm automatically detects outliers based on the dynamics of the underlying system enabling both an order of magnitude reduction in ensemble size and continuation of systems at much higher levels of noise than classical approaches we demonstrate our method with application to several abm models revealing parameter dependence bifurcation and stability analysis of these complex systems giving a deep understanding of the dynamical behaviour of the models in a way that is not otherwise easily obtainable in each case we demonstrate our systematic parameter determination stage for configuring the system specific ef parameters

treating inhomogeneous essential boundary conditions in finite element methods and the calculation of boundary stresses finite element approximations of the stokes and navier stokes equations with in homogeneous essential boundary conditions are considered boundary conditions are enforced weakly by introducing lagrange multipliers optimal error estimates including some for the stress vector on the boundary are derived under minimal regularity assumptions on the data particular attention is paid to the analysis of a practical choice of finite element spaces for which the lagrange multiplier calculation uncouples from that for the velocity and pressure the results are also applicable to general second order elliptic systems

hp adaptive discontinuous galerkin methods for bifurcation phenomena in open flows we consider the application of high order hp version adaptive discontinuous galerkin finite element methods for the discretization of the bifurcation problem associated with the steady incompressible navier stokes equations based on exploiting the dual weighted residual approach reliable and efficient a posteriori estimates of the error in the computed critical reynolds number at which a steady pitchfork bifurcation occurs when the underlying physical system possesses either reflectional z 2 symmetry or rotational and reflectional o 2 symmetry are derived numerical experiments highlighting the practical performance of the proposed a posteriori error indicator on hp adaptively refined computational meshes are presented for both two and three dimensional problems

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

optimal homotopy methods for solving nonlinear systems this paper deals with the monotone homotopy methods for solving the system of nonlinear equations in sect 2 the homotopy with a distancemonotone homotopy path is discussed and it is proved that a homotopy with a given structure has a distance monotone homotopy path under some regular conditions then two structure variable homotopy algorithms called local straighten up method and global straighten up method are developed to approximate the optimal homotopy and an adaptive step size control strategy for these algorithms is proposed respectively a new method for analyzing the convergence is presented which is based on the geometrical properties of the algorithm then the convergence of the algorithm is proved under certain regular conditions finally two numerical examples are given to illustrate the effectiveness of the above two algorithms

the numerical solution of nonlinear equations having several parameters part iii equations with z 2 symmetry the computation of symmetry breaking bifurcation points of nonlinear multiparameter problems with z 2 reflectional symmetry is considered the numerical approach is based on recent work in singularity theory which is used to construct systems of equations and inequalities characterising various types of symmetry breaking bifurcation points numerical continuation methods are then used to follow paths of symmetry breaking bifurcations and hence compute regions in parameter space for which a problem has qualitatively similar bifurcation diagrams the power of the numerical approach is illustrated by computations of axisymmetric flows in the finite taylor problem

an analysis of upwind schemes for the navier stokes equations we state a general result on approximation of branches of nonsingular solutions of nonlinear problems the abstract theory is then applied to the study of two mixed finite element schemes for the navier stokes equations using an upwind discretization of the convective terms

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

hellinger transform of gaussian autoregressive processes abstract the limit of the hellinger transform of measures related to multidimensional stationary gaussian autoregressive processes is obtained

piecewise linear methods for nonlinear equations and optimization piecewise linear methods had their beginning in the mid 1960s with lemke s algorithm for calculating solutions to linear complementarity problems in the 1970s and 1980s activity moved on to computing fixed points of rather general maps and economic equilibria more recently they have been used to approximate implicitly defined manifolds with applications being made to computer graphics and approximations of integral over implicitly defined manifolds in this paper we present the basic ideas of piecewise linear algorithms and a selection of applications further references to the literature on piecewise linear algorithms are indicated

a posteriori error estimators for the stokes equations ii non conforming discretizations we present an a posteriori error estimator for the non conforming crouzeix raviart discretization of the stokes equations which is based on the local evaluation of residuals with respect to the strong form of the differential equation the error estimator yields global upper and local lower bounds for the error of the finite element solution it can easily be generalized to the stationary incompressible navier stokes equations and to other non conforming finite element methods numerical examples show the efficiency of the proposed error estimator

une methode numerique pour le calcul des points de retournement application a un probleme aux limites non lineaire an iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter one shows that the convergence is superlinear and that this method may constitute a complement for the newton method with a finite element method numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch

a covariance free iterative algorithm for distributed principal component analysis on vertically partitioned data in this paper a covariance free iterative algorithm is developed to achieve distributed principal component analysis on high dimensional data sets that are vertically partitioned we have proved that our iterative algorithm converges monotonously with an exponential rate different from existing techniques that aim at approximating the global pca our covariance free iterative distributed pca cidpca algorithm can estimate the principal components directly without computing the sample covariance matrix therefore a significant reduction on transmission costs can be achieved furthermore in comparison to existing distributed pca techniques cidpca can provide more accurate estimations of the principal components and classification results we have demonstrated the superior performance of cidpca through the studies of multiple real world data sets

a dlm fd method for fluid flexible body interactions in this study we extended the distributed lagrange multiplier fictitious domain dlm fd formulation of glowinski et al int j multiphase flow 25 1999 755 for the fluid rigid body interactions to deal with the fluid flexible body interactions by replacing newton s equations of motion for the rigid body with the continuum equations for the general solid material similar to the rigid body case where the dlm is introduced as a pseudo body force to enforce the constraint of rigid body motion of the fictitious fluid in the solid domain the lagrange multiplier in our formulation is to enforce the fictitious fluid to move at the same velocity as the solid for our computational scheme a first order accurate fractional step scheme is employed to decouple the entire system into three sub systems a fluid problem a solid problem and a lagrange multiplier problem the flow problem is solved with the projection method on half staggered grids the solid problem is solved with the lagrangian finite element method and the newton iterative method and the incompressibility of the material is implemented with the penalty function method the proposed method is applied to two typical fluid structure interaction problems the flow driven oscillation of a flexible plate along the flow direction and the self sustained oscillation across the flow direction both results compare favorably with previously reported numerical and experimental results and show that our method is suited to the simulation of the motion of an incompressible non linear elastic material in a fluid

certified reduced basis vms smagorinsky model for natural convection flow in a cavity with variable height abstract in this work we present a reduced basis vms smagorinsky boussinesq model applied to natural convection problems in a variable height cavity in which the buoyancy forces are involved we take into account in this problem both physical and geometrical parametrizations considering the rayleigh number as a parameter so as the height of the cavity we perform an empirical interpolation method to approximate the sub grid eddy viscosity term that lets us obtain an affine decomposition with respect to the parameters we construct an a posteriori error estimator based upon the brezzi rappaz raviart theory used in the greedy algorithm for the selection of the basis functions finally we present several numerical tests for different parameter configuration

stability analysis of preconditioned approximations of the euler equations on unstructured meshes this paper analyses the stability of a discretisation of the euler equations on 3d unstructured grids using an edge based data structure first order characteristic smoothing a block jacobi preconditioner and runge kutta timemarching this is motivated by multigrid navier stokes calculations in which this inviscid discretisation is the dominant component on coarse grids the analysis uses algebraic stability theory which allows at worst a bounded linear growth in a suitably defined perturbation energy provided the range of values of the preconditioned spatial operator lies within the stability region of the runge kutta algorithm the analysis also includes consideration of the effect of solid wall boundary conditions and the addition of a low mach number preconditioner to accelerate compressible flows in which the mach number is very low in a significant portion of the flow numerical results for both inviscid and viscous applications confirm the effectiveness of the numerical algorithm and show that the analysis provides accurate stability bounds

preconditioned methods for solving the incompressible low speed compressible equations abstract acceleration methods are presented for solving the steady state incompressible equations these systems are preconditioned by introducing artificial time derivatives which allow for a faster convergence to the steady state we also consider the compressible equations in conservation form with slow flow two arbitrary functions and are introduced in the general preconditioning an analysis of this system is presented and an optimal value for is determined given a constant it is further sown that the resultant incompressible equations form a symmetric hyperbolic system and so are well posed several generalizations to the compressible equations are presented which extend previous results

an efficient parallel unstructured multigrid preconditioned implicit method for simulating 3d unsteady compressible flows with moving objects this paper presents the development and validation of a new parallel unstructured multi grid preconditioned implicit method for the simulation of three dimensional 3d unsteady compressible flows using a so called immersed membrane method imm g h xia y zhao j h yeo an immersed membrane method for simulation of flui structure interaction in bio fluid flows in 1st international bioengineering conference ibec 2004 08 10 september 2004 the novel feature of the method presented is that it employs a unique combination of a parallel multi grid scheme with low mach number preconditioning and the imm so that 3d unsteady low mach number flows with arbitrarily moving objects can be efficiently simulated

primal mixed finite element approximation of elliptic equations with gradient nonlinearities we study the primal mixed finite element approximation of the second order elliptic problem with gradient nonlinearities existence and uniqueness of the approximate solution are proved and optimal order a priori error estimates are obtained also reliable a posteriori error estimators are derived

probabilistic principal component analysis principal component analysis pca is a ubiquitous technique for data analysis and processing but one which is not based upon a probability model in this paper we demonstrate how the principal axes of a set of observed data vectors may be determined through maximum likelihood estimation of parameters in a latent variable model closely related to factor analysis we consider the properties of the associated likelihood function giving an em algorithm for estimating the principal subspace iteratively and discuss the advantages conveyed by the definition of a probability density function for pca

candid covariance free incremental principal component analysis appearance based image analysis techniques require fast computation of principal components of high dimensional image vectors we introduce a fast incremental principal component analysis ipca algorithm called candid covariance free ipca ccipca used to compute the principal components of a sequence of samples incrementally without estimating the covariance matrix so covariance free the new method is motivated by the concept of statistical efficiency the estimate has the smallest variance given the observed data to do this it keeps the scale of observations and computes the mean of observations incrementally which is an efficient estimate for some well known distributions e g gaussian although the highest possible efficiency is not guaranteed in our case because of unknown sample distribution the method is for real time applications and thus it does not allow iterations it converges very fast for high dimensional image vectors some links between ipca and the development of the cerebral cortex are also discussed

a three dimensional computational analysis of fluid structure interaction in the aortic valve numerical analysis of the aortic valve has mainly been focused on the closing behaviour during the diastolic phase rather than the kinematic opening and closing behaviour during the systolic phase of the cardiac cycle moreover the fluid structure interaction in the aortic valve system is most frequently ignored in numerical modelling the effect of this interaction on the valve s behaviour during systolic functioning is investigated the large differences in material properties of fluid and structure and the finite motion of the leaflets complicate blood valve interaction modelling this has impeded numerical analyses of valves operating under physiological conditions a numerical method known as the lagrange multiplier based fictitious domain method is used to describe the large leaflet motion within the computational fluid domain this method is applied to a three dimensional finite element model of a stented aortic valve the model provides both the mechanical behaviour of the valve and the blood flow through it results show that during systole the leaflets of the stented valve appear to be moving with the fluid in an essentially kinematical process governed by the fluid motion

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

a geometric framework for the numerical study of singular points while bifurcation theory has developed rapidly in recent years there appears to be a need for a tighter framework for the numerical analysis of bifurcation problems this paper presents such a mathematical framework for the numerical study of the bifurcation phenomena associated with a parameter dependent equation f z lambda 0 the presentation draws from differential geometry and singularity theory and provides a basis for various numerical methods used to detect and compute certain types of bifurcation points

a priori error estimates of finite element solutions of parametrized strongly nonlinear boundary value problems abstract nonlinear boundary value problems with parameters are called parametrized nonlinear boundary problems this paper studies a priori error estimates of finite element solutions of second order parametrized strongly nonlinear boundary value problems in divergence form on one dimensional bounded intervals the banach space w 0 1 is chosen in formulation of the error analysis so that the nonlinear differential operators defined by the differential equations are nonlinear fredholm operators of index 1 finite element solutions are defined in a natural way and several a priori estimates are proved on regular branches and on branches around turning points in the proofs the extended implicit function theorem due to brezzi et al 1980 plays an essential role

on the calculation of limit and bifurcation points in stability problems of elastic shells abstract numerical methods for the computation of singular points of nonlinear equations g u 0 are discussed where and are real parameters simple and double limit points are treated in some detail and numerical algorithms are presented and applied to elastic shell stability problems the case of simple symmetry breaking bifurcation points is also treated with applications to nonsymmetric bifurcation from axisymmetric states of deformation of shells of revolution

perturbation scheme for online learning of features incremental principal component analysis a new incremental online feature extraction approach is proposed based on principal component analysis in conjunction with perturbation theory which for its validity requires the perturbation parameter to be small our approach is found to be computationally more efficient in comparison to batch method which for its applicability requires simultaneous availability of all observations for computation of features it is found on the basis of numerical experiments that the results based on our approach besides being in good agreement with the batch method and other incremental methods are also computationally more efficient to demonstrate the efficacy of the proposed scheme experiments have been performed on randomly generated datasets as well as on low and high dimensional datasets i e uci and face datasets which are available in public domain

svd gfd scheme to simulate complex moving body problems in 3d space abstract the present paper presents a hybrid meshfree and cartesian grid method for simulating moving body incompressible viscous flow problems in 3d space the method combines the merits of cost efficient and accurate conventional finite difference approximations on cartesian grids with the geometric freedom of generalized finite difference gfd approximations on meshfree grids error minimization in gfd is carried out by singular value decomposition svd the arbitrary lagrangian eulerian ale form of the navier stokes equations on convecting nodes is integrated by a fractional step projection method the present hybrid grid method employs a relatively simple mode of nodal administration nevertheless it has the geometrical flexibility of unstructured mesh based finite volume and finite element methods boundary conditions are precisely implemented on boundary nodes without interpolation the present scheme is validated by a moving patch consistency test as well as against published results for 3d moving body problems finally the method is applied on low reynolds number flapping wing applications where large boundary motions are involved the present study demonstrates the potential of the present hybrid meshfree and cartesian grid scheme for solving complex moving body problems in 3d

decomposable principal component analysis in this paper we consider principal component analysis pca in decomposable gaussian graphical models we exploit the prior information in these models in order to distribute pca computation for this purpose we reformulate the pca problem in the sparse inverse covariance concentration domain and address the global eigenvalue problem by solving a sequence of local eigenvalue problems in each of the cliques of the decomposable graph we illustrate our methodology in the context of decentralized anomaly detection in the abilene backbone network based on the topology of the network we propose an approximate statistical graphical model and distribute the computation of pca

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

iterative kernel principal component analysis for image modeling in recent years kernel principal component analysis kpca has been suggested for various image processing tasks requiring an image model such as e g denoising or compression the original form of kpca however can be only applied to strongly restricted image classes due to the limited number of training examples that can be processed we therefore propose a new iterative method for performing kpca the kernel hebbian algorithm which iteratively estimates the kernel principal components with only linear order memory complexity in our experiments we compute models for complex image classes such as faces and natural images which require a large number of training examples the resulting image models are tested in single frame super resolution and denoising applications the kpca model is not specifically tailored to these tasks in fact the same model can be used in super resolution with variable input resolution or denoising with unknown noise characteristics in spite of this both super resolution and denoising performance are comparable to existing methods

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

an algorithm for piecewise linear approximation of an implicitly defined manifold a simplicial method is used to approximate the solution manifold to a system of nonlinear equations h x theta where h mathbb r n k to mathbb r n the method begins at a point x 0 in the solution set where the derivative dh x 0 is of full rank given any varepsilon 0 a piecewise linear manifold is constructed along which h x infty varepsilon an algorithm is presented to carry out this construction in an efficient fashion

manpak a set of algorithms for computations on implicitly defined manifolds abstract mathematical models often involve differentiable manifolds that are implicitly defined as the solution sets of systems of nonlinear equations the resulting computational tasks differ considerably from those arising for manifolds defined in parametric form here a collection of algorithms is presented for performing a range of essential tasks on general implicitly specified submanifolds of a finite dimensional space this includes algorithms for determining local parametrizations and their derivatives and for evaluating quantities related to the curvature with sensitivity measures the methods have been implemented as a fortran 77 package called manpak

analysis and finite element approximation of optimal control problems for the stationary navier stokes equations with distributed and neumann controls we examine certain analytic and numerical aspects of optimal control problems for the stationary navier stokes equations the controls considered may be of either the distributed or neumann type the functionals minimized are either the viscous dissipation or the l4 distance of candidate flows to some desired flow we show the existence of optimal solutions and justify the use of lagrange multiplier techniques to derive a system of partial differential equations from which optimal solutions may be deduced we study the regularity of solutions of this system then we consider the approximation by finite element methods of solutions of the optimality system and derive optimal error estimates

efficient numerical path following beyond critical points the paper presents a numerical pathfollowing technique based on an implicit change of parametrization using qr decomposition an automatic steplength control for tangent continuation is derived on a theoretical basis in the suggested approach turning points do not play any exceptional role for the actual computation of simple bifurcation points a special algorithm based on the augmented system of moore is developed this treatment includes imperfect bifurcations and permits an easy way of branch switching an efficient linear equation solver is worked out in connection with the qr decomposition finally numerical comparisons are included that clearly document the efficiency of the newly developed algorithms

on the sensitivity of solutions of parameterized equations the sensitivity of a solution of a parameterized equation f z lambda 0 with respect to the parameter vector lambda is usually defined as the change of the state z in dependence of lambda in other words for any solution expressible in the form z lambda lambda with some smooth function z z lambda the sensitivity is the derivative dz lambda typically the solutions form a manifold m in the product of 1the state space and the parameter space and this sensitivity is available only at those points of m where the parameters can be used to define a local coordinate system this paper introduces a general sensitivity concept which applies at all solutions on m and which includes the earlier definition some general geometric interpretations of the new measure are presented and it is shown that the sensitivity analysis can be easily integrated into the solution process the theory also suggests the introduction of a readily computable second order sensitivity measure reflectin

on the theory and error estimation of the reduced basis method for multi parameter problems abstract in an earlier paper zamm 63 1983 21 j p fink and w c rheinboldt developed a priori local error estimates for the scalar parameter case of the reduced basis method by considering the method in a differential geometric setting here it is shown that an analogous setting can be used for the analysis of the method applied to problems with a multidimensional parameter vector and that this leads to a corresponding local error theory also in this general case

a fictitious domain method for external incompressible viscous flow modeled by navier stokes equations abstract in this article we discuss a fictitious domain method for the numerical solutions of three dimensional elliptic problems with dirichlet boundary conditions and also of the navier stokes equations modeling incompressible viscous flow the methodology for the navier stokes equations described here takes a systematic advantage of time discretization by operator splitting in order to treat separately advection imbedding and incompressibility due to the decoupling fast elliptic solvers can be used to treat the incompressibility condition even if the original problem is taking place on a nonregular geometry the resulting methodology is applied to two dimensional unsteady external incompressible viscous flow problems and three dimensional stokes problems

a low numerical dissipation immersed interface method for the compressible navier stokes equations a numerical method to solve the compressible navier stokes equations around objects of arbitrary shape using cartesian grids is described the approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite difference discretization the method is suitable for compressible flows without shocks and can be classified as an immersed interface method the objects are sharply captured by the cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes located around the boundary in contrast with available methods no jump conditions are used or explicitly derived from the boundary conditions although a number of elements are adopted from previous immersed interface approaches a new element in the present approach is the use of the summation by parts formalism to develop stable non stiff first order derivative approximations at the irregular grid points second order derivative approximations as those appearing in the transport terms can be stiff when irregular grid points are located too close to the boundary this is addressed using a semi implicit time integration method moreover it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different mach and reynolds numbers

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

analysis of the l2 least squares finite element method for the velocity vorticity pressure stokes equations with velocity boundary conditions a theoretical analysis of the l 2 least squares finite element method lsfem for solving the stokes equations in the velocity vorticity pressure vvp first order system formulation with the dirichlet velocity boundary condition is given the least squares energy functional is defined to be the sum of the squared l 2 norms of the residuals in the partial differential equations weighted appropriately by the viscosity constant n it is shown that with many advantages the method is stable and convergent without requiring extra smoothness of the exact solution and the piecewise linear finite elements can be used to approximate all the unknowns furthermore with respect to the order of approximation for smooth exact solutions the method exhibits an optimal rate of convergence in the h 1 norm for velocity and a suboptimal rate of convergence in the l 2 norm for vorticity and pressure some numerical experiments in two and three dimensions are given which confirm the a priori error estimates since the boundary of the bounded domain under consideration is polygonal in r 2 or polyhedral in r 3 instead of c 1 smooth the authors adopt the more direct technique of bramble pasciak and cai manteuffel mccormick which departs from the agmon douglis nirenberg theory in showing the coercivity bound of the least squares functional

analysis of piecewise linear approximations to the generalized stokes problem in the velocity stress pressure formulation in this paper we study continuous piecewise linear polynomial approximations to the generalized stokes equations in the velocity stress pressure first order system formulation by using a cell vertex finite volume least squares scheme this method is composed of a direct cell vertex finite volume discretization step and an algebraic least squares step where the least squares procedure is applied after the discretization process is accomplished this combined approach has the advantages of both finite volume and least squares approaches an error estimate in the h1 product norm for continuous piecewise linear approximating functions is derived it is shown that with respect to the order of approximation for h2 regular exact solutions the method exhibits an optimal rate of convergence in the h1 norm for all unknowns velocity stress and pressure

preconditioning applied to variable and constant density flows a time derivative preconditioning of the navier stokes equations suitable for both variable and constant density fluids is developed the ideas of low mach number preconditioning and artificial compressibility are combined into a unified approach designed to enhance convergence rates of density based time marching schemes for solving flows of incompressible and variable density fluids at all speeds the preconditioning is coupled with a dual time stepping scheme implemented within an explicit multistage algorithm for solving time accurate flows the resultant time integration scheme is used in conjunction with a finite volume discretization designed for unstructured solution adaptive mesh topologies this method is shown to provide accurate steady state solutions for transonic and low speed flow of variable density fluids the time accurate solution of unsteady incompressible flow is also demonstrated