Showing papers in "Multiscale Modeling & Simulation in 2003"
TL;DR: A new model for image restoration and image decomposition into cartoon and texture is proposed, based on the total variation minimization of Rudin, Osher, and Fatemi, and on oscillatory functions, which follows results of Meyer.
Abstract: In this paper, we propose a new model for image restoration and image decomposition into cartoon and texture, based on the total variation minimization of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259--268], and on oscillatory functions, which follows results of Meyer [Oscillating Patterns in Image Processing and Nonlinear Evolution Equations, Univ. Lecture Ser. 22, AMS, Providence, RI, 2002]. This paper also continues the ideas introduced by the authors in a previous work on image decomposition models into cartoon and texture [L. Vese and S. Osher, J. Sci. Comput., to appear]. Indeed, by an alternative formulation, an initial image f is decomposed here into a cartoon part u and a texture or noise part v. The u component is modeled by a function of bounded variation, while the v component is modeled by an oscillatory function, bounded in the norm dual to $|\cdot|_{H^1_0}$. After some transformation, the resulting PDE is of fourth order, envolving the Laplacian of the curvature of level lines. Fina...
580 citations
TL;DR: This paper proposes to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic process that varies on several characteristic time scales.
Abstract: In this paper we propose to use a combination of regular and singular perturbations to analyze parabolic PDEs that arise in the context of pricing options when the volatility is a stochastic proces...
224 citations
TL;DR: A theoretical analysis heterogeneous model coupling ordinary differential equations and partial differential equations providing a local-in-time existence result for the solution, based on the successive solution of the subproblems.
Abstract: The purpose of the present article is to provide a theoretical analysis heterogeneous model coupling ordinary differential equations (ODEs) and partial differential equations (PDEs), providing a local-in-time existence result for the solution. The heterogeneous problem will actually be split into subproblems. The solution of the original problem will be regarded as the solution of a suitable fixed point problem, based on the successive solution of the subproblems. Moreover, the authors study the role of matching conditions between the two submodels for the numerical simulation.
200 citations
TL;DR: This work investigates the concept of dual-weighted residuals for measuring model errors in the numerical solution of nonlinear partial differential equations and derives the method first derived in the case where the model errors are nonlinear.
Abstract: We investigate the concept of dual-weighted residuals for measuring model errors in the numerical solution of nonlinear partial differential equations. The method is first derived in the case where...
177 citations
TL;DR: A model describing rate-independent hysteretic response of shape-memory alloys under slow external forcing under natural assumptions is formulated and it is proved that this model has a solution.
Abstract: We formulate a model describing rate-independent hysteretic response of shape-memory alloys under slow external forcing. Under natural assumptions we prove that this model has a solution. The microstructure is treated on a "mesoscopic" level, described by volume fractions of particular phases in terms of Young measures. The whole formulation is based on energetic functionals for energy storage and energy dissipation. The latter is built into the model by a dissipation distance between different values of these volume fractions.
109 citations
TL;DR: A model hierarchy for queuing networks and supply chains, analogous to the hierarchy leading from the many body problem to the equations of gas dynamics, is presented.
Abstract: We present a model hierarchy for queuing networks and supply chains, analogous to the hierarchy leading from the many body problem to the equations of gas dynamics. Various possible mean field models for the interaction of individual parts in the chain are presented. For the case of linearly ordered queues the mean field models and fluid approximations are verified numerically.
107 citations
TL;DR: This work studies the residual-free bubbles (RFB) finite element method for solv- ing second order elliptic equations with rapidly varying coefficients and introduces a vari- ation of the RFB method, based on macrobubbles and referred to as the RFMB method, which gives more accurate numerical solutions.
Abstract: In this work we study the residual-free bubbles (RFB) finite element method for solv- ing second order elliptic equations with rapidly varying coefficients. The RFB technique is closely related to both the multiscale finite element method (MsFEM) introduced by Hou, Wu, and Cai (Math. Comp., 68 (1999), pp. 913-943) and the upscaling procedures which are very common in the engineering literature for solving this kind of partial differential equation. We also introduce a vari- ation of the RFB method, based on macrobubbles and referred to as the residual-free macrobubbles (RFMB) method, which gives more accurate numerical solutions. In the case of periodic coefficients we are able to prove a priori error estimates for the methods. Eventually, we test the numerical methods on model problems.
94 citations
TL;DR: Kef is given by the well-known self-consistent or effective medium argument by computing it at the dilute limit of n at infinity.
Abstract: The study aims at deriving the effective conductivity Kef of a three-dimensional heterogeneous medium whose local conductivity K(x) is a stationary and isotropic random space function of lognormal distribution and finite integral scale IY. We adopt a model of spherical inclusions of different K, of lognormal pdf, that we coin as a multi-indicator structure. The inclusions are inserted at random in an unbounded matrix of conductivity K 0 within a sphere $\Omega $, of radius R 0, and they occupy a volume fraction n. Uniform flow of flux $% U_{\infty }$ prevails at infinity. The effective conductivity is defined as the equivalent one of the sphere $\Omega ,$ under the limits $n\rightarrow 1$ and $R_{0}/I_{Y}\rightarrow \infty .$ Following a qualitative argument, we derive an exact expression of Kef by computing it at the dilute limit $% n\rightarrow 0.$ It turns out that Kef is given by the well-known self-consistent or effective medium argument. The above result is validated by accurate numerical simulation...
82 citations
TL;DR: An extremely fast graph drawing algorithm for very large graphs, which is term ACE (for Algebraic multigrid Computation of Eigenvectors), which finds an optimal drawing by minimizing a quadratic energy function.
Abstract: We present an extremely fast graph drawing algorithm for very large graphs, which we term ACE (for Algebraic multigrid Computation of Eigenvectors). ACE exhibits a vast improvement over the fastest algorithms we are currently aware of; using a serial PC, it draws graphs of millions of nodes in less than a minute. ACE finds an optimal drawing by minimizing a quadratic energy function. The minimization problem is expressed as a generalized eigenvalue problem, which is solved rapidly using a novel algebraic multigrid technique. The same generalized eigenvalue problem seems to come up also in other fields; hence ACE appears to be applicable outside graph drawing too.
76 citations
TL;DR: This work addresses the problem of actually finding these conditional expectations in the Hamiltonian case, and starts from Hald's observation that if the full system is Hamiltonian, then so is the reduced system whose right-hand sides are conditional expectations.
Abstract: Suppose one wants to approximate m components of an n-dimensional system of nonlinear differential equations (m < n) without solving the full system. In general, a smaller system of m equations has right-hand sides which depend on all of the n variables. The simplest approximation is the replacement of those right-hand sides by their conditional expectations given the values of the m variables that are kept. It is assumed that an initial probability density of all the variables is known. This construction is first-order optimal prediction. We here address the problem of actually finding these conditional expectations in the Hamiltonian case.We start from Hald's observation that if the full system is Hamiltonian, then so is the reduced system whose right-hand sides are conditional expectations. The relation between the Hamiltonians of the full system and those of the reduced system is the same as the relation between a renormalized and a bare Hamiltonian in a renormalization group (RNG) transformation. Thi...
66 citations
TL;DR: A generalized nonlinear convection-diffusion model for subgrid transport in two-dimensional systems is developed and applied and results are shown to be in consistently better agreement with reference fine scale solutions than are coarse scale results using standard subgrid treatments.
Abstract: Models for transport in heterogeneous subsurface formations usually require some type of treatment for subgrid effects. In this work, a generalized nonlinear convection-diffusion model for subgrid transport in two-dimensional systems is developed and applied. The model, although somewhat heuristic, is motivated by previous findings within both the stochastic and the deterministic frameworks. The numerical calculation of the diffusive and convective subgrid terms is described. The model is applied to several example cases involving heterogeneous permeability fields and different global boundary conditions. Both linear and nonlinear fine scale flux functions are considered. Coarse scale results for oil cut (fraction of oil in the produced fluid) and the global saturation field generated using the new subgrid model are shown to be in consistently better agreement with reference fine scale solutions than are coarse scale results using standard subgrid treatments. Extensions of the method required to treat more realistic subsurface systems are discussed.
TL;DR: The solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse equations is proposed.
Abstract: Recent developments in multiscale computation allow the solution of coarse equations for the expected macroscopic behavior of microscopically evolving particles without ever obtaining these coarse ...
TL;DR: This work considers the steady flow transport through highly heterogeneous porous media driven by extraction wells and develops a new upscaling technique which lumps the small-scale details of the medium property into a few representative macroscopic parameters on a coarse scale that preserve the large-scale behavior of theMedium and are more appropriate for simulations.
Abstract: We consider the steady flow transport through highly heterogeneous porous media driven by extraction wells. We develop a new upscaling technique which lumps the small-scale details of the medium property into a few representative macroscopic parameters on a coarse scale that preserve the large-scale behavior of the medium and are more appropriate for simulations. The method is based on the recently introduced oversampling multiscale finite element method and the introduction of new base functions that locally resolve the well singularities. The modeling error which reduces the original problem having wells into problems with Dirac sources is carefully analyzed. We also provide a detailed multiscale convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. New homogenization results for Green f...
TL;DR: Some important properties of generalized polarization tensors, such as symmetry, positivity, and their relations to the Dirichlet-to-Neumann map and weighted volume are investigated.
Abstract: Generalized polarization tensors (GPTs) are employed for the derivation of full asymptotic expansions of the perturbations in the steady-state voltage potentials that are due to the presence of dielectric inhomogeneities of small diameter. In this paper we investigate some important properties of these GPTs such as symmetry, positivity, and their relations to the Dirichlet-to-Neumann map and weighted volume. It is expected that these results will find important applications not only for developing efficient algorithms for identifying dielectric inhomogeneities of small diameter but as well for optimizing properties of dilute composite materials.
TL;DR: The mathematical arguments leading to a unique definition of planar shape elements are reviewed, which leads to a single possibility: shape elements as the normalized, affine smoothed pieces of level lines of the image.
Abstract: One of the aims of computer vision in the past 30 years has been to recognize shapes by numerical algorithms. Now, what are the geometric features on which shape recognition can be based? In this paper, we review the mathematical arguments leading to a unique definition of planar shape elements. This definition is derived from the invariance requirement to not less than five classes of perturbations, namely noise, affine distortion, contrast changes, occlusion, and background. This leads to a single possibility: shape elements as the normalized, affine smoothed pieces of level lines of the image. As a main possible application, we show the existence of a generic image comparison technique able to find all shape elements common to two images.
TL;DR: A numerical homogenization technique for nonlinear elliptic equations is constructed and a stochastic two-scale corrector is proposed where one of the scales is a numerical scale and the other is a physical scale to calculate fine scale oscillations of the solutions.
Abstract: In this paper we construct a numerical homogenization technique for nonlinear elliptic equations. In particular, we are interested in when the elliptic flux depends on the gradient of the solution in a nonlinear fashion which makes the numerical homogenization procedure nontrivial. The convergence of the numerical procedure is presented for the general case using G-convergence theory. To calculate the fine scale oscillations of the solutions we propose a stochastic two-scale corrector where one of the scales is a numerical scale and the other is a physical scale. The analysis of the convergence of two-scale correctors is performed under the assumption that the elliptic flux is strictly stationary with respect to spatial variables. The nonlinear multiscale finite element method has been proposed and analyzed.
TL;DR: This work considers solutions to the unforced incompressible Navier-Stokes equations in a $2\pi-periodic box and attempts to reconstruct the small scale by incorporating the large-scale solution as known forcing into the equations governing the evolution of the small Scale.
Abstract: We consider solutions to the unforced incompressible Navier-Stokes equations in a $2\pi$-periodic box. We split the solution into two parts representing the large-scale and small-scale motions. We define the large scale as the sum of the first kc Fourier modes in each direction and the small scale as the sum of the remaining modes. We attempt to reconstruct the small scale by incorporating the large-scale solution as known forcing into the equations governing the evolution of the small scale. We want to find the smallest value of kc for which the time evolution of the large scale sets up the dissipative structures so that the small scale is determined to a significant degree. Existing theory based on energy estimates gives a pessimistic estimate for kc that is inversely proportional to the smallest length scale of the flow. At this value of kc the energy in the small scale is exponentially small. In contrast, numerical calculations indicate that kc can often be chosen remarkably small. We attempt to expla...
TL;DR: The asymptotics of two space dimensional reaction-diffusion front speeds through mean zero space-time periodic shears are studied using both analytical and numerical methods to derive estimates based on qualitative properties such as monotonicity and a priori integral inequalities.
Abstract: We study the asymptotics of two space dimensional reaction-diffusion front speeds through mean zero space-time periodic shears using both analytical and numerical methods. The analysis hinges on traveling fronts and their estimates based on qualitative properties such as monotonicity and a priori integral inequalities. The computation uses an explicit second order upwind finite difference method to provide more quantitative information. At small shear amplitudes, front speeds are enhanced by an amount proportional to shear amplitude squared. The proportionality constant has a closed form expression. It decreases with increasing shear temporal frequency and is independent of the form of the known reaction nonlinearities. At large shear amplitudes and for all reaction nonlinearities, the enhanced speeds grow proportional to shear amplitude and are again decreasing with increasing shear temporal frequencies. The results extend previous ones in the literature on front speeds through spatially periodic shears ...
TL;DR: A rigorous analysis of a scaling limit related to the motion of an inertial particle in a Gaussian random field that leads to a white noise limit for the fluid velocity, which balances particle inertia and the friction term.
Abstract: In this paper we present a rigorous analysis of a scaling limit related to the motion of an inertial particle in a Gaussian random field. The mathematical model comprises Stokes's law for the particle motion and an infinite dimensional Ornstein-Uhlenbeck process for the fluid velocity field. The scaling limit studied leads to a white noise limit for the fluid velocity, which balances particle inertia and the friction term. Strong convergence methods are used to justify the limiting equations. The rigorously derived limiting equations are of physical interest for the concrete problem under investigation and facilitate the study of two-point motions in the white noise limit. Furthermore, the methodology developed may also prove useful in the study of various other asymptotic problems for stochastic differential equations in infinite dimensions.
TL;DR: By including a kinetic model for the structure and evolution of step edges, the island dynamics model is made mathematically well-posed and a simplified model of one-dimensional surface diffusion and kink convection is derived and found to be linearly stable.
Abstract: This work is concerned with analysis and refinement for a class of island dynamics models for epitaxial growth of crystalline thin films. An island dynamics model consists of evolution equations for step edges (or island boundaries), coupled with a diffusion equation for the adatom density, on an epitaxial surface. The island dynamics model with irreversible aggregation is confirmed to be mathematically ill-posed, with a growth rate that is approximately linear for large wavenumbers. By including a kinetic model for the structure and evolution of step edges, the island dynamics model is made mathematically well-posed. In the limit of small edge Peclet number, the edge kinetics model reduces to a set of boundary conditions, involving line tension and one-dimensional surface diffusion, for the adatom density. Finally, in the infinitely fast terrace diffusion limit, a simplified model ofone-dimensional surface diffusion and kink convection is derived and found to be linearly stable.
TL;DR: The repulsive interaction of one-dimensional Neel walls and the internal length scale of the cross-tie wall is studied to yield a specific prediction for the internallength scale of a cross- tie wall.
Abstract: Neel walls and cross-tie walls are two structures commonly seen in ferromagnetic thin films. They are interesting because their internal length scales are not determined by dimensional analysis alone. This paper studies (a) the repulsive interaction of one-dimensional Neel walls and (b) the internal length scale of the cross-tie wall. Our analysis of (a) is mathematically rigorous; it provides, roughly speaking, the first two terms of an asymptotic expansion for the energy of a pair of interacting walls. Our analysis of (b) is heuristic, since it rests on an analogy between the cross-tie wall and an ensemble of Neel walls. This analogy, combined with our results on Neel walls and a judicious choice of parameter regime, yields a specific prediction for the internal length scale of a cross-tie wall. This prediction is consistent with the experimentally observed trends.
TL;DR: A new multiple time stepping (MTS) multiscale integrator with stochasticity built in for constant temperature molecular dynamics simulations, called the targeted mollified impulse method (TM).
Abstract: Molecular dynamics (MD) is widely used in simulations of biomolecular systems such as DNA and proteins, systems which are multiscale in nature. However, current time stepping integrators are not able to address the time scale problems. Multiscale integrators, in which the presence of "fast" modes does not affect the time integration of "slow" modes, are pressingly needed in light of the fast growing biological data generated from the many genome sequencing projects. In this paper, we present a new multiple time stepping (MTS) multiscale integrator with stochasticity built in for constant temperature molecular dynamics simulations, called the targeted mollified impulse method (TM). TM combines the mollified impulse method, which is a stabler version of Verlet-I/r-RESPA ({\bf r}eversible {\bf RE}ference {\bf S}ystem {\bf P}ropagator {\bf A}lgorithm), and a self-consistent dissipative leapfrog integrator commonly used in dissipative particle dynamics. TM introduces the Langevin coupling in a targeted manner ...
TL;DR: A variational approach for filling in regions ofMissing data in two-dimensional and three-dimensional digital images, based on a joint interpolation of the image gray levels and gradient/isophotes directions, smoothly extending the isophote lines into the holes of missing data.
Abstract: In this paper we study a variational approach for filling in regions of missing data in two-dimensional and three-dimensional digital images. Applications of this technique include the restoration of old photographs and removal of superimposed text like dates, subtitles, or publicity, or the zooming of images. The approach presented here, initially introduced in [IEEE Trans. Image Process., 10 (2001), pp. 1200--1211] is based on a joint interpolation of the image gray levels and gradient/isophotes directions, smoothly extending the isophote lines into the holes of missing data. The process underlying this approach can be considered as an interpretation of the Gestaltist's principle of good continuation. We study the existence of minimizers of our functional and its approximation by minima of smoother functionals. Then we present the numerical algorithm used to minimize it and display some numerical experiments.
TL;DR: This work considers the problem of high-frequency asymptotics for the time-dependent one-dimensional Schrodinger equation with rapidly oscillating initial data and employs the semiclassical Wigner function, which is a formal asymPTotic approximation of the scaled WIGNer function but also a regularization of the limit Wigners measure.
Abstract: We consider the problem of high-frequency asymptotics for the time-dependent one-dimensional Schrodinger equation with rapidly oscillating initial data. This problem is commonly studied via the WKB method. An alternative method is based on the limit Wigner measure. This approach recovers geometrical optics, but, like the WKB method, it fails at caustics. To remedy this deficiency we employ the semiclassical Wigner function which is a formal asymptotic approximation of the scaled Wigner function but also a regularization of the limit Wigner measure. We obtain Airy-type asymptotics for the semiclassical Wigner function. This representation is shown to be exact in the context of concrete examples. In these examples we compute both the semiclassical and the limit Wigner function, as well as the amplitude of the wave field near a fold or a cusp caustic, which evolve naturally from suitable initial data.
TL;DR: This paper gives a complete analysis of time- reversal of waves emanating from a point source and propagating in a randomly layered medium and shows that random medium fluctuations actually enhance the spatial refocusing around the initial source position.
Abstract: Time-reversal refocusing for waves propagating in inhomogeneous media have re- cently been observed and studied experimentally in various contexts (ultrasound, underwater acous- tics, ... ); see, for instance, (M. Fink, Scientific American, November (1999), pp. 63-97). Important potential applications have been proposed in various fields, for instance in imaging or communica- tion. However, the full mathematical analysis, meaning both modeling of the physical problem and derivation of the time-reversal effect, is a deep and complex problem. Two cases that have been considered in depth recently correspond to one-dimensional media and the parabolic approximation regime where the backscattering is negligible. In this paper we give a complete analysis of time- reversal of waves emanating from a point source and propagating in a randomly layered medium. The wave transmitted through the random medium is recorded on a small time-reversal mirror and sent back into the medium, time-reversed. Our analysis enables us to contrast the refocusing proper- ties of a homogeneous medium and a random medium. We show that random medium fluctuations actually enhance the spatial refocusing around the initial source position. We consider a regime where the correlation length of the medium is much smaller than the pulse width, which itself is much smaller than the distance of propagation. We derive asymptotic formulas for the refocused pulse which we interpret in terms of an enhanced effective aperture. This interpretation is, in fact, comparable to the superresolution effect obtained in the other extreme regime corresponding to the parabolic approximation. However, as we discuss, the mechanism that generates the superresolution is very different in these two extreme situations.
TL;DR: Comparison to Monte Carlo simulation shows that the MDE approach gives a good approximation to total oil production, and for such spatially integrated or averaged quantities MDEs may be substantially more efficient than MCS.
Abstract: We solve statistical moment differential equations (MDEs) for immiscible flow in porous media in the limit of zero capillary pressure, with application to secondary oil recovery. Closure is achieved by Taylor expansion of the fractional flow function and a perturbation argument. Previous results in one dimension are extended to two dimensions. Comparison to Monte Carlo simulation (MCS) shows that the MDE approach gives a good approximation to total oil production. For such spatially integrated or averaged quantities MDEs may be substantially more efficient than MCS.
TL;DR: The Taylor approximation of the macroscopic filtration law written in the form of a nonlinear Darcy's law is studied and it is shown that the usual engineer's approach to approximate the permeability function is a first order approximation.
Abstract: This paper deals with filtration laws for polymericflow in porous media. General global filtration laws that are obtained by mathematical homogenization suffer from the fact that the (sufficiently ...
TL;DR: In this paper, from a two-fluid isentropic system coupled with the Poisson equation, a formal asymptotic analysis is performed, which leads to a quasi-neutral model for the plasma region and a Child--Langmuir models for the beam region.
Abstract: We consider a system consisting of a vacuum gap delimited by two electrodes. A quasi-neutral plasma (ions and electrons) is injected from the cathode and expands. Electrons are emitted from the plasma-vacuum interface to the anode forming a beam. In this paper, from a two-fluid isentropic system coupled with the Poisson equation, we perform a formal asymptotic analysis. This leads to a quasi-neutral model for the plasma region and a Child--Langmuir model for the beam region. The main point of the analysis is the connection between these two models. This is done by studying a transmission layer problem. Finally, we numerically show the accuracy of the asymptotic model, comparing it to the original two-fluid model.
TL;DR: A diffusion model for the gas density in a channel of constant and small section is derived from a kinetic description of the gas, using the methodology established in H. Babovsky, C. Bardos, and T. Platkowski.
Abstract: In this paper, we derive a convection-diffusion model for a rarefied gas flow in a channel of small section. The flow is driven by temperature gradients along the channel walls. The channel itself is made of segments of different sections. Both the channel section and the temperature profile have periodic variations along the channel axis. This situation is that of an experiment reported in, e.g., [Y. Sone and K. Sato, Phys. Fluids, 12 (2000), pp. 1864--1868]. The model is obtained through a three-step asymptotic procedure. First, a diffusion model for the gas density in a channel of constant and small section is derived from a kinetic description of the gas, using the methodology established in [H. Babovsky, C. Bardos, and T. Platkowski, Asymptot. Anal., 3 (1991), pp. 265--289]. Second, connection conditions for the gas density at the junction between two segments of different sections are obtained. Third, by using homogenization techniques, a convection-diffusion equation is deduced for the periodic cha...
TL;DR: A brief mathematical review of the time-reversal (in reflection) theory is presented in the context of the linear shallow water equations, and the numerically refocused pulse is compared with the theoretical predicted shape.
Abstract: A time-reversal mirror is, roughly speaking, a device which is capable of receiving a signal in time, keeping it in memory, and sending it back into the medium in the reversed direction of time. A brief mathematical review of the time-reversal (in reflection) theory is presented in the context of the linear shallow water equations. In particular, an explicit expression is given for the refocused pulse in the simplest time-reversal case. The explicit expression for the power spectral density of the reflection process is used to construct the highpass filter, which controls the refocusing process. Time-reversal numerical experiments in the (effectively) linear regime are used to validate the nonlinear shallow water code. The numerically refocused pulse is compared with the theoretical predicted shape. Further numerical experiments illustrate the robustness of the theory, in particular the time-reversal refocusing with smaller cutoff windows, the self-averaging property, and finally refocusing when the nonli...