the geometry of r adaptive meshes generated using optimal transport methods the principles of mesh equidistribution and alignment play a fundamental role in the design of adaptive methods and a metric tensor and mesh metric are useful theoretical tools for understanding a method s level of mesh alignment or anisotropy we consider a mesh redistribution method based on the monge ampere equation which combines equidistribution of a given scalar density function with optimal transport it does not involve explicit use of a metric tensor although such a tensor must exist for the method and an interesting question to ask is whether or not the alignment produced by the metric gives an anisotropic mesh for model problems with a linear feature and with a radially symmetric feature we derive the exact form of the metric which involves expressions for its eigenvalues and eigenvectors the eigenvectors are shown to be orthogonal and tangential to the feature and the ratio of the eigenvalues corresponding to the level of anisotropy is shown to depend both locally and globally on the value of the density function and the amount of curvature we thereby demonstrate how the optimal transport method produces an anisotropic mesh along a given feature while equidistributing a suitably chosen scalar density function numerical results are given to verify these results and to demonstrate how the analysis is useful for problems involving more complex features including for a non trivial time dependant nonlinear pde which evolves narrow and curved reaction fronts

iterative bregman projections for regularized transportation problems this article details a general numerical framework to approximate so lutions to linear programs related to optimal transport the general idea is to introduce an entropic regularization of the initial linear program this regularized problem corresponds to a kullback leibler bregman di vergence projection of a vector representing some initial joint distribu tion on the polytope of constraints we show that for many problems related to optimal transport the set of linear constraints can be split in an intersection of a few simple constraints for which the projections can be computed in closed form this allows us to make use of iterative bregman projections when there are only equality constraints or more generally bregman dykstra iterations when inequality constraints are in volved we illustrate the usefulness of this approach to several variational problems related to optimal transport barycenters for the optimal trans port metric tomographic reconstruction multi marginal optimal trans port and in particular its application to brenier s relaxed solutions of in compressible euler equations partial un balanced optimal transport and optimal transport with capacity constraints

convergent finite difference solvers for viscosity solutions of the elliptic monge ampere equation in dimensions two and higher the elliptic monge ampere equation is a fully nonlinear partial differential equation that originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail novel solution methods are required for stability and convergence to the weak viscosity solution in this article we build a wide stencil finite difference discretization for the monge ampere equation the scheme is monotone so the barles souganidis theory allows us to prove that the solution of the scheme converges to the unique viscosity solution of the equation solutions of the scheme are found using a damped newton s method we prove convergence of newton s method and provide a systematic method to determine a starting point for the newton iteration computational results are presented in two and three dimensions which demonstrates the speed and accuracy of the method on a number of exact solutions which range in regularity from smooth to nondifferentiable

convergent difference schemes for degenerate elliptic and parabolic equations hamilton jacobi equations and free boundary problems convergent numerical schemes for degenerate elliptic partial differential equations are constructed and implemented simple conditions are identified which ensure that nonlinear finite difference schemes are monotone and nonexpansive in the maximum norm explicit schemes endowed with an explicit cfl condition are built for time dependent equations and are used to solve stationary equations iteratively explicit and implicit formulations of monotonicity for first and second order equations are unified bounds on orders of accuracy are established an example of a scheme which is stable but nonmonotone and nonconvergent is presented schemes for hamilton jacobi equations obstacle problems one phase free boundary problems and stochastic games are built and computational results are presented

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

a continuous linear optimal transport approach for pattern analysis in image datasets we present a new approach to facilitate the application of the optimal transport metric to pattern recognition on image databases the method is based on a linearized version of the optimal transport metric which provides a linear embedding for the images hence it enables shape and appearance modeling using linear geometric analysis techniques in the embedded space in contrast to previous work we use monge s formulation of the optimal transport problem which allows for reasonably fast computation of the linearized optimal transport embedding for large images we demonstrate the application of the method to recover and visualize meaningful variations in a supervised learning setting on several image datasets including chromatin distribution in the nuclei of cells galaxy morphologies facial expressions and bird species identification we show that the new approach allows for high resolution construction of modes of variations and discrimination and can enhance classification accuracy in a variety of image discrimination problems highlightsa continuous version of the lot framework is described that bypasses many of the difficulties associated with the discrete formulation using continuous transport maps a forward and inverse transform operation is defined for images an improved reference average image estimation algorithm is proposed the range within which points in lot space are invertible according to the continuous formulation is calculated we show that our method significantly speeds up the computation of the lot embedding

optimal mass transport for higher dimensional adaptive grid generation in this work we describe an approach for higher dimensional adaptive grid generation based on solving the l 2 monge kantorovich problem mkp which is a special case of the classical optimal mass transportation problem two methods are developed for computing the coordinate transformation used to define the grid adaptation for the first method the transformation is determined by solving a parabolic monge ampere equation for a steady state solution for the second method the grid movement is determined from the velocity field obtained by solving a fluid dynamics formulation of the l 2 mkp several numerical experiments are presented to demonstrate the performance of the mkp methods and to compare them with some related adaptive grid methods the experimental results demonstrate that the mkp methods show promise as effective and reliable methods for higher dimensional adaptive grid generation

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

the ideal pseudo rigid continuum in the literature on pseudo rigid bodies and their applications it is generally assumed that these bodies can undergo only a restricted class of motions without questioning how this restriction is to be strictly enforced in 2004 i proposed in these proceedings that such a restriction may be regarded as a global constraint on a deformable continuum and influenced by ideas of antman marlow from the early 1990s i assumed that the constraint is enforced by a field of reactive stresses and i constructed a mathematical model that idealizes pseudo rigid bodies as globally constrained continua of finite size in a recent article in proceedings of the royal society a the validity of this model was challenged essentially the controversy revolves around the issue of working definitions versus idealized mathematical models of pseudo rigid bodies

on moment sequences and infinitely divisible sequences we consider infinitely divisible sequences in the context of moment sequences in the same vein as r a horn j math anal appl 31 1970 130 135 considered renewal sequences and we compare the results

moving mesh generation using the parabolic monge ampere equation this article considers a new method for generating a moving mesh which is suitable for the numerical solution of partial differential equations pdes in several spatial dimensions the mesh is obtained by taking the gradient of a scalar mesh potential function which satisfies an appropriate nonlinear parabolic partial differential equation this method gives a new technique for performing r adaptivity based on ideas from optimal transportation combined with the equidistribution principle applied to a time varying scalar monitor function used successfully in moving mesh methods in one dimension detailed analysis of this new method is presented in which the convergence regularity and stability of the mesh is studied additionally this new method is shown to be straightforward to program and implement requiring the solution of only one simple scalar time dependent equation in arbitrary dimension with adaptivity along the boundaries handled automatically we discuss three preexisting methods in the context of this work examples are presented in which either the monitor function is prescribed in advance or it is given by the solution of a partial differential equation

the dirichlet problem for nonlinear second order elliptic equations i monge ampegre equation on considere le probleme de dirichlet pour des equations elliptiques non lineaires pour une fonction reelle u definie dans la fermeture d un domaine borne dans r n avec une frontiere c

optimal mass transport for registration and warping image registration is the process of establishing a common geometric reference frame between two or more image data sets possibly taken at different times in this paper we present a method for computing elastic registration and warping maps based on the monge kantorovich theory of optimal mass transport this mass transport method has a number of important characteristics first it is parameter free moreover it utilizes all of the grayscale data in both images places the two images on equal footing and is symmetrical the optimal mapping from image a to image b being the inverse of the optimal mapping from b to a the method does not require that landmarks be specified and the minimizer of the distance functional involved is uniques there are no other local minimizers finally optimal transport naturally takes into account changes in density that result from changes in area or volume although the optimal transport method is certainly not appropriate for all registration and warping problems this mass preservation property makes the monge kantorovich approach quite useful for an interesting class of warping problems as we show in this paper our method for finding the registration mapping is based on a partial differential equation approach to the minimization of the l2 kantorovich wasserstein or earth mover s distance under a mass preservation constraint we show how this approach leads to practical algorithms and demonstrate our method with a number of examples including those from the medical field we also extend this method to take into account changes in intensity and show that it is well suited for applications such as image morphing

a class of infinitely divisible distributions connected to branching processes and random walks a class of infinitely divisible distributions on 0 1 2 is defined by requiring the discrete levy function to be equal to the probability function except for a very simple factor these distributions turn out to be special cases of the total offspring distributions in sub critical branching processes and can also be interpreted as first passage times in certain random walks there are connections with lambert s w function and generalized negative binomial convolutions

an efficient numerical method for the solution of the l_2 optimal mass transfer problem in this paper we present a new computationally efficient numerical scheme for the minimizing flow approach for the computation of the optimal l 2 mass transport mapping in contrast to the integration of a time dependent partial differential equation proposed in s angenent s haker and a tannenbaum siam j math anal 35 2003 pp 61 97 we employ in the present work a direct variational method the efficacy of the approach is demonstrated on both real and synthetic data

interior w 2 p estimates for solutions of the monge ampere equation in this work we adapt the techniques developed in c1 to prove interior estimates for solutions of perturbations of the monge ampere equation

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

a fast approach to optimal transport the back and forth method we present a method to efficiently solve the optimal transportation problem for a general class of strictly convex costs given two probability measures supported on a discrete grid with n points we compute the optimal transport map between them in o n log n operations and o n storage space our approach allows us to solve optimal transportation problems on spatial grids as large as 4096 times 4096 and 384 times 384 times 384 in a matter of minutes

generalised lagrangian solutions for atmospheric and oceanic flows atmospheric or oceanic flows strongly constrained by rotation and stratification can be described by a set of lagrangian partial differential equations called the semigeostrophic equations in these equations the trajectories must be determined implicitly generalised solutions of these equations are defined as a sequence of rearrangements of the fluid which need not be smooth these solutions are closely related to generalised solutions of the monge ampere equation existence and uniqueness of such solutions is proved the evolution is shown to be a sequence of minimum energy states of the fluid giving strong physical plausibility to the solutions

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

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

application of optimal transport and the quadratic wasserstein metric to full waveform inversion conventional full waveform inversion fwi using the least squares norm as a misfit function is known to suffer from cycle skipping issues which increases the risk of computing a local rather than the global minimum of the misfit the quadratic wasserstein metric has been proved to have many ideal properties with regards to convexity and insensitivity to noise when the observed and predicted seismic data are considered to be two density functions the quadratic wasserstein metric corresponds to the optimal cost of rearranging one density into the other where the transportation cost is quadratic in distance unlike the least squares norm the quadratic wasserstein metric measures not only amplitude differences but also global phase shifts which helps to avoid cycle skipping issues we propose a new way of using the quadratic wasserstein metric trace by trace in fwi and compare it to the global quadratic wasserstein metric via the solution of the monge ampere equation we incorporate the quadratic wasser

convergent filtered schemes for the monge ampere partial differential equation the theory of viscosity solutions has been effective for representing and approximating weak solutions to fully nonlinear partial differential equations such as the elliptic monge ampere equation the approximation theory of barles and souganidis asymptotic anal 4 1991 pp 271 283 requires that numerical schemes be monotone or elliptic in the sense of a m oberman siam j numer anal 44 2006 pp 879 895 but such schemes have limited accuracy in this article we establish a convergence result for filtered schemes which are nearly monotone this allows us to construct finite difference discretizations of arbitrarily high order we demonstrate that the higher accuracy is achieved when solutions are sufficiently smooth in addition the filtered scheme provides a natural detection principle for singularities we employ this framework to construct a formally second order scheme for the monge ampere equation and present computational results on smooth and singular solutions

fast finite volume simulation of 3d electromagnetic problems with highly discontinuous coefficients we consider solving three dimensional electromagnetic problems in parameter regimes where the quasi static approximation applies and the permeability permittivity and conductivity may vary significantly the difficulties encountered include handling solution discontinuities across interfaces and accelerating convergence of traditional iterative methods for the solution of the linear systems of algebraic equations that arise when discretizing maxwell s equations in the frequency domain the present article extends methods we proposed earlier for constant permeability e haber u ascher d aruliah and d oldenburg j comput phys 163 2000 pp 150 171 d aruliah u ascher e haber and d oldenburg math models methods appl sci to appear to handle also problems in which the permeability is variable and may contain significant jump discontinuities in order to address the problem of slow convergence we reformulate maxwell s equations in terms of potentials applying a helmholtz decomposition to either the electric field or the magnetic field the null space of the curl operators can then be annihilated by adding a stabilizing term using a gauge condition and thus obtaining a strongly elliptic differential operator a staggered grid finite volume discretization is subsequently applied to the reformulated pde system this scheme works well for sources of various types even in the presence of strong material discontinuities in both conductivity and permeability the resulting discrete system is amenable to fast convergence of ilu preconditioned krylov methods we test our method using several numerical examples and demonstrate its robust efficiency we also compare it to the classical yee method using similar iterative techniques for the resulting algebraic system and we show that our method is significantly faster especially for electric sources

finite difference methods for the infinity laplace and p laplace equations abstract we build convergent discretizations and semi implicit solvers for the infinity laplacian and the game theoretical p laplacian the discretizations simplify and generalize earlier ones we prove convergence of the solution of the wide stencil finite difference schemes to the unique viscosity solution of the underlying equation we build a semi implicit solver which solves the laplace equation as each step it is fast in the sense that the number of iterations is independent of the problem size this is an improvement over previous explicit solvers which are slow due to the cfl condition

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

convergence framework for the second boundary value problem for the monge ampere equation it is well known that the quadratic cost optimal transportation problem is formally equivalent to the second boundary value problem for the monge ampere equation viscosity solutions are a powerfu

a convergent difference scheme for the infinity laplacian construction of absolutely minimizing lipschitz extensions this article considers the problem of building absolutely minimizing lipschitz extensions to a given function these extensions can be characterized as being the solution of a degenerate elliptic partial differential equation the infinity laplacian for which there exist unique viscosity solutions a convergent difference scheme for the infinity laplacian equation is introduced which arises by minimizing the discrete lipschitz constant of the solution at every grid point existence and uniqueness of solutions to the scheme is shown directly solutions are also shown to satisfy a discrete comparison principle solutions are computed using an explicit iterative scheme which is equivalent to solving the parabolic version of the equation

the monge ampere trajectory correction for semi lagrangian schemes requiring that numerical estimates of the flow trajectories comply with the fundamental euler expansion formula that governs the evolution of a volume of fluid leads to a second order nonlinear monge ampere partial differential equation mae in cossette and smolarkiewicz 2011 15 a numerical algorithm based on solving the mae with an inexact newton krylov solver has been developed and used to correct standard estimates of the departure points of flow trajectories provided by a classical semi lagrangian scheme in the context of an incompressible fluid here we extend the theoretical analysis of the elemental rotational and deformational motions presented in cossette and smolarkiewicz 2011 15 in particular closed form analytic solutions are derived for both cases that serve to illustrate the mechanics of the enhanced trajectory scheme and to address the issues of existence and uniqueness scalar advection shows that the ma correction improves mass conservation substantially by suppressing anomalous fluid contraction the impact of the ma correction on complex flows is studied in the framework of an ideal magneto fluid in which the formation of current sheets leads to topological changes and reconnection of field lines the use of the ma correction improves the numerical stability of the solutions and prevents trajectory intersections as well as the spurious growth of the magnetic energy 2d and 3d examples are presented and the computational performance of the solver is documented

construction of simple stable and convergent high order schemes for steady first order hamilton jacobi equations we develop a very simple algorithm that permits to construct compact high order schemes for steady first order hamilton jacobi equations the algorithm relies on the blending of a first order scheme and a compact high order scheme the blending is conducted in such a way that the scheme is formally high order accurate a convergence proof without error estimate is given we provide several numerical illustrations that demonstrate the effective accuracy of the scheme the numerical examples use triangular unstructured meshes but our method may be applied to other kind of meshes several implementation remarks are also given

a least squares method for optimal transport using the monge ampere equation in this article we introduce a novel numerical method to solve the problem of optimal transport and the related elliptic monge ampere equation it is one of the few numerical algorithms capable of solving this problem efficiently with the proper transport boundary condition the computation time scales well with the grid size and has the additional advantage that the target domain may be nonconvex we present the method and several numerical experiments

the monge ampere equation various forms and numerical solution we present three novel forms of the monge ampere equation which is used e g in image processing and in reconstruction of mass transportation in the primordial universe the central role in this paper is played by our fourier integral form for which we establish positivity and sharp bound properties of the kernels this is the basis for the development of a new method for solving numerically the space periodic monge ampere problem in an odd dimensional space convergence is illustrated for a test problem of cosmological type in which a gaussian distribution of matter is assumed in each localised object and the right hand side of the monge ampere equation is a sum of such distributions

a least squares method for the inverse reflector problem in arbitrary orthogonal coordinates abstract in this article we solve the inverse reflector problem for a light source emitting a parallel light bundle and a target in the far field of the reflector by use of a least squares method we derive the monge ampere equation expressing conservation of energy while assuming an arbitrary coordinate system we generalize a cartesian coordinate least squares method presented earlier by prins et al 13 to arbitrary orthogonal coordinate systems this generalized least squares method provides us the freedom to choose a coordinate system suitable for the shape of the light source this results in significantly increased numerical accuracy decrease of errors by factors up to 104 is reported we present the generalized least squares method and compare its numerical results with the cartesian version for a disk shaped light source

numerical methods for fully nonlinear elliptic equations of the monge ampere type abstract in this article we discuss the numerical solution of the dirichlet problem for the monge ampere equation in two dimensions the solution of closely related problems is also discussed these include a family of pucci s equations the equation prescribing the harmonic mean of the eigenvalues of the hessian of a smooth function of two variables and a minimization problem from nonlinear elasticity where the cost functional involves the determinant of the gradient of vector valued functions to solve the monge ampere equation we consider two methods the first one reduces the monge ampere equation to a saddle point problem for a well chosen augmented lagrangian to solve this saddle point problem we advocate an uzawa douglas rachford algorithm the second method combines nonlinear least squares and operator splitting this second method being simpler to implements we apply variants of it to the solution of the other problems for the space discretization we use mixed finite element approximations closely related to methods already used for the solution of linear and nonlinear bi harmonic problems through these approximations the solution of the above problems is essentially reduced to the solution of discrete poisson problems the methods discussed in this article are validated by the results of numerical experiments

a lagrangian scheme for the solution of the optimal mass transfer problem a lagrangian method to numerically solve the l 2 optimal mass transfer problem is presented the initial and final density distributions are approximated by finite mass particles having a gaussian kernel mass conservation and the hamilton jacobi equation for the potential are identically satisfied by constant mass transport along straight lines the scheme is described in the context of existing methods to solve the problem and a set of numerical examples including applications to medical imagery are presented

on the convergence of augmented lagrangian method for optimal transport between nonnegative densities the dynamical formulation of the optimal transport problem introduced by j d benamou and y brenier corresponds to the time space search of a density and a momentum minimizing a transport energy between two densities in order to solve this problem an algorithm has been proposed to estimate a saddle point of a lagrangian we will study the convergence of this algorithm to a saddle point of the lagrangian in the most general conditions particularly in cases where initial and final densities vanish on some areas of the transportation domain the principal difficulty of our study will consist in the proof under these conditions of the existence of a saddle point and especially in the uniqueness of the density momentum component indeed these conditions imply to have to deal with non regular optimal transportation maps for these reasons a detailed study of the properties of the velocity field associated to an optimal transportation map in quadratic space is required

a numerical method for the elliptic monge ampere equation with transport boundary conditions the problem of optimal mass transport arises in numerous applications including image registration mesh generation reflector design and astrophysics one approach to solving this problem is via the monge ampere equation while recent years have seen much work in the development of numerical methods for solving this equation very little has been done on the implementation of the transport boundary condition in this paper we propose a method for solving the transport problem by iteratively solving a monge ampere equation with neumann boundary conditions to enable mappings between variable densities we extend an earlier discretization of the equation to allow for right hand sides that depend on gradients of the solution b d froese and a m oberman siam j numer anal 49 2011 pp 1692 1714 this discretization provably converges to the viscosity solution the resulting system is solved efficiently with newton s method we provide several challenging computational examples that demonstra

the boundary method for semi discrete optimal transport partitions and wasserstein distance computation abstract we introduce a new technique which we call the boundary method for solving semi discrete optimal transport problems with a wide range of cost functions the boundary method reduces the effective dimension of the problem thus improving complexity for cost functions equal to a p norm with p 1 we provide mathematical justification convergence analysis and algorithmic development our testing supports the boundary method with these p norms as well as other more general cost functions

pseudo rigid continua basic theory and a geometrical derivation of lagrange s equations pseudorigid bodies are regarded here as globally constrained threedimensional homogeneous continua the constraint reaction stresses play a fundamental role in maintaining the homogeneity of the de

mixed finite element methods for the fully nonlinear monge ampere equation based on the vanishing moment method this paper studies mixed finite element approximations of the viscosity solution to the dirichlet problem for the fully nonlinear monge ampere equation det d 2u 0 f 0 based on the vanishing moment method which was proposed recently by the authors in x feng and m neilan j scient comp doi 10 1007 s10915 008 9221 9 2008 in this approach the second order fully nonlinear monge ampere equation is approximated by the fourth order quasilinear equation varepsilon delta 2 u varepsilon det d 2u varepsilon f it was proved in x feng trans ams submitted that the solution u varepsilon converges to the unique convex viscosity solution u 0 of the dirichlet problem for the monge ampere equation this result then opens a door for constructing convergent finite element methods for the fully nonlinear second order equations a task which has been impracticable before the goal of this paper is threefold first we develop a family of hermann miyoshi type mixed finite element methods for approximating the solution u varepsilon of the regularized fourth order problem which computes simultaneously u varepsilon and the moment tensor sigma varepsilon d 2u varepsilon second we derive error estimates which track explicitly the dependence of the error constants on the parameter varepsilon for the errors u varepsilon u varepsilon h and sigma 0 sigma h varepsilon finally we present a detailed numerical study on the rates of convergence in terms of powers of varepsilon for the error u 0 u h varepsilon and sigma varepsilon sigma h varepsilon and numerically examine what is the best mesh size h in relation to varepsilon in order to achieve these rates due to the strong nonlinearity of the underlying equation the standard perturbation argument for error analysis of finite element approximations of nonlinear problems does not work for the problem to overcome the difficulty we employ a fixed point technique which strongly relies on the stability of the linearized problem and its mixed finite element approximations

numerical solution of the optimal transportation problem using the monge ampere equation a numerical method for the solution of the elliptic monge ampere partial differential equation with boundary conditions corresponding to the optimal transportation ot problem is presented a local representation of the ot boundary conditions is combined with a finite difference scheme for the monge ampere equation email protected s method is implemented leading to a fast solver comparable to solving the laplace equation on the same grid several times theoretical justification for the method is given by a convergence proof in the companion paper 4 solutions are computed with densities supported on non convex and disconnected domains computational examples demonstrate robust performance on singular solutions and fast computational times

designing freeform lenses for intensity and phase control of coherent light with help from geometry and mass transport we study here the problem of determining a system of two refractive interfaces transforming a plane wavefront of a given shape and radiation intensity into a coherent output plane wavefront with prescribed output position shape and intensity such interfaces can be refracting surfaces of two different lenses or of one lens in geometrical optics approximation the analytic formulation of this problem in both cases requires construction of maps with controlled jacobian though this jacobian can be expressed as a second order partial differential equation of monge ampere type for a scalar function defining one of the refracting surfaces its analysis is not straightforward in this paper we use a geometric approach for reformulating the problem in certain associated measures and defining weak solutions existence and uniqueness of weak solutions in lipschitz classes for both cases are established by variational methods our results show in particular that two types of interfaces exist in each case for the same data one of these types always consists of two interfaces one of which is concave or convex and the second convex or concave while the interfaces of the second type may be neither convex nor concave the availability of a design with convex concave lenses is particularly important for fabrication the truly geometric nature of this problem permits its statement and investigation in mathbb r n 1 n geqq 1

fast finite difference solvers for singular solutions of the elliptic monge ampere equation the elliptic monge ampere equation is a fully nonlinear partial differential equation which originated in geometric surface theory and has been applied in dynamic meteorology elasticity geometric optics image processing and image registration solutions can be singular in which case standard numerical approaches fail in this article we build a finite difference solver for the monge ampere equation which converges even for singular solutions regularity results are used to select a priori between a stable provably convergent monotone discretization and an accurate finite difference discretization in different regions of the computational domain this allows singular solutions to be computed using a stable method and regular solutions to be computed more accurately the resulting nonlinear equations are then solved by newton s method computational results in two and three dimensions validate the claims of accuracy and solution speed a computational example is presented which demonstrates the necessity of the use of the monotone scheme near singularities

coping with model error in variational data assimilation using optimal mass transport classical variational data assimilation methods address the problem of optimally combining model predictions with observations in the presence of zero mean gaussian random errors however in many natural systems uncertainty in model structure and or model parameters often results in systematic errors or biases prior knowledge about such systematic model error for parametric removal is not always feasible in practice limiting the efficient use of observations for improved prediction the main contribution of this work is to advocate the relevance of transportation metrics for quantifying nonrandom model error in variational data assimilation for nonnegative natural states and fluxes transportation metrics also known as wasserstein metrics originate in the theory of optimal mass transport omt and provide a nonparametric way to compare distributions which is natural in the sense that it penalizes mismatch in the values and relative position of masses in the two distributions we demonstrate the promise of the proposed methodology using 1 d and 2 d advection diffusion dynamics with systematic error in the velocity and diffusivity parameters moreover we combine this methodology with additional regularization functionals such as the l1 norm of the state in a properly chosen domain to incorporate both model error and potential prior information in the presence of sparsity or sharp fronts in the underlying state of interest

data driven optimal transport the problem of optimal transport between two distributions x and y is extended to situations where the distributions are only known through a finite number of samples xi and yj a weak formulation is proposed based on the dual of the kantorovich formulation with two main modifications replacing the expected values in the objective function by their empirical means over the xi and yj and restricting the dual variables u x and v y to a suitable set of test functions adapted to the local availability of sample points a procedure is proposed and tested for the numerical solution of this problem based on a fluidlike flow in phase space where the sample points play the role of active lagrangian markers 2016 wiley periodicals inc

explanation of variability and removal of confounding factors from data through optimal transport a methodology based on the theory of optimal transport is developed to attribute variability in data sets to known and unknown factors and to remove such attributable components of the variability from the data denoting by x the quantities of interest and by z the explanatory factors the procedure transforms x into filtered variables y through a z dependent map so that the conditional probability distributions x z are pushed forward into a target distribution y independent of z among all maps and target distributions that achieve this goal the procedure selects the one that minimally distorts the original data the barycenter of the x z connections are found to unsupervised learning and to fundamental problems in statistics such as conditional density estimation and sampling particularly simple instances of the methodology are shown to be equivalent to k means and principal component analysis an application is shown to a time series of ground temperature hourly data across the united states 2017 wiley periodicals inc

optimal transport for particle image velocimetry particle image velocimetry piv is a method used to measure the velocity field of a fluid flow traditionally cross correlation algorithms are used to retrieve the flow velocity recently a new method for piv based on the l 2 optimal mass transportation problem ot piv was introduced and analyzed in l p saumier b khouider and m agueh comm math sci 13 2015 pp 269 296 for the case of synthetically generated data and for particles of equal mass or brightness here we extend the work of saumier et al to the case of particles with different masses and randomly seeded particles more importantly we compare the ot piv method with a typical cross correlation algorithm for the case of real data using a combination of theory and numerical experiments we demonstrate that in the presence of particles with different weights brightness the ot method is more accurate for the largest brightest particles and it is more faithful when the particles are far enough from each other this makes it more suitable for the so called particle tracking regime of piv i e when the seeding density is low we demonstrate in particular that for low seeding densities the ot method performs better than a typical cross correlation algorithm based on these new results and the previous ones of saumier et al we propose a suite of postprocessing algorithms for the ot piv method it is found that the ot piv with postprocessing performs very well on the synthetic and real data when compared to the cross correlation method

c 0 penalty methods for the fully nonlinear monge ampere equation in this paper we develop and analyze c 0 penalty methods for the fully nonlinear monge ampere equation det d 2 u f in two dimensions the key idea in designing our methods is to build discretizations such that the resulting discrete linearizations are symmetric stable and consistent with the continuous linearization we are then able to show the well posedness of the penalty method as well as quasi optimal error estimates using the banach fixed point theorem as our main tool numerical experiments are presented which support the theoretical results

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

mesh independence for an augmented lagrangian sqp method in hilbert spaces an augmented lagrangian sqp algorithm for optimal control of differential equations in hilbert spaces is analyzed this algorithm has second order convergence rate provided that a second order sufficient optimality condition is satisfied the internal approximation of this method is investigated and convergence results are presented a mesh independence principle for the augmented lagrangian sqp method is proved which asserts that asymptotically the infinite dimensional algorithm and finite dimensional discretizations have the same convergence property more precisely for sufficiently small mesh size there is at most a difference of one iteration step between the number of steps required by the infinite dimensional method and its discretization to converge within a given tolerance varepsilon 0 the theoretical results are demonstrated by two optimal control problems for the burgers equation

adaptive solution to two dimensional partial differential equations in curved domains using the monge ampere equation we present a finite difference method for mesh generation and adaptation with application to the solution of partial differential equations in curved domains for mesh generation we construct a ma

the sinkhorn algorithm parabolic optimal transport and geometric monge ampere equations we show that the discrete sinkhorn algorithm as applied in the setting of optimal transport on a compact manifold converges to the solution of a fully non linear parabolic pde of monge ampere type in a large scale limit the latter evolution equation has previously appeared in different contexts e g on the torus it can be be identified with the ricci flow this leads to algorithmic approximations of the potential of the optimal transport map as well as the optimal transport distance with explicit bounds on the arithmetic complexity of the construction and the approximation errors as applications we obtain explicit schemes of nearly linear complexity at each iteration for optimal transport on the torus and the two sphere as well as the far field antenna problem connections to quasi monte carlo methods are exploited

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

the geometry of optimal transportation introduction 113 1 summary of main results 120 2 background on optimal measures 126 part i strictly convex costs 133 3 existence and uniqueness of optimal maps 133 4 characterization of the optimal map 137 part ii costs which are strictly concave as a function of d i s t a n c e 141 5 the role of optimal maps 141 6 uniqueness of optimal solutions 144 part iii appendices 148 a legendre transforms and conjugate costs 148 b examples of c concave potentials 152 c regularity of c concave potentials 154 references 159