fast structured direct spectral methods for differential equations with variable coefficients i the one dimensional case we study the rank structures of the matrices in fourier and chebyshev spectral methods for differential equations with variable coefficients in one dimension we show analytically that these matrices have a so called low rank property not only for constant or smooth variable coefficients but also for coefficients with steep gradients and or high variations large ratios in their maximum minimum function values we develop a matrix free direct spectral solver which uses only a small number of matrix vector products to construct a structured approximation to the original discretized matrix a without the need to explicitly form a this is followed by fast structured matrix factorizations and solutions the overall direct spectral solver has o n log 2 n complexity and o n memory requirement numerical tests for several important but notoriously difficult problems show the superior efficiency and accuracy of our direct spectral solver especially when iterative methods have severe difficulties

a penalty function promoting sparsity within and across groups we introduce a new penalty function that promotes signals composed of a small number of active groups where within each group only a few high magnitude coefficients are nonzero we derive the threshold function associated with the proposed penalty and study its properties we discuss how the proposed penalty threshold function can be useful for signals with isolated nonzeros such as audio with isolated harmonics along the frequency axis or reflection functions in exploration seismology where the nonzeros occur on the boundaries of subsoil layers we demonstrate the use of the proposed penalty threshold functions in a convex denoising and a nonconvex deconvolution formulation we provide convergent algorithms for both formulations and compare the performance with state of the art methods

generalized convexities and generalized gradients based on algebraic operations abstract in this paper we investigate properties of generalized convexities based on algebraic operations introduced by ben tal a ben tal on generalized means and generalized convex functions j optim theory appl 21 1977 1 13 and relations between these generalized convexities and generalized monotonicities we also discuss the h generalized directional derivative and gradient and explore the relation between this gradient and the clarke generalized gradient definitions of some generalized averages of the values of a generalized convex function at n equally spaced points based on the algebraic operations are also presented and corresponding results are obtained finally the convexity is defined and some properties of convex functions are derived

optimality and duality for nondifferentiable multiobjective variational problems abstract the concept of efficiency is used to formulate duality for nondifferentiable multiobjective variational problems wolfe and mond weir type vector dual problems are formulated by using the generalized schwarz inequality and a characterization of efficient solution we established the weak strong and converse duality theorems under generalized f convexity assumptions

approximate convexity and submonotonicity abstract it is shown that a locally lipschitz function is approximately convex if and only if its clarke subdifferential is a submonotone operator consequently in finite dimensions the class of locally lipschitz approximately convex functions coincides with the class of lower c 1 functions directional approximate convexity is introduced and shown to be a natural extension of the class of lower c 1 functions in infinite dimensions the following characterization is established a multivalued operator is maximal cyclically submonotone if and only if it coincides with the clarke subdifferential of a locally lipschitz directionally approximately convex function which is unique up to a constant furthermore it is shown that in asplund spaces every regular function is generically approximately convex

preservation of prox regularity of sets with applications to constrained optimization in this paper we first provide counterexamples showing that sublevels of prox regular functions and levels of differentiable mappings with lipschitz derivatives may fail to be prox regular then we prove the uniform prox regularity of such sets under usual verifiable qualification conditions the preservation of uniform prox regularity of intersection and inverse image under usual qualification conditions is also established applications to constrained optimization problems are given

mixed type duality for multiobjective variational problems with generalized f ρ convexity a mixed type dual for multiobjective variational problems is formulated several duality theorems are established relating properly efficient solutions of the primal and dual variational problems under generalized f convexity static mixed type dual multiobjective problems are particular cases of these problems

duality for nondifferentiable static multiobjective variational problems involving generalized f ϱ convex functions using the generalized schwarz inequality we establish some weak duality theorems for nondifferentiable static multiobjective variational problems involving generalized f convex functions later in the sequel we introduce three dual models for the nondifferentiable static multiobjective fractional variational problems and derive weak duality results for such programs

weakly convex sets and modulus of nonconvexity we consider a definition of a weakly convex set which is a generalization of the notion of a weakly convex set in the sense of vial and a proximally smooth set in the sense of clarke from the case of the hilbert space to a class of banach spaces with the modulus of convexity of the second order using the new definition of the weakly convex set with the given modulus of nonconvexity we prove a new retraction theorem and we obtain new results about continuity of the intersection of two continuous set valued mappings one of which has nonconvex images and new affirmative solutions of the splitting problem for selections we also investigate relationship between the new definition and the definition of a proximally smooth set and a smooth set

new metric properties for prox regular sets in this paper we present diverse new metric properties that prox regular sets shared with convex ones at the heart of our work lie the legendre fenchel transform and complements of balls first we show that a connected prox regular set is completely determined by the legendre fenchel transform of a suitable perturbation of its indicator function then we prove that such a function is also the right tool to extend to the context of prox regular sets the famous connection between the distance function and the support function of a convex set on the other hand given a prox regular set we examine the intersection of complements of open balls containing the set we establish that the distance of a point to a prox regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point this is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes to the best of our knowledge these results are new in the literature and show that the class of prox regular sets have good properties known in convex analysis mathematics subject classification 2010 49j52 49j53

a locally constrained watershed transform the watershed transform from mathematical morphology is a powerful and flexible tool for segmentation however it does not allow a priori knowledge relating to characteristics of region boundaries to be included in the way that other approaches do this paper introduces the locally constrained watershed transform which includes border constraints by modifying the underlying path definition upon which the watershed transform depends this approach maintains many of the desirable properties of the watershed transform such as well defined stopping conditions and efficient implementation while offering more stable segmentation in the presence of noisy or incomplete boundaries

sufficiency and duality in multiobjective programming involving generalized f ρ convexity abstract new classes of generalized f convexity are introduced for vector valued functions examples are given to show their relations with f pseudoconvex f quasiconvex and strictly f pseudoconvex vector valued functions the sufficient optimality conditions and duality results are obtained for multiobjective programming involving generalized f convex vector valued functions

snakes active contour models a snake is an energy minimizing spline guided by external constraint forces and influenced by image forces that pull it toward features such as lines and edges snakes are active contour models they lock onto nearby edges localizing them accurately scale space continuation can be used to enlarge the capture region surrounding a feature snakes provide a unified account of a number of visual problems including detection of edges lines and subjective contours motion tracking and stereo matching we have used snakes successfully for interactive interpretation in which user imposed constraint forces guide the snake near features of interest

a legendre petrov galerkin method for fourth order differential equations in this paper we present a legendre petrov galekin method for one dimensional linear fourth order differential equations a legendre petrov galerkin and chebyshev collocation method is developed for the nonlinear kuramoto sivashinsky equation numerical results are presented to demonstrate the efficiency of the proposed schemes and optimal rates of convergence in the l 2 norm are rigorously derived

morse theory for some lower c 2 functions in finite dimension using inf regularization methods we prove that morse inequalities hold for some lower c2 functions for this purpose we first recall some properties of the class of lower c2 functions and of their moreau yosida approximations then we establish under some qualification conditions on the critical points that it is possible to define a morse index for a lower c2 functionf this index is preserved by the moreau yosida approximation process we prove in particular that the moreau yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space in a last step morse inequalities are written for moreau yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower c2 functions

subsmooth sets functional characterizations and related concepts prox regularity of a set poliquin rockafellar thibault 2000 or its global version proximal smoothness clarke stern wolenski 1995 plays an important role in variational analysis not only because it is associated with some fundamental properties as the local continuous differentiability of the function dist c or the local uniqueness of the projection mapping but also because in the case where c is the epigraph of a locally lipschitz function it is equivalent to the weak convexity lower c 2 property of the function in this paper we provide an adapted geometrical concept called sub smoothness which permits an epigraphic characterization of the approximate convex functions or lower c 1 property subsmooth sets turn out to be naturally situated between the classes of prox regular and of nearly radial sets this latter class has been recently introduced by lewis in 2002 we hereby relate it to the mifflin semismooth functions

robustness of optimal portfolios under risk and stochastic dominance constraints abstract solutions of portfolio optimization problems are often influenced by a model misspecification or by errors due to approximation estimation and incomplete information the obtained results recommendations for the risk and portfolio manager should be then carefully analyzed we shall deal with output analysis and stress testing with respect to uncertainty or perturbations of input data for static risk constrained portfolio optimization problems by means of the contamination technique dependence of the set of feasible solutions on the probability distribution rules out the straightforward construction of convexity based global contamination bounds results obtained in our paper dupacova j kopa m 2012 robustness in stochastic programs with risk constraints annals of operations research 200 55 74 were derived for the risk and second order stochastic dominance constraints under suitable smoothness and or convexity assumptions that are fulfilled e g for the markowitz mean variance model in this paper we relax these assumptions having in mind the first order stochastic dominance and probabilistic risk constraints local bounds for problems of a special structure are obtained under suitable conditions on the structure of the problem and for discrete distributions we shall exploit the contamination technique to derive a new robust first order stochastic dominance portfolio efficiency test

efficiency and generalized convex duality for multiobjective programs abstract the concept of efficiency pareto optimum is used to formulate duality for multiobjective non linear programs the results are obtained for convex functions convex i e weakly convex and strongly convex functions and pseudoconvex functions for the convex and convex functions a wolfe type of dual is formulated while for the pseudoconvex and convex functions a mond weir type dual is proposed

about the lipschitz property of the metric projection in the hilbert space abstract we consider the metric projection operator from the real hilbert space onto a strongly convex set we prove that the restriction of this operator on the complement of some neighborhood of the strongly convex set is lipschitz continuous with the lipschitz constant strictly less than 1 this property characterizes the class of strongly convex sets and to a certain degree the hilbert space we apply the results obtained to the question concerning the rate of convergence for the gradient projection algorithm with differentiable convex function and strongly convex set

on efficiency and duality for multiobjective programs abstract for a multiobjective nonlinear program which involved inequality and equality constraints wolfe mond weir and general mond weir type duals are formulated and the concept of efficiency pareto optimum is used to state some duality results under generalized f convexity assumptions

vector variational inequality as a tool for studying vector optimization problems the paper aims to show that a vector variational inequality can be an useful tool for studying a vector optimization problem

explicit bounds for rational points near planar curves and metric diophantine approximation abstract the primary goal of this paper is to complete the theory of metric diophantine approximation initially developed in beresnevich et al 2007 10 for c 3 non degenerate planar curves with this goal in mind here for the first time we obtain fully explicit bounds for the number of rational points near planar curves further introducing a perturbational approach we bring the smoothness condition imposed on the curves down to c 1 lowest possible this way we broaden the notion of non degeneracy in a natural direction and introduce a new topologically complete class of planar curves to the theory of diophantine approximation in summary our findings improve and complete the main theorems of beresnevich et al 2007 10 and extend the celebrated theorem of kleinbock and margulis 1998 20 in dimension 2 beyond the notion of non degeneracy

watersnakes energy driven watershed segmentation the watershed algorithm from mathematical morphology is powerful for segmentation however it does not allow incorporation of a priori information as segmentation methods that are based on energy minimization in particular there is no control of the smoothness of the segmentation result in this paper we show how to represent watershed segmentation as an energy minimization problem using the distance based definition of the watershed line a priori considerations about smoothness can then be imposed by adding the contour length to the energy function this leads to a new segmentation method called watersnakes integrating the strengths of watershed segmentation and energy based segmentation experimental results show that when the original watershed segmentation has noisy boundaries or wrong limbs attached to the object of interest the proposed method overcomes those drawbacks and yields a better segmentation

second order f α ρ d convexity and duality in multiobjective programming a class of second order f a r d convex functions and their generalizations is introduced using the assumptions on the functions involved weak strong and strict converse duality theorems are established for a second order mond weir type multiobjective dual

generalized convexity and concavity of the optimal value function in nonlinear programming in this paper we consider generalized convexity and concavity properties of the optimal value functionf for the general parametric optimization problemp of the form minxf x s t x r many results on convexity and concavity characterizations off were presented by the authors in a previous paper such properties off and the solution set maps form an important part of the theoretical basis for sensitivity stability and parametric analysis in mathematical optimization we give sufficient conditions for several types of generalized convexity and concavity off in terms of respective generalized convexity and concavity assumptions onf and convexity and concavity assumptions on the feasible region point to set mapr specializations of these results to the parametric inequality equality constrained nonlinear programming problem are provided

fouriertransformation auf dunnen gittern mit hierarchischen basen an algorithm for the trigonometric interpolation of functions ofn variables on a sparse grid is described this discrete fourier transform based on fft has a greatly reduced complexity in comparison to the fourier transform on a regular grid whereas the approximation quality is only slightly reduced if the function belongs to a korobov space the transformation is easily invertible es wird ein auf der schnellen fouriertransformation beruhender algorithmus zur trigonometrischen interpolation von multivariaten funktionen auf einem dunnen gitter beschrieben bei nur geringfugig schlechterer approximationsgenauigkeit fur funktionen aus korobovraumen hat dieser algorithmus eine viel kleinere komplexitat als die algorithmen die auf gewohnlichen gittern arbeiten die transformation ist auf einfache weise umkehrbar

metric projection and compatibly parameterized families of prox regular sets in hilbert space abstract the aim of the paper is to prove under appropriate conditions that the metric projection to a parameterized closed set c u of a hilbert space is near a point u x with x c u single valued and jointly continuous in the parameter variable u and the variable x whose projection is taken on c u it is also shown how the single valuedness of the metric projection and its above continuity property near the point u x is related to similar properties of a suitable truncation of a set valued mapping associated with the normal cone of c u the above properties are obtained under the attouch wets continuity of c near u and the assumption that the set c u is prox regular in x at x with a compatible parameterization by u at u the paper thus extends previous results established by robinson in finite dimensional euclidean spaces to hilbert spaces

an ellipsoidal branch and bound algorithm for global optimization a branch and bound algorithm is developed for global optimization branching in the algorithm is accomplished by subdividing the feasible set using ellipses lower bounds are obtained by replacing the concave part of the objective function by an affine underestimate a ball approximation algorithm obtained by generalizing of a scheme of lin and han is used to solve the convex relaxation of the original problem the ball approximation algorithm is compared to sedumi as well as to gradient projection algorithms using randomly generated test problems with a quadratic objective and ellipsoidal constraints

the lower semicontinuity of optimal solution sets abstract we provide a sufficient condition which guarantees the lower semicontinuity of the optimal solution set for a general nonlinear programming problem it is shown that the same condition is both sufficient and necessary for convex programming problems that satisfy the slater condition

on minimax fractional optimality conditions with f ρ convexity abstract we establish the sufficient conditions for generalized fractional programming in the framework of f convex functions when the sufficient conditions are utilized one parametric dual problem and two parametric free dual problems may be formulated and duality results are derived

adaptive wavelet algorithms for elliptic pde s on product domains with standard isotropic approximation by piecewise polynomials of fixed order in a domain d subset of r d the convergence rate in terms of the number n of degrees of freedom is inversely proportional to the space dimension d this so called curse of dimensionality can be circumvented by applying sparse tensor product approximation when certain high order mixed derivatives of the approximated function happen to be bounded in l 2 it was shown by nitsche 2006 that this regularity constraint can be dramatically reduced by considering best n term approximation from tensor product wavelet bases when the function is the solution of some well posed operator equation dimension independent approximation rates can be practically realized in linear complexity by adaptive wavelet algorithms assuming that the infinite stiffness matrix of the operator with respect to such a basis is highly compressible applying piecewise smooth wavelets we verify this compressibility for general non separable elliptic pdes in tensor domains applications of the general theory developed include adaptive galerkin discretizations of multiple scale homogenization problems and of anisotropic equations which are robust i e independent of the scale parameters resp of the size of the anisotropy

sufficiency criteria and duality for nonlinear programs involving n set functions abstract several sufficiency criteria and duality results are established under generalized convexity conditions for a class of nonlinear programming problems involving differentiable n set functions

fast geodesic active contours we use an unconditionally stable numerical scheme to implement a fast version of the geodesic active contour model the proposed scheme is useful for object segmentation in images like tracking moving objects in a sequence of images the method is based on the weickert romeney viergever additive operator splitting aos scheme it is applied at small regions motivated by the adalsteinsson sethian level set narrow band approach and uses sethian s 1996 fast marching method for re initialization experimental results demonstrate the power of the new method for tracking in color movies

sparse pseudospectral approximation method multivariate global polynomial approximations such as polynomial chaos or stochastic collocation methods are now in widespread use for sensitivity analysis and uncertainty quantification the pseudospectral variety of these methods uses a numerical integration rule to approximate the fourier type coefficients of a truncated expansion in orthogonal polynomials for problems in more than two or three dimensions a sparse grid numerical integration rule offers accuracy with a smaller node set compared to tensor product approximation however when using a sparse rule to approximately integrate these coefficients one often finds unacceptable errors in the coefficients associated with higher degree polynomials by reexamining smolyak s algorithm and exploiting the connections between interpolation and projection in tensor product spaces we construct a sparse pseudospectral approximation method that accurately reproduces the coefficients for basis functions that naturally correspond to the sparse grid integration rule the compelling numerical results show that this is the proper way to use sparse grid integration rules for pseudospectral approximation

sparse grid discontinuous galerkin methods for high dimensional elliptic equations this paper constitutes our initial effort in developing sparse grid discontinuous galerkin dg methods for high dimensional partial differential equations pdes over the past few decades dg methods have gained popularity in many applications due to their distinctive features however they are often deemed too costly because of the large degrees of freedom of the approximation space which are the main bottleneck for simulations in high dimensions in this paper we develop sparse grid dg methods for elliptic equations with the aim of breaking the curse of dimensionality using a hierarchical basis representation we construct a sparse finite element approximation space reducing the degrees of freedom from the standard o h d to o h 1 log 2 h d 1 for d dimensional problems where h is the uniform mesh size in each dimension our method based on the interior penalty ip dg framework can achieve accuracy of o h k log 2 h d 1 in the energy norm where k is the degree of polynomials used error estimates are provided and confirmed by numerical tests in multi dimensions

digital topology and geometry in medical imaging a survey digital topology and geometry refers to the use of topologic and geometric properties and features for images defined in digital grids such methods have been widely used in many medical imaging applications including image segmentation visualization manipulation interpolation registration surface tracking object representation correction quantitative morphometry etc digital topology and geometry play important roles in medical imaging research by enriching the scope of target outcomes and by adding strong theoretical foundations with enhanced stability fidelity and efficiency this paper presents a comprehensive yet compact survey on results principles and insights of methods related to digital topology and geometry with strong emphasis on understanding their roles in various medical imaging applications specifically this paper reviews methods related to distance analysis and path propagation connectivity surface tracking image segmentation boundary and centerline detection topology preservation and local topological properties skeletonization and object representation correction and quantitative morphometry a common thread among the topics reviewed in this paper is that their theory and algorithms use the principle of digital path connectivity path propagation and neighborhood analysis

approximations by orthonormal mapped chebyshev functions for higher dimensional problems in unbounded domains this paper is concerned with approximation properties of orthonormal mapped chebyshev functions omcfs in unbounded domains unlike the usual mapped chebyshev functions which are associated with weighted sobolev spaces the omcfs are associated with the usual non weighted sobolev spaces this leads to particularly simple stiffness and mass matrices for higher dimensional problems the approximation results for both the usual tensor product space and hyperbolic cross space are established with the latter particularly suitable for higher dimensional problems

sufficiency and duality in nondifferentiable multiobjective programming involving generalized type i functions abstract recently hachimi and aghezzaf defined generalized f d type i functions a new class of functions that unifies several concepts of generalized type i functions in this paper the generalized f d type i functions are extended to nondifferentiable functions by utilizing the new concepts we obtain several sufficient optimality conditions and prove mixed type and mond weir type duality results for the nondifferentiable multiobjective programming problem

mixed type duality in multiobjective programming problems two mixed type duals are introduced for multiobjective programming and for multiobjective fractional programming respectively results about weak duality strong duality and strictly reverse duality are presented under more suitable conditions

second order global optimality conditions for convex composite optimization in recent years second order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established in this paper second order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition this result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification in particular second order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained 1998 the mathematical programming society inc published by elsevier science b v

nonconvex sparse logistic regression with weakly convex regularization in this paper we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem the idea is based on the finding that a weakly convex function as an approximation of the l 0 pseudo norm is able to better induce sparsity than the commonly used l 1 norm for a class of weakly convex sparsity inducing functions we prove the nonconvexity of the corresponding problem and study its local optimality conditions and the choice of the regularization parameter despite the nonconvexity a method based on proximal gradient descent is used to solve the general weakly convex sparse logistic regression and its convergence behavior is studied theoretically then the general framework is applied to a specific weakly convex function and a local optimality condition and a bound on the logistic loss at a local optimum are provided the solution method is instantiated in this case as an iterative firm shrinkage algorithm and a nesterov acceleration is used with a convergence guarantee its effectiveness is demonstrated in numerical experiments by both randomly generated and real datasets

classical and strong convexity of sublevel sets and application to attainable sets of nonlinear systems necessary and sufficient conditions for convexity and strong convexity respectively of connected sublevel sets that are defined by finitely many real valued c 1 1 maps are presented a novel characterization of strongly convex sets in terms of the so called local quadratic support is proved the results concerning strong convexity are used to derive sufficient conditions for attainable sets of continuous time nonlinear systems to be strongly convex an application of these conditions is a novel method to overapproximate attainable sets when strong convexity is present

efficient spectral sparse grid methods and applications to high dimensional elliptic problems we develop in this paper some efficient algorithms which are essential to implementations of spectral methods on the sparse grid by smolyak s construction based on a nested quadrature more precisely we develop a fast algorithm for the discrete transform between the values at the sparse grid and the coefficients of expansion in a hierarchical basis and by using the aforementioned fast transform we construct two very efficient sparse spectral galerkin methods for a model elliptic equation in particular the chebyshev legendre galerkin method leads to a sparse matrix with a much lower number of nonzero elements than that of low order sparse grid methods based on finite elements or wavelets and can be efficiently solved by a suitable sparse solver ample numerical results are presented to demonstrate the efficiency and accuracy of our algorithms

optimality and duality for generalized fractional programming involving nonsmooth f ρ convex functions using a parametric approach we establish necessary and sufficient conditions and derive duality theorems for a class of nonsmooth generalized minimax fractional programming problems containing f r convex functions

generalized convex duality for multiobjective fractional programs abstract egudo derived some duality theorems for multi objective programs using the concept of efficiency coupled with some generalized convexity assumptions on the objective and constraint functions the main results of the present work can be thought of as extensions of the results of egudo in the context of multi objective fractional programs

proximal methods for cohypomonotone operators conditions are given for the viability and the weak convergence of an inexact relaxed proximal point algorithm for finding a common zero of countably many cohypomonotone operators in a hilbert space in turn new convergence results are obtained for an extended version of the proximal method of multipliers in nonlinear programming

currents carried by the graphs of semi monotone maps in this paper we study the structure weak continuity and approximability properties for the integer multiplicity rectifiable currents carried by the graphs of maximal semi monotone set valued maps on an n dimensional convex domain especially we give an enhanced version of approximation theorem for the subgradients of semi convex functions

approximately convex functions and approximately monotonic operators abstract we present characterizations of some generalized convexity properties of functions with the help of a general subdifferential we stress the case of lower semicontinuous functions we also study the important case of marginal functions and we provide representation results

nonsmooth minimax programming under locally lipschitz φ ρ invexity abstract minimax programming problems involving locally lipschitz invex functions are considered the parametric and non parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable invexity assumption imposed on objective and constraint functions when the sufficient conditions are utilized parametric and non parametric dual problems in the sense of mond weir and wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem with the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems

softness sleekness and regularity properties in nonsmooth analysis abstract we study continuity properties of tangent and normal cones to a closed subset of a banach space such a study can be seen as a nonsmooth set theoretic analogue of the study of continuously differentiable functions we give particular attention to the case where different concepts coincide a desirable feature such properties are able to offer we identify a wide class of sets and functions for which this property occurs

fourier pseudospectral method on generalized sparse grids for the space fractional schrodinger equation abstract in this paper the time splitting fourier pseudospectral method on the generalized sparse grids is applied to solve the space fractional schrodinger equation we give a containment relation between different level index sets of the generalized sparse grids and it can be used in designing the reference generalized sparse grids which are finer than other considered grids thus the numerical solution on the reference generalized sparse grids can be used as the reference true solution of the equation then the fully discrete algorithm is obtained in the numerical experiments we compare the numerical results on the generalized sparse grids with those on the full grids for the interpolation of the gaussian multiplied by a factor and for the computation of the schrodinger equation with two kinds of non smooth potentials the advantages of the fourier pseudospectral method on the generalized sparse grids with the level index set of parameter k 1 2 3 are manifest in the approximation with high resolution here the sparsity of the generalized sparse grids will become weak when the parameter k becomes large moreover the advantage of the generalized sparse grids is more pronounced in solving the schrodinger equation with the higher dimension the square well potential or the fractional laplacian

area theorem and smoothness of compact cauchy horizons we obtain an improved version of the area theorem for not necessarily differentiable horizons which in conjunction with a recent result on the completeness of generators allows us to prove that under the null energy condition every compactly generated cauchy horizon is smooth and compact we explore the consequences of this result for time machines topology change black holes and cosmic censorship for instance it is shown that compact cauchy horizons cannot form in a non empty spacetime which satisfies the stable dominant energy condition wherever there is some source content

error analysis of the wiener askey polynomial chaos with hyperbolic cross approximation and its application to differential equations with random input abstract it is well known that sparse grid algorithm has been widely accepted as an efficient tool to overcome the curse of dimensionality in some degree in this note we give the error estimate of hyperbolic cross hc approximations with all sorts of askey polynomials these polynomials are useful in generalized polynomial chaos gpc in the field of uncertainty quantification the exponential convergences in both regular and optimized hc approximations have been shown under the condition that the random variable depends on the random inputs smoothly in some degree moreover we apply gpc to numerically solve the ordinary differential equations with slightly higher dimensional random inputs both regular and optimized hc have been investigated with laguerre chaos charlier chaos and hermite chaos in the numerical experiment the discussion of the connection between the standard anova approximation and galerkin approximation is in the appendix

convergence of quasi optimal sparse grid approximation of hilbert space valued functions application to random elliptic pdes in this work we provide a convergence analysis for the quasi optimal version of the sparse grids stochastic collocation method we presented in a previous work on the optimal polynomial approximation of stochastic pdes by galerkin and collocation methods beck et al math models methods appl sci 22 09 2012 the construction of a sparse grid is recast into a knapsack problem a profit is assigned to each hierarchical surplus and only the most profitable ones are added to the sparse grid the convergence rate of the sparse grid approximation error with respect to the number of points in the grid is then shown to depend on weighted summability properties of the sequence of profits this is a very general argument that can be applied to sparse grids built with any uni variate family of points both nested and non nested as an example we apply such quasi optimal sparse grids to the solution of a particular elliptic pde with stochastic diffusion coefficients namely the inclusions problem we detail the convergence estimates obtained in this case using polynomial interpolation on either nested clenshaw curtis or non nested gauss legendre abscissas verify their sharpness numerically and compare the performance of the resulting quasi optimal grids with a few alternative sparse grid construction schemes recently proposed in the literature

optimality and mixed saddle point criteria in multiobjective optimization in this paper higher order strong convexity for lipschitz functions is introduced and is utilized to derive the optimality conditions for the new concept of strict minimizer of higher order for a multiobjective optimization problem variational inequality problem is introduced and its solutions are related to the strict minimizers of higher order for a multiobjective optimization problem the notion of vector valued partial lagrangian is also introduced and equivalence of the mixed saddle points of higher order and higher order minima are provided

mixed type duality for multiobjective variational problems the concept of mixed type duality has been extended to the class of multiobjective variational problems a number of duality relations are proved to relate the efficient solutions of the primal and its mixed type dual problems the results are obtained for convex generalized convex functions these studies have been generalized to the case of invex generalized invex functions our results apparently generalize a fairly large number of duality results previously obtained for finite dimensional nonlinear programming problems under various convexity assumptions

generalized sufficiency criteria in continuous time programming with application to a class of variational type inequalities abstract in this paper we establish a number of sufficiency criteria under generalized invexity conditions for general continuous time programming problems with nonlinear operator equality and inequality constraints as an application of these results we establish some existence criteria for solutions of a class of variational type inequalities

sparse spectral approximations of high dimensional problems based on hyperbolic cross hyperbolic cross approximations by some classical orthogonal polynomials functions in both bounded and unbounded domains are considered in this paper optimal error estimates in proper anisotropic weighted korobov spaces for both regular hyperbolic cross approximations and optimized hyperbolic cross approximations are established these fundamental approximation results indicate that spectral methods based on hyperbolic cross approximations can be effective for treating certain high dimensional problems and will serve as basic tools for analyzing sparse spectral methods in high dimensions

strong and weak convexity of sets and functions in this paper we study two classes of sets strongly and weakly convex sets for each class we derive a series of properties which involve either the concept of supporting ball an obvious extension of the concept of supporting hyperplane or the normal cone to the set we also study a class of functions denoted convex which satisfy for arbitrary points x1 and x2 and any value 0 1 the classical inequality of convex functions up to a term 1 x1 x2 2 depending on the sign of the constant the function is said to be strongly or weakly convex we provide characteristic properties of this class of sets and we relate it to strongly and weakly convex sets via the epigraph and the level sets finally we give three applications a separation theorem a sufficient condition for global optimum of a nonconvex programming problem and a sufficient geometrical condition for a set to be a manifold

matrix criteria for the pseudo p convexity of a quadratic form abstract except for the family of quasiconvex quadratic functions no characterization of generalized convex quadratic forms on convex sets in r k exists we characterize pseudo p convex quadratic functions i e bi pseudoconvex quadratics for which every stationary point is a global minimum on solid convex sets in r n r m

approximation and decomposition properties of some classes of locally d c function we study the connexion between local and global decompositions of some important subclasses of locally d c functions functions which locally split as a difference of two convex functions then we tackle the problem of regularizing such functions by the moreau yosida process and prove in particular that the class of lower c2 functions fits well this approximation procedure

on the degree and separability of nonconvexity and applications to optimization problems we study qualitative indications for d c representations of closed sets in and functions on hilbert spaces the first indication is an index of nonconvexity which can be regarded as a measure for the degree of nonconvexity we show that a closed set is weakly closed if this indication is finite using this result we can prove the solvability of nonconvex minimization problems by duality a minimization problem on a feasible set in which this indication is low can be reduced to a quasi concave minimization over a convex set in a low dimensional space the second indication is the separability which can be incorporated in solving dual problems both the index of nonconvexity and the separability can be characteristics to good d c representations for practical computation we present a notion of clouds which enables us to obtain a good d c representation for a class of nonconvex sets using a generalized caratheodory s theorem we present various applications of clouds

local differentiability of distance functions recently clarke stern and wolenski characterized in a hilbert space the closed subsets c for which the distance function dc is continuously differentiable everywhere on an open tube of uniform thickness around c here a corresponding local theory is developed for the property of dc being continuously differentiable outside of c on some neighborhood of a point x c this is shown to be equivalent to the prox regularity of c at x which is a condition on normal vectors that is commonly fulfilled in variational analysis and has the advantage of being verifiable by calculation additional characterizations are provided in terms of dc being locally of class c 1 or such that dc 2 is convex around x for some 0 prox regularity of c at x corresponds further to the normal cone mapping nc having a hypomonotone truncation around x and leads to a formula for pc by way of nc the local theory also yields new insights on the global level of the clarke stern wolenski results and on a property of sets introduced by shapiro as well as on the concept of sets with positive reach considered by federer in the finite dimensional setting

on the convergence of the iterative shrinkage thresholding algorithm with a weakly convex penalty we consider the iterative shrinkage thresholding algorithm ista applied to a cost function composed of a data fidelity term and a penalty term the penalty is nonconvex but the concavity of the penalty is accounted for by the data fidelity term so that the overall cost function is convex we provide a generalization of the convergence result for ista viewed as a forward backward splitting algorithm we also demonstrate experimentally that for the current setup using large stepsizes in ista can accelerate convergence more than existing schemes proposed for the convex case like twist or fista

efficient spectral galerkin method i direct solvers of second and fourth order equations using legendre polynomials this paper presents some efficient algorithms based on the legendre galerkin approximations for the direct solution of the second and fourth order elliptic equations the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with sparse matrices for the discrete variational formulations the complexities of the algorithms are a small multiple of n d 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions in addition the algorithms can be effectively parallelized since the bottlenecks of the algorithms are matrix matrix multiplications

uniform prox regularity of functions and epigraphs in hilbert spaces abstract we study the local uniform prox regularity of functions make the connection with the lower c 2 property and essentially with the prox regularity of epigraphs