solving second order initial value problems by a hybrid multistep method without predictors abstract a three step seventh order hybrid linear multistep method hlmm with three non step points is proposed for the direct solution of the special second order initial value problems ivps of the form y f x y with an extension to y f x y y the main method and additional methods are obtained from the same continuous scheme derived via interpolation and collocation procedures the stability properties of the methods are discussed by expressing them as a one step method in higher dimension the methods are then applied in block form as simultaneous numerical integrators over non overlapping intervals numerical results obtained using the proposed block form reveal that it is highly competitive with existing methods in the literature

a tenth order a stable two step hybrid block method for solving initial value problems of odes in this article a new two step hybrid block method for the numerical integration of ordinary differential initial value systems is presented the method is obtained after considering two intermediate points and the approximation of the true solution by an adequate polynomial and imposing collocation conditions the proposed method has the tenth algebraic order of convergence and is a stable the numerical experiments considered revealed the superiority of the new method for solving this kind of problems in comparison with methods of similar characteristics appeared in the literature

complementarity constraint qualification via the theory of 2 regularity we exhibit certain second order regularity properties of parametric complementarity constraints which are notorious for being irregular in the classical sense our approach leads to a constraint qualification in terms of 2 regularity of the mapping corresponding to the subset of constraints which must be satisfied as equalities around the given feasible point while no qualification is required for the rest of the constraints under this 2 regularity assumption we derive constructive sufficient conditions for tangent directions to feasible sets defined by complementarity constraints a special form of primal dual optimality conditions is also obtained we further show that our 2 regularity condition always holds under the piecewise mangasarian fromovitz constraint qualification but not vice versa relations with other constraint qualifications and optimality conditions are also discussed it is shown that our approach can be useful when alternative ones are not applicable

nonrigid coregistration of diffusion tensor images using a viscous fluid model and mutual information in this paper a nonrigid coregistration algorithm based on a viscous fluid model is proposed that has been optimized for diffusion tensor images dti in which image correspondence is measured by the mutual information criterion several coregistration strategies are introduced and evaluated both on simulated data and on brain intersubject dti data two tensor reorientation methods have been incorporated and quantitatively evaluated simulation as well as experimental results show that the proposed viscous fluid model can provide a high coregistration accuracy although the tensor reorientation was observed to be highly sensitive to the local deformation field nevertheless this coregistration method has demonstrated to significantly improve spatial alignment compared to affine image matching

spatial normalization of diffusion tensor mri using multiple channels diffusion tensor mri dt mri can provide important in vivo information for the detection of brain abnormalities in diseases characterized by compromised neural connectivity to quantify diffusion tensor abnormalities based on voxel based statistical analysis spatial normalization is required to minimize the anatomical variability between studied brain structures in this article we used a multiple input channel registration algorithm based on a demons algorithm and evaluated the spatial normalization of diffusion tensor image in terms of the input information used for registration registration was performed on 16 dt mri data sets using different combinations of the channels including a channel of t2 weighted intensity a channel of the fractional anisotropy a channel of the difference of the first and second eigenvalues two channels of the fractional anisotropy and the trace of tensor three channels of the eigenvalues of the tensor and the six channel tensor components to evaluate the registration of tensor data we defined two similarity measures i e the endpoint divergence and the mean square error which we applied to the fiber bundles of target images and registered images at the same seed points in white matter segmentation we also evaluated the tensor registration by examining the voxel by voxel alignment of tensors in a sample of 15 normalized dt mris in all evaluations nonlinear warping using six independent tensor components as input channels showed the best performance in effectively normalizing the tract morphology and tensor orientation we also present a nonlinear method for creating a group diffusion tensor atlas using the average tensor field and the average deformation field which we believe is a better approach than a strict linear one for representing both tensor distribution and morphological distribution of the population

a 6 4 optimized embedded runge kutta nystrom pair for the numerical solution of periodic problems in this paper an optimization of the non fsal embedded rkn 6 4 pair with six stages of moawwad el mikkawy el desouky rahmo is presented the new method is derived after applying phase fitting and amplification fitting and has variable coefficients the preservation of the algebraic order is verified and the principal term of the local truncation error is evaluated furthermore periodicity analysis is performed which reveals that the new method is almost p stable the efficiency of the new method is measured via the integration of several initial value problems

variable stepsize implementation of multistep methods for y f x y y the falkner method is a multistep scheme intended for the numerical solution of second order initial value problems where the first derivative does appear explicitly in this paper we develop a procedure to obtain k step falkner methods explicit and implicit in their variable step size versions providing recurrence formulas to compute the coefficients efficiently considering a pair of explicit and implicit formulae these may be implemented in predictor corrector mode

multiframe temporal estimation of cardiac nonrigid motion a robust flexible system for tracking the point to point nonrigid motion of the left ventricular lv endocardial wall in image sequences has been developed this system is unique in its ability to model motion trajectories across multiple frames the foundation of this system is an adaptive transversal filter based on the recursive least squares algorithm this filter facilitates the integration of models for periodicity and proximal smoothness as appropriate using a contour based description of the object s boundaries a set of correspondences between contours and an associated set of correspondence quality measures comprise the input to the system frame to frame relationships from two different frames of reference are derived and analyzed using synthetic and actual images two multiframe temporal models both based on a sum of sinusoids are derived illustrative examples of the system s output are presented for quantitative analysis validation of the system is performed by comparing computed trajectory estimates with the trajectories of physical markers implanted in the lv wall sample case studies of marker trajectory comparisons are presented ensemble statistics from comparisons with 15 marker trajectories are acquired and analyzed a multiframe temporal model without spatial periodicity constraints was determined to provide excellent performance with the least computational cost a multiframe spatiotemporal model provided the best performance based on statistical standard deviation although at significant computational expense

generalized convexity and efficiency for non regular multiobjective programming problems with inequality type constraints abstract for multiobjective problems with inequality type constraints the necessary conditions for efficient solutions are presented these conditions are applied when the constraints do not necessarily satisfy any regularity assumptions and they are based on the concept of 2 regularity introduced by izmailov in general the necessary optimality conditions are not sufficient and the efficient solution set is not the same as the karush kuhn tucker points set so it is necessary to introduce generalized convexity notions in the multiobjective non regular case we give the notion of 2 kkt pseudoinvex ii problems this new concept of generalized convexity is both necessary and sufficient to guarantee the characterization of all efficient solutions based on the optimality conditions

generalized convexity in non regular programming problems with inequality type constraints abstract convexity plays a very important role in optimization for establishing optimality conditions different works have shown that the convexity property can be replaced by a weaker notion the invexity in particular for problems with inequality type constraints martin defined a weaker notion of invexity the karush kuhn tucker invexity hereafter kkt invexity that is both necessary and sufficient to obtain karush kuhn tucker type optimality conditions it is well known that for this result to hold the problem has to verify a constraint qualification i e it must be regular or non degenerate in non regular problems the classical optimality conditions are totally inapplicable meaningful results were obtained for problems with inequality type constraints by izmailov they are based on the 2 regularity condition of the constraints at a feasible point in this work we generalize martin s result to non regular problems by defining an analogous concept the 2 kkt invexity and using the characterization of the tangent cone in the 2 regular case and the necessary optimality condition given by izmailov

piecewise linearized methods for initial value problems piecewise linearized methods for the solution of initial value problems in ordinary differential equations are developed by approximating the right hand sides of the equations by means of a taylor polynomial of degree one the resulting approximation can be integrated analytically to obtain the solution in each interval and yields the exact solution for linear problems three adaptive methods based on the norm of the jacobian matrix maintaining constant the value of the approximation errors incurred by the linearization of the right hand sides of the ordinary differential equations and richardson s extrapolation are developed numerical experiments with some nonstiff first and second order ordinary differential equations indicate that the accuracy of piecewise linearized methods is in general superior to those of the explicit modified second order accurate euler method and the implicit trapezoidal rule but lower than that of the explicit fourth order accurate runge kutta technique it is also shown that piecewise linearized methods do not exhibit computational i e spurious modes for the relaxation oscillations of the van der pol oscillator and for those systems of equations which satisfy certain conservation principles conserve more accurately the invariants than the trapezoidal rule an error bound for piecewise linearized methods is provided for ordinary differential equations whose right hand sides satisfy certain lipschitz conditions

left ventricular deformation recovery from cine mri using an incompressible model this paper presents a method for 3d deformation recovery of the left ventricular lv wall from anatomical cine magnetic resonance imaging mri the method is based on a de formable model that is incompressible a desired property since the myocardium has been shown to be nearly incompressible the lv wall needs to be segmented in an initial frame after which the method automatically determines the deformation everywhere in the lv wall throughout the cardiac cycle two studies were conducted to validate the method in the first study the deformation recovered from a 3d anatomical cine mri of a healthy volunteer was compared against the manual segmentation of the lv wall and against the corresponding 3d tagged cine mri the average volume agreement between the model and the manual segmentation had a false positive rate of 3 false negative rate of 3 and true positive rate of 93 the average distance between the model and manually determined intersections of perpendicular tag planes was 1 6 mm 1 1 pixel another set of 3d anatomical and tagged mri scans was taken of the same volunteer four months later the method was applied to the second set and the recovered deformation was very similar to the one obtained from the first set in the second study the method was applied to 3d anatomical cine mri scans of three patients with ventricular dyssynchrony and three age matched healthy volunteers the lv wall deformations recovered for the three normals agreed well and the recovered strains were similar to those reported by other researchers for normal subjects strains and displacements of the three patients were clearly smaller than those of the three normals indicating reduced cardiac function the deformation recovered for the three normals and the three patients was validated against manual segmentation and corresponding tag cine mri scans and the agreement was similar to that of the first validation study

first and second order necessary conditions for control problems with constraints second order necessary conditions are developed for an abstract nonsmooth control problem with mixed state control equality and inequality constraints as well as a constraint of the form g x u e if where f is a closed convex set of a banach space with nonempty interior the inequality constraints g s x u 0 depend on a parameter s belonging to a compact metric space s the equality constraints are split into two sets of equations k x u 0 and h x u 0 where the first equation is an abstract control equation and h is assumed to have a full rank property in u the objective function is maxtet f t x u where t is a compact metric space f is upper semicontinuous in t and lipschitz in x u the results are in terms of a function a that disappears when the parameter spaces t and s are discrete we apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints thus we obtain a generalization and extension of the existing results on this problem

revised trigonometrically fitted two step hybrid methods with equation dependent coefficients for highly oscillatory problems for the numerical integration of highly oscillatory problems revised trigonometrically fitted two step hybrid methods rtftsh with equation dependent coefficients are considered the local truncation errors stability and phase properties of the new method are analyzed a feature of the new type of the methods is that the errors in the internal stages are assumed to contribute to the accuracy of the update a new revised method rtftsh4 of algebraic order four and phase lag order four is derived numerical experiments are reported to show that the new method rtftsh4 is much more efficient and robust than the standard fourth order method stftsh4

registration of 4d cardiac ct sequences under trajectory constraints with multichannel diffeomorphic demons we propose a framework for the nonlinear spatiotemporal registration of 4d time series of images based on the diffeomorphic demons dd algorithm in this framework the 4d spatiotemporal registration is decoupled into a 4d temporal registration defined as mapping physiological states and a 4d spatial registration defined as mapping trajectories of physical points our contribution focuses more specifically on the 4d spatial registration that should be consistent over time as opposed to 3d registration that solely aims at mapping homologous points at a given time point first we estimate in each sequence the motion displacement field which is a dense representation of the point trajectories we want to register then we perform simultaneously 3d registrations of corresponding time points with the constraints to map the same physical points over time called the trajectory constraints under these constraints we show that the 4d spatial registration can be formulated as a multichannel registration of 3d images to solve it we propose a novel version of the diffeomorphic demons dd algorithm extended to vector valued 3d images the multichannel diffeomorphic demons mdd for evaluation this framework is applied to the registration of 4d cardiac computed tomography ct sequences and compared to other standard methods with real patient data and synthetic data simulated from a physiologically realistic electromechanical cardiac model results show that the trajectory constraints act as a temporal regularization consistent with motion whereas the multichannel registration acts as a spatial regularization finally using these trajectory constraints with multichannel registration yields the best compromise between registration accuracy temporal and spatial smoothness and computation times a prospective example of application is also presented with the spatiotemporal registration of 4d cardiac ct sequences of the same patient before and after radiofrequency ablation rfa in case of atrial fibrillation af the intersequence spatial transformations over a cardiac cycle allow to analyze and quantify the regression of left ventricular hypertrophy and its impact on the cardiac function

high order stability conditions for degenerate optimization problems elements of p regularity theory abstract this paper is devoted to sensitivity analysis of degenerate equality constrained optimization problems when the customary regularity of constraints may be violated under the p regularity assumptions we develop p order stability theory for this class of problems subject to small perturbations and present the estimations of asymptotic behavior of the solutions to the perturbed problems we derive conditions that guarantee the existence of a solution for perturbed optimization problems and substantiate the assertion of natural character of these conditions all results are illustrated by classical calculus of variations problems

third derivative modification of k step block falkner methods for the numerical solution of second order initial value problems abstract this paper is devoted to the development and analysis of a modified family of falkner type methods for solving differential systems of second order initial value problems the approaches of collocation and interpolation are adopted to derive the new methods these modified methods are implemented in block form to obtain the numerical solutions to the considered problems the study of the properties of the proposed block falkner type methods reveals that they are consistent and zero stable and thus convergent from the stability analysis it could be seen that the proposed falkner methods have non empty stability regions for k 2 3 4 some numerical test are presented to illustrate the efficiency of the proposed family

dt refind diffusion tensor registration with exact finite strain differential in this paper we propose the dt refind algorithm for the diffeomorphic nonlinear registration of diffusion tensor images unlike scalar images deforming tensor images requires choosing both a reorientation strategy and an interpolation scheme current diffusion tensor registration algorithms that use full tensor information face difficulties in computing the differential of the tensor reorientation strategy and consequently these methods often approximate the gradient of the objective function in the case of the finite strain fs reorientation strategy we borrow results from the pose estimation literature in computer vision to derive an analytical gradient of the registration objective function by utilizing the closed form gradient and the velocity field representation of one parameter subgroups of diffeomorphisms the resulting registration algorithm is diffeomorphic and fast we contrast the algorithm with a traditional fs alternative that ignores the reorientation in the gradient computation we show that the exact gradient leads to significantly better registration at the cost of computation time independently of the choice of euclidean or log euclidean interpolation and sum of squared differences dissimilarity measure the exact gradient achieves better alignment over an entire spectrum of deformation penalties alignment quality is assessed with a battery of metrics including tensor overlap fractional anisotropy inverse consistency and closeness to synthetic warps the improvements persist even when a different reorientation scheme preservation of principal directions is used to apply the final deformations

fast geodesic active fields for image registration based on splitting and augmented lagrangian approaches in this paper we present an efficient numerical scheme for the recently introduced geodesic active fields gaf framework for geometric image registration this framework considers the registration task as a weighted minimal surface problem hence the data term and the regularization term are combined through multiplication in a single parametrization invariant and geometric cost functional the multiplicative coupling provides an intrinsic spatially varying and data dependent tuning of the regularization strength and the parametrization invariance allows working with images of nonflat geometry generally defined on any smoothly parametrizable manifold the resulting energy minimizing flow however has poor numerical properties here we provide an efficient numerical scheme that uses a splitting approach data and regularity terms are optimized over two distinct deformation fields that are constrained to be equal via an augmented lagrangian approach our approach is more flexible than standard gaussian regularization since one can interpolate freely between isotropic gaussian and anisotropic tv like smoothing in this paper we compare the geodesic active fields method with the popular demons method and three more recent state of the art algorithms nl optical flow mrf image registration and landmark enhanced large displacement optical flow thus we can show the advantages of the proposed fastgaf method it compares favorably against demons both in terms of registration speed and quality over the range of example applications it also consistently produces results not far from more dedicated state of the art methods illustrating the flexibility of the proposed framework

dissipative chebyshev exponential fitted methods for numerical solution of second order differential equations a family of implicit methods based on intra step chebyshev interpolation has been developed to integrate oscillatory second order initial value problems of the form y t 2g y t g2 w2 y t f t y t the procedure integrates the homogeneous part exactly in the absence of round off errors the chebyshev approach uses stepsizes that are considerably larger than those typically used in runge kutta or multistep methods computational overheads are comparable to those incurred by high order conventional procedures chebyshev interpolation coupled with the exponential fitted nature of the method substantially reduces local errors global error propagation rates are also reduced making these procedures good candidates to be used in long term simulations of perturbed oscillatory systems with a dissipative term

second order necessary conditions for nonlinear optimization problems with abstract constraints the degenerate case in this paper we derive second order necessary conditions for optimality for an optimization problem with abstract constraints in banach spaces results for the nondegenerate case derived earlier h gfrerer siam j control optim 45 2006 pp 972 997 are extended to the degenerate case for the mathematical programming problem where the constraints are given by equality and finitely many inequality constraints our approach applies to the degenerate case when the equality constraints are not regular our results appear to be new even in this special case our second order necessary conditions are contained in the gap between the standard necessary and sufficient conditions where the only difference is the change from a nonstrict to a strict inequality our results are formulated in such a way to be applicable also to vector optimization problems

regression based cardiac motion prediction from single phase cta state of the art cardiac computed tomography ct enables the acquisition of imaging data of the heart over the entire cardiac cycle at concurrent high spatial and temporal resolution however in clinical practice acquisition is increasingly limited to 3 d images estimating the shape of the cardiac structures throughout the entire cardiac cycle from a 3 d image is therefore useful in applications such as the alignment of preoperative computed tomography angiography cta to intra operative x ray images for improved guidance in coronary interventions we hypothesize that the motion of the heart is partially explained by its shape and therefore investigate the use of three regression methods for motion estimation from single phase shape information quantitative evaluation on 150 4 d cta images showed a small but statistically significant increase in the accuracy of the predicted shape sequences when using any of the regression methods compared to shape independent motion prediction by application of the mean motion the best results were achieved using principal component regression resulting in point to point errors of 2 3 0 5 mm compared to values of 2 7 0 6 mm for shape independent motion estimation finally we showed that this significant difference withstands small variations in important parameter settings of the landmarking procedure

necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications we derive first and second order necessary optimality conditions for set constrained optimization problems under the constraint qualification type conditions significantly weaker than robinson s constraint qualification our development relies on the so called 2 regularity concept and unifies and extends the previous studies based on this concept specifically in our setting constraints are given by an inclusion with an arbitrary closed convex set on the right hand side thus for the second order analysis some curvature characterizations of this set near the reference point must be taken into account

a new class of two step p stable tfpl methods for the numerical solution of second order ivps with oscillating solutions abstract a new class of two step linear symmetric methods is introduced for the numerical solution of second order initial value problems that have highly oscillatory solutions in this class for the first time in the literature we calculate the coefficients of the method by a combination of a trigonometrically fitted tf method and a vanishing of the phase lag and some of its derivatives vsdpl method and we construct a new class of methods that has the properties of tf methods and vsdpl methods which we call the tfpl method this method is of algebraic order 8 and has an important p stability property the main structure of the method is multiderivative and the combined phases were applied to expand the stability interval and to achieve p stability the advantage of the method in comparison with similar methods in terms of efficiency accuracy and stability is shown by its implementation in some important problems the undamped duffing equation etc

solving nonstiff higher order odes directly by the direct integration method the variable order and stepsize direct integration di method for solving systems of nonstiff higher order odes directly is reviewed mention is also made of the current inroads in solving stiff higher order odes directly the advantages of the di method over the corresponding adams method when used to solve the same problem as a first order system are pointed out numerical results are presented comparing the two methods in terms of execution times for each tolerance used and execution times for each solution of the same range of accuracy

optimality conditions for irregular inequality constrained problems we consider feasible sets given by conic constraints where the cone defining the constraints is convex with nonempty interior we study the case where the feasible set is not assumed to be regular in the classical sense of robinson and obtain a constructive description of the tangent cone under a certain new second order regularity condition this condition contains classical regularity as a special case while being weaker when constraints are twice differentiable assuming that the cone defining the constraints is finitely generated we also derive a special form of primal dual optimality conditions for the corresponding constrained optimization problem our results subsume optimality conditions for both the classical regular and second order regular cases while still being meaningful in the more general setting in the sense that the multiplier associated with the objective function is nonzero

computing large deformation metric mappings via geodesic flows of diffeomorphisms this paper examine the euler lagrange equations for the solution of the large deformation diffeomorphic metric mapping problem studied in dupuis et al 1998 and trouve 1995 in which two images i 0 i 1 are given and connected via the diffeomorphic change of coordinates i 0 1 i 1 where 1 is the end point at t 1 of curve t t 0 1 satisfying t v t t t 0 1 with 0 id the variational problem takes the form mathop arg text m in limits upsilon dot phi t upsilon t left dot phi right left int 0 1 left upsilon t right 2 text d t left i 0 circ phi 1 1 i 1 right l 2 2 right where v t v is an appropriate sobolev norm on the velocity field v t and the second term enforces matching of the images with l 2 representing the squared error norm in this paper we derive the euler lagrange equations characterizing the minimizing vector fields v t t 0 1 assuming sufficient smoothness of the norm to guarantee existence of solutions in the space of diffeomorphisms we describe the implementation of the euler equations using semi lagrangian method of computing particle flows and show the solutions for various examples as well we compute the metric distance on several anatomical configurations as measured by 0 1 v t v dt on the geodesic shortest paths

parametric excitation of computational mode of the leapfrog scheme applied to the van der pol equation the leapfrog scheme is applied to the van der pol equation when the amplitude of oscillation of the physical mode exceeds a critical value the computational mode is parametrically excited by the physical mode the growth of the computational mode interrupts the integration based on the leapfrog scheme the critical amplitude of the physical mode is determined by the linear stability analysis and the parametric excitation theory the runge kutta smoother eliminating the computational mode enables the longtime integration based on the leapfrog scheme

fluid registration of diffusion tensor images using information theory we apply an information theoretic cost metric the symmetrized kullback leibler skl divergence or j divergence to fluid registration of diffusion tensor images the difference between diffusion tensors is quantified based on the skl divergence of their associated probability density functions pdfs three dimensional dti data from 34 subjects were fluidly registered to an optimized target image to allow large image deformations but preserve image topology we regularized the flow with a large deformation diffeomorphic mapping based on the kinematics of a navier stokes fluid a driving force was developed to minimize the j divergence between the deforming source and target diffusion functions while reorienting the flowing tensors to preserve fiber topography in initial experiments we showed that the skl divergence based on full diffusion pdfs is adaptable to higher order diffusion models such as high angular resolution diffusion imaging hardi the skl divergence was sensitive to subtle differences between two diffusivity profiles showing promise for nonlinear registration applications and multisubject statistical analysis of hardi data

the solution of high order nonlinear ordinary differential equations by chebyshev series by the use of the chebyshev series a direct computational method for solving the higher order nonlinear differential equations has been developed in this paper this method transforms the nonlinear differential equation into the matrix equation which corresponds to a system of nonlinear algebraic equations with unknown chebyshev coefficients via chebyshev collocation points the solution of this system yields the chebyshev coefficients of the solution function an algorithm for this nonlinear system is also proposed in this paper the method is valid for both initial value and boundary value problems several examples are presented to illustrate the accuracy and effectiveness of the method

a continuous two step method of order 8 with a block extension for y f x y y abstract we derive a piecewise continuous hybrid third derivative approximation chtda that is defined for all values of the independent variable on the range of interest this continuous approximation has the ability to provide a continuous solution between all the grid points with a uniform accuracy comparable to that obtained at the grid points furthermore the chtda is used to produce several implicit hybrid third derivative formulas ihtdfs which are expressed in block form and applied as a block hybrid third derivative algorithm bhtda for the numerical solution of second order initial value problems the stability of the block extension is discussed numerical examples are given to illustrate the accuracy and efficiency of the bhtda

how many k step linear block methods exist and which of them is the most efficient and simplest one abstract there have appeared in the literature a lot of k step block methods for solving initial value problems the methods consist in a set of k simultaneous multistep formulas over k non overlapping intervals a feature of block methods is that there is no need of other procedures to provide starting approximations and thus the methods are self starting sharing this advantage of runge kutta methods all the formulas are usually obtained from a continuous approximation derived via interpolation and collocation at k 1 points nevertheless all the k step block methods thus obtained may be considered as different formulations of one of them which results to be the most efficient and simple formulation of all of them the theoretical analysis and the numerical experiments presented support this claim

the pth order optimality conditions for inequality constrained optimization problems in this paper we present pth order necessary conditions for optimality for an optimization problem with inequality constraints in the finite dimensional spaces models of this type arise as the discretization of optimal control problems calculus of variations problems and other problems the paper addresses the degenerate nonregular case when the active constraint gradients are linearly dependent in this case the classical necessary conditions are either inapplicable or trivially satisfied with the zero multiplier corresponding to the objective function the proposed high order optimality conditions give new and nontrivial conditions for nonregular cases and reduce to the classical conditions for regular cases

high order approximations and generalized necessary conditions for optimality in this paper we derive generalized necessary conditions for optimality for an optimization problem with equality and inequality constraints in a banach space the equality constraints are given in operator form as q x in x f x 0 where f x rightarrow y is an operator between banach spaces the inequality constraints are given by smooth functionals or by closed convex sets models of this type are common in the optimal control problem the paper addresses the case when the frechet derivative f x is not onto and hence the classical lyusternik theorem does not apply to describe the tangent space to q in this case the classical euler lagrange type necessary conditions are trivially satisfied generating abnormal cases a high order generalization of the lyusternik theorem derived earlier u ledzewicz and h schattler nonlinear anal 34 1998 pp 793 815 is used to calculate high order tangent cones to the equality constraint at points x in q where f x is not onto combining these with high order approximating cones related to the other constraints of the problem feasible cones respectively cones of decrease a high order generalization of the dubovitskii milyutin theorem is given and then applied to derive generalized necessary conditions for optimality these conditions reduce to classical conditions for normal cases but they give new and nontrivial conditions for abnormal cases

phase vector incompressible registration algorithm for motion estimation from tagged magnetic resonance images tagged magnetic resonance imaging has been used for decades to observe and quantify motion and strain of deforming tissue it is challenging to obtain 3 d motion estimates due to a tradeoff between image slice density and acquisition time typically interpolation methods are used either to combine 2 d motion extracted from sparse slice acquisitions into 3 d motion or to construct a dense volume from sparse acquisitions before image registration methods are applied this paper proposes a new phase based 3 d motion estimation technique that first computes harmonic phase volumes from interpolated tagged slices and then matches them using an image registration framework the approach uses several concepts from diffeomorphic image registration with a key novelty that defines a symmetric similarity metric on harmonic phase volumes from multiple orientations the material property of harmonic phase solves the aperture problem of optical flow and intensity based methods and is robust to tag fading a harmonic magnitude volume is used in enforcing incompressibility in the tissue regions the estimated motion fields are dense incompressible diffeomorphic and inverse consistent at a 3 d voxel level the method was evaluated using simulated phantoms human brain data in mild head accelerations human tongue data during speech and an open cardiac data set the method shows comparable accuracy to three existing methods while demonstrating low computation time and robustness to tag fading and noise

a fourth order method for y ƒ x y y abstract a fourth order numerical method for solving the general second order differential equation y x y y is suggested this method is self starting p stable when applied to y x y and it involves only two function evaluations per step easily calculable and simple truncation error estimates are given which can be used for automatic error control in computations the method is illustrated on numerical examples

the theory of 2 regularity for mappings with lipschitzian derivatives and its applications to optimality conditions we study local structure of a nonlinear mapping near points where standard regularity and or smoothness assumptions need not be satisfied we introduce a new concept of 2 regularity a certain kind of second order regularity for a once differentiable mapping whose derivative is lipschitz continuous under this 2 regularity condition we obtain the representation theorem and the covering theorem i e stability with respect to right hand side perturbations under assumptions that are weaker than those previously employed in the literature for results of this type these results are further used to derive a constructive description of the tangent cone to a set defined by 2 regular equality constraints and optimality conditions for related optimization problems the class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation based reformulation optimality conditions for mathematical programs with equivalently reformulated complementarity constraints are also discussed

a new four step p stable obrechkoff method with vanished phase lag and some of its derivatives for the numerical solution of radial schrodinger equation abstract in this paper we present a new four step implicit p stable obrechkoff method of tenth algebraic order for solving one dimensional second order linear periodic and oscillatory initial value problems of ordinary differential equations such as reasonable schrodinger equation numerical stability and phase properties of the new methods are analyzed numerical experiments are carried out to show the efficiency and robustness of our new methods in comparison with the well known codes proposed in the scientific literature

a unified approach for the development of k step block falkner type methods for solving general second order initial value problems in odes a family of k step block multistep methods where the main formulas are of falkner type is proposed for the direct integration of the general second order initial value problem where the differential equation is of the general form y f x y y the two main falkner formulas and the additional ones to complete the block procedure are obtained from a continuous approximation derived via interpolation and collocation at k 1 points the main characteristics of the methods are discussed through their formulation in vector form each method is formulated as a group of 2 k simultaneous formulas over k non overlapping intervals in this way the method produces the approximation of the solution simultaneously at k points on these intervals as in other block methods there is no need of other procedures to provide starting approximations and thus the methods are self starting sharing this advantage of runge kutta methods the resulting family is efficient and competitive compared with other existing methods in the literature as may be seen from the numerical results

spatio temporal nonrigid registration for ultrasound cardiac motion estimation we propose a new spatio temporal elastic registration algorithm for motion reconstruction from a series of images the specific application is to estimate displacement fields from two dimensional ultrasound sequences of the heart the basic idea is to find a spatio temporal deformation field that effectively compensates for the motion by minimizing a difference with respect to a reference frame the key feature of our method is the use of a semi local spatio temporal parametric model for the deformation using splines and the reformulation of the registration task as a global optimization problem the scale of the spline model controls the smoothness of the displacement field our algorithm uses a multiresolution optimization strategy to obtain a higher speed and robustness we evaluated the accuracy of our algorithm using a synthetic sequence generated with an ultrasound simulation package together with a realistic cardiac motion model we compared our new global multiframe approach with a previous method based on pairwise registration of consecutive frames to demonstrate the benefits of introducing temporal consistency finally we applied the algorithm to the regional analysis of the left ventricle displacement and strain parameters were evaluated showing significant differences between the normal and pathological segments thereby illustrating the clinical applicability of our method

a family of improved falkner type methods for oscillatory systems for solving general second order initial value problems u t f t u t u t the classical falkner methods can date back to falkner s work in 1936 in this paper we propose and study a family of improved falkner type methods for the oscillatory system u t m u t g t u t where g r rd rd in which the first derivative does not appear explicitly and m rd d is a symmetric positive semi definite matrix the new methods take into account the oscillatory structures of the problem and exactly integrate the unperturbed problem u t m u t 0 the global error bounds of the new methods are presented numerical experiments are performed to show that the new methods are more efficient than other effective methods appeared in the scientific literature

bilinear models for spatio temporal point distribution analysis in this work we describe the usage of bilinear statistical models as a means of factoring the shape variability into two components attributed to inter subject variation and to the intrinsic dynamics of the human heart we show that it is feasible to reconstruct the shape of the heart at discrete points in the cardiac cycle provided we are given a small number of shape instances representing the same heart at different points in the same cycle we can use the bilinear model to establish this using a temporal and a spatial alignment step in the preprocessing of the shapes around half of the reconstruction errors were on the order of the axial image resolution of 2 mm and over 90 was within 3 5 mm from this we conclude that the dynamics were indeed separated from the inter subject variability in our dataset

variable order variable step methods for second order initial value problems abstract we develop a variable order variable step algorithm for solving second order initial value problems y f t y y 0 y 0 given this algorithm employs a family of implicit two step methods in which the differential equation is evaluated at a number of off step points we have derived the order conditions for this family of methods in previously published work and we have also shown that the remaining free parameters of the method may be chosen to given p stability here the emphasis is on the combination of second fourth sixth and eighth order methods from this family in a variable order variable step algorithm the aim is to choose particular p stable methods from the family in such a way that a saving can be made in the amount of computational effort involved when they are combined together

control of local error stabilizes integrations codes for the solution of the initial value problem for a system of ordinary differential equations adjust their step sizes so as to keep the estimated local errors smaller than a given tolerance and at the same time solve the problem as efficiently as possible step sizes providing the desired accuracy may be too large for stability yet this does not cause the popular codes to become unstable the theory explaining why this is so holds only in very restricted circumstances this paper broadens and deepens the theory for explicit runge kutta methods

nonsmooth optimum problems with constraints this paper develops second order necessary conditions for nonsmooth infinite dimensional optimization problems with banach space valued equality and real valued inequality constraints another constraint in the form of a closed convex set is also present the objective function is the maximum over a parameter of functions f t z that are lipschitz in z and upper semicontinuous in t the inequality constraints g s z depend on a parameter s the technique we use is a generalization of that of dubovitskii and milyutin the second order conditions obtained here are in terms of a certain function sigma that disappears when the parameters and a certain set that derives from the given convex set are absent the presence of the function sigma and that set is due to the envelope like effect discovered by kawasaki

a fast diffeomorphic image registration algorithm abstract this paper describes dartel which is an algorithm for diffeomorphic image registration it is implemented for both 2d and 3d image registration and has been formulated to include an option for estimating inverse consistent deformations nonlinear registration is considered as a local optimisation problem which is solved using a levenberg marquardt strategy the necessary matrix solutions are obtained in reasonable time using a multigrid method a constant eulerian velocity framework is used which allows a rapid scaling and squaring method to be used in the computations dartel has been applied to intersubject registration of 471 whole brain images and the resulting deformations were evaluated in terms of how well they encode the shape information necessary to separate male and female subjects and to predict the ages of the subjects

ilogdemons a demons based registration algorithm for tracking incompressible elastic biological tissues tracking soft tissues in medical images using non linear image registration algorithms requires methods that are fast and provide spatial transformations consistent with the biological characteristics of the tissues logdemons algorithm is a fast non linear registration method that computes diffeomorphic transformations parameterised by stationary velocity fields although computationally efficient its use for tissue tracking has been limited because of its ad hoc gaussian regularisation which hampers the implementation of more biologically motivated regularisations in this work we improve the logdemons by integrating elasticity and incompressibility for soft tissue tracking to that end a mathematical justification of demons gaussian regularisation is proposed building on this result we replace the gaussian smoothing by an efficient elastic like regulariser based on isotropic differential quadratic forms of vector fields the registration energy functional is finally minimised under the divergence free constraint to get incompressible deformations as the elastic regulariser and the constraint are linear the method remains computationally tractable and easy to implement tests on synthetic incompressible deformations showed that our approach outperforms the original logdemons in terms of elastic incompressible deformation recovery without reducing the image matching accuracy as an application we applied the proposed algorithm to estimate 3d myocardium strain on clinical cine mri of two adult patients results showed that incompressibility constraint improves the cardiac motion recovery when compared to the ground truth provided by 3d tagged mri

obrechkoff methods having additional parameters for general second order differential equations abstract a class of two step implicit methods involving higher order derivatives of y for initial value problems of the form y f t y y is developed the methods involve arbitrary parameters p and q which are determined so that the methods become absolutely stable when applied to the test equation y y y 0 numerical results for bessel s and general second order differential equations are presented to illustrate that the methods are absolutely stable and are of order o h 4 o h 6 and o h 8

embedded runge kutta formulae with stable equilibrium states the equilibrium theory of hall and higham 1988 can be used to determine whether a runge kutta algorithm will perform smoothly when stability restricts the stepsize in this paper we show that current high quality order 4 5 pairs do not behave well in this respect and we determine the extent to which the overall quality must be compromised in order for the equilibrium conditions to be satisfied three new formulae are presented and their properties are compared with those of existing formulae

variational methods for multimodal image matching matching images of different modalities can be achieved by the maximization of suitable statistical similarity measures within a given class of geometric transformations handling complex nonrigid deformations in this context turns out to be particularly difficult and has attracted much attention in the last few years the thrust of this paper is that many of the existing methods for nonrigid monomodal registration that use simple criteria for comparing the intensities e g ssd can be extended to the multimodal case where more complex intensity similarity measures are necessary to this end we perform a formal computation of the variational gradient of a hierarchy of statistical similarity measures and use the results to generalize a recently proposed and very effective optical flow algorithm l alvarez j weickert and j sanchez 2000 technical report and ijcv 39 1 41 56 to the case of multimodal image registration our method readily extends to the case of locally computed similarity measures thus providing the flexibility to cope with spatial non stationarities in the way the intensities in the two images are related the well posedness of the resulting equations is proved in a complementary work o d faugeras and g hermosillo 2001 technical report 4235 inria using well established techniques in functional analysis we briefly describe our numerical implementation of these equations and show results on real and synthetic data