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

generic representation theory of quivers with relations abstract the irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra are explored with regard to both their geometry and the structure of the modules they encode provided that the base field is algebraically closed and has infinite transcendence degree over its prime field we establish the existence and uniqueness not up to isomorphism but in a strong sense to be specified of a generic module for any irreducible component c that is of a module which displays all categorically defined generic properties of the modules parametrized by c the crucial point of the existence statement a priori almost obvious lies in the description of such a module in a format accessible to representation theoretic techniques our approach to generic modules over path algebras modulo relations by way of minimal projective resolutions is largely constructive it is explicit for large classes of algebras of wild type we follow with an investigation of the properties of such generic modules in terms of quiver and relations the sharpest specific results on all fronts are obtained for truncated path algebras that is for path algebras of quivers modulo ideals generated by all paths of a fixed length this class of algebras extends the comparatively thoroughly studied hereditary case displaying many novel features

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

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

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

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

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

towards the smoothness of optimal maps on riemannian submersions and riemannian products of round spheres in particular the variant a3w of ma trudinger and wang s condition for regularity of optimal transportation maps is implied by the non negativity of a pseudo riemannian curvature which we call cross curvature induced by the transportation cost for the riemannian distance squared cost it is shown that 1 cross curvature non negativity is preserved for products of two manifolds 2 both a3w and cross curvature non negativity are inherited by riemannian submersions as is domain convexity for the exponential maps and 3 the n dimensional round sphere satisfies cross curvature non negativity from these results a large new class of riemannian manifolds satisfying cross curvature non negativity thus a3w is obtained including many whose sectional curvature is far from constant all known obstructions to the regularity of optimal maps are absent from these manifolds making them a class for which it is natural to conjecture that regularity holds this conjecture is confirmed for certain riemannian submersions of the sphere such as the complex projective spaces cp n

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

high order essentially nonsocillatory schemes for hamilton jacobi equations hamilton jacobi h j equations are frequently encountered in applications e g in control theory and differential games h j equations are closely related to hyperbolic conservation laws in one space dimension the former is simply the integrated version of the latter similarity also exists for the multidimensional case and this is helpful in the design of difference approximations in this paper high order essentially nonoscillatory eno schemes for h j equations are investigated which yield uniform high order accuracy in smooth regions and sharply resolve discontinuities in the derivatives the eno scheme construction procedure is adapted from that for hyperbolic conservation laws the schemes are numerically tested on a variety of one dimensional and two dimensional problems including a problem related to control optimization and high order accuracy in smooth regions good resolution of discontinuities in the derivatives and convergence to viscosity solutions are observed

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

a monge ampere solver for free form reflector design in this article we present a method for the design of fully free form reflectors for illumination systems we derive an elliptic partial differential equation of the monge ampere type for the surface of a reflector that converts an arbitrary parallel beam of light into a desired intensity output pattern the differential equation has an unusual boundary condition known as the transport boundary condition we find a convex or concave solution to the equation using a state of the art numerical method the method uses a nonstandard discretization based on the diagonalization of the hessian the discretized system is solved using standard newton iteration the method was tested for a circular beam with uniform intensity a street light and a uniform beam that is transformed into a famous dutch painting the reflectors were verified using commercial ray tracing software

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

classifying representations by way of grassmannians let a be a finite dimensional algebra over an algebraically closed field criteria are given which characterize existence of a fine or coarse moduli space classifying up to isomorphism the representations of a with fixed dimension d and fixed squarefree top t next to providing a complete theoretical picture some of these equivalent conditions are readily checkable from quiver and relations of a in the case of existence of a moduli space unexpectedly frequent in light of the stringency of fine classification this space is always projective and in fact arises as a closed subvariety brassy of a classical grassmannian even when the full moduli problem fails to be solvable the variety grass t d is seen to have distinctive properties recommending it as a substitute for a moduli space as an application a characterization of the algebras having only finitely many representations with fixed simple top is obtained in this case of finite local representation type at a given simple t the radical layering j l m j l 1 m l 0 is shown to be a classifying invariant for the modules with top t this relies on the following general fact obtained as a byproduct proper degenerations of a local module m never have the same radical layering as m

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

irreducible components of varieties of representations the local case abstract for any positive integer d we determine the irreducible components of the varieties that parametrize the d dimensional representations of a local truncated path algebra here is a quotient k q the paths of length l 1 of a path algebra kq where k is an algebraically closed field l is a positive integer and q is the quiver with a single vertex and a finite number r of loops the components are determined in both the classical and the grassmannian settings rep d and grass d our method is to corner the components by way of a twin pair of upper semicontinuous maps from rep d to a poset consisting of sequences of semisimple modules an excerpt of the main result is as follows given a sequence s s 0 s l of semisimple modules with dim 0 l l s l d let rep s be the subvariety of rep d consisting of the points that parametrize the modules with radical layering s the radical layering of a module m is the sequence j l m j l 1 m 0 l l where j is the jacobson radical of suppose the quiver q has r 2 loops if d l 1 the variety rep d is irreducible and generically its modules are uniserial if on the other hand d l 1 then the irreducible components of rep d are the closures of the subvarieties rep s for those sequences s which satisfy the inequalities dim s l r dim s l 1 and dim s l 1 r dim s l for 0 l l generically the modules in any such component have socle layering s l s 0 as a byproduct the main result provides further installments of generic information on the modules corresponding to the irreducible components of the parametrizing varieties

module varieties over canonical algebras abstract the main purpose of this paper is the study of module varieties over the class of canonical algebras providing a rich source of examples of varieties with interesting properties our main tool is a stratification of module varieties which was recently introduced by richmond this stratification does not require a precise knowledge of the module category if it is finite then it provides a method to classify irreducible components we determine the canonical algebras for which this stratification is finite in this case we describe the algorithm for calculating the dimension of the variety and the number of irreducible components of maximal dimension for an infinite family of examples we give easy combinatorial criteria for irreducibility cohen macaulay and normality

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

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

general representations of quivers we investigate general representations of a quiver q there are two main questions that we shall address the first which was raised by kac is to find the dimension vectors of the summands of a general representation of dimension vector a the second is to find the dimension vectors of the subrepresentations of a general representation of dimension vector a these two questions are closely related one consequence of our investigation is to find an algorithm to compute the canonical decomposition of an arbitrary dimension vector in terms of the euler form of the quiver

on the second boundary value problem for lagrangian mean curvature flow abstract we consider a fully nonlinear parabolic equation with nonlinear neumann type boundary condition and show the long time existence and convergence of the flow finally we apply this study to the boundary value problem for minimal lagrangian graphs

the neumann problem of special lagrangian equations with supercritical phase abstract in this paper we establish the global c 2 estimates of the neumann problem of special lagrangian equations with supercritical phase and the existence theorem by the method of continuity

irreducible components of varieties of representations the acyclic case the goals of this article are as follows 1 to determine the irreducible components of the affine varieties mathbf rep mathbf d lambda parametrizing the representations with dimension vector mathbf d where lambda traces a major class of finite dimensional algebras 2 to generically describe the representations encoded by the components the target class consists of those truncated path algebras lambda over an algebraically closed field k which are based on a quiver q without oriented cycles the main result characterizes the irreducible components of mathbf rep mathbf d lambda in representation theoretic terms and provides a means of listing them from quiver and loewy length of lambda combined with existing theory this classification moreover yields an array of generic features of the modules parametrized by the components such as generic minimal projective presentations generic sub and quotient modules etc our second principal result pins down the generic socle series of the modules in the components it does so for more general lambda in fact the information on truncated path algebras of acyclic quivers supplements the theory available in the special case where lambda kq filling in generic data on the mathbf d dimensional representations of q with any fixed loewy length

extension orthogonal components of preprojective varieties let q be a dynkin quiver and let a be the corresponding preprojective algebra let c c i i i be a set of pairwise different indecomposable irreducible components of varieties of a modules such that generically there are no extensions between c i and cj for all i j we show that the number of elements in c is at most the number of positive roots of q furthermore we give a module theoretic interpretation of leclerc s counterexample to a conjecture of berenstein and zelevinsky

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

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

periodicity algorithms and a conjecture on overlaps in partial words we propose an algorithm that given as input a full word w of length n and positive integers p and d outputs if any exists a maximal p periodic partial word contained in w with the property that no two holes are within distance d so called d valid our algorithm runs in o nd time and is used for the study of repetition freeness of partial words furthermore we construct an infinite word over a five letter alphabet that is overlap free even after holes are inserted in arbitrary 2 valid positions answering affirmatively a conjecture from blanchet sadri mercas and scott

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

the geometry of finite dimensional algebras with vanishing radical square abstract let be a basic finite dimensional algebra over an algebraically closed field with the property that the square of the jacobson radical j vanishes we determine the irreducible components of the module variety rep d for any dimension vector d our description leads to a count of the components in terms of the underlying gabriel quiver a closed formula for the number of components when is local extends existing counts for the two loop quiver to quivers with arbitrary finite sets of loops for any algebra with j 2 0 our criteria for identifying the components of rep d permit us to characterize the modules parametrized by the individual irreducible components focusing on such a component we explore generic properties of the corresponding modules by establishing a geometric bridge between the algebras with zero radical square on the one hand and their stably equivalent hereditary counterparts on the other the bridge links certain closed subvarieties of grassmannians parametrizing the modules with fixed top over the two types of algebras by way of this connection we transfer results of kac and schofield from the hereditary case to algebras of loewy length 2 finally we use the transit of information to show that any algebra of loewy length 2 which enjoys the dense orbit property in the sense of chindris kinser and weyman has finite representation type

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

all highest scoring paths in weighted grid graphs and their application to finding all approximate repeats in strings weighted paths in directed grid graphs of dimension m x n can be used to model the string edit problem which consists of obtaining optimal weighted alignments between substrings of a a m and substrings of b b n we build a data structure in o mn log m time that supports o log m time queries about the weight of any of the o m2n best paths from the vertices in column 0 of the graph to all other vertices using these techniques we present a simple o n2 log n time and theta n 2 space algorithm to find all the locally optimal approximate tandem or nontandem repeats xy within a string of size n this improves by a factor of log n upon several previous algorithms for this problem and is the first algorithm to find all locally optimal repeats for edit graphs with weights in 0 1 1 a slight modification of our techniques yields an o n2 algorithm for the cyclic string comparison problem as compared to o n2 log n for the case of general weights

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

optimal transportation between unequal dimensions we establish that solving an optimal transportation problem in which the source and target densities are defined on spaces with different dimensions is equivalent to solving a new nonlocal analog of the monge ampere equation introduced here for the first time under suitable topological conditions we also establish that solutions are smooth if and only if a local variant of the same equation admits a smooth and uniformly elliptic solution we show that this local equation is elliptic and c 2 alpha solutions can therefore be bootstrapped to obtain higher regularity results assuming smoothness of the corresponding differential operator which we prove under simplifying assumptions for one dimensional targets our sufficient criteria for regularity of solutions to the resulting ode are considerably less restrictive than those required by earlier works

continuity of optimal transport maps and convexity of injectivity domains on small deformations of 2 given a compact riemannian manifold we study the regularity of the optimal transport map between two probability measures with cost given by the squared riemannian distance our strategy is to define a new form of the so called ma trudinger wang condition and to show that this condition together with the strict convexity on the nonfocal domains implies the continuity of the optimal transport map moreover our new condition again combined with the strict convexity of the nonfocal domains allows us to prove that all injectivity domains are strictly convex too these results apply for instance on any small c4 deformation of the 2 sphere 2009 wiley periodicals inc

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

meshfree finite difference approximations for functions of the eigenvalues of the hessian we introduce meshfree finite difference methods for approximating nonlinear elliptic operators that depend on second directional derivatives or the eigenvalues of the hessian approximations are defined on unstructured point clouds which allows for very complicated domains and a non uniform distribution of discretisation points the schemes are monotone which ensures that they converge to the viscosity solution of the underlying pde as long as the equation has a comparison principle numerical experiments demonstrate convergence for a variety of equations including problems posed on random point clouds complex domains degenerate equations and singular solutions

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

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

regularity of optimal maps on the sphere the quadratic cost and the reflector antenna building on the results of ma et al in arch rational mech anal 177 2 151 183 2005 and of the author loeper in acta math to appear we study two problems of optimal transportation on the sphere the first corresponds to the cost function d2 x y where d is the riemannian distance of the round sphere the second corresponds to the cost function log x y known as the reflector antenna problem we show that in both cases the cost sectional curvature is uniformly positive and establish the geometrical properties so that the results of loeper in acta math to appear and ma et al in arch rational mech anal 177 2 151 183 2005 can apply global smooth solutions exist for arbitrary smooth positive data and optimal maps are holder continuous under weak assumptions on the data

freeness of partial words the paper approaches the classical combinatorial problem of freeness of words in the more general case of partial words first we propose an algorithm that tests efficiently whether a partial word is k free or not for a given k then we show that there exist arbitrarily many k free infinite partial words over a binary alphabet containing an infinite number of holes for k 3 moreover we present an efficient algorithm for the construction of a cube free partial word with a given number of holes over a binary alphabet in the final section of the paper we show that there exists an infinite word over a four symbol alphabet in which we can substitute randomly one symbol with a hole and still obtain a cube free word we show that such a word does not exist for alphabets with fewer symbols further we prove that in this word we can replace arbitrarily many symbols with holes such that each two consecutive holes are separated by at least two symbols and obtain a cube free partial word this result seems interesting because any partial word containing two holes with less than two symbols between them is not cube free finally we modify the previously presented algorithm to construct over a four symbol alphabet a cube free partial word with exactly n holes having minimal length among all the possible cube free partial words with at least n holes

partial w 2 p regularity for optimal transport maps abstract we prove that in the optimal transportation problem with general costs and positive continuous densities the potential function is always of class w l o c 2 p for any p 1 outside of a closed singular set of measure zero we also establish global w 2 p estimates when the cost is a small perturbation of the quadratic cost the latter result is new even when the cost is exactly the quadratic cost

regularity of optimal transport between planar convex domains for 0 p we prove a global w2 p estimate for potentials of optimal transport maps between convex domains in the plane among the tools developed for that purpose are obliqueness in general convex domains and estimates for the growth of eccentricity of sections of the potentials

overlap freeness in infinite partial words we prove that there exist infinitely many infinite overlap free binary partial words containing at least one hole moreover we show that these words cannot contain more than one hole and the only hole must occur either in the first or in the second position we define that a partial word is k overlap free if it does not contain a factor of the form xyxyx where the length of x is at least k we prove that there exist infinitely many 2 overlap free binary partial words containing an infinite number of holes

regularity of optimal transport in curved geometry the nonfocal case we explore some geometric and analytic consequences of a curvature condition introduced by ma trudinger and wang in relation to the smoothness of optimal transport in curved geometry we discuss a conjecture according to which a strict version of the ma trudinger wang condition is sufficient to prove regularity of optimal transport on a riemannian manifold we prove this conjecture under a somewhat restrictive additional assumption of nonfocality at the same time we establish the striking geometric property that the tangent cut locus is the boundary of a convex set partial extensions are presented to the case when there is no pure focalization on the tangent cut locus

holder continuity and injectivity of optimal maps consider transportation of one distribution of mass onto another chosen to optimize the total expected cost where cost per unit mass transported from x to y is given by a smooth function c x y if the source density f x is bounded away from zero and infinity in an open region u subset mathbf r n and the target density f y is bounded away from zero and infinity on its support overline v subset mathbf r n which is strongly c convex with respect to u and the transportation cost c satisfies the mathbf a3 rm w condition of trudinger and wang ann sc norm super pisa cl sci 5 8 1 143 174 2009 we deduce the local holder continuity and injectivity of the optimal map inside u so that the associated potential u belongs to c 1 alpha loc u here the exponent 0 depends only on the dimension and the bounds on the densities but not on c our result provides a crucial step in the low interior regularity setting in a sequel figalli et al j eur math soc jems 1131 1166 2013 we use it to establish regularity of optimal maps with respect to the riemannian distance squared on arbitrary products of spheres three key tools are introduced in the present paper namely we first find a transformation that under mathbf a3 rm w makes c convex functions level set convex as was also obtained independently from us by liu calc var partial diff eq 34 435 451 2009 we then derive new alexandrov type estimates for the level set convex c convex functions and a topological lemma showing that optimal maps do not mix the interior with the boundary this topological lemma which does not require mathbf a3 rm w is needed by figalli and loeper calc var partial diff eq 35 537 550 2009 to conclude the continuity of optimal maps in two dimensions in higher dimensions if the densities f are holder continuous our result permits continuous differentiability of the map inside u in fact c 2 alpha loc regularity of the associated potential to be deduced from the work of liu et al comm partial diff eq 35 1 165 184 2010

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

irreducible components of module varieties an example abstract we determine the irreducible components of the variety of d dimensional modules over the algebra k 2 2 q with q k for any d and we describe any intersection of irreducible components as the zero set of some ideal we show that any such intersection is irreducible to our knowledge this is the first time irreducible components are described by equations for modules over an algebra which is neither representation finite nor hereditary

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

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

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

boundary e regularity in optimal transportation the second author has been partially supported by nsf grant dms 1262411 this material is based upon work supported by the national science foundation under grant no 0932078 000

on strict convexity and continuous differentiability of potential functions in optimal transportation this note concerns the relationship between conditions on cost functions and domains the convexity properties of potentials in optimal transportation and the continuity of the associated optimal mappings in particular we prove that if the cost function satisfies the condition a3 introduced in our previous work with xinan ma the densities and their reciprocals are bounded and the target domain is convex with respect to the cost function then the potential is continuously differentiable and its dual potential strictly concave with respect to the cost function our results extend by different and more direct proof similar results of loeper proved by approximation from our earlier work on regularity

a generalization of thue freeness for partial words this paper approaches the combinatorial problem of thue freeness for partial words partial words are sequences over a finite alphabet that may contain a number of holes first we give an infinite word over a three letter alphabet which avoids squares of length greater than two even after we replace an infinite number of positions with holes then we give an infinite word over an eight letter alphabet that avoids longer squares even after an arbitrary selection of its positions are replaced with holes and show that the alphabet size is optimal we find similar results for overlap free partial words

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

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

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

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

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

weingarten hypersurfaces with prescribed gradient image we prove the existence of globally smooth convex solutions u of a class of curvature equations subject to the boundary condition du omega omega where omega and omega are smooth uniformly convex domains in mathbb r n the results generalize some of our previous work on the two dimensional case and on hessian equations in all dimensions

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

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

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 neumann problem for equations of monge ampere type we prove the existence of a classical solution to a neumann type problem for nonlinear elliptic equations of monge ampere type the methods depend upon the establishment of a priori derivative estimates up to order two and yield a sharp result for the equation of prescribed gauss curvature we also present applications to asymptotic problems which approach the non unique case and maximum principles for linear equations of aleksandrov bakelman type

boundary c2 α estimates for monge ampere type equations abstract in this paper we obtain global second derivative estimates for solutions of the dirichlet problem of certain monge ampere type equations under some structural conditions while the inhomogeneous term is only assumed to be holder continuous and bounded away from zero and infinity these estimates correspond to those for the standard monge ampere equation obtained by trudinger and wang 2008 28 and by savin 2013 24 and have natural applications in optimal transportation and prescribed jacobian equations

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