numerical studies of discrete approximations to the allen cahn equation in the sharp interface limit the numerical approximations to the allen cahn type diffuse interface models are studied with a particular focus on their performance in the sharp interface limit and the effectiveness of high order discretization schemes different spatial discretizations of an energy functional in the diffuse interface framework are compared first discretizations of the time dependent equation using various different time stepping schemes and an adaptive finite element spatial approximations are then analyzed assessments of the numerical accuracy in different parameter regimes are carried out the analysis and computational findings provide insight on the effectiveness of the discretization schemes and offer guidance to parameter choices when numerical simulations are conducted within the diffuse interface framework

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

model categories of diagram spectra working in the category mathcal t of based spaces we give the basic theory of diagram spaces and diagram spectra these are functors mathcal d longrightarrow mathcal t for a suitable small topological category mathcal d when mathcal d is symmetric monoidal there is a smash product that gives the category of mathcal d spaces a symmetric monoidal structure examples include begin enumerate item prespectra as defined classically item symmetric spectra as defined by jeff smith item orthogonal spectra a coordinate free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category item gamma spaces as defined by graeme segal item mathcal w spaces an analogue of gamma spaces with finite sets replaced by finite cw complexes in the domain category end enumerate we construct and compare model structures on these categories with the caveat that gamma spaces are always connective these categories and their simplicial analogues are quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category monoids in these categories are strict ring spectra often the subcategories of ring spectra module spectra over a ring spectrum and commutative ring spectra are also model categories when this holds the respective categories of ring and module spectra are quillen equivalent and thus have equivalent homotopy categories this allows interchangeable use of these categories in applications 2000 mathematics subject classification primary 55p42 secondary 18a25 18e30 55u35

proper orthogonal decomposition reduced order extrapolation continuous space time finite element method for the two dimensional unsteady stokes equation abstract in this study we mainly consider the order reduction for the solution coefficient vectors of the classical continuous space time finite element ccstfe method for the two dimensional 2d unsteady stokes equation first we construct the ccstfe model for the 2d unsteady stokes equation about vorticity stream functions and show the existence stability and convergence of the ccstfe solutions we then apply proper orthogonal decomposition pod to develop a reduced order extrapolation continuous space time finite element roecstfe model as well as discussing the existence stability and convergence of the roecstfe solutions based on matrix analysis so the theoretical analysis is highly concise and we use a flowchart to search for the roecstfe solutions finally we provide two numerical examples to confirm the correctness of the theoretical results obtained we also show that the computational results agree well with the theoretical results thereby demonstrating the effectiveness and feasibility of the proposed roecstfe method therefore both the theory and method are totally different from the existing pod reduced order methods

numerical analysis of the leray reduced order model abstract standard roms generally yield spurious numerical oscillations in the simulation of convection dominated flows regularized roms use explicit rom spatial filtering to decrease these spurious numerical oscillations the leray rom is a recently introduced regularized rom that utilizes explicit rom spatial filtering of the convective term in the navier stokes equations this paper presents the numerical analysis of the finite element discretization of the leray rom error estimates for the rom differential filter which is the explicit rom spatial filter used in the leray rom are proved these rom filtering error estimates are then used to prove error estimates for the leray rom finally both the rom filtering error estimates and the leray rom error estimates are numerically investigated in the simulation of the two dimensional navier stokes equations with an analytic solution

uniqueness of the multiplicative cyclotomic trace abstract making use of the theory of noncommutative motives we characterize the topological dennis trace map as the unique multiplicative natural transformation from algebraic k theory to topological hochschild homology thh and the cyclotomic trace map as the unique multiplicative lift through topological cyclic homology tc moreover we prove that the space of all multiplicative structures on algebraic k theory is contractible we also show that the algebraic k theory functor from small stable categories to spectra is lax symmetric monoidal which in particular implies that e n ring spectra give rise to e n 1 ring algebraic k theory spectra along the way we develop a multiplicative morita theory establishing a symmetric monoidal equivalence between the category of small idempotent complete stable categories and the morita localization of the category of spectral categories

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

mathematical and numerical results on the sensitivity of the pod approximation relative to the burgers equation from mathematical and numerical points of view we study the sensitivity with respect to parametric evolutions of the error obtained by approximating a given parametric partial differential equation using a proper orthogonal decomposition pod basis determined once and for all with association to a fixed parameter more precisely we will be considering the one dimensional burgers equation with a source term

pod based feedback control of the burgers equation by solving the evolutionary hjb equation a new approach for solving finite time horizon feedback control problems for distributed parameter systems is proposed it is based on model reduction by proper orthogonal decomposition combined with efficient numerical methods for solving the resulting low order evolutionary hamilton jacobi bellman hjb equation the feasibility of the proposed methodology is demonstrated by means of optimal feedback control for the burgers equation the method for solving the hjb equation is first tested on several 1 d problems and then successfully applied to the control of the reduced order burgers equation the effect of noise is investigated and parallelism is used for computational speedup

the pod deim reduced order method for stochastic allen cahn equations with multiplicative noise abstract in this paper we propose a reduced order method rom based on the monte carlo finite difference method fdm or monte carlo finite element method fem for the stochastic allen cahn sac equation with multiplicative noise the reduction method is a combination of proper orthogonal decomposition pod and discrete empirical interpolation method deim in the theoretical part the error bounds of full order method fom and pod deim are provided as well as the computational complexity of fom and pod deim in the numerical experiments section several two dimensional sac examples with multiplicative noise are considered using the monte carlo fdm and monte carlo fem as the reference methods the pod deim has obvious advantage in the computational efficiency and with the encryption in the space or time direction this advantage is more prominent

galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics error estimates for galerkin proper orthogonal decomposition pod methods for nonlinear parabolic systems arising in fluid dynamics are proved for the time integration the backward euler scheme is considered the asymptotic estimates involve the singular values of the pod snapshot set and the grid structure of the time discretization as well as the snapshot locations

a reduced order extrapolated crank nicolson finite spectral element method based on pod for the 2d non stationary boussinesq equations abstract in this paper we mainly utilize proper orthogonal decomposition pod to reduce the order for the coefficient vector of the classical crank nicolson finite spectral element ccnfse method of the two dimensional 2d non stationary boussinesq equations about vorticity stream functions so that the reduced order method maintains all the advantages of the ccnfse method toward this end we first establish a ccnfse format with the second order time accuracy for the two dimensional 2d non stationary boussinesq equations about vorticity stream functions and analyze the existence stability and convergence of the ccnfse solutions and then by pod we establish a reduced order extrapolated crank nicolson finite spectral element roecnfse method and analyze the existence stability and convergence of the roecnfse solutions as well as offer the flowchart for seeking roecnfse solutions finally we use two sets of numerical experiments to validate that the numerical computational consequences are accorded with the theoretical ones such that the effectiveness and feasibility of the roecnfse method are further verified both theory and method in this paper is new and completely different from the existing reduced order methods

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

a semi continuous formulation for goal oriented reduced order models 1d problems a semi continuous formulation is introduced for finding bases that minimise the error in a specific output functional of a reduced order model the formulation is advantageous in that it can be used with arbitrary reference data and can be easily applied to nonlinear models and functionals a general description of the approach is given then explicit formulations are derived for the advection diffusion and burgers equations numerical results are given for both linear and nonlinear functionals these show substantial reductions in error when compared with pod modes depending on the functional considered the optimisation of bases for a reduced order model using an approximated governing equation is also described for which large increases in accuracy are obtained relative to pod modes copyright c 2015 john wiley sons ltd

noisy dynamic simulations in the presence of symmetry data alignment and model reduction abstract we process snapshots of trajectories of evolution equations with intrinsic symmetries and demonstrate the use of recently developed eigenvector based techniques to successfully quotient out the degrees of freedom associated with the symmetries in the presence of noise our illustrative examples include a one dimensional evolutionary partial differential the kuramoto sivashinsky equation with periodic boundary conditions as well as a stochastic simulation of nematic liquid crystals which can be effectively modeled through a nonlinear smoluchowski equation on the surface of a sphere this is a useful first step towards data mining the symmetry adjusted ensemble of snapshots in search of an accurate low dimensional parametrization and the associated reduction of the original dynamical system we also demonstrate a technique vector diffusion maps that combines in a single formulation the symmetry removal step and the dimensionality reduction step

mixed finite element formulation and error estimates based on proper orthogonal decomposition for the nonstationary navier stokes equations in this paper proper orthogonal decomposition pod is used for model reduction of mixed finite element mfe for the nonstationary navier stokes equations and error estimates between a reference solution and the pod solution of reduced mfe formulation are derived the basic idea of this reduction technique is that ensembles of data are first compiled from transient solutions computed equation system derived with the usual mfe method for the nonstationary navier stokes equations or from physics system trajectories by drawing samples of experiments and interpolation or data assimilation and then the basis functions of the usual mfe method are substituted with the pod basis functions reconstructed by the elements of the ensemble to derive the pod reduced mfe formulation for the nonstationary navier stokes equations it is shown by considering numerical simulation results obtained for the illustrating example of cavity flows that the error between pod solution of reduced mfe formulation and the reference solution is consistent with theoretical results moreover it is also shown that this result validates the feasibility and efficiency of the pod method

an hp proper orthogonal decomposition moving least squares approach for molecular dynamics simulation abstract the application of model order reduction for molecular dynamic systems exhibits intrinsic complexities due to the highly nonlinear and nonlocal properties of such systems the most costly computational part of such work is the calculation of potential energy and inter atomic force fields we introduce the so called hp proper orthogonal decomposition moving least squares hp pod mls method not only to tackle this force computation but also to improve the accuracy of reduced displacement approximations in an adaptive and controllable manner the approach combines the proper orthogonal decomposition pod method with the moving least squares mls one to interpolate the inter atomic force field at an arbitrary time instance this approximated force field is then fed into the reduced molecular dynamic md governing equation to compute the reduced order displacement field in fact this pod mls approach is already able to handle efficiently the force computation in the online computational stage i e the online force computational cost is independent of the dimension of the full md space however its corresponding approximated displacement error will be very large if the considered total time interval is long the proposed hp pod mls algorithm will divide adaptively and automatically the global temporal interval into many smaller local subintervals h refinement and then build the sets of local reduced basis functions over each of these subintervals p refinement the goal is to reduce the corresponding displacement errors to a desired tolerance over these subintervals the effectiveness of the algorithm is demonstrated by performing reduced order simulations of several molecular dynamic systems

an unconditionally energy stable second order finite element method for solving the allen cahn equation abstract in this paper we design analyze and numerically validate an unconditionally energy stable second order numerical method for solving the allen cahn equation which represents a model for anti phase domain coarsening in a binary alloy the proposed scheme inherits the property of the decrease of the total energy from the allen cahn equation an error estimate for the fully discretized scheme is also established numerical examples are presented to confirm the accuracy efficiency and stability of the proposed method in particular the method is shown to be unconditionally energy stable and second order accurate in both time and space discretizations

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

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

preserving symmetries in the proper orthogonal decomposition the proper orthogonal decomposition pod also called karhunen loeve expansion has been recently used in turbulence to derive optimally fast converging bases of spatial functions leading to efficient finite truncations whether a finite number of these modes can be used in numerical simulations to derive an accurate finite set of ordinary differential equations over a certain range of bifurcation parameter values still remains an open question it is shown here that a necessary condition for achieving this goal is that the truncated system inherit the symmetry properties of the original infinite dimensional system in most cases this leads to a systematic involvement of the symmetry group in deriving a new expansion basis called the symmetric pod basis the kuramoto sivashinsky equation with periodic boundary conditions is used as a paradigm to illustrate this point of view however the conclusion is general and can be applied to other equations such as the navier stokes equations the complex g

the smash product for derived categories in stable homotopy theory abstract an e 1 or a ring spectrum r has a derived category of modules d r an e 2 structure on r endows d r with a monoidal product r an e 3 structure on r endows r with a braiding if the e 3 structure extends to an e 4 structure then the braided monoidal product r is symmetric monoidal

novel modal methods for transient analysis with a reduced order model based on enhanced craig bampton formulation abstract for transient analysis of structural dynamics with reduced order model rom data recovery procedures that use modal methods such as the classical mode acceleration ma and modal displacement md methods are an important step in order to increase the convergence and accuracy of the solution in this work we propose novel ma and md methods for highly accurate transient analysis with a reduced order model based on enhanced craig bampton ecb formulation which is an extension of the classical craig bampton cb method that is a way to reduce the size of a finite element fe model the performance of the proposed data recovery approach is demonstrated with two numerical examples we also investigate the relation between the proposed and classical ma and md methods

a reduced order method for allen cahn equations in this article we present a reduced order method for modeling and computing allen cahn equations a global basis method is used in the discretized system of the allen cahn equations and proper orthogonal decomposition pod method is utilized to reduce the global basis to treat the difficulty of nonlinearity for allen cahn equations we apply discrete empirical interpolation method deim to the nonlinear term from the discretization system a reduced order method is developed by integrating pod and deim it is well known that the allen cahn equations have a nonlinear stability property i e the free energy functional decreases with respect to time the discretized allen cahn system modeled by the pod deim reduced order method can inherit the nonlinear stability of the continuous model the computation efficiency is significantly enhanced by using the reduced order method a few numerical results are presented to illustrate the performance of the reduced order method for deterministic allen cahn equations and stochastic allen cahn equations

hdg pod reduced order model of the heat equation abstract we propose a new hybridizable discontinuous galerkin hdg model order reduction technique based on proper orthogonal decomposition pod we consider the heat equation as a test problem and prove error bounds that converge to zero as the number of pod modes increases we present 2d and 3d numerical results to illustrate the convergence analysis

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

remarks on the internal exponential stabilization to a nonstationary solution for 1d burgers equations the feedback stabilization of the burgers system to a nonstationary solution using finite dimensional internal controls is considered estimates for the dimension of the controller are derived in the particular case of no constraint on the support of the control a better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed some numerical examples are presented illustrating the stabilizing effect of the feedback control and suggesting that the existence of an estimate in the general case analogous to that in the particular one is plausible

a reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for burgers equation in this article a reduced optimizing finite difference scheme fds based on singular value decomposition svd and proper orthogonal decomposition pod for burgers equation is presented also the error estimates between the usual finite difference solution and the pod solution of reduced optimizing fds are analyzed it is shown by considering the results obtained for numerical simulations of cavity flows that the error between the pod solution of reduced optimizing fds and the solution of the usual fds is consistent with theoretical results moreover it is also shown that the reduced optimizing fds is feasible and efficient

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

a ray based ipdg method for high frequency time domain acoustic wave propagation in inhomogeneous media abstract the numerical approximation of high frequency wave propagation in inhomogeneous media is a challenging problem in particular computing high frequency solutions by direct simulations requires several points per wavelength for stability and usually requires many points per wavelength for a satisfactory accuracy in this paper we propose a new method for the acoustic wave equation in inhomogeneous media in the time domain to achieve superior accuracy and stability without using a large number of unknowns the method is based on a discontinuous galerkin discretization together with carefully chosen basis functions to obtain the basis functions we use the idea from geometrical optics and construct the basis functions by using the leading order term in the asymptotic expansion also we use a wavefront tracking method and a dimension reduction procedure to obtain dominant rays in each cell we show numerically that the accuracy of the numerical solutions computed by our method is significantly higher than that computed by the ipdg method using polynomials moreover the relative errors of our method grow only moderately as the frequency increases

iterated monoidal categories abstract we develop a notion of an n fold monoidal category and show that it corresponds in a precise way to the notion of an n fold loop space specifically the group completion of the nerve of such a category is an n fold loop space and free n fold monoidal categories give rise to a finite simplicial operad of the same homotopy type as the classical little cubes operad used to parametrize the higher h space structure of an n fold loop space we also show directly that this operad has the same homotopy type as the n th smith filtration of the barratt eccles operad and the n th filtration of berger s complete graph operad moreover this operad contains an equivalent preoperad which gives rise to milgram s small model for 2 2 x when n 2 and is very closely related to milgram s model of n n x for n 2

a reduced order extrapolation technique for solution coefficient vectors in the mixed finite element method for the 2d nonlinear rosenau equation abstract this study focuses mainly on a reduced order method for the coefficient vectors in the crank nicolson mixed finite element cnmfe solutions for the two dimensional nonlinear rosenau equation we first provide a cnmfe method for the rosenau equation and give relative theoretical results as well as rewrite the cnmfe method in matrix form we then produce two sets of proper orthogonal decomposition pod bases by means of the first few coefficient vectors of the cnmfe solutions and set up a reduced order extrapolating cnmfe roecnmfe model with the pod bases as well as discuss the existence and stability as well as errors of the roecnmfe solutions by means of the matrix tool resulting in the theoretical analysis becoming very succinct lastly we present some numerical experiments to validate the correctness of the theoretical results finding that the superiority of the roecnmfe method is further verified

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

a galerkin free model reduction approach for the navier stokes equations galerkin projection of the navier stokes equations on proper orthogonal decomposition pod basis is predominantly used for model reduction in fluid dynamics the robustness for changing operating conditions numerical stability in long term transient behavior and the pressure term consideration are generally the main concerns of the galerkin reduced order models rom in this article we present a novel procedure to construct an off reference solution state by using an interpolated pod reduced basis a linear interpolation of the pod reduced basis is performed by using two reference solution states the pod basis functions are optimal in capturing the averaged flow energy the energy dominant pod modes and corresponding base flow are interpolated according to the change in operating parameter the solution state is readily built without performing the galerkin projection of the navier stokes equations on the reduced pod space modes as well as the following time integration of the resulted ordinary differential equations ode to obtain the pod time coefficients the proposed interpolation based approach is thus immune from the numerical issues associated with a standard pod galerkin rom in addition a posteriori error estimate and a stability analysis of the obtained rom solution are formulated a detailed case study of the flow past a cylinder at low reynolds numbers is considered for the demonstration of proposed method the rom results show good agreement with the high fidelity numerical flow simulation

model reduction for compressible flows using pod and galerkin projection abstract we present a framework for applying the method of proper orthogonal decomposition pod and galerkin projection to compressible fluids for incompressible flows only the kinematic variables are important and the techniques are well known in a compressible flow both the kinematic and thermodynamic variables are dynamically important and must be included in the configuration space we introduce an energy based inner product which may be used to obtain pod modes for this configuration space we then obtain an approximate version of the navier stokes equations valid for cold flows at moderate mach number and project these equations onto a pod basis the resulting equations of motion are quadratic and are much simpler than projections of the full compressible navier stokes equations

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

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

galerkin proper orthogonal decomposition methods for parabolic problems in this work error estimates for galerkin proper orthogonal decomposition pod methods for linear and certain non linear parabolic systems are proved the resulting error bounds depend on the number of pod basis functions and on the time discretization numerical examples are included

an unconditionally gradient stable numerical method for solving the allen cahn equation abstract we consider an unconditionally gradient stable scheme for solving the allen cahn equation representing a model for anti phase domain coarsening in a binary mixture the continuous problem has a decreasing total energy we show the same property for the corresponding discrete problem by using eigenvalues of the hessian matrix of the energy functional we also show the pointwise boundedness of the numerical solution for the allen cahn equation we describe various numerical experiments we performed to study properties of the allen cahn equation

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

a reduced order extrapolation algorithm based on sfve method and pod technique for non stationary stokes equations in this paper a reduced order extrapolation algorithm roea based on proper orthogonal decomposition pod technique and classical stabilized finite volume element sfve method for non stationary stokes equations is established the error estimates between the roea solutions and the classical sfve solutions and the implementation for solving roea are provided some numerical examples are used to verify that the results of numerical computation are consistent with theoretical conclusions moreover it is shown that roea based on sfve formulation and pod method is feasible and efficient for solving non stationary stokes equations

pod galerkin reduced order methods for combined navier stokes transport equations based on a hybrid fv fe solver abstract the purpose of this work is to introduce a novel pod galerkin strategy for the semi implicit hybrid high order finite volume finite element solver introduced in bermudez et al 2014 and busto et al 2018 the interest is into the incompressible navier stokes equations coupled with an additional transport equation the full order model employed in this article makes use of staggered meshes this feature will be conveyed to the reduced order model leading to the definition of reduced basis spaces in both meshes the reduced order model presented herein accounts for velocity pressure and a transport related variable the pressure term at both the full order and the reduced order level is reconstructed making use of a projection method more precisely a poisson equation for pressure is considered within the reduced order model results are verified against three dimensional manufactured test cases moreover a modified version of the classical cavity test benchmark including the transport of a species is analysed

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

nonlinear model reduction via discrete empirical interpolation a dimension reduction method called discrete empirical interpolation is proposed and shown to dramatically reduce the computational complexity of the popular proper orthogonal decomposition pod method for constructing reduced order models for time dependent and or parametrized nonlinear partial differential equations pdes in the presence of a general nonlinearity the standard pod galerkin technique reduces dimension in the sense that far fewer variables are present but the complexity of evaluating the nonlinear term remains that of the original problem the original empirical interpolation method eim is a modification of pod that reduces the complexity of evaluating the nonlinear term of the reduced model to a cost proportional to the number of reduced variables obtained by pod we propose a discrete empirical interpolation method deim a variant that is suitable for reducing the dimension of systems of ordinary differential equations odes of a certain type as presented here it is applicable to odes arising from finite difference discretization of time dependent pdes and or parametrically dependent steady state problems however the approach extends to arbitrary systems of nonlinear odes with minor modification our contribution is a greatly simplified description of the eim in a finite dimensional setting that possesses an error bound on the quality of approximation an application of deim to a finite difference discretization of the one dimensional fitzhugh nagumo equations is shown to reduce the dimension from 1024 to order 5 variables with negligible error over a long time integration that fully captures nonlinear limit cycle behavior we also demonstrate applicability in higher spatial dimensions with similar state space dimension reduction and accuracy results

balanced pod for model reduction of linear pde systems convergence theory we consider convergence analysis for a model reduction algorithm for a class of linear infinite dimensional systems the algorithm computes an approximate balanced truncation of the system using solution snapshots of specific linear infinite dimensional differential equations the algorithm is related to the proper orthogonal decomposition and it was first proposed for systems of ordinary differential equations by rowley int j bifurc chaos appl sci eng 15 3 997 1013 2005 for the convergence analysis we consider the algorithm in terms of the hankel operator of the system rather than the product of the system gramians as originally proposed by rowley for exponentially stable systems with bounded finite rank input and output operators we prove that the balanced realization can be expressed in terms of balancing modes which are related to the hankel operator the balancing modes are required to be smooth and this can cause computational difficulties for pde systems we show how this smoothness requirement can be lessened for parabolic systems and we also propose a variation of the algorithm that avoids the smoothness requirement for general systems we prove entrywise convergence of the matrices in the approximate reduced order models in both cases and present numerical results for two example pde systems

implicit explicit runge kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit we consider implicit explicit imex runge kutta r k schemes for hyperbolic systems with stiff relaxation in the so called diffusion limit in such a regime the system relaxes towards a convection diffusion equation the first objective of this paper is to show that traditional partitioned imex r k schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step the main goal of this paper is to present an approach based on imex r k schemes that in the diffusion limit relaxes to an imex r k scheme for the convection diffusion equation in which the diffusion is treated implicitly this is achieved by a novel reformulation of the problem and subsequent application of imex r k schemes to it an analysis of such schemes to the reformulated problem shows that the schemes reduce to imex r k schemes for the limit equation unde

proper orthogonal decomposition reduced order model for nonlinear aeroelastic oscillations in this study the proper orthogonal decomposition method in conjunction with a galerkin projection scheme is employed to solve the title problem of a fluttering plate in both two and three dimensions undergoing supersonic flow using von karman s large deflection plate theory and quasi steady supersonic aerodynamic theory proper orthogonal decomposition based reduced order models are constructed by using the responses as snapshots from the conventional galerkin method which also plays a role as a reference method in this paper results for the buckled limit cycle oscillation and chaotic responses of the simply supported plate are presented and compared with the galerkin solutions numerical examples demonstrate that the proper orthogonal decomposition reduced order model permits a much lower dimensional model as compared to that obtainable via the galerkin approach for example for a plate length to width ratio equal to 4 only two proper orthogonal decomposition modes are required to describe the pa

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

reduced order modelling of parameter dependent linear and nonlinear dynamic partial differential equation models in this paper we develop reduced order models for dynamic parameter dependent linear and nonlinear partial differential equations using proper orthogonal decomposition pod the main challenges are to accurately and efficiently approximate the pod bases for new parameter values and in the case of nonlinear problems to efficiently handle the nonlinear terms we use a bayesian nonlinear regression approach to learn the snapshots of the solutions and the nonlinearities for new parameter values computational efficiency is ensured by using manifold learning to perform the emulation in a low dimensional space the accuracy of the method is demonstrated on a linear and a nonlinear example with comparisons with a global basis approach

a reduced order crank nicolson finite volume element formulation based on pod method for parabolic equations in this paper a proper orthogonal decomposition pod method is used to deal with a classical crank nicolson finite volume element cnfve method for two dimensional parabolic equations a reduced order cnfve formulation with lower dimensions and sufficiently high accuracy based on pod technique is established the error estimates between reduced order cnfve solutions based on the pod method and classical cnfve solutions are provided and the extrapolation algorithm for solving reduced order cnfve formulation is implemented some numerical examples show that the results of numerical computation are consistent with previous theoretical conclusions moreover it is shown that the reduced order cnfve formulation based on pod method is feasible and efficient for solving two dimensional parabolic equations

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

an object oriented framework for reduced order models using proper orthogonal decomposition pod this work presents an object oriented design for reduced order modeling using the proper orthogonal decomposition pod technique object oriented design attributes such as data encapsulation inheritance and polymorphism are shown to produce efficient implementations that optimize code reuse and maintainability in reduced order models this paper demonstrates how existing finite element libraries can be used to efficiently carry out integration of field quantities and their gradients over complex domains for producing reduced order models that are based on the weak form galerkin method and pod furthermore numerical examples are used to show how reduced order models are computationally feasible for reducing cpu time as long as the computational expense incurred in solving the associated systems of linear equations is significantly higher than the computational expense related to the assembly of the system of equations

a reduced order dg formulation based on pod method for the time domain maxwell s equations in dispersive media abstract in this work a proper orthogonal decomposition pod method is applied to time domain maxwell s equations coupled to a drude dispersion model which are discretized in space by a discontinuous galerkin dg method an auxiliary differential equation ade method is used to represent the constitutive relation for the dispersive medium a pod dgtd formulation with lower dimension and sufficiently high accuracy is established together with the description of the pod reduced order basis its construction from a snapshot set and its application to the solution of the time domain maxwell s equations the overall goal is to reduce the computational time while maintaining an acceptable level of accuracy in order to obtain an efficient time domain solver to be used as a starting point for an optimization strategy we provide results from numerical experiments for two dimensional problems that illustrate the capabilities of the proposed pod dgtd formulation and assess its efficiency

supremizer stabilization of pod galerkin approximation of parametrized steady incompressible navier stokes equations in this work we present a stable proper orthogonal decomposition pod galerkin approximation for parametrized steady incompressible navier stokes equations with low reynolds number supremizers solutions are added to the reduced velocity space in order to obtain a stable reduced order system considering in particular the fulfillment of an inf sup condition the stability analysis is first carried out from a theoretical standpoint then confirmed by numerical tests performed on a parametrized two dimensional backward facing step flow

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

projection based reduced order models for a cut finite element method in parametrized domains abstract this work presents a reduced order modeling technique built on a high fidelity embedded mesh finite element method such methods and in particular the cutfem method are attractive in the generation of projection based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains and in general to handle topological changes the combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems avoiding remeshing as well as the reference domain formulation often used in the reduced order modeling for boundary fitted finite element formulations the resulting novel methodology is presented on linear elliptic and stokes problems together with several test cases to assess its capability the role of a proper extension and transport of embedded solutions to a common background is analyzed in detail

new pod error expressions error bounds and asymptotic results for reduced order models of parabolic pdes the derivations of existing error bounds for reduced order models of time varying partial differential equations pdes constructed using proper orthogonal decomposition pod have relied on bounding the error between the pod data and various pod projections of that data furthermore the asymptotic behavior of the model reduction error bounds depends on the asymptotic behavior of the pod data approximation error bounds we consider time varying data taking values in two different hilbert spaces h and v with v subset h and prove exact expressions for the pod data approximation errors considering four different pod projections and the two different hilbert space error norms furthermore the exact error expressions can be computed using only the pod eigenvalues and modes and we prove the errors converge to zero as the number of pod modes increases we consider the pod error estimation approaches of kunisch and volkwein siam j numer anal 40 2002 pp 492 515 and chapelle gariah an

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 numerical solutions for the energy dissipative and mass conservative allen cahn equation abstract there are three criteria when we model with phase field equations the first one is the evolution of phase separation the second one is energy dissipation law the last one is whether we have the law of conservation of mass or not if we minimize the ginzburg landau free energy functional the kinetics of phase separation inherently follows however the other properties such as energy dissipation and mass conservation are more difficult to model hitherto there have been a few proposals to conserve mass using the allen cahn ac equation but we do not have the energy dissipation law in these models when considered in light of local effects of mass correction for the reasons certain researchers still prefer to use the cahn hilliard equation e g biharmonic equation instead of using the allen cahn equation when we consider a mass conserving case this paper is concerned with numerical solutions of the ac equation in the mass conserving space where the solution is corrected at each iteration for mass conservation we introduce two different methods for solving the conservative ac equation with periodic boundary conditions in this work we have the ac equation in common but use two different mass correction operators note that both of mass correction operators satisfy the discrete energy dissipation property first we project the solution of the ac equation to the mass conserving space and then recover the linear convex splitting scheme for the conventional mass conserving equation the linear convex splitting scheme is well known but the convergence of numerical solutions is elaborately studied in terms of gradient projection next we propose the other energy minimizing method on the mass conserving space it translates the solution of the ac equation to the mass conserving space via an energy dissipation operator the dissipation of discrete energy functional for the proposed method is also established as far as the author knows no report has been presented for the numerical scheme having discrete energy dissipation property of the mass conserving ac equation without using the nonlocal and constant multiplier to verify the common properties of mass projection and proposed methods numerically we simulate the evolutions for phase separation dissipation of energy and mass conservation we moreover consider the short and long time stages of evolutions to see the difference between two methods these several tests are provided for its applications of the proposed method for modeling natural phenomena

parametric domain decomposition for accurate reduced order models applications of mp lrom methodology abstract the multivariate predictions of local reduced order models mp lrom methodology recently proposed by the authors moosavi et al 0000 uses machine learning based regression methods to predict the errors of reduced order models this study considers two applications of mp lrom first the error model is used in conjunction with a greedy sampling algorithm to generate decompositions of one dimensional parametric domains with overlapping regions such that the associated local reduced order models meet the prescribed accuracy requirements once a parametric domain decomposition is constructed any parametric configuration belongs to at least one of the partitions the local reduced order model associated with that partition approximates the full order model at the given parameters within an accuracy level that is estimated a priori the parameter domain decomposition creates a database of the available local bases local reduced order and high fidelity models and identifies the most accurate solutions for an arbitrary parametric configuration next this database is used to enhance the accuracy of the reduced order models using 1 lagrange interpolation of reduced bases in the matrix space 2 lagrange interpolation of reduced bases in the tangent space of the grassmann manifold 3 concatenation of reduced bases followed by a gram schmidt orthogonalization process and 4 lagrange interpolation of high fidelity model solutions numerical results with a viscous burgers model illustrate the potential of the mp lrom methodology to improve the design of parametric reduced order models

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

efficient model reduction of parametrized systems by matrix discrete empirical interpolation in this work we apply a matrix version of the so called discrete empirical interpolation mdeim for the efficient reduction of nonaffine parametrized systems arising from the discretization of linear partial differential equations dealing with affinely parametrized operators is crucial in order to enhance the online solution of reduced order models roms however in many cases such an affine decomposition is not readily available and must be recovered through often intrusive procedures such as the empirical interpolation method eim and its discrete variant deim in this paper we show that mdeim represents a very efficient approach to deal with complex physical and geometrical parametrizations in a non intrusive efficient and purely algebraic way we propose different strategies to combine mdeim with a state approximation resulting either from a reduced basis greedy approach or proper orthogonal decomposition a posteriori error estimates accounting for the mdeim error are also developed in the case of parametrized elliptic and parabolic equations finally the capability of mdeim to generate accurate and efficient roms is demonstrated on the solution of two computationally intensive classes of problems occurring in engineering contexts namely pde constrained shape optimization and parametrized coupled problems