regularity of potential functions of the optimal transportation problem the potential function of the optimal transportation problem satisfies a partial differential equation of monge ampere type in this paper we introduce the notion of a generalized solution and prove the existence and uniqueness of generalized solutions of the problem we also prove the solution is smooth under certain structural conditions on the cost function

on the monotonicity of optimal transportation plans abstract we obtain analytical properties of the maps that give optimal transportation plans for the l 2 wasserstein distance we take advantage of the monotonicity of such optimal transportation plans to discuss their measurability and continuity as a main result of independent interest we obtain the a e continuity of monotone in the sense of zarantonello operators

optimal maps for the multidimensional monge kantorovich problem let 1 n be borel probability measures on d denote by 1 n the set of all n tuples t t1 tn such that ti d d i 1 n are borel measurable and satisfy 1 i v for all borel v d the multidimensional monge kantorovich problem investigated in this paper consists of finding s s1 sn 1 n minimizing over the set 1 n we study the case where the i s have finite second moments and vanish on d 1 rectifiable sets we prove existence and uniqueness of optimal maps s when we impose that s1 x x and give an explicit form of the maps si the result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality as a consequence we obtain uniqueness and characterization of an optimal measure for the multidimensional kantorovich problem 1998 john wiley sons inc

on approximation of non newtonian fluid flow by the finite element method in this paper the problem of numerical approximation of non newtonian fluid flow with free surface is considered namely the flow of fresh concrete is addressed industrial mixtures often behaves like non newtonian fluids exhibiting a yield stress that needs to be overcome for the flow to take place cf r b bird r c armstrong o hassager dynamics of polymeric liquids vol 1 fluid mechanics wiley new york 1987 r p chhabra j f richardson non newtonian flow in the process industries butterworth heinemann london 1999 the main interest is paid to the mathematical formulation of the problem and to discretization with the aid of finite element method the described numerical procedure is applied onto the solution of several problems

an elementary proof of the polar factorization of vector valued functions we present an elementary proof of an important result of y brenier br1 br2 namely that vector fields in d satisfying a nondegeneracy condition admit the polar factorization u x s x where is a convex function and s is a measure preserving mapping brenier solves a minimization problem using monge kantorovich theory whereas we turn our attention to a dual problem whose euler lagrange equation turns out to be

geometric properties of the sections of solutions to the monge ampere equation in this paper we establish several geometric properties of the cross sections of generalized solutions 4 to the monge ampere equation det d2 when the measure t satisfies a doubling property a main result is a characterization of the doubling measures t in terms of a geometric property of the cross sections of 4 this is used to obtain estimates of the shape and invariance properties of the cross sections that are valid under appropriate normalizations 0 introduction let ri r be a convex function a supporting hyperplane to q at the point xo q xo is an affine function x q xo p x xo such that q x x for all x e r given t 0 a section of 0 at height t is the convex set so xo p t x e r x p ri defined by vq xo p q x xo p x xo vx e rn if e c rn then v e uxee vq x by a classical theorem of aleksandrov the class of sets e such that v e is lebesgue measurable is a o algebra that contains the borel sets and one can define the monge ampere measure associated with q as the borel measure u given by pt v e see ch y the purpose in this paper to analyze in detail several important geometric properties of the sections of the convex function q when its associated monge ampere measure u satisfies a doubling condition the interest in these properties comes from the study of the solutions of the monge ampere equation and its linearizations both elliptic and parabolic and from real harmonic analysis see cl c2 c g1 c g2 and h some of these properties have been used in those references to prove a lemma of besicovitch s type and a calderon zygmund decomposition ir terms of sections which allows us to establish estimates of the solutions to the linearized monge ampere equation also some of these properties imply that r received by the editors june 9 1997 1991 mathematics subject classification primary 35j60 35d10 secondary 26b25

a variational approach for the 2 dimensional semi geostrophic shallow water equations existence of weak solutions to the 3 d semi geostrophic equations with rigid boundaries was proved by benamou and brenier 3 using monge transport theory this paper extends the results to a free surface boundary condition which is more physically appropriate this extension is at present for the 2 d shallow water case only in addition we establish stronger time regularity than was possible in 3

notes on the wasserstein metric in hilbert spaces let x y be a pair of hilbert valued random variables for which the wasserstein distance between the marginal distributions is reached we prove that the mapping omega rightarrow x omega y omega is increasing in a certain sense moreover if y satisfies a nondegeneration condition we can take x t y with t monotone in the sense of zarantarello we apply these results to obtain a proof of the central limit theorem clt in hilbert spaces which does not make use of the clt for real valued random variables

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

a retinex model based on absorbing markov chains the retinex algorithm developed by land and mccann provides an abstract model of the mechanism of color sensation in the human vision system at the basis of model lies the fact that the color appearance of a point does not depend only on its color value but rather on the comparison among itself and other pixels according to the model separately for each chromatic channel an image pixel receives suitably filtered information about the brightness of other image regions based on which its own brightness is eventually re scaled the original formulation land and mccann 1971 uses a path based sampling approach the information is transported by memoryless random walks starting from randomly chosen points along the path the information is filtered based on the brightness of the travelled regions by a specific path function computed through chains of ratios of pixel intensities such a function is path dependent and retains the value of the brightest point found along the path the overall correction to a pixel depends on the specific realizations of two sampling processes the starting point sampling process and the path sampling process as a consequence of the sampling this algorithm is known to be intrinsically noisy this draw back can be overcome by passing from the path sampling algorithm to the probabilistic representation of the corresponding diffusion process in this paper we start from the random path simulative model of retinex we respell the standard path based sampling process representation of the retinex model as formalized in provenzi et al 2005 and we show that despite the overall path dependence the model can be given a representation in terms of absorbing markov chains by means of the embedding into a suitable state space we derive the corresponding analytic model accounting for the combined effects of path function path sampling process and starting point sampling process finally we provide a numerical algorithm for working out its solution using such a model the output brightness of a pixel can be computed based on the solution of a simple sparse linear system we show that the output of the random walk sampling algorithm and the markov chain based algorithm agree to an extent that can be controlled by few model parameters we have found also that the markov chain based algorithm is more efficient than the basic random path sampling in obtaining noise free images those analytic probabilistic models and simulative models can be used as complementary tools for studying the retinex mechanism and for identifying and comparing variants

issues about retinex theory and contrast enhancement we present an interpretation of land s retinex theory that we show to be consistent with the original formulation the proposed model relies on the computation of the expectation value of a suitable random variable weighted with a kernel function thus the name kernel based retinex kbr for the corresponding algorithm kbr shares the same intrinsic characteristics of the original retinex it can reduce the effect of a color cast and enhance details in low key images but since it can only increase pixel intensities it is not able to enhance over exposed pictures comparing the analytical structure of kbr with that of a recent variational model of color image enhancement we are able to perform an analysis of the action of kbr on contrast showing the need to anti symmetrize its equation in order to produce a two sided contrast modification able to enhance both under and over exposed pictures the anti symmetrized kbr equations show clear correspondences with other existing color correction models in particular ace whose relationship with retinex has always been difficult to clarify finally from an image processing point of view we mention that both kbr and its antisymmetric version are free from the chromatic noise due to the use of paths in the original retinex implementation and that they can be suitably approximated in order to reduce their computational complexity from mathcal o n 2 to mathcal o n log n being n the number of input pixels

the regularity of mappings with a convex potential in this work we apply the techniques developed in cl to the problem of mappings with a convex potential between domains that is given two bounded domains q q2 of rn and two nonnegative real functions fi defined in qi that are bounded away from zero and infinity we want to study the map vf v w for a lipschitz convex v such that v maps ql onto q 2 in the a e sense and in some weak sense 1 f2 vyv det dij v f1 x in recent work y brenier showed existence and uniqueness of such a map provided that jaqil 0 under the obvious compatibility condition

constructing optimal maps for monge s transport problem as a limit of strictly convex costs the monge kantorovich problem is to move one distribution of mass onto another as efficiently as possible where monge s original criterion for efficiency 19 was to minimize the average distance transported subsequently studied by many authors it was not until 1976 that sudakov showed solutions to be realized in the original sense of monge i e as mappings from rn to rn 23 a second proof of this existence result formed the subject of a recent monograph by evans and gangbo 7 who avoided sudakov s measure decomposition results by using a partial differential equations approach in the present manuscript we give a third existence proof for optimal mappings which has some advantages and disadvantages relative to existing approaches it requires no continuity or separation of the mass distributions yet our explicit construction yields more geometrical control than the abstract method of sudakov indeed this control turns out to be essential for addressing a gap which has recently surfaced in sudakov s approach to the problem in dimensions n 3 see the remarks at the end of this section it is also shorter and more flexible than either and can be adapted to handle transportation on riemannian manifolds or around obstacles as we plan to show in a subsequent work 13 the problem considered here is the classical one problem 1 monge fix a norm d x y llx yll on rn and two densities non negative borel functions f fe ll rn satisfying the mass balance condition

free boundaries in optimal transport and monge ampere obstacle problems given compactly supported 0 f g 2 l 1 r n the problem of trans porting a fraction m min kfkl1 kgkl1 of the mass of f onto g as cheaply as possible is considered where cost per unit mass transported is given by a cost function c typically quadratic c x y x y 2 2 this question is shown to be equivalent to a double obstacle problem for the monge ampere equation for which sucient conditions are given to guarantee uniqueness of the solution such as f vanishing on sptg in the quadratic case the part of f to be transported increases monotonically with m and if sptf and sptg are separated by a hyperplane h then this part will separated from the balance of f by a semiconcave lipschitz graph over the hyperplane if f f and g g are bounded away from zero and infinity on separated strictly convex domains r n for the quadratic cost this graph is shown to be a c 1 loc hypersurface in whose normal coincides with the direction transported the optimal map between f and g is shown to be holder continuous up to this free bound ary and to those parts of the fixed boundary which map to locally convex parts of the path connected target region

optimal pricing policies for public transportation networks in this paper we study the problem of finding an optimal pricing policy for the use of the public transportation network in a given populated area the transportation network modeled by a borel set sigma subset r n of finite length the densities of the population and of the services or workplaces modeled by the respective finite borel measures varphi 0 and varphi 1 and the effective cost a t for a citizen to cover a distance t without the use of the transportation network are assumed to be given the pricing policy to be found is then a cost b t to cover a distance t with the use of the transportation network i e the price of the ticket for a distance t and it has to provide an equilibrium between the needs of the population hence minimizing the total cost of transportation of the population to the services workplaces and that of the owner of the transportation network hence maximizing the total income of the latter we present a model for such a choice and discuss the existence as well as some qualitative properties of the resulting optimal pricing policies

suboptimal and optimal convergence in mixed finite element methods an elliptic partial differential equation may be formulated in different but equivalent ways and the mixed finite element methods derived from these formulations have different properties we give general error estimates for two such methods which are always optimal for the raviart thomas elements but which are suboptimal for the brezzi douglas marini elements in one of the methods computational experiments show that this suboptimal estimate is sharp

regularity of solutions for some variational problems subject to a convexity constraint we first study the minimizers in the class of convex functions of an elliptic functional with nonhomogeneous dirichlet boundary conditions we prove c1 regularity of the minimizers under the assumption that the upper envelope of admissible functions is c1 this condition is optimal at least when the functional depends only on the gradient 3 we then give various extensions of this result in particular we consider a problem without boundary conditions arising in an economic model introduced by rochet and chone in 4

a perceptually inspired variational framework for color enhancement basic phenomenology of human color vision has been widely taken as an inspiration to devise explicit color correction algorithms the behavior of these models in terms of significative image features such as e g contrast and dispersion can be difficult to characterize to cope with this we propose to use a variational formulation of color contrast enhancement that is inspired by the basic phenomenology of color perception in particular we devise a set of basic requirements to be fulfilled by an energy to be considered as perceptually inspired showing that there is an explicit class of functionals satisfying all of them we single out three explicit functionals that we consider of basic interest showing similarities and differences with existing models the minima of such functionals is computed using a gradient descent approach we also present a general methodology to reduce the computational cost of the algorithms under analysis from o n2 to o n logn being n the number of pixels of the input image

stability of higher order hood taylor methods the stability of a higher order hood taylor method for the approximation of the stationary stokes equations using continuous piecewise polynomials of degree 3 to approximate velocities and continuo

a finite element formulation for the stokes problem allowing equal velocity pressure interpolation abstract in this paper we study a variational formulation of the stokes problem that accommodates the use of equal velocity pressure finite element interpolations the motivation of this method relies on the analysis of a class of fractional step methods for the navier stokes equations for which it is known that equal interpolations yield good numerical results the reason for this turns out to be the difference between two discrete laplacian operators computed in a different manner the formulation of the stokes problem considered here aims to reproduce this effect from the analysis of the finite element approximation of the problem we obtain stability and optimal error estimates using velocity pressure interpolations satisfying a compatibility condition much weaker than the inf sup condition of the standard formulation in particular this condition is fulfilled by the most common equal order interpolations

analysis of a pressure stabilized finite element approximation of the stationary navier stokes equations the purpose of this paper is to analyze a finite element approximation of the stationary navier stokes equations that allows the use of equal velocity pressure interpolation the idea is to introduce as unknown of the discrete problem the projection of the pressure gradient multiplied by suitable algorithmic parameters onto the space of continuous vector fields the difference between these two vectors pressure gradient and projection is introduced in the continuity equation the resulting formulation is shown to be stable and optimally convergent both in a norm associated to the problem and in the l 2 norm for both velocities and pressure this is proved first for the stokes problem and then it is extended to the nonlinear case all the analysis relies on an inf sup condition that is much weaker than for the standard galerkin approximation in spite of the fact that the present method is only a minor modification of this

an anisotropic gls stabilized finite element method for incompressible flow problems abstract we consider a stabilized finite element approximation of incompressible flow problems in primitive velocity pressure variables which is based on a gls formulation equal order interpolation for the velocity and the pressure can be employed with this formulation the method introduced here is specially developed to be used on anisotropic finite element meshes with large element aspect ratios rather than using a single stabilization parameter within each element the jacobian matrix of the affine transformation from the reference element is employed for stabilization purposes stability and anisotropic error estimates in a suitable mesh dependent norm are proved for the linear stokes problem numerical results confirm the accuracy and robustness of the proposed method compared to similar previous stabilized schemes

on the regularity of solutions of optimal transportation problems we give a necessary and sufficient condition on the cost function so that the map solution of monge s optimal transportation problem is continuous for arbitrary smooth positive data this condition was first introduced by ma trudinger and wang 24 30 for a priori estimates of the corresponding monge ampere equation it is expressed by a socalled cost sectional curvature being non negative we show that when the cost function is the squared distance of a riemannian manifold the cost sectional curvature yields the sectional curvature as a consequence if the manifold does not have non negative sectional curvature everywhere the optimal transport map cannot be continuous for arbitrary smooth positive data the non negativity of the cost sectional curvature is shown to be equivalent to the connectedness of the contact set between any cost convex function the proper generalization of a convex function and any of its supporting functions when the cost sectional curvature is uniformly positive we obtain that optimal maps are continuous or holder continuous under quite weak assumptions on the data compared to what is needed in the euclidean case this case includes the quadratic cost on the round sphere

perceptual color correction through variational techniques in this paper we present a discussion about perceptual based color correction of digital images in the framework of variational techniques we propose a novel image functional whose minimization produces a perceptually inspired color enhanced version of the original the variational formulation permits a more flexible local control of contrast adjustment and attachment to data we show that a numerical implementation of the gradient descent technique applied to this energy functional coincides with the equation of automatic color enhancement ace a particular perceptual based model of color enhancement moreover we prove that a numerical approximation of the euler lagrange equation reduces the computational complexity of ace from o n2 to o nlogn where n is the total number of pixels in the image

properties of the solutions to the monge ampere equation abstract we consider solutions to the equation det d 2 when has a doubling property we prove new geometric characterizations for this doubling property by means of the so called engulfing property and deduce the quantitative behaviour of also a constructive approach to the celebrated c1 estimates proved by l caffarelli is presented settling one of the open questions posed by villani amer math soc 58 2003

existence of optimal transport maps for crystalline norms we show the existence of optimal transport maps in the case when the cost function is the distance induced by a crystalline norm in n assuming that the initial distribution of mass is absolutely continuous with respect to mathcal l n the proof is based on a careful decomposition of the space in transport rays induced by a secondary variational problem having the euclidean distance as cost function moreover improving a construction by larman we show the existence of a nikodym set in 3 having full measure in the unit cube intersecting each element of a family of pairwise disjoint open lines only in one point this example can be used to show that the regularity of the decomposition in transport rays plays an essential role in sudakov type arguments for proving the existence of optimal transport maps

the wasserstein gradient flow of the fisher information and the quantum drift diffusion equation we prove the global existence of non negative variational solutions to the drift diffusion evolution equation partial t u div left u mathrm d left 2 frac delta sqrt u sqrt u f right right 0 under variational boundary condition despite the lack of a maximum principle for fourth order equations non negative solutions can be obtained as a limit of a variational approximation scheme by exploiting the particular structure of this equation which is the gradient flow of the perturbed fisher information functional fancyscript f f u frac 12 int left mathrm d log u right 2 u dx int fu dx with respect to the kantorovich rubinstein wasserstein distance between probability measures we also study long time behavior of the solutions proving their exponential decay to the equilibrium state g e v characterized by delta v frac12 left mathrm d v right 2 f quad int rm e v dx int u 0 dx when the potential v is uniformly convex in this case the functional fancyscript f f coincides with the relative fisher information fancyscript f f u frac12 fancyscript i u g int left mathrm d log u g right 2u dx

equilibrium shapes for planar crystals in an external field the equilibrium shape of a two dimensional crystal in a convex background potential g x is analyzed for g 0 the shape of minimum energy may be deduced from surface tension via the wulff construction but if g is not constant little is known beyond the case of a crystal sitting in a uniform field only an unpublished result of okikiolu shows each connected component of the equilibrium crystal to be convex here it will be shown that any such component minimizes energy uniquely among convex sets of its area if the wulff shape and g x are symmetric under x x it follows that the equilibrium crystal is unique convex and connected this last result leads to a new proof that convex crystals away from equilibrium remain convex as they evolve by curvature driven flow subsequent work with felix otto shows without assuming symmetry that no equilibrium crystal has more than two convex components

a pde formalization of retinex theory in 1964 edwin h land formulated the retinex theory the first attempt to simulate and explain how the human visual system perceives color his theory and an extension the reset retinex were further formalized by land and mccann several retinex algorithms have been developed ever since these color constancy algorithms modify the rgb values at each pixel to give an estimate of the color sensation without a priori information on the illumination unfortunately the retinex land mccann original algorithm is both complex and not fully specified indeed this algorithm computes at each pixel an average of a very large set of paths on the image for this reason retinex has received several interpretations and implementations which among other aims attempt to tune down its excessive complexity in this paper it is proved that if the paths are assumed to be symmetric random walks the retinex solutions satisfy a discrete poisson equation this formalization yields an exact and fast implementation using only two ffts several experiments on color images illustrate the effectiveness of the retinex original theory

analysis of locally stabilized mixed finite element methods for the stokes problem in this paper a locally stabilized finite element formulation of the stokes problem is analyzed a macroelement condition which is sufficient for the stability of locally stabilized mixed methods based on a piecewise constant pressure approximation is introduced by satisfying this condition the stability of the qi po quadrilateral and the pi po triangular element can be established

stabilization of low order cross grid pkql mixed finite elements abstract in this paper we analyze a low order family of mixed finite element methods for the numerical solution of the stokes problem and a second order elliptic problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals for the lowest order p 1 q 0 a global spurious pressure mode is shown to exist and so this element as p 1 q 1 case analyzed in armentano and blasco 2010 is unstable however following the ideas given in bochev et al 2006 a simple stabilization procedure can be applied when we approximate the solution of the stokes problem such that the new p 1 q 0 and p 1 q 1 methods are unconditionally stable and achieve optimal accuracy with respect to solution regularity with simple and straightforward implementations moreover we analyze the application of our p 1 q 1 element to the mixed formulation of the elliptic problem in this case by introducing the modified mixed weak form proposed in brezzi et al 1993 optimal order of accuracy can be obtained with our stabilized p 1 q 1 elements numerical results are also presented which confirm the existence of the spurious pressure mode for the p 1 q 0 element and the excellent stability and accuracy of the new stabilized methods

contrast enhancement by nonlinear diffusion filtering to enhance the visual quality of an image that is degraded by uneven light an effective method is to estimate the illumination component and compress it some previous methods have either defects of halo artifacts or contrast loss in the enhanced image due to incorrect estimation in this paper we discuss this problem and propose a novel method to estimate the illumination the illumination is obtained by iteratively solving a nonlinear diffusion equation during the diffusion process surround suppression is embedded in the conductance function to specially enhance the diffusive strength in textural areas of the image the proposed estimation method has the following two merits 1 the boundary areas are preserved in the illumination and thus halo artifacts are prevented and 2 the textural details are preserved in the reflectance to not suffer from illumination compression which contributes to the contrast enhancement in the result experimental results show that the proposed algorithm achieves excellent performance in artifact removal and local contrast enhancement

an adaptive stabilized finite element method for the generalized stokes problem in this work we present an adaptive strategy based on an a posteriori error estimator for a stabilized finite element method for the stokes problem with and without a reaction term the hierarchical type estimator is based on the solution of local problems posed on appropriate finite dimensional spaces of bubble like functions an equivalence result between the norm of the finite element error and the estimator is given where the dependence of the constants on the physics of the problem is explicited several numerical results confirming both the theoretical results and the good performance of the estimator are given

a family of mixed finite elements for the elasticity problem a new mixed finite element formulation for the equations of linear elasticity is considered in the formulation the variables approximated are the displacement the unsymmetric stress tensor and the rotation the rotation act as a lagrange multiplier introduced in order to enforce the symmetry of the stress tensor based on this formulation a new family of both two and three dimensional mixed methods is defined optimal error estimates which are valid uniformly with respect to the poisson ratio are derived finally a new postprocessing scheme for improving the displacement is introduced and analyzed

monge s transport problem on a riemannian manifold monge s problem refers to the classical problem of optimally transporting mass given borel probability measures find the measurepreserving map s m m between them which minimizes the average distance transported set on a complete connected riemannian manifold m and assuming absolute continuity of an optimal map will be shown to exist aspects of its uniqueness are also established

stable and unstable cross grid pkql mixed finite elements for the stokes problem in this paper we develop and analyze a family of mixed finite element methods for the numerical solution of the stokes problem in two space dimensions in these schemes the pressure is interpolated on a mesh of rectangular elements while the velocity is approximated on a triangular mesh obtained by dividing each rectangle into four triangles by its diagonals continuous interpolations of degrees k for the velocity and l for the pressure are considered so the new finite elements are called cross grid p kq l a stability analysis of these approximations is provided based on the macroelement technique of stenberg the lowest order p 1q 1 and p 2q 1 cases are analyzed in detail in the first case a global spurious pressure mode is shown to exist so this element is unstable in the second case however stability is rigorously proved numerical results obtained in these two cases are also presented which confirm the existence of the spurious pressure mode for the p 1q 1 element and the stability of the p 2q 1 element

a wavelet perspective on variational perceptually inspired color enhancement the issue of perceptually inspired correction of color and contrast in digital images has been recently analyzed with the help of variational principles these techniques allowed building a general framework in which the action of many already existing algorithms can be more easily understood and compared in terms of intensification of local contrast and control of dispersion around the average intensity value in this paper we analyze this issue from the dual perspective of wavelet theory showing that it is possible to build energy functionals of wavelet coefficients that lead to a multilevel perceptually inspired color correction by computing the euler lagrange equations associated to the wavelet based functionals we were able to find an analytical formula for the modification of wavelet detail coefficients that overcomes the problem of an ad hoc selection based on empirical considerations besides these theoretical results the wavelet perspective provides the computational advantage of generating much faster algorithms in comparison with the spatial variational framework

analysis of some mixed elements for the stokes problem abstract in this paper we discuss some mixed finite element methods related to the reduced integration penalty method for solving the stokes problem we prove optimal order error estimates for bilinear constant and biquadratic bilinear velocity pressure finite element solutions the result for the biquadratic bilinear element is new while that for the bilinear constant element improves the convergence analysis of johnson and pitkaranta 1982 in the degenerate case when the penalty parameter is set to be zero our results reduce to some related known results proved in by brezzi and fortin 1991 for the bilinear constant element and bercovier and pironneau 1979 for the biquadratic bilinear element our theoretical results are consistent with the numerical results reported by carey and krishnan 1982 and oden et al 1982

analysis of an upwind mixed hybrid finite element method for transport problems we prove optimal order convergence of an upwind mixed hybrid finite element scheme for linear parabolic advection diffusion reaction problems it was introduced in radu et al adv water resources 34 2011 pp 47 61 and is based on an euler implicit mixed hybrid finite element discretization of the problem in fully mass conservative form using the raviart thomas mixed finite element of lowest order on triangular meshes optimal order convergence in time and space is obtained for the fully discrete formulation the scheme provides the same order of convergence as the standard upwind mixed method while it is more efficient since a local elimination of variables is possible with our choice of the upwind weights the theoretical findings are sustained by a numerical experiment

a probabilistic method for image enhancement with simultaneous illumination and reflectance estimation in this paper a new probabilistic method for image enhancement is presented based on a simultaneous estimation of illumination and reflectance in the linear domain we show that the linear domain model can better represent prior information for better estimation of reflectance and illumination than the logarithmic domain a maximum a posteriori map formulation is employed with priors of both illumination and reflectance to estimate illumination and reflectance effectively an alternating direction method of multipliers is adopted to solve the map problem the experimental results show the satisfactory performance of the proposed method to obtain reflectance and illumination with visually pleasing enhanced results and a promising convergence rate compared with other testing methods the proposed method yields comparable or better results on both subjective and objective assessments

stabilization of low order mixed finite elements for the stokes equations we present a new family of stabilized methods for the stokes problem the focus of the paper is on the lowest order velocity pressure pairs while not lbb compliant their simplicity and attractive computational properties make these pairs a popular choice in engineering practice our stabilization approach is motivated by terms that characterize the lbb deficiency of the unstable spaces the stabilized methods are defined by using these terms to modify the saddle point lagrangian associated with the stokes equations the new stabilized methods offer a number of attractive computational properties in contrast to other stabilization procedures they are parameter free do not require calculation of higher order derivatives or edge based data structures and always lead to symmetric linear systems furthermore the new methods are unconditionally stable achieve optimal accuracy with respect to solution regularity and have simple and straightforward implementations we present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods

proximal operator of quotient functions with application to a feasibility problem in query optimization in this paper we determine the proximity functions of the sum and the maximum of componentwise reciprocal quotients of positive vectors for the sum of quotients denoted by q 1 the proximity function is just a componentwise shrinkage function which we call q shrinkage this is similar to the proximity function of the 1 norm which is given by componentwise soft shrinkage for the maximum of quotients q the proximal function can be computed by first order primal dual methods involving epigraphical projections the proximity functions of q 1 are applied to solve convex problems of the form argmin x q a x b subject to x 0 1 x 1 such problems are of interest in selectivity estimation for cost based query optimizers in database management systems

duality theorems for marginal problems given topological spaces x1 xn with product space x probability measures i on xi together with a real function h on x define a marginal problem as well as a dual problem using an extended version of choquet s theorem on capacities an analogue of the classical duality theorem of linear programming is established imposing only weak conditions on the topology of the spaces xi and the measurability resp boundedness of the function h applications concern among others measures with given support stochastic order and general marginal problems