an invariant imbedding method for computation of eigenlengths of singular two point boundary value problems in this article we have described an invariant imbedding method for calculating the smallest eigenlength of a singular tpbvp with the singularity at the origin the invariant imbedding yields a first order nonlinear equation called a riccati equation and also gives the initial conditions at the origin for this equation with the aid of theorem 8 in section 3 we numerically integrate the riccati equation to blowup which gives our computed eigenlength in closing we would like to comment on the numerical merits of the integration to blowup technique on the basis of the examples presented it appears that this technique combined with the available numerical integrators with variable step size is capable of producing accurate results the feature of a variable step size is essential as the value of z approaches the actual eigenlength however it is desirable to have a priori estimate or bounds of the eigenlength similar to those of boland and nelson 2 for the nonsingular case the singular system however presents difficulties due to the lack of sign conditions on the coefficient matrices in obtaining such bounds hopefully an investigation of the matrix r z will yield these results

quasilinear multistep methods and variable step predictor corrector methods for neutral functional differential equations the theory of quasilinear multistep methods and predictor corrector methods for neutral functional differential equations is developed the theorem about the asymptotic behaviour of the global discretization error of quasilinear methods which forms a theoretical basis for step control strategy for predictor corrector methods for neutral equations is given some problems of implementation of these methods are also addressed and the results of numerical experiments are presented

variable step size predictor corrector schemes for second kind volterra integral equations in this paper a family of implicit multistep methods for the solution of volterra integral equations is derived these methods together with an explicit euler predictor permit the use of a variable step size when solving integral equations means of controlling the error and stability by varying the step size and the order of the method are described extensive numerical results are presented

the sdfem for a convection diffusion problem with a boundary layer optimal error analysis and enhancement of accuracy the streamline diffusion finite element method sdfem is applied to a convection diffusion problem posed on the unit square using a shishkin rectangular mesh with piecewise bilinear trial functions the hypotheses of the problem exclude interior layers but allow exponential boundary layers an error bound is proved for u i u n sd where u i is the interpolant of the solution u u n is the sdfem solution and cdot sd is the streamline diffusion norm this bound implies that u u n l 2 is of optimal order thereby settling an open question regarding the l 2 accuracy of the sdfem on rectangular meshes furthermore the bound shows that u n is superclose to u i which allows the construction of a simple postprocessing that yields a more accurate solution enhancement of the rate of convergence by using a discrete streamline diffusion norm is also discussed finally the verification of these rates of convergence by numerical experiments is examined and it is shown that this practice is less reliable than was previously believed

linear conservation laws for odes abstract some physical systems described by odes have quantities that are conserved as the system evolves runge kutta formulas and linear multistep methods preserve linear conservation laws automatically a code for the integration of odes involves a number of algorithms in addition to the basic formula it is shown here that popular codes preserve linear conservation laws if they are used properly an application is made to differential algebraic equations of index 2 arising in the solution of the navier stokes equation by the method of lines

a multi purpose system for the numerical integration of ode s in this paper we describe the development of an experimental multi purpose integration system strongly modular the system allows the use of wide classes of multistep methods for solving ode s singulsr perturbation problems as well as implicit differential algebraic equations we focus our attention on a new family of collocation methods for nonstiff problems while these methods are strongly related to the stiffly stable bdf methods they are particularly well suited for the implementation of a type insensitive integration algorithm compound discretizations in which certain variables are integrated with the first method and others with the second can also be used a stability analysis is given and some implementation issues are discussed

the modified newton method in the solution of stiff ordinary differential equations this paper presents an analysis of the modified newton method as it is used in codes implementing implicit formulae for integrating stiff ordinary differential equations we prove that near a smooth solution of the differential system when the jacobian is essentially negative dominant and slowly varying the modified newton iteration is contractive converging to the locally unique solution whose existence is hereby demonstrated of the implicit equations this analysis eliminates several common restrictive or unrealistic assumptions and provides insight for the design of robust codes 1 background results significance 1 1 prototype stiff problems their salient properties note the structure o the solution of the model differential equation 1 1 y ay cost a 1 namely y t eit y 0 2 1 2 sin t a cos t there is an initial transient of duration o ja 1 l log ji after which the tern eit is not active and the solution is as smooth as cos t under suitable conditions see 29 and infra solutions of the stiff time varying linear system n 1 2 y b t y g t y e r have the same structure y t is the sum of a smooth particular solution y t and a transient v t the transient a solution of the homogeneous equation v t b t v t v o y o 0 expires after a short time meanwhile ys t and its derivative have bounds expressible in terms of 1 dvb dvg 1 3 b t w b t v o 1 dtv received september 28 1989 revised august 10 1990 1980 mathematics subject classification 1985 revision primary 65l05 65h10

a singular nonlinear boundary value problem membrane response of a spherical cap the nonlinear boundary value problem y n x 2 32y 2 lambda 2 8 quad 0 0 nu 0 are constants models the large deflection membrane response of a spherical cap and has been solved numerically by various techniques in the engineering literature we prove existence and uniqueness of a solution of this problem provide qualitative information about the solution and use this information to suggest an approach to numerical computation

stiffness characteristic and the applications of the method of lines on nonuniform grids the method of lines is investigated for the numerical solution of the stream function and vorticity form of the navier stokes equations on nonuniform grids stiffness characteristics of a linear one dimensional model equation are examined to establish the feasibility of applying the method to the vorticity equation in two dimensions the governing equations are transformed from the physical domain with a highly variable grid to a computational domain with a uniform grid the method of lines is used to solve only the vorticity equation and the successive over relaxation technique is used to solve the stream function equation it is observed that the transformed governing equations become stiffer with increased concentration of grid points and also as the number of grid points increases it is also observed that the differencing technique affects the stiffness characteristics the use of forward differencing is not feasible and backward differencing is preferable to central differencing for high reynolds numbers the results of specific applications for the solution of flow in curved wall diffusers and a driven cavity demonstrate that the method of lines under certain circumstances is feasible for the numerical solution of physical problems on domains covered with variable grids

finite element galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems abstract this paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or more generally galerkin methods suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let m lambda denote the space of eigenvectors corresponding to lambda keywords hilbert space

on the exponential fitting of composite multiderivative linear multistep methods exponentially fitted composite multiderivative linear multistep methods of orders up to 5 are derived a numerical investigation of these formulae indicates that they may be a stable for all choices of the fitting parameters an important aspect of the class of integration formulae considered is that each formula contains a built in local error estimate a facility lacking in most other exponentially fitted formulae some preliminary numerical results are given and these show that for problems for which exponential fitting is appropriate the new formulae provide a large saving in computational effort compared with conventional ones

numerical treatment of o d es the theory ofa methods almost all commonly used methods for o d es and their most miscellaneous compositions area methods i e they can be reduced toz o z j az j 1 h x j 1 z j 1 z j h z j s a s s j 1 1 m this paper presents a general theory fora methods and discusses its practical consequences an analysis of local discretization error l d e accumulation results in a general order criterium and reveals which part of the l d e effectively influences the global error this facilitates the comparison of methods and generalizes considerably the concept of error constants it is shown as a consequence that the global error cannot be safely controlled by the size of the l d e and that the conventional error control may fail in important cases furthermore butcher s effective order methods the concept of nordsieck forms and gear s interpretation of lineark step schemes as relaxation methods are generalized the stability of step changing is shortly discussed

variable order adams codes abstract variable step size variable order vsvo adams codes are very effective for solving initial value problems for first order systems of ordinary differential equations the theory of fixed order codes is classical but when the order is varied there is no theory explaining fundamental issues with realistic assumptions about order and step size selection we prove convergence approximate locally the behavior of the error and justify standard error estimators

numerical solution of differential algebraic equations in mechanical systems simulation abstract the numerical solution of the differential algebraic equations of motion of mechanical systems offers many computational challenges in this paper we describe progress which has been made in understanding the formulation of the equations of motion from the viewpoint of numerical stability and outline some of the difficulties which must be resolved for efficient and reliable numerical methods in real time simulation of mechanical systems

type insensitive ode codes based on implicit a stable formulas a special concept of stiffness is appropriate for implict a stable formulas it is possible to recognize this kind of stiffness economically and reliably using information readily available during the integration of an ode using this development a variety of effective ode solvers could be made insensitive to the type of problem i e the code would automatically recognize and alter automatically its algorithm at any step depending on whether the problem is stiff there

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

hp adaptive time integration based on the bdf for viscous flows this paper presents a procedure based on the backward differentiation formulas of order 1 to 5 to obtain efficient time integration of the incompressible navier stokes equations the adaptive algorithm performs both stepsize and order selections to control respectively the solution accuracy and the computational efficiency of the time integration process the stepsize selection h adaptivity is based on a local error estimate and an error controller to guarantee that the numerical solution accuracy is within a user prescribed tolerance the order selection p adaptivity relies on the idea that low accuracy solutions can be computed efficiently by low order time integrators while accurate solutions require high order time integrators to keep computational time low the selection is based on a stability test that detects growing numerical noise and deems a method of order p stable if there is no method of lower order that delivers the same solution accuracy for a larger stepsize hence it guarantees both that 1 the used method of integration operates inside of its stability region and 2 the time integration procedure is computationally efficient the proposed time integration procedure also features a time step rejection and quarantine mechanisms a modified newton method with a predictor and dense output techniques to compute solution at off step points

efficient and robust implementation of the tlismni method abstract the dynamics of large scale and complex multibody systems mbs that include flexible bodies and contact impact pairs is governed by stiff equations because explicit integration methods can be inefficient and often fail in the case of stiff problems the use of implicit numerical integration methods is recommended in this case this paper presents a new and efficient implementation of the two loop implicit sparse matrix numerical integration tlismni method proposed for the solution of constrained rigid and flexible mbs differential and algebraic equations the tlismni method has desirable features that include avoiding numerical differentiation of the forces allowing for an efficient sparse matrix implementation and ensuring that the kinematic constraint equations are satisfied at the position velocity and acceleration levels in this method a sparse lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is used to solve for all the accelerations and lagrange multipliers the generalized coordinate partitioning or recursive methods can be used to satisfy the constraint equations at the position and velocity levels in order to improve the efficiency and robustness of the tlismni method the simple iteration and the jacobian free newton krylov approaches are used in this investigation the new implementation is tested using several low order formulas that include hilber hughes taylor hht l stable park a stable trapezoidal and a stable bdf methods the hht method allows for including numerical damping discussion on which method is more appropriate to use for a certain application is provided the paper also discusses tlismni implementation issues including the step size selection the convergence criteria the error control and the effect of the numerical damping the use of the computer algorithm described in this paper is demonstrated by solving complex rigid and flexible tracked vehicle models railroad vehicle models and very stiff structure problems the results obtained using these low order formulas are compared with the results obtained using the explicit adams bashforth predictor corrector method using the tlismni method which does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation for solving complex and stiff structure problems leads to significant computational cost saving as demonstrated in this paper in some problems it was found that the new tlismni implementation is 35 times faster than the explicit adams bashforth method

a sparse finite element method with high accuracy part i in this paper we develop and analyze a new finite element method called the sparse finite element method for second order elliptic problems this method involves much fewer degrees of freedom than the standard finite element method we show nevertheless that such a sparse finite element method still possesses the superconvergence and other high accuracy properties same as those of the standard finite element method the main technique in our analysis is the use of some integral identities

run time estimation of the spectral radius of jacobians it is useful for ordinary differential equation ode solvers to include an estimator of the spectral radius of the jacobian matrix of the system of ode s since this determines the numerical stability of the method hence a knowledge of spectral radius enables the run time selection of a more efficient integrator some techniques for estimating spectral radius are described and compared they include methods suitable for use with any ode solvers but which require additional computation other methods are described which are suitable with runge kutta or rosenbrock methods and which require little extra computation

on the computation of the eigenproblems of hydrogen helium in strong magnetic and electric fields with the sparse grid combination technique abstract we introduce the combination technique for the numerical solution of d dimensional eigenproblems on sparse grids here o d log n d 1 different problems each of size o n have to be solved independently this is in contrast to the one problem of size o n d for a conventional finite element discretization where n denotes the number of grid points in one coordinate direction therefore also higher dimensional eigenvalue problems can be treated by our sparse grid combination approach we apply this method to solve the three dimensional schrodinger equation for hydrogen one electron problem and the six dimensional schrodinger equation for helium two electron problem in strong magnetic and electric fields

an algorithm for starting multistep methods the present paper deals with the problem of starting multistep methods we take into consideration an adams bashforth moulton pece pair with the predictor of order q and the corrector of order q 1 to start this method q 1 starting values are necessary in addition to y0 a well known result from theory says that the order of convergence of the whole integration is q 1 if all those starting values are accurate of that same order present production codes start with a predictor of order 1 and a corrector of order 2 at the first step and then proceed step by step each time raising the order by 1 until all the necessary starting values have been obtained but in this manner all the starting errors keep of order 3 and so the whole integration converges no faster than that order this drawback is normally compensated for by taking very small step sizes in the starting phase the general algorithm we propose furnishes at a reasonably low cost the necessary number q 1 of starting values each of the appropriate order q 1 whatever q might be it is independent of the particular multistep formula considered and is mainly designed to be used for high values of q q 10 where the alternative strategies are too expensive or do not exist at all the numerical results reported show the validity of our approach 2003 elsevier science ltd all rights reserved

sensitivity analysis of flexible multibody systems using composite materials components in this paper a general formulation for the computation of the first order analytical sensitivities based on the direct method using automatic differentiation of flexible multibody systems is presented the direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible multibody systems of composite materials with respect to the design variables which are the ply orientations of the laminated in order to appraise the benefits of the approach suggested and to highlight the risks of the procedure the analytical sensitivities are compared with the numerical results obtained by using the finite difference method for the beam composite material elements the section properties and their sensitivities are found using an asymptotic procedure that involves a two dimensional 2 d finite element analysis of their cross section the equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems the equations of motion and the sensitivities of the flexible multibody system are solved and the accelerations velocities and the sensitivities of accelerations and velocities are integrated in time using a multi step multi order integration algorithm through the application of the methodology to two simple flexible multibody systems the difficulties and benefits of the procedure with respect to finite difference approaches or to the direct implementation of the analytic sensitivities are discussed copyright 2008 john wiley sons ltd

a note on the relative efficiency of adams methods abstract in several pieces of the most effective mathematical software for non stiff and mildly stiff initial value problems in ordinary differential equations the classical adams methods are implemented in a particular mode one reason for the choice of mode is sometimes explained in terms of the associated scaled absolute stability regions for the methods in this note it is shown that the same choice of mode remains appropriate from this point of view when certain families of generalized adams type methods are considered

error analysis of the combination technique the combination technique is a method to reduce the computational time in the numerical approximation of partial differential equations in this paper we present a new technique to analyze the convergence rate of the combination technique this technique is applied to general second order elliptic differential equations in two dimensions furthermore it is proved that the combination technique for poisson s equation convergences in arbitrary dimensions

approximation of weak adjoints by reverse automatic differentiation of bdf methods we shed light on the relation between the discrete adjoints of multistep backward differentiation formula bdf methods and the solution of the adjoint differential equation to this end we develop a functional analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations we devise a petrov galerkin finite element fe interpretation of the bdf method and its discrete adjoint scheme obtained by reverse internal numerical differentiation we show how the fe approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation we also show convergence of the discrete adjoints to the classical adjoints on the inner time interval finally we give numerical results for non adaptive and fully adaptive bdf schemes the presented framework opens the way to carry over techniques on global error estimation from fe methods to bdf methods

non rigid visible and infrared face registration via regularized gaussian fields criterion registration of multi sensor data particularly visible color sensors and infrared sensors is a prerequisite for multimodal image analysis such as image fusion typically the relationships between image pairs are modeled by rigid or affine transformations however this cannot produce accurate alignments when the scenes are not planar for example face images in this paper we propose a regularized gaussian fields criterion for non rigid registration of visible and infrared face images the key idea is to represent an image by its edge map and align the edge maps by a robust criterion with a non rigid model we model the transformation between images in a reproducing kernel hilbert space and a sparse approximation is applied to the transformation to avoid high computational complexity moreover a coarse to fine strategy by applying deterministic annealing is used to overcome local convergence problems the qualitative and quantitative comparisons on two publicly available databases demonstrate that our method significantly outperforms the state of the art method with an affine model as a result our method will be beneficial for fusion based face recognition highlightswe analyze the robustness of gaussian fields criterion both in theory and experiment the gaussian fields criterion is generalized from rigid to the non rigid case we propose a new non rigid registration method to deal with more real world problems a sparse approximation on the transformation is applied to speed up the method we customize and apply the proposed method to visible thermal ir face registration

icp registration using invariant features investigates the use of euclidean invariant features in a generalization of iterative closest point icp registration of range images pointwise correspondences are chosen as the closest point with respect to a weighted linear combination of positional and feature distances it is shown that under ideal noise free conditions correspondences formed using this distance function are correct more often than correspondences formed using the positional distance alone in addition monotonic convergence to at least a local minimum is shown to hold for this method when noise is present a method that automatically sets the optimal relative contribution of features and positions is described this method trades off the error in feature values due to noise against the error in positions due to misalignment experimental results suggest that using invariant features decreases the probability of being trapped in a local minimum and may be an effective solution for difficult range image registration problems where the scene is very small compared to the model

efficient kernel density estimation using the fast gauss transform with applications to color modeling and tracking many vision algorithms depend on the estimation of a probability density function from observations kernel density estimation techniques are quite general and powerful methods for this problem but have a significant disadvantage in that they are computationally intensive in this paper we explore the use of kernel density estimation with the fast gauss transform fgt for problems in vision the fgt allows the summation of a mixture of ill gaussians at n evaluation points in o m n time as opposed to o mn time for a naive evaluation and can be used to considerably speed up kernel density estimation we present applications of the technique to problems from image segmentation and tracking and show that the algorithm allows application of advanced statistical techniques to solve practical vision problems in real time with today s computers

varying the componentwise order of the multistep methods in solving odes and its absolute stability most variable order variable stepsize codes for solving odes vary the order along the interval of integration but keep the order constant across the system however some codes that do partitioning of the odes into stiff and nonstiff subsystems vary the order for each equation in the system in this paper the reasons for varying the order componentwise are given and some of its salient features are illustrated using a numerical example also given is the direct proof of the existence of a region of absolute stability when different order adams explicit methods are used to solve a system of odes

solving delay differential equations using componentwise partitioning by runge kutta method embedded singly diagonally implicit runge kutta sdirk method is used to solve stiff systems of delay differential equations ddes the delay argument is approximated using hermite interpolation initially the whole system is considered as nonstiff and solved using simple iteration when stiffness is indicated the appropriate equation is placed into the stiff subsystem and solved using newton iteration this type of partitioning is called componentwise partitioning the process is continued until all the equations have been placed in the right subsystem numerical results based on componentwise partitioning and intervalwise partitioning are tabulated and compared

numerical treatment of large scale air pollution models abstract studying air pollution phenomena over a large space domain by a fairly general mathematical model is discussed the space discretization of such a model leads to huge system of odes the integration algorithm used in the solution of these systems must be efficient if a 3 d model is considered then a vector processor is to be used moreover the most time consuming parts of the code are to be vectorized the calculated results concentrations of different air pollutants must be reliable because they have to be used by other specialists therefore different checks concerningthe reliability of the results are carried out in this paper it is shown that an efficient and reliable code for studying both sulphur and nitrogen pollution has been developed at the danish agency of environmental protection some simulation processes are also described breifly

a two grid discretization scheme for eigenvalue problems a two grid discretization scheme is proposed for solving eigenvalue problems including both partial differential equations and integral equations with this new scheme the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy

conservation laws and the numerical solution of odes ii abstract many physical systems described by an initial value problem for a system of ordinary differential equations odes conserve physical quantities such as the net charge or total energy as the system evolves typical codes for the numerical solution of the odes will not conserve these quantities and this can lead to solutions which are not even qualitatively correct one way to impose conservation laws is to perturb the numerical solution at each step of the integration a simple theory is developed in this paper which tells how this should be done so as to guarantee convergence of codes based on one step methods it is also easy to interpret the effect on the accuracy of the perturbations

strategies for solving index one dae with non negative constraints liquid liquid extraction modeling leads to solve an index one dae system for the sake of robustness it is desirable to account for non negative constraints based on the dassl architecture a classical index one dae solver we propose and compare three different strategies to implement these bound constraints each of these strategies corresponds to a different newton modification clipping damping or interior point method the comparisons are made on two test cases the robertson ode problem and an example from liquid liquid extraction modeling

efficient solution of nonlinear ordinary differential equations expressed in s system canonical form ordinary differential equations can be recast into a nonlinear canonical form called an s system evidence for the generality of this class comes from extensive empirical examples that have been recast and from the discovery that sets of differential equations and functions recognized as among the most general are special cases of s systems identification of this nonlinear canonical form suggests a radically different approach to numerical solution of ordinary differential equations by capitalizing on the regular structure of s systems efficient formulas for a variable order variable step taylor series method are developed the minimum improvement in efficiency over conventional runge kutta methods is shown to be more than 20 to 60 percent for equations expressed in s system form tests with implemented methods demonstrate that 10 to 100 fold reductions in time are actually realized for solution of s systems to examine the potential benefit of recasting ordinary differential equations and then sol

using nonconvergence of iteration to partition odes given is a variation of partitioning odes when using the multistep method essentially the nonconvergence property is made use of for partitioning the criteria used are explained by using a test system of two linear equations and generalisation is made to a general system of first order odes finally some numerical results are given that include systems greater than two

rkn type parallel block pc methods with lagrange type predictors abstract this paper describes the construction of block predictor corrector methods based on runge kutta nystrom correctors our approach is to apply the predictor corrector method not only at step points but also at off step points block points so that in each step a whole block of approximations to the exact solution at off step points is computed in the next step these approximations are used to obtain a high order predictor formula using lagrange interpolation by suitable choice of the abscissas of the off step points a much more accurately predicted value is obtained than by predictor formulas based on last step values since the block of approximations at the off step points can be computed in parallel the sequential costs of these block predictor corrector methods are comparable with those of a conventional predictor corrector method furthermore by using runge kutta nystrom corrector methods the computation of the approximation at each off step point is also highly parallel application of the resulting block predictor corrector methods to a few widely used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature

postprocessed two scale finite element discretizations part i this paper studies a simple postprocessing of two scale finite element discretizations of elliptic partial differential operators where boundary value problems and eigenvalue problems on tensor product domains omega subset mathbb r d are both examined and conforming piecewise d linear elements are used for d geq2 suppose that the two scale method is applied using a coarse mesh and some univariate fine meshes satisfying h mathcal o h 2 3 where h and h are the fine and coarse mesh widths it is shown for both classes of problem that the postprocessed two scale finite element solution achieves the same order of accuracy in the h 1 omega norm as the postprocessed standard finite element solution but the former solution uses only mathcal o h 2d 1 3 degrees of freedom compared with the mathcal o h d degrees of freedom required by the latter

gaussian fields a new criterion for 3d rigid registration abstract this paper introduces a new and simple criterion for rigid registration based on gaussian fields the criterion is always differentiable and convex in a large neighborhood of the alignment parameters allowing for the use of well proven optimization techniques using this method we can extend the size of the region of convergence so that no close initialization is needed thus overcoming local convergence problems of iterative closest point algorithms furthermore the gaussian energy function can be evaluated with linear complexity using the fast gauss transform which permits efficient implementation of the registration algorithm experimental analysis on real world data sets shows the usefulness and points the limits of the approach

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

construction of variable stepsize multistep formulas a systematic way of extending a general fixed stepsize multistep formula to a minimum storage variable stepsize formula has been discovered that encompasses fixed coefficient interpolatory variable coefficient variable step and fixed leading coefficient as special cases in particular it is shown that the interpolatory stepsize changing technique of nordsieck leads to a truly variable stepsize multistep formula which has implications for local error estimation and formula changing and it is shown that the variable step stepsize changing technique applicable to the adams and backward differentiation formulas has a reasonable generalization to the general multistep formula in fact it is shown how to construct a variable order family of variable coefficient formulas finally it is observed that the first dahlquist barrier does not apply to adaptable multistep methods if storage rather than stepnumber is the key consideration

implicit schemes for differential equations abstract implicit high order accurate temporal integration schemes are developed to reduce partial differential equations of evolution type to a sequence of ordinary differential equations involving the spatial directions only the ordinary differential equations are solved using various accurate integration methods the schemes are free from extraneous boundary and or initial conditions and are stable for large courant number

the projected gradient methods for least squares matrix approximations with spectral constraints the problems of computing least squares approximations for various types of real and symmetric matrices subject to spectral constraints share a common structure this paper describes a general procedure in using the projected gradient method it is shown that the projected gradient of the objective function on the manifold of constraints usually can be formulated explicitly this gives rise to the construction of a descent flow that can be followed numerically the explicit form also facilitates the computation of the second order optimality conditions examples of applications are discussed with slight modifications the procedure can be extended to solve least squares problems for general matrices subject to singular value constraints

efficient calculation of chemically reacting flow a semi implicit finite volume formulation is used to jrvstudy flow with chemical reactions in this formulation the source terms resulting from the chemical reaction are treated implicitly and the resulting system of partial differential equations is sovled using two time stepping schemes the first is based on the runge kutta method and the second on an adams predictor corrector method results show that improvements in computational efficiency depend to a large extent on the manner in which the source term is treated furthermore analysis and computation indicate that the rungekutta method is more efficient than the adams methods for these systems of differential equations finally an adaptive time stepping scheme is developed to study problems involving shock ignition calculations for a hydrogen air system agree well with other methods

on noncharacteristic boundary conditions for discrete hyperbolic initial boundary value problems abstract the stability properties of some discrete hyperbolic initial boundary value problems involving two dependent variables and noncharacteristic boundary conditions are examined approximations based on the lax wendroff leap frog and crank nicolson methods are considered the most accurate approximations are found to be the least stable the results for the model problems can be used in the stability analysis of larger problems where characteristics occur singly or in pairs in particular discretizations of the shallow water and gas dynamics equations which involve boundary conditions written in natural variables are considered some computations demonstrating the stability results are discussed an appendix discussing numerical stability verification techniques is included

convergence of the combination technique for second order elliptic differential equations the combination technique is an algorithm for the approximate solution of partial differential equations on sparse grids that has to be combined with a suitable standard discretization the advantage of the combination technique compared to the standard discretization is that the same accuracy is achieved with many fewer grid points in this paper the combination technique is used with a bilinear finite element discretization depending on the smoothness of the solution and the coefficients it is proved for general second order elliptic differential equations on the unit square that the combined solution converges with order o h or o h log h 1 in the energy norm and with order o h2 log h 1 or o h3 2 in the l2 norm respectively this holds even if the bilinear form corresponding to the elliptic equation is not symmetric positive definite the proof does not use an asymptotic error expansion but sobolev space techniques

an efficient approach for the numerical simulation of multibody systems the field of kinematics and dynamics of mechanical systems has progressed from a manual graphics art to a highly developed discipline in analytical geometry and dynamics various general purpose formulations for the dynamic analysis of constrained mechanical systems cms lead to mixed differential algebraic equations daes called the euler lagrange equations during the past fifteen years many contributions have been made to the theory of computational kinematics and dynamics of cms also called mutibody dynamics the recent advances in computer hardware and software have tremendously revolutionized the analysis of cms there are various numerical approaches for solving general vector fields in the previous paper appl math comput 92 1998 153 193 a complete and detailed analysis of various approaches for the numerical solution of vector fields is given in this paper we extend the algorithms presented in the above mentioned reference to solve the euler lagrange equations of motion for cms the numerical experiments suggest that perturbation approach performs better than the other approaches

a two loop sparse matrix numerical integration procedure for the solution of differential algebraic equations application to multibody systems abstract in this paper a two loop implicit sparse matrix numerical integration tlismni procedure for the solution of constrained rigid and flexible multibody system differential and algebraic equations is proposed the proposed method ensures that the kinematic constraint equations are satisfied at the position velocity and acceleration levels in this method a sparse lagrangian augmented form of the equations of motion that ensures that the constraints are satisfied at the acceleration level is first used to solve for all the accelerations and lagrange multipliers the independent coordinates and velocities are then identified and integrated using htt or newmark formulas expressed in this paper in terms of the independent accelerations only the constraint equations at the position level are then used within an iterative newton raphson procedure to determine the dependent coordinates the dependent velocities are determined by solving a linear system of algebraic equations in order to effectively exploit efficient sparse matrix techniques and have minimum storage requirements a two loop iterative method is proposed equally important the proposed method avoids the use of numerical differentiation which is commonly associated with the use of implicit integration methods in multibody system algorithms numerical examples are presented in order to demonstrate the use of the new integration procedure

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

extrapolation combination and sparse grid techniques for elliptic boundary value problems several variants of extrapolation can be used for the solution of partial differential equations there are richardson extrapolation truncation error extrapolation and extrapolation of the functional in multi dimensional problems multivariate error expansions can be exploited by multivariate extrapolation where the asymptotic expansions in different mesh parameters are exploited particularly interesting cases are the combination technique that uses all the grids that have a constant product of the meshspacings in the different coordinate directions another related technique is the sparse grid finite element technique that can be interpreted as a combination extrapolation of the functional

removing mismatches for retinal image registration via multi attribute driven regularized mixture model in order to address the problem of retinal image registration this paper proposes and analyzes a novel and general matching algorithm called multi attribute driven regularized mixture model mad rmm mismatches removal can play a key role in image registration which refers to establish reliable matches between two point sets here the presented approach starts from multi feature attributes which are used to guide the feature matching to identify inliers correct matches from outliers incorrect matches and then estimates the spatial transformation in this paper motivated by the problem of feature matching that the initial correspondence is always contaminated by outliers thereby we formulate this issue as a probability deformable mixture model which consists of gaussian components for inliers and uniform components for outliers moreover the algorithm takes full advantage of using multiple attributes for better general matching performance here we are assuming all inliers are mapped into a high dimensional feature space namely reproducing kernel hilbert space rkhs and the closed form solution to the mapping function is given by the representation theorem with l2 norm regularization under the expectation maximization em algorithm finally we evaluate the performance of the algorithm by applying it to retinal image registration on several datasets where experimental results demonstrate that the mad rmm outperforms current state of the art methods and shows the robustness to outliers on real retinal images

partitioning ordinary differential equations using runge kutta methods two techniques for detecting stiffness when using runge kutta type of methods are discussed and compared and a partitioning strategy for first order system of equations into stiff and nonstiff subsystems is proposed a few problems are solved using three stage semi implicit runge kutta method newton iteration is used for the stiff part and simple iteration for the nonstiff finally numerical results based on different criteria to detect stiffness are compared

local and parallel finite element algorithms based on two grid discretizations a number of new local and parallel discretization and adaptive nite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems these algorithms are motivated by the observation that for a solution to some elliptic problems low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a ne grid by some local and parallel procedure the theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for nite element solutions on general shape regular grids some numerical experiments are also presented to support the theory in this paper we will propose some new parallel techniques for nite element computation these techniques are based on our understanding of the local and global properties of a nite element solution to some elliptic problems simply speaking the global behavior of a solution is mostly governed by low frequency components while the local behavior is mostly governed by high frequency compo nents the main idea of our new algorithms is to use a coarse grid to approximate the low frequencies and then to use a ne grid to correct the resulted residue which contains mostly high frequencies by some local parallel procedures let us now give a somewhat more detailed but informal and hopefully infor mative description of the main ideas and results in this paper we consider the following very simple model problem posed on a convex polygonal domain r 2 u bru f in

the combination technique and some generalisations the combination technique has repeatedly been shown to be an effective tool for the approximation with sparse grid spaces little is known about the reasons of this effectiveness and in some cases the combination technique can even break down it is known however that the combination technique produces an exact result in the case of a projection into a sparse grid space if the involved partial projections commute the performance of the combination technique is analysed using a projection framework and the c s decomposition error bounds are given in terms of angles between the spanning subspaces or the projections onto these subspaces based on this analysis modified combination coefficients are derived which are optimal in a certain sense and which can substantially extend the applicability and performance of the combination technique

error estimates and adaptive time step control for a class of one step methods for stiff ordinary differential equations we prove new optimal a priori error estimates for a class of implicit one step methods for stiff ordinary differential equations obtained by using the discontinuous galerkin method with piecewise polynomials of degree zero and one starting from these estimates we propose a new algorithm for automatic time step control and we discuss the relation between this algorithm and earlier algorithms implemented in packages for the numerical solution of stiff ordinary differential equations

recent advances in methods for numerical solution of o d e initial value problems abstract this is a review paper which describes recent advances in numerical methods and computer codes for solving initial value problems of ordinary differential equations particular emphasis is placed upon stiff systems

a type insensitive ode code based on second derivative formulas abstract several effective codes for the solution of stiff ordinary differential equations odes are based on second derivative formulas they are inefficient for non stiff problems it is shown how to modify such codes to make them reasonably efficient the modifications make them more efficient for stiff problems as well users do not have the information to determine stiffness reliably the modified codes recognize the type automatically at each step and respond appropriately it is a great convenience for a user to have one code efficient regardless of the type