polar factorization and monotone rearrangement of vector valued functions given a probability space x p and a bounded domain r in rd equipped with the lebesgue measure 1 i normalized so that 10 i i it is shown under additional technical assumptions on x and q that for every vector valued function u e lp x p rd there is a unique polar factorization u v s where is a convex function defined on r and s is a measure preserving mapping from x p into q i i provided that u is nondegenerate in the sense that p u e 0 for each lebesgue negligible subset e of rd through this result the concepts of polar factorization of real matrices helmholtz decomposition of vector fields and nondecreasing rearrangements of real valued functions are unified the monge amgre equation is involved in the polar factorization and the proof relies on the study of an appropriate monge kantorovich problem

pointwise rates of convergence for the oliker prussner method for the monge ampere equation we study the oliker prussner method exploiting its geometric nature we derive discrete stability and continuous dependence estimates in the max norm by using a discrete alexandroff estimate and the brunn minkowski inequality we show that the method is exact for all convex quadratic polynomials provided the underlying set of nodes is translation invariant within the domain nodes still conform to the domain boundary this gives a suitable notion of operator consistency which combined with stability leads to pointwise rates of convergence for classical and non classical solutions of the monge ampere equation

conforming approximation of convex functions with the finite element method we consider the interior approximation of convex functions with convex finite element functions the main motivation for this study is the investigation of a novel discretization of optimization problems with convexity constraints by the finite element method under a mild assumption on the family of meshes we show that the conforming approximation is convergent if the finite elements are at least piecewise quadratic we further provide similar results under additional constraints on the function values or on the gradient the theoretical findings are illustrated by numerical examples

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

stickiness in generic low dimensional hamiltonian systems a recurrence time statistics approach we analyze the structure and stickiness in the chaotic components of generic hamiltonian systems with divided phase space following the method proposed recently in lozej and robnik phys rev e 98 022220 2018 2470 004510 1103 physreve 98 022220 the sticky regions are identified using the statistics of recurrence times of a single chaotic orbit into cells dividing the phase space into a grid we perform extensive numerical studies of three example systems the chirikov standard map the family of robnik billiards and the family of lemon billiards the filling of the cells is compared to the random model of chaotic diffusion introduced in robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 for the description of transport in the phase spaces of ergodic systems the model is based on the assumption of completely uncorrelated cell visits because of the strongly chaotic dynamics of the orbit and the distribution of recurrence times is exponential in generic systems the stickiness induces correlations in the cell visits the distribution of recurrence times exhibits a separation of timescales because of the dynamical trapping we model the recurrence time distributions to cells inside sticky areas as a mixture of exponential distributions with different decay times we introduce the variable s which is the ratio between the standard deviation and the mean of the recurrence times as a measure of stickiness we use s to globally assess the distributions of recurrence times we find that in the bulk of the chaotic sea s 1 while s 1 in areas of stickiness we present the results in the form of animated grayscale plots of the variable s in the largest chaotic component for the three example systems included as supplemental material to this paper

secret symmetries of type iib superstring theory on rm ad rm s _ 3 s 3 m 4 we establish features of so called yangian secret symmetries for ads3 type iib superstring backgrounds thus verifying the persistence of such symmetries to this new instance of the ads cft correspondence specifically we find two a priori different classes of secret symmetry generators one class of generators anticipated from the previous literature is more naturally embedded in the algebra governing the integrable scattering problem the other class of generators is more elusive and somewhat closer in its form to its higher dimensional ads5 counterpart all of these symmetries respect left right crossing in addition by considering the interplay between left and right representations we gain a new perspective on the ads5 case we also study the realisation of the yangian in ads3 backgrounds thus establishing a new incarnation of the beisert de leeuw construction

the total variation flow in rn the purpose of this chapter is to prove existence and uniqueness of the minimizing total variation flow in n frac partial u partial t div left frac du left du right right in left 0 infty right times mathbb r n 3 1 coupled with the initial condition u 0 x u 0 x text x in mathbb r n 3 2 when u0 l loc 1 l n

ironing sweeping and multidimensional screening we provide existence proofs and characterization results for the multidimensional version of the multiproduct monopolist problem of mussa and rosen 1978 these results are also directly applicable to the multidimensional nonlinear pricing problems studied by wilson 1993 and armstrong 1996 we establish that bunching is robust in these multidimensional screening problems even with very regular distributions of types this comes from a strong conflict between participation constraints and second order incentive compatibility conditions we consequently design a new technique the sweeping procedure for dealing with bunching in multidimensional contexts this technique extends the ironing procedure of mussa and rosen 1978 to several dimensions we illustrate it on two examples we first solve a linear quadratic version of the bidimensional nonlinear pricing problem where consumers types are exponentially distributed the solution involves pure bundling for consumers with low demands the second example is the bidimensional version of the mussa and rosen problem when consumers types are uniformly distributed on a square the solution is such that the seller offers a full differentiation of products in the upper part of the qualities spectrum but only limited choice for lower qualities this seems to be a quite general pattern for multidimensional screening problems the sweeping procedure is potentially applicable to other multidimensional screening problems

comments on world sheet form factors in ads cft we study form factors in the light cone gauge world sheet theory for strings in ads5 s5 we perturbatively calculate the two particle form factor in a closed sector to one loop in the near plane wave limit and to two loops in the maldacena swanson limit we also perturbatively solve the functional equation which follows from the form factor axioms for the world sheet theory and show that the minimal solution correctly reproduces the discontinuities of the perturbative calculations finally we propose a prescription valid for polynomial orders of the inverse world sheet length for extracting the finite volume world sheet matrix element from the form factors and show that the two excitation matrix element matches with the thermodynamic limit of the spin chain description of certain tree level sym structure constants

approximating optimization problems over convex functions many problems of theoretical and practical interest involve finding an optimum over a family of convex functions for instance finding the projection on the convex functions in hk and optimizing functionals arising from some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and functions with positive semidefinite discrete hessian need not be convex in a discrete sense previous work has concentrated on non local descriptions of convexity making the number of constraints to grow super linearly with the number of nodes even in dimension 2 and these descriptions are very difficult to extend to higher dimensions in this paper we propose a finite difference approximation using positive semidefinite programs and discrete hessians and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to problems in two and three dimensions

handling convexity like constraints in variational problems we provide a general framework to construct finite dimensional approximations of the space of convex functions which also applies to the space of c convex functions and to the space of support functions of convex bodies we give precise estimates of the distance between the approximation space and the admissible set this framework applies to the approximation of convex functions by piecewise linear functions on a mesh of the domain and by other finite dimensional spaces such as tensor product splines we show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints our implementation relies on proximal algorithms and can be easily parallelized thus making it applicable to large scale problems in dimension two and three we illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation three dimensional denoising the principal agent problem and optimizat

on the numerical solution of the equation frac partial 2 z partial x 2 frac partial 2 z partial y 2 left frac partial 2 z partial x partial y right 2 f and its discretizations i the equation indicated in the title is the simplest representative of the class of nonlinear equations of monge ampere type equations with such nonlinearities arise in dynamic meteorology geometric optics elasticity and differential geometry in some special cases heuristic procedures for numerical solution are available but in order for them to be successful a good initial guess is required for a bounded convex domain nonnegativef and dirichlet data we consider a special discretization of the equation based on its geometric interpretation for the discrete version of the problem we propose an iterative method that produces a monotonically convergent sequence no special information about an initial guess is required and to initiate the iterates a routine step is made the method is self correcting and is structurally suitable for a parallel computer the computer program modules and several examples are presented in two appendices

foundations of the ads5 s5 superstring i we review the recent advances towards finding the spectrum of the ads5 s5 superstring we thoroughly explain the theoretical techniques which should be useful for the ultimate solution of the spectral problem in certain cases our exposition is original and cannot be found in the existing literature the present part i deals with foundations of classical string theory in ads5 s5 light cone perturbative quantization and the derivation of the exact light cone world sheet scattering matrix

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

billiards and two dimensional problems of optimal resistance a body moves in a medium composed of noninteracting point particles the interaction of the particles with the body is completely elastic the problem is find the body s shape that minimizes or maximizes resistance of the medium to its motion this is the general setting of the optimal resistance problem going back to newton here we restrict ourselves to the two dimensional problems for rotating generally non convex bodies the main results of the paper are the following first to any compact connected set with piecewise smooth boundary b subset mathbb r 2 we assign a measure b on conv b 2 2 generated by the billiard in mathbb r 2 setminus b and characterize the set of measures b second using this characterization we solve various problems of minimal and maximal resistance of rotating bodies by reducing them to special monge kantorovich problems

the dirichlet problem for the convex envelope the convex envelope of a given function was recently characterized as the solution of a fully nonlinear partial differential equation pde in this article we study a modified problem the dirichlet problem for the underlying pde the main result is an optimal regularity result differentiability c1 regularity of the boundary data implies the corresponding result for the solution in the interior despite the fact that the solution need not be continuous up to the boundary secondary results are the characterization of the convex envelope as i the value function of a stochastic control problem and ii the optimal underestimator for a class of nonlinear elliptic pdes

convergence of adaptive finite element methods adaptive finite element methods fems have been widely used in applications for over 20 years now in practice they converge starting from coarse grids although no mathematical theory has been able to prove this assertion ensuring an error reduction rate based on a posteriori error estimators together with a reduction rate of data oscillation information missed by the underlying averaging process we construct a simple and efficient adaptive fem for elliptic partial differential equations we prove that this algorithm converges with linear rate without any preliminary mesh adaptation nor explicit knowledge of constants any prescribed error tolerance is thus achieved in a finite number of steps a number of numerical experiments in two and three dimensions yield quasi optimal meshes along with a competitive performance extensions to higher order elements and applications to saddle point problems are discussed as well

approximated convex envelope of a function the goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope such a problem arises in various physical situations where the function considered is some energy that has to be minimized this study is a first step toward understanding how to approximate the quasi convex envelope of a function the importance of this issue is due to the various applications that are encountered in particular in the field of material science 1994 copyright society for industrial and applied mathematics

uniqueness of the cheeger set of a convex body abstract we prove that if c r n is an open bounded convex set then there is only one cheeger set inside c and it is convex a cheeger set of c is a set which minimizes the ratio perimeter over volume among all subsets of c

scalar products of bethe vectors in models with mathfrak gl 2 1 symmetry 1 super analog of reshetikhin formula we study scalar products of bethe vectors in integrable models solvable by nested algebraic bethe ansatz and possessing mathfrak gl 2 1 symmetry using explicit formulas of the monodromy matrix entries multiple actions onto bethe vectors we obtain a representation for the scalar product in the most general case this explicit representation appears to be a sum over partitions of the bethe parameters it can be used for the analysis of scalar products involving on shell bethe vectors as a by product we obtain a determinant representation for the scalar products of generic bethe vectors in integrable models with mathfrak gl 1 1 symmetry

embedding sn into rn 1 with given integral gauss curvature and optimal mass transport on sn in a d aleksandrov convex polyhedra gittl moscow ussr 1950 in russian english translation a d aleksandrov convex polyhedra springer berlin new york 2005 a d aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data the first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in euclidean space with prescribed integral gauss curvature our solution includes the case of a convex polytope this problem was also first considered by aleksandrov and below it is referred to as aleksandrov s problem the second goal of this paper is to show that in variational form the aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations

on convex functions and the finite element method many problems of theoretical and practical interest involve finding a convex or concave function for instance optimization problems such as finding the projection on the convex functions in h k omega or some problems in economics in the continuous setting and assuming smoothness the convexity constraints may be given locally by asking the hessian matrix to be positive semidefinite but in making discrete approximations two difficulties arise the continuous solutions may be not smooth and an adequate discrete version of the hessian must be given in this paper we propose a finite element description of the hessian and prove convergence under very general conditions even when the continuous solution is not smooth working on any dimension and requiring a linear number of constraints in the number of nodes using semidefinite programming codes we show concrete examples of approximations to optimization problems

discretization of functionals involving the monge ampere operator gradient flows in the wasserstein space have become a powerful tool in the analysis of diffusion equations following the seminal work of jordan kinderlehrer and otto jko the numerical applications of this formulation have been limited by the difficulty to compute the wasserstein distance in dimension geqslant 2 one step of the jko scheme is equivalent to a variational problem on the space of convex functions which involves the monge ampere operator convexity constraints are notably difficult to handle numerically but in our setting the internal energy plays the role of a barrier for these constraints this enables us to introduce a consistent discretization which inherits convexity properties of the continuous variational problem we show the effectiveness of our approach on nonlinear diffusion and crowd motion models

two dimensional body of maximum mean resistance a two dimensional body exhibiting a slight rotational movement moves in a rarefied medium of particles which collide with it in a perfectly elastic way in previously realized investigations by the first two authors alexander yu plakhov paulo d f gouveia problems of maximal mean resistance on the plane nonlinearity 20 2007 2271 2287 shapes of nonconvex bodies were sought which would maximize the braking force of the medium on their movement giving continuity to this study new investigations have been undertaken which culminate in an outcome which represents a large qualitative advance relative to that which was achieved earlier this result now presented consists of a two dimensional shape which confers on the body a resistance which is very close to its theoretical supremum value but its interest does not lie solely in the maximization of newtonian resistance on regarding its characteristics other areas of application are seen to begin to appear which are thought to be capable of having great utility the optimal shape which has been encountered resulted from numerical studies thus it is the object of additional study of an analytical nature where it proves some important properties which explain in great part its effectiveness

structure size and statistical properties of chaotic components in a mixed type hamiltonian system we perform a detailed study of the chaotic component in mixed type hamiltonian systems on the example of a family of billiards introduced by robnik in j phys a math gen 16 3971 1983 jphac50305 447010 1088 0305 4470 16 17 014 the phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times the structure of the chaotic component is discerned from the cells visited by the chaotic orbit the fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box counting method the cell filling dynamics is compared to a model of uncorrelated motion the so called random model robnik et al j phys a math gen 30 l803 1997 jphac50305 447010 1088 0305 4470 30 23 003 and deviations attributed to sticky objects in the phase space are found the statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run the stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times the recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed the standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter lastly the decay of correlations and the diffusion of momenta is analyzed

calculation of norms of bethe wave functions a class of two dimensional completely integrable models of statistical mechanics and quantum field theory is considered eigenfunctions of the hamiltonians are known for these models norms of these eigenfunctions in the finite box are calculated in the present paper these models include in particular the quantum nonlinear schrodinger equation and the heisenbergxxz model

an algorithm for computing solutions of variational problems with global convexity constraints we present an algorithm to approximate the solutions to variational problems where set of admissible functions consists of convex functions the main motivation behind the numerical method is to compute solutions to adverse selection problems within a principal agent framework problems such as product lines design optimal taxation structured derivatives design etc can be studied through the scope of these models we develop a method to estimate their optimal pricing schedules

convergence of the mass transport steepest descent scheme for the subcritical patlak keller segel model variational steepest descent approximation schemes for the modified patlak keller segel equation with a logarithmic interaction kernel in any dimension are considered we prove the convergence of the suitably interpolated in time implicit euler scheme defined in terms of the euclidean wasserstein distance associated with this equation for subcritical masses as a consequence we recover the recent result about the global in time existence of weak solutions to the modified patlak keller segel equation for the logarithmic interaction kernel in any dimension in the subcritical case moreover we show how this method performs numerically in dimension one in this particular case this numerical scheme corresponds to a standard implicit euler method for the pseudoinverse of the cumulative distribution function we demonstrate its capabilities to reproduce the blow up of solutions for supercritical masses easily without the need of mesh refinement

adaptive anisotropic and hierarchical cones of discrete convex functions we introduce a new class of adaptive methods for optimization problems posed on the cone of convex functions among the various mathematical problems which possess such a formulation the monopolist problem rochet and chone econometrica 66 783 826 1998 ekeland and moreno bromberg numer math 115 45 69 2010 arising in economics is our main motivation consider a two dimensional domain omega sampled on a grid x x of n points we show that the cone mathrm conv x conv x of restrictions to x x of convex functions on omega is typically characterized by approx n 2 n2 linear inequalities a direct computational use of this description therefore has a prohibitive complexity we thus introduce a hierarchy of sub cones mathrm conv mathcal v conv v of mathrm conv x conv x associated to stencils mathcal v v which can be adaptively locally and anisotropically refined we show using the arithmetic structure of the grid that the trace u x u x of any convex function u on omega is contained in a cone mathrm conv mathcal v conv v defined by only mathcal o n ln 2 n o nln2n linear constraints in average over grid orientations numerical experiments for the monopolist problem based on adaptive stencil refinement strategies show that the proposed method offers an unrivaled accuracy complexity trade off in comparison with existing methods we also obtain as a side product of our theory a new average complexity result on edge flipping based mesh generation

efficient fast marching with finsler metrics we study the discretization of the escape time problem find the length of the shortest path joining an arbitrary point z z of a domain omega to the boundary partial omega path length is measured locally via a finsler metric potentially asymmetric and strongly anisotropic this optimal control problem can be reformulated as a static hamilton jacobi partial differential equation or as a front propagation model it has numerous applications ranging from motion planning to image segmentation we introduce a new algorithm fast marching using anisotropic stencil refinement fm asr which addresses this problem on a two dimensional domain discretized on a cartesian grid the local stencils used in our discretization are produced by arithmetic means like in the fm lbr mirebeau in anisotropic fast marching on cartesian grids using lattice basis reduction preprint 2012 a method previously introduced by the author in the special case of riemannian metrics the complexity of the fm asr in an average sense over all grid orientations only depends poly logarithmically on the anisotropy ratio of the metric while most alternative approaches have a polynomial dependence numerical experiments show in several occasions that the accuracy complexity compromise is improved by an order of magnitude or more

the numerical solution of newton s problem of least resistance in this paper we consider newton s problem of finding a convex body of least resistance this problem could equivalently be written as a variational problem over concave functions in mathbb r 2 r 2 we propose two different methods for solving it numerically first we discretize this problem by writing the concave solution function as a infimum over a finite number of affine functions the discretized problem could be solved by standard optimization software efficiently second we conjecture that the optimal body has a certain structure we exploit this structure and obtain a variational problem in mathbb r 1 r 1 deriving its euler lagrange equation yields a program with two unknowns which can be solved quickly

potpourri of conjectures and open questions in nonlinear analysis and optimization we present a collection of fourteen conjectures and open problems in the fields of nonlinear analysis and optimization these problems can be classified into three groups problems of pure mathematical interest problems motivated by scientific computing and applications and problems whose solutions are known but for which we would like to know better proofs for each problem we provide a succinct presentation a list of appropriate references and a view of the state of the art of the subject

a finite element method for nonlinear elliptic problems we present a galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations a key tool is the discretization proposed in lakkis and pryer 2011 allowing us to work directly on the strong form of a linear pde an added benefit to making use of this discretization method is that a recovered finite element hessian is a byproduct of the solution process we build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear pdes benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the monge ampere equation and the pucci equation

a numerical approach to variational problems subject to convexity constraint we describe an algorithm to approximate the minimizer of an elliptic functional in the form int omega j x u nabla u on the set cal c of convex functions u in an appropriate functional space x such problems arise for instance in mathematical economics 4 a special case gives the convex envelope u 0 of a given function u 0 let t n be any quasiuniform sequence of meshes whose diameter goes to zero and i n the corresponding affine interpolation operators we prove that the minimizer over cal c is the limit of the sequence u n where u n minimizes the functional over i n cal c we give an implementable characterization of i n cal c then the finite dimensional problem turns out to be a minimization problem with linear constraints

a convexity principle for interacting gases a new set of inequalities is introduced based on a novel but natural interpolation between borel probability measures on r d using these estimates in lieu of convexity or rearrangement inequalities the existence and uniqueness problems are solved for a family of attracting gas models in these models the gas interacts with itself through a force which increases with distance and is governed by an equation of state p p relating pressure to density p d 1 d is assumed non decreasing for a d dimensional gas by showing that the internal and potential energies for the system are convex functions of the interpolation parameter an energy minimizing stateunique up to translationis proven to exist the concavity established for t pd q as a function of t 0 1 generalizes the brunnminkowski inequality from sets to measures 1997 academic press

multifractality stickiness and recurrence time statistics we identify the fine structure of resonance islands and the stickiness in chaos through recurrence time statistics rts which is based on the concept of poincare recurrences the projection of recurrence time statistics onto the phase space does give relevant information on the hierarchical and microstructures of the chaotic beach around the islands of a near integrable system the annular billiard these microstructures interfere in the effective transport of a particle in the phase space which can be observed through rts this technique proves also to be a powerful tool to describe the homoclinic tangle of the manifolds within the chaotic sea

norms and scalar products for ads 3 we compute scalar products and norms of bethe vectors in the massless sector of ads3 integrable superstring theories by exploiting the general difference form of the s matrix of massless excitations in the pure ramond ramond case and the difference form valid only in the bmn limit in the mixedflux case we obtain determinant like formulas for the scalar products generalising a procedure developed in previous literature for standard r matrices to the present non conventional situation we verify our expressions against explicit calculations using bethe vectors for chains of small length and perform some computer tests of the exact formulas as far as numerical accuracy sustains us this should be the first step towards the derivation of integrable form factors and correlation functions for the ads3 s matrix theory

scalar products of bethe vectors in models with gl 2 1 symmetry 2 determinant representation we study integrable models with mathfrak gl 2 1 symmetry and solvable by nested algebraic bethe ansatz we obtain a determinant representation for scalar products of bethe vectors when the bethe parameters obey some relations weaker than the bethe equations this representation allows us to find the norms of on shell bethe vectors and obtain determinant formulas for form factors of the diagonal entries of the monodromy matrix

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

surface fitting using convex tensor product splines a construction of linear sufficient convexity conditions for polynomial tensor product spline functions is presented as the main new feature of this construction the obtained conditions are asymptotically necessary increasing the number of linear inequalities in a suitable manner adapts them to any finite set of strongly convex spline surfaces based on the linear constraints we formulate least squares approximation of scattered data by spline surfaces as a quadratic programming problem

optimal transport and cournot nash equilibria we study a class of games with a continuum of players for which a cournot nash equilibria can be obtained by the minimisation of some cost related to optimal transport this cost is not convex in the usual sense in general but it turns out to have hidden strict convexity properties in many relevant cases this enables us to obtain new uniqueness results and a characterisation of equilibria in terms of some partial differential equations a simple numerical scheme in dimension one as well as an analysis of the inefficiency of equilibria

numerical simulation of diffusive and aggregation phenomena in nonlinear continuity equations by evolving diffeomorphisms we propose a numerical algorithm for solving nonlinear continuity equations written in lagrangian coordinates this transformation is intimately related to variational approaches for the well posedness of gradient flows of energy functionals with respect to the quadratic transportation distance in optimal transport theory these schemes allow the numerical approximation of both diffusive and aggregation regimes of different models positivity energy decreasing and mesh adaptation are built in in the numerical scheme and thus we are capable of capturing blow up densities and of dealing with vacuum regions and merging of mass patches in a natural way

finite element methods for fully nonlinear second order pdes based on a discrete hessian with applications to the monge ampere equation the purpose of this paper is twofold first we modify a method due to lakkis and pryer where the notion of a discrete hessian is introduced to compute fully nonlinear second order pdes the discrete hessian used in our approach is entirely local making the resulting linear system within the newton iteration much easier to solve the second contribution of this paper is to analyze both lakkis and pryer s method and its modification in parallel applied to the two dimensional monge ampere equation in both cases we show the well posedness of the methods as well as derive optimal error estimates numerical experiments are presented which i back up the theoretical findings and ii indicate that the methods are able to capture weak viscosity solutions

a numerical method for variational problems with convexity constraints we consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex this problem is at the interface between convex analysis convex optimization variational problems and partial differential equation techniques the approach is to approximate the nonpolyhedral cone of convex functions by a polyhedral cone which can be represented by linear inequalities this approach leads to an optimization problem with linear constraints which can be computed efficiently hundreds of times faster than existing methods

problems of optimal transportation on the circle and their mechanical applications abstract we consider a mechanical problem concerning a 2d axisymmetric body moving forward on the plane and making slow turns of fixed magnitude about its axis of symmetry the body moves through a medium of non interacting particles at rest and collisions of particles with the body s boundary are perfectly elastic billiard like the body has a blunt nose a line segment orthogonal to the symmetry axis it is required to make small cavities with special shape on the nose so as to minimize its aerodynamic resistance this problem of optimizing the shape of the cavities amounts to a special case of the optimal mass transportation problem on the circle with the transportation cost being the squared euclidean distance we find the explicit solution for this problem when the amplitude of rotation is smaller than a fixed critical value and give a numerical solution otherwise as a by product we get explicit description of the solution for a class of optimal transportation problems on the circle

minimizing within convex bodies using a convex hull method we present numerical methods to solve optimization problems on the space of convex functions or among convex bodies hence convexity is a constraint on the admissible objects whereas the functionals are not required to be convex to deal with this our method mixes geometrical and numerical algorithms we give several applications arising from classical problems in geometry and analysis alexandrov s problem of finding a convex body of prescribed surface function cheeger s problem of a subdomain minimizing the ratio surface area on volume newton s problem of the body of minimal resistance in particular for the latter application the minimizers are still unknown except in some particular classes we give approximate solutions better than the theoretical known ones hence demonstrating that the minimizers do not belong to these classes

anisotropic fast marching on cartesian grids using lattice basis reduction we introduce a modification of the fast marching algorithm which solves the anisotropic eikonal equation associated to an arbitrary continuous riemannian metric cal m on a two or three dimensional domain the algorithm has a complexity cal o n ln n n ln kappa cal m where n is the discrete domain cardinality the logarithmic dependency in the maximum anisotropy ratio kappa cal m of the riemannian metric allows us to handle extreme anisotropies for a limited numerical cost we prove the convergence of the algorithm and illustrate its efficiency by numerical experiments the algorithm relies on the computation at each grid point z of a special system of coordinates a reduced basis of the lattice bbb z m with respect to the symmetric positive definite matrix cal m z encoding the desired anisotropy at this point

the affine plateau problem in this paper we study a second order variational problem for locally convex hypersurfaces which is the affine invariant analogue of the classical plateau problem for minimal surfaces we prove existence regularity and uniqueness results for hypersurfaces maximizing affine area under appropriate boundary conditions