adaptive grid generation from harmonic maps on reimannian manifolds abstract in this paper we describe a new method for generation of solution adaptive grids based on harmonic maps on riemannian manifolds the reliability of the method is assured by an existence and uniqueness theorem for one to one maps between multidimensional multiconnected domains we formulate an adaptive riemannian metric consistent with this theorem several examples demonstrating application of the developed procedure are provided

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

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

finite difference simulations of transient radiation from a finite length pipe the far field radiation originating from a finite length pipe is well studied especially for steady state conditions however because all physical systems do not begin in steady state these radiation characteristics are only valid after the transient portion of the solution has decayed understanding transient radiation characteristics may be important particularly for systems transmitting very short duration signals as they can differ quite significantly a numerical complication to this problem involves dealing with a sharp corner in the domain of interest while many numerical studies have attempted to couple solutions from the domains inside and outside a pipe the analysis presented in this work treats the computational domain as a single region by expressing the entire physical domain as a map from a simple rectangular domain in generalized curvilinear coordinates this method will be introduced in detail and general results of transient radiation will be presented for an infinitely baffled finite length pipe using the finite difference method expressed in generalized curvilinear coordinates comparison will be made to previous results p stepanishen and r a tougas j acoust soc am 93 3074 3084 1993 that used a semi analytic approach with certain assumptions the far field radiation originating from a finite length pipe is well studied especially for steady state conditions however because all physical systems do not begin in steady state these radiation characteristics are only valid after the transient portion of the solution has decayed understanding transient radiation characteristics may be important particularly for systems transmitting very short duration signals as they can differ quite significantly a numerical complication to this problem involves dealing with a sharp corner in the domain of interest while many numerical studies have attempted to couple solutions from the domains inside and outside a pipe the analysis presented in this work treats the computational domain as a single region by expressing the entire physical domain as a map from a simple rectangular domain in generalized curvilinear coordinates this method will be introduced in detail and general results of transient radiation will be presented for an infinitely baffled f

on semilinear δλ laplace equation abstract we prove some existence nonexistence and regularity results for the boundary value problem u f u 0 in u 0 where is a bounded subset of r n n 2 and is a laplacian i e a degenerate elliptic operator of the kind i 1 n x i i 2 x x i 1 n together with some assumptions made in franchi and lanconelli 1984 1 the family is supposed to verify a condition making homogeneous of degree two with respect to a group of dilations in r n

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

finite difference on grids with nearly uniform cell area and line spacing for the wave equation on complex domains a finite difference time dependent numerical method for the wave equation supported by recently derived novel elliptic grids is analyzed the method is successfully applied to single and multiple two dimensional acoustic scattering problems including soft and hard obstacles with complexly shaped boundaries the new grids have nearly uniform cell area j grids and nearly uniform grid line spacing email protected numerical experiments reveal the positive impact of these two grid properties on the scattered field convergence to its harmonic steady state the restriction imposed by stability conditions on the time step size is relaxed due to the near uniformity cell areas and grid line spacing as a consequence moderately large time steps can be used for relatively fine spatial grids resulting in greater accuracy at a lower computational cost also numerical solutions for wave problems inside annular regions of complex shapes are obtained the use of the new grids results in late time stability in contrast with other classical finite difference time dependent methods

coupling of dirichlet to neumann boundary condition and finite difference methods in curvilinear coordinates for multiple scattering the applicability of the dirichlet to neumann technique coupled with finite difference methods is enhanced by extending it to multiple scattering from obstacles of arbitrary shape the original boundary value problem bvp for the multiple scattering problem is reformulated as an interface bvp a heterogenous medium with variable physical properties in the vicinity of the obstacles is considered a rigorous proof of the equivalence between these two problems for smooth interfaces in two and three dimensions for any finite number of obstacles is given the problem is written in terms of generalized curvilinear coordinates inside the computational region then novel elliptic grids conforming to complex geometrical configurations of several two dimensional obstacles are constructed and approximations of the scattered field supported by them are obtained the numerical method developed is validated by comparing the approximate and exact far field patterns for the scattering from two circular obstacles in this case for a second order finite difference scheme a second order convergence of the numerical solution to the exact solution is easily verified

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

time dependent numerical method with boundary conforming curvilinear coordinates applied to wave interactions with prototypical antennas a new unstaggered finite difference time dependent technique to accurately model scattring from prototypical 2d antenna structures is devised the unbounded boundary value problems defining these phenomena are redefined over bounded domains using appropriate radiation operators over finite artificial boundaries generalized curvilinear coordinates are generated such that physical boundaries correspond to coordinate lines a numerical procedure to generate almost orthogonal boundary conforming fine grids over these bounded regions is developed once the governing equations are written in terms of the new curvilinear coordinates a time dependent numerical method is applied to obtain time harmonic steady state solutions to these problems the electric field wave amplitude as well as the wave pattern inside the waveguide and the scattered field from the prototypical antennas are obtained accuracy and computational cost are compared when almost orthogonal and nonorthogonal grids are in use an optical theorem for a flanged waveguide antenna with perfect electrical conductor walls is derived it is verified that the approximate solutions obtained by application of the time dependent numerical method with boundary conforming curvilinear coordinates satisfy the optical theorem

monge ampere simulation of fourth order pdes in two dimensions with application to elastic electrostatic contact problems abstract we present an efficient moving mesh method for the simulation of fourth order nonlinear partial differential equations pdes in two dimensions using the parabolic monge ampere pma equation pma methods have been successfully applied to the simulation of second order problems but not on systems with higher order equations which arise in many topical applications our main application is the resolution of fine scale behavior in pdes describing elastic electrostatic interactions the pde system considered has multiple parameter dependent singular solution modalities including finite time singularities and sharp interface dynamics we describe how to construct a dynamic mesh algorithm for such problems which incorporates known self similar or boundary layer scalings of the underlying equation to locate and dynamically resolve fine scale solution features in these singular regimes we find a key step in using the pma equation for mesh generation in fourth order problems is the adoption of a high order representation of the transformation from the computational to physical mesh we demonstrate the efficacy of the new method on a variety of examples and establish several new results and conjectures on the nature of self similar singularity formation in higher order pdes

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

generation of boundary conforming grids around wing body configurations using transfinite interpolation a computational procedure for generating three dimensional nonorthogonal surface fitted mesh systems around wing fuselage configurations is presented the method is based on the concept of transfinite interpolation which has been extended to handle very general mapping function specifications at the boundaries thereby making it possible to generate single block mappings with geometry data specified only at the outer boundaries of the computational domain since it is a direct algebraic mapping technique the method is very inexpensive in terms of computer cost different types of possible mappings are compared with respect to resolution and economy of nodal points a procedure for a novel type of mapping designated type o o is described and several plots of generated grids demonstrate the capabilities of the method the singular lines inherent in every three dimensional mesh for this type of surface geometry are also discussed

an explicit fourth order orthogonal curvilinear staggered grid fdtd method for maxwell s equations the explicit fourth order staggered finite difference time domain scheme previously proposed for a cartesian grid is extended to maxwell s equations in an orthogonal curvilinear coordinate system and applied to electromagnetic wave problems a simple technique is also presented for generating orthogonal curvilinear grids that conform to the material boundaries and interfaces of the problem numerical experiments are presented to illustrate the efficiency and accuracy of the method

the enhancement of an elliptic grid using appropriate control functions three dimensional structured grid generation methodology based on elliptic partial differential equation pde method is reviewed it will be shown that the quality of a grid depends on how to treat the second normal derivative terms involved in control functions a 7 point finite analytic scheme based on one dimensional analytic solution will be introduced to discretize the governing equations of the elliptic grid generation

moving mesh methods based on moving mesh partial differential equations several versions of a moving mesh method are developed based on a mesh spatial smoothing technique and on the moving mesh pdes derived in a previous paper these versions are quite simple and easy to program they are applied to three bench mark one dimensional problems which show different solution behaviour the numerical results clearly demonstrate that the present methods are capable of accurately tracking rapid spatial and temporal transitions

an adaptive grid with directional control abstract a variational formulation of winslow s variable diffusion method is combined with a directional control functional to derive an adaptive grid generator for adaptive node movement the addition of directional control allows one to align mesh lines with a prescribed vector field while adapting the spacing to resolve the data it is shown that harmonic function theory can be applied to define the conditions under which unique solutions of the resulting elliptic equations exist the grid generator is applied to the complex problem of magnetic field reconnection driven by fluid instability the results demonstrate that the cost of a computation with an adaptive grid with directional control is only one tenth of the cost of either a calculation on an eulerian grid or on an adaptive grid without directional control

adaptive zoning for singular problems in two dimensions abstract winslow s method for the automatic generation of computation meshes is extended to adaptively vary the zone sizes and orthogonality of grid lines in the resulting mesh through simple analysis and numerical examples the adaptive mesh is shown to give significant increases in accuracy in the computation of singular problems

liouville type theorems for stable solutions of the weighted elliptic system abstract we examine the weighted elliptic system u 1 x 2 2 v v 1 x 2 2 u p in r n and prove liouville type theorems for the classical positive and nonnegative stable solutions in higher dimension in particular there are no positive stable solutions for any n 12 5 p 1 and 0 our proof is based on a combination of the bootstrap argument souplet s inequality and intermediate stability criterion which is used to obtain sharp results

nonlinear liouville theorems for grushin and tricomi operators the aim of this paper is to study necessary conditions for existence of weak solutions of the inequality l x y dx dy u u q x 1 y 2 x rd y rk where l is a quasi homogeneous differential operator q 1 1 2 r and k d 1 as special cases our results can be applied when l is tricomi or grushin type operators

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

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 stability of numerical boundary treatments for compact high order finite difference schemes the stability characteristics of various compact fourth and sixth order spatial operators are assessed with the theory of gustafsson kreiss and sundstrom g k s for the semidiscrete initial boundary value problem these results are generalized to the fully discrete case with a recently developed theory of kreiss and wu in all cases favorable comparisons are obtained between g k s theory eigenvalue determination and numerical simulation the conventional definition of stability then is sharpened to include only those spatial discretizations that are asymptotically stable bounded left half plane eigenvalues many of the higher order schemes that are g k s stable are not asymptotically stable a series of compact fourth and sixth order schemes that are both asymptotically and g k s stable for the scalar case are then developed

numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence abstract adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near singular solutions adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers see for example mulholland et al siam j sci comput 19 4 1998 1261 1289 much is now being done in developing error analyes for methods that are based on adaptivity in this paper we present a rigorous error analysis for the solution of a singularly perturbed two point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution the discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc length an error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter epsilon the epsilon uniform convergence is confirmed by numerical computations

grading functions and mesh redistribution we consider the problem of determining an appropriate grading of a mesh for piecewise polynomial interpolation and for approximate solution of two point boundary value problems by finite difference or finite element methods an analysis of optimality of the mesh and optimal grading functions in various norms and seminorms for the interpolation problem leads to the formulation of an adaptive mesh redistribution algorithm for boundary value problems in one dimension error estimates are given and numerical results presented to demonstrate the performance of the scheme and compare alternative redistribution criteria

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

piecewise convex function estimation pilot estimators given noisy data function estimation is considered when the unknown function is known a priori to be either convex or concave on each of a small number of regions where the function when the number of regions is unknown the model selection problem is to determine the number of convexity change points for kernel estimates in gaussian noise the number of false change points is evaluated as a function of the smoothing parameter to insure that the number of false convexity change points tends to zero the smoothing level must be larger than is generically optimal for minimizing the mean integrated square error mise a two stage estimator is proposed and shown to achieve the optimal rate of convergence of the mise in the first stage the number and location of the change points is estimated using strong smoothing in the second stage a constrained smoothing spline fit is performed with the smoothing level chosen to minimize the mise the imposed constraint is that a single change point occur in a region about each empirical change point from the first stage estimate this constraint is equivalent to the requirement that the third derivative of the second stage estimate has a single sign in a small neighborhood about each first stage change point the change points from the second stage are first stage change points but need not be at the identical locations

on the classification of positive supersolutions of elliptic systems involving the advection terms abstract in this paper we establish some liouville type theorems for positive classical supersolutions in the whole space r n of the elliptic equation u a u u p and of the elliptic system u a u v p v a v u q here p q r and a is a smooth vector field satisfying some growth condition at infinity

finite difference time domain modeling of curved surfaces em scattering the finite difference time domain fdtd method is generalized to include the accurate modeling of curved surfaces this generalization the contour path cp method accurately models the illumination of bodies with curved surfaces yet retains the ability to model corners and edges cp modeling of two dimensional electromagnetic wave scattering from objects of various shapes and compositions is presented

numerical solution of the quasilinear poisson equation in a nonuniform triangle mesh a finite difference method using a nonuniform triangle mesh is described for the numerical solution of the nonlinear two dimensional poisson equation s 0 where is a function of or its derivatives sis a function of position and or its normal derivative is specified on the boundary the finite difference equations are solved by successive overrelaxation the triangle mesh which is constructed numerically by solving laplace s equation is easily adapted to nonrectangular boundaries and interfaces examples of numerical results are given for the magnetostatic problem with iron and other possible applications are mentioned

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

adaptive mesh movement the mmpde approach and its applications a class of moving mesh algorithms based upon a so called moving mesh partial differential equation mmpde is reviewed various forms for the mmpde are presented for both the simple one and the higher dimensional cases additional practical features such as mesh movement on the boundary selection of the monitor function and smoothing of the monitor function are addressed the overall discretization and solution procedure including for unstructured meshes are briefly covered finally we discuss some physical applications suitable for mmpde techniques and some challenges facing mmpde methods in the future

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

some liouville theorems for henon type elliptic equations abstract we investigate here the nonlinear elliptic equations u x e u and u x u p 1 u with 2 p 1 and n 2 in particular we prove some liouville type theorems for weak solutions with finite morse index in the low dimensional euclidean spaces or half spaces

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

the dtn nonreflecting boundary condition for multiple scattering problems in the half plane abstract the multiple dirichlet to neumann multiple dtn non reflecting boundary condition is adapted to acoustic scattering from obstacles embedded in the half plane the multiple dtn map is coupled with the method of images as an alternative model for multiple acoustic scattering in the presence of acoustically soft and hard plane boundaries as opposed to the current practice of enclosing all obstacles with a large semicircular artificial boundary that contains portion of the plane boundary the proposed technique uses small artificial circular boundaries that only enclose the immediate vicinity of each obstacle in the half plane the adapted multiple dtn condition is simultaneously imposed in each of the artificial circular boundaries as a result the computational effort is significantly reduced a computationally advantageous boundary value problem is numerically solved with a finite difference method supported on boundary fitted grids approximate solutions to problems involving two scatterers of arbitrary geometry are presented the proposed numerical method is validated by comparing the approximate and exact far field patterns for the scattering from a single and from two circular obstacles in the half plane

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

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

generation of curvilinear coordinates on multiply connected regions with boundary singularities a new algorithm to generate smooth two dimensional boundary conforming coordinates with grid lines control on multiply connected regions including boundary singularities is devised the technique fits into the category of elliptic grid generators since it is based on the numerical solution of poisson equations the physical domain d is transformed into a topologically equivalent connected rectangular domain d by defining a branch cut inside d control of the grid line spacing over the multiply connected regions is established by appropriately distributing nodal points on the branch cut and on other boundary curves simple and computationally convenient expressions for the control functions present in the poisson system are obtained from this initial distribution of points thus a natural link between clustering properties and the control functions is created fixing the nodal points on the branch cut causes the convergence of the numerical method about the singular boundary points an iterative smoothing process relocating the branch cut is described as a result smooth grids are obtained grid quality analysis shows that coordinate line orthogonality in the new meshes is superior than in similar grids obtained from a well known commercial software the algorithm is successfully tested over a variety of multiply connected domains the computational advantage of this novel grids is revealed by using them to numerically model the vibrations of complexly shaped annular membranes

measuring mesh qualities and application to variational mesh adaptation the mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior three elementwise mesh quality measures characterizing the shape alignment and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh an adaptive mesh is assessed by an overall quality measure defined as a weighted lebesgue norm of a product of the three elementwise quality measures it is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the so called roughness measure of the solution defined as the ratio of two lebesgue norms of a derivative of the solution the definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh as an immediate result the task of mesh adaptation becomes to control the overall mesh quality this idea is applied to variational mesh adaptation to develop two functionals one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality

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

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

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

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

a three dimensional fdtd algorithm in curvilinear coordinates em scattering the finite difference time domain fdtd algorithm for the solution of electromagnetic scattering problems is formulated in numerically defined generalized coordinates in three dimensions and implemented in a code with the lowest order bayliss turkel radiation boundary condition expressed in spherical coordinates it is shown that the algorithm is capable of accurately tracking the progress of a pulse of electromagnetic radiation through the curvilinear mesh generated by a body of revolution the only problems occurring in the vicinity of the rotation axis which represents a coordinate singularity a simple method to deal with this singular line is presented and discussed and its is shown that at least for the test problem this approximation is sufficient the algorithm discussed is useful for the solution of the exterior problem in the presence of conductors and dielectrics with complicated shapes and electrical compositions and for near field problems such as cavity penetration problems the far fields are obtained by replacing the scatterer with a virtual surface enclosing all sources

signs of divided differences yield least squares data fitting with constrained monotonicity or convexity methods are presented for least squares data smoothing by using the signs of divided differences of the smoothed values professor m j d powell initiated the subject in the early 1980s and since then theory algorithms and fortran software make it applicable to several disciplines in various ways let us consider n data measurements of a univariate function which have been altered by random errors then it is usual for the divided differences of the measurements to show sign alterations which are probably due to data errors we make the least sum of squares change to the measurements by requiring the sequence of divided differences of order m to have at most q sign changes for some prescribed integer q the positions of the sign changes are integer variables of the optimization calculation which implies a combinatorial problem whose solution can require about o nq quadratic programming calculations in n variables and n m constraints suitable methods have been developed for the following cases it has been found that a dynamic programming procedure can calculate the global minimum for the important cases of piecewise monotonicity m 1 q 1 and piecewise convexity concavity m 2 q 1 of the smoothed values the complexity of the procedure in the case of m 1 is o n2 qn log2 n computer operations while it is reduced to only o n when q 0 monotonicity and q 1 increasing decreasing monotonicity the case m 2 q 1 requires o qn2 computer operations and n2 quadratic programming calculations which is reduced to one and n 2 quadratic programming calculations when m 2 q 0 i e convexity and m 2 q 1 i e convexity concavity respectively unfortunately the technique that receives this efficiency cannot generalize for the highly nonlinear case m 3 q 2 however the case m 3 q 0 is solved by a special strictly convex quadratic programming calculation and the case m 3 q 1 can be solved by at most 2 n m applications of the previous algorithm also as m gets larger large sets of active constraints usually occur at the optimal approximation which makes the calculation for higher values of q less expensive than what it seems to further attention is given to the sensitivity of the solution with respect to changes in the constraints and the data the smoothing technique is an active research topic and there seems to be room for further developments one strong reason for studying methods that make use of divided differences for data smoothing is that whereas only the data are provided the achieved accuracy goes much beyond the accuracy of the data at an order determined automatically by the chosen divided differences

new self similar solutions of the nonlinear schrodinger equation with moving mesh computations we study the blow up self similar solutions of the radially symmetric nonlinear schrodinger equation nls given by iut urr d 1rur u u 2 with dimension d 2 these solutions become infinite in a finite time t by a series of careful numerical computations partly supported by analytic results we demonstrate that there is a countably infinite set of blow up self similar solutions which satisfy a second order complex ordinary differential equation with an integral constraint these solutions are characterised by the number of oscillations in their amplitude when d is close to 2 the solutions are computed as functions of d and their behaviour in the critical limit as d 2 is investigated the stability of these solutions is then studied by solving the nls by using an adaptive numerical method this method uses moving mesh partial differential equations and exploits the scaling invariance properties of the underlying equation we demonstrate that the single humped self similar solution is globally stable whereas the multi humped solutions all appear to be unstable

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

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 data smoothing technique for piecewise convex concave curves a data smoothing technique for piecewise convex concave curves is developed and its computational aspects are discussed some of its statistical properties are determined and the difficulty in computing the general expression for bias and the mean squared error are illustrated through an example a variety of examples are considered to demonstrate the potential of the proposed smoothing technique and the behavior of the piecewise convex concave smoother is shown by monte carlo simulations we also give a detailed comparison between the proposed smoothing technique and spline smoothing

an efficient dynamically adaptive mesh for potentially singular solutions abstract we develop an efficient dynamically adaptive mesh generator for time dependent problems in two or more dimensions the mesh generator is motivated by the variational approach and is based on solving a new set of nonlinear elliptic pdes for the mesh map when coupled to a physical problem the mesh map evolves with the underlying solution and maintains high adaptivity as the solution develops complicated structures and even singular behavior the overall mesh strategy is simple to implement avoids interpolation and can be easily incorporated into a broad range of applications the efficacy of the mesh is first demonstrated by two examples of blowing up solutions to the 2 d semilinear heat equation these examples show that the mesh can follow with high adaptivity a finite time singularity process the focus of applications presented here is however the baroclinic generation of vorticity in a strongly layered 2 d boussinesq fluid a challenging problem the moving mesh follows effectively the flow resolving both its global features and the almost singular shear layers developed dynamically the numerical results show the fast collapse to small scales and an exponential vorticity growth

divergence preserving discrete surface integral methods for maxwell s curl equations using non orthogonal unstructured grids abstract several new discrete surface integral methods for solving maxwell s equations in the time domain are presented these methods which allow the use of general non orthogonal mixed polyhedral unstructured grids are direct generalizations of the canonical staggered grid finite difference method these methods are conservative in that they locally preserve divergence or charge employing mixed polyhedral cells hexahedral tetrahedral etc these methods allow more accurate modeling of non rectangular structures and objects because the traditional stair stepped boundary approximations associated with the orthogonal grid based finite difference methods can be avoided numerical results demonstrating the accuracy of these new methods are presented

partial regularity of finite morse index solutions to the lane emden equation we prove regularity and partial regularity results for finite morse index solutions u h1 lp to the lane emden equation u u p 1u in

a fast higher order solver for scattering by penetrable bodies in three dimensions in this paper we introduce a new fast higher order solver for scattering by inhomogeneous media in three dimensions as in previously existing methods the low complexity of our integral equation method o nlogn operations for an n point discretization is obtained through extensive use of the fast fourier transform fft for the evaluation of convolutions however the present approach obtains significantly higher order accuracy than these previous approaches yielding at worst third order far field accuracy or substantially better for smooth scatterers even for discontinuous and complex refractive index distributions possibly containing severe geometric singularities such as corners and cusps the increased order of convergence of our method results from i a partition of unity decomposition of the green s function into a smooth part with unbounded support and a singular part with compact support and ii replacement of the possibly discontinuous scatterer by an appropriate fourier smoothed scatterer the resulting convolutions can then be computed with higher order accuracy by means of o nlogn ffts we present a parallel implementation of our approach and demonstrate the method s efficiency and accuracy through a variety of computational examples for a very large scatterer considered earlier in the literature with a volume of 3648 l 3 where l is the wavelength using the same number of points per wavelength and in computing times comparable to those required by the previous approach the present algorithm produces far field values whose errors are two orders of magnitude smaller than those reported previously

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

regularized model of post touchdown configurations in electrostatic mems equilibrium analysis abstract in canonical models of micro electro mechanical systems mems an event called touchdown whereby the electrical components of the device come into contact is characterized by a blow up in the governing equations and a non physical divergence of the electric field in the present work we propose novel regularized governing equations whose solutions remain finite at touchdown and exhibit additional dynamics beyond this initial event before eventually relaxing to new stable equilibria we employ techniques from variational calculus dynamical systems and singular perturbation theory to obtain a detailed understanding of the properties and equilibrium solutions of the regularized family of equations

liouville type results for semi stable solutions of the weighted lane emden system abstract we examine the weighted lane emden system u 1 x 2 2 v p v 1 x 2 2 u q in r n where 1 p q and 0 and the weighted lane emden equation u 1 x 2 2 u p in r n where p 4 3 and 0 we prove that the system equation does not have the classical positive semi stable solution in dimension n 2 4 2 q 1 p q 1 0 n 2 2 2 p 1 p p 2 p in particular there is no positive semi stable solution of the weighted lane emden system if n 10 4 and 2 p q

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

a study of monitor functions for two dimensional adaptive mesh generation in this paper we study the problem of two dimensional adaptive mesh generation using a variational approach and specifically the effect that the monitor function has on the resulting mesh behavior the basic theoretical tools employed are green s function for elliptic problems and the eigendecomposition of symmetric positive definite matrices based upon this study a general strategy is suggested for how to choose the monitor function and numerical results are presented for illustrative purposes the three dimensional case is also briefly discussed it is noted that the strategy used here can be applied to other elliptic mesh generation techniques as well

on the use of higher order finite difference schemes on curvilinear and deforming meshes this study enables the use of very high order finite difference schemes for the solution of conservation laws on stretched curvilinear and deforming meshes to illustrate these procedures we focus on up to 6th order pade type spatial discretizations coupled with up to 10th order low pass filters these are combined with explicit and implicit time integration methods to examine wave propagation and wall bounded flows described by the navier stokes equations it is shown that without the incorporation of the filter application of the high order compact scheme to nonsmooth meshes results in spurious oscillations which inhibit their applicability inclusion of the discriminating low pass high order filter restores the advantages of high order approach even in the presence of large grid discontinuities when three dimensional curvilinear meshes are employed the use of standard metric evaluation procedures significantly degrades accuracy since freestream preservation is violated to overcome this problem a simple technique is adopted which ensures metric cancellation and thus ensures freestream preservation even on highly distorted curvilinear meshes for dynamically deforming grids an effective numerical treatment is described to evaluate expressions containing the time varying transformation metrics with these techniques metric cancellation is guaranteed regardless of the manner in which grid speeds are defined the efficacy of the new procedures is demonstrated by solving several model problems as well as by application to flow past a rapidly pitching airfoil and past a flexible panel

elliptic grid generation based on laplace equations and algebraic transformations abstract an elliptic grid generation method is presented to generate boundary conforming grids in domains in 2d and 3d physical space and on minimal surfaces and parametrized surfaces in 3d physical space the elliptic grid generation method is based on the use of a composite mapping this composite mapping consists of a nonlinear transfinite algebraic transformation and an elliptic transformation the elliptic transformation is based on the laplace equations for domains or on the laplace beltrami equations for surfaces the algebraic transformation maps the computational space one to one onto a parameter space the elliptic transformation maps the parameter space one to one onto the domains or surfaces the composition of these two mappings is a differentiable one to one mapping from computational space onto the domains or surfaces and has a nonvanishing jacobian this composite mapping defines the grid point distribution in the interior of the domains or surfaces for domains and minimal surfaces the composite mapping obeys a nonlinear elliptic poisson system with control functions completely defined by the algebraic transformation the solution of the poisson systems is obtained by picard iteration and black box multigrid solvers for parametrized curved surfaces it is not necessary to define and solve a nonlinear elliptic poisson system instead a linear elliptic system and an inversion problem is solved to generate the grid in the interior of the surface

liouville type theorem for nonlinear elliptic system involving grushin operator abstract we study the degenerate elliptic system of the form g u v p g v u q on r n r n 1 r n 2 where g x x 2 y is the grushin operator 0 and p q 1 we establish some liouville type results for stable solutions of the system in particular we prove the comparison principle a crucial step to establish such results as consequences we obtain a liouville type theorem for the scalar equation and provide a counterpart of the previous result in c cowan 2013 7

monge ampere based moving mesh methods for numerical weather prediction with applications to the eady problem we derive a moving mesh method based upon ideas from optimal transport theory which is suited to solving pde problems in meteorology in particular we show how the parabolic monge ampere method for constructing a moving mesh in two dimensions can be coupled successfully to a pressure correction method for the solution of incompressible flows with significant convection and subject to coriolis forces this method can be used to resolve evolving small scale features in the flow in this paper the method is then applied to the computation of the solution to the eady problem which is observed to develop large gradients in a finite time the moving mesh method is shown to work and be stable and to give significantly better resolution of the evolving singularity than a fixed uniform mesh

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

parabolic monge ampere methods for blow up problems in several spatial dimensions this paper constructs and analyses an adaptive moving mesh scheme for the numerical simulation of singular pdes in one or more spatial dimensions the scheme is based on computing a legendre transformation from a regular to a spatially non uniform mesh via the solution of a relaxed form of the monge ampere equation the method is shown to preserve the inherent scaling properties of the pde and to identify natural computational coordinates numerical examples are presented in one and two dimensions which demonstrate the effectiveness of this approach

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