Showing papers in "Siam Journal on Applied Mathematics in 2002"
TL;DR: The broad applications of the inpainting models are demonstrated through restoring scratched old photos, disocclusion in vision analysis, text removal, digital zooming, and edge-based image coding.
Abstract: Dedicated to Stanley Osher on the occasion of his 60th birthday. Abstract. Inspired by the recent work of Bertalmio et al. on digital inpaintings (SIGGRAPH 2000), we develop general mathematical models for local inpaintings of nontexture images. On smooth regions, inpaintings are connected to the harmonic and biharmonic extensions, and inpainting orders are analyzed. For inpaintings involving the recovery of edges, we study a variational model that is closely connected to the classical total variation (TV) denoising model of Rudin, Osher, and Fatemi (Phys. D, 60 (1992), pp. 259-268). Other models are also discussed based on the Mumford-Shah regularity (Comm. Pure Appl. Math., XLII (1989), pp. 577-685) and curvature driven diffusions (CDD) of Chan and Shen (J. Visual Comm. Image Rep., 12 (2001)). The broad applications of the inpainting models are demonstrated through restoring scratched old photos, disocclusion in vision analysis, text removal, digital zooming, and edge-based image coding.
1,189 citations
TL;DR: The diffusion-limit expansion of transport equations developed earlier are used to study the limiting equation under a variety of external biases imposed on the motion and it is shown that the classical chemotaxis equation---which is called the Patlak--Keller--Segel--Alt (PKSA) equation---arises only when the bias is sufficiently small.
Abstract: In this paper, we use the diffusion-limit expansion of transport equations developed earlier [T. Hillen and H. G. Othmer, SIAM J. Appl. Math., 61 (2000), pp. 751--775] to study the limiting equation under a variety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modification of the turning rate, the movement speed, or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic diffusion, to a drift term in the flux, or to both, in the parabolic limit. We show that the classical chemotaxis equation---which we call the Patlak--Keller--Segel--Alt (PKSA) equation---arises only when the bias is sufficiently small. Using this general framework, we derive phenomenological models for chemotaxis of flagellated bacteria, of slime molds, and of myxobacteria. We also show that certain results derived earlier for one-dimensional motion can easily be generalized to two- or three-dimensional motion as well.
385 citations
TL;DR: This work proposes a theoretically exact formula for inversion of data obtained by a spiral computed tomography (CT) scan with a two-dimensional detector array that can be implemented in a truly filtered backprojection fashion.
Abstract: Proposed is a theoretically exact formula for inversion of data obtained by a spiral computed tomography (CT) scan with a two-dimensional detector array. The detector array is supposed to be of limited extent in the axial direction. The main property of the formula is that it can be implemented in a truly filtered backprojection fashion. First, one performs shift-invariant filtering of a derivative of the cone beam projections, and, second, the result is backprojected in order to form an image. Another property is that the formula solves the so-called long object problem. Limitations of the algorithm are discussed. Results of numerical experiments are presented.
373 citations
TL;DR: An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network by constraining the form of the velocity profile across the vessel radius.
Abstract: An efficient finite difference model of blood flow through the coronary vessels is developed and applied to a geometric model of the largest six generations of the coronary arterial network. By constraining the form of the velocity profile across the vessel radius, the three-dimensional Navier--Stokes equations are reduced to one-dimensional equations governing conservation of mass and momentum. These equations are coupled to a pressure-radius relationship characterizing the elasticity of the vessel wall to describe the transient blood flow through a vessel segment. The two step Lax--Wendroff finite difference method is used to numerically solve these equations. The flow through bifurcations, where three vessel segments join, is governed by the equations of conservation of mass and momentum. The solution to these simultaneous equations is calculated using the multidimensional Newton--Raphson method. Simulations of blood flow through a geometric model of the coronary network are presented demonstrating phy...
329 citations
TL;DR: It is shown rigorously that, at least in the homogeneous case, the macroscopic model can be viewed as the limit of the time discretization of the microscopic model as the number of vehicles increases, with a scaling in space and time.
Abstract: In this paper we establish a connection between a microscopic follow-the-leader model based on ordinary differential equations and a semidiscretization of a macroscopic continuum model based on a c...
281 citations
TL;DR: The effects of coupling between units in the network are analyzed, showing that if the connection strengths decay monotonically with distance, then no more than one region of high activity canpersist, whereas if they decay inanoscillatory fashion, then multiple regions can persist.
Abstract: We study a partial integro-differential equation defined on a spatially extended domain that arises from the modeling of "working" or short-term memory in a neuronal network The equation is capable of supporting spatially localized regions of high activity which can be switched "on" and "off" by transient external stimuli We analyze the effects of coupling between units in the network, showing that if the connection strengths decay monotonically with distance, then no more thanon e regionof high activity canpersist, whereas if they decay inanoscillatory fashion , then multiple regions can persist
266 citations
TL;DR: A simple single substrate limiting model of a growing biofilm layer and the nonlinear evolution of the fingering instabilities is tracked numerically using a level set method, leading to the observation of mushroom-like structures.
Abstract: A simple single substrate limiting model of a growing biofilm layer is presented. One-dimensional moving front solutions are analyzed. Under certain conditions these solutions are shown to be linearly unstable to fingering instabilities. Scaling laws for the biofilm growth rate and length scale are derived. The nonlinear evolution of the fingering instabilities is tracked numerically using a level set method, leading to the observation of mushroom-like structures.
243 citations
TL;DR: Self-focusing and singularity formation in the nonlinear Schrodinger equation (NLS) with high-order dispersion is analyzed in the isotropic mixed-dispersion NLS equations which model propagation in fiber arrays.
Abstract: We analyze self-focusing and singularity formation in the nonlinear Schrodinger equation (NLS) with high-order dispersion $i \psi_t \pm \Delta^q \psi + |\psi|^{2 \sigma} \psi = 0,$ in the isotropic mixed-dispersion NLS $i \psi_t + \Delta \psi +\epsilon \Delta^2 \psi + |\psi|^{2 \sigma} \psi = 0$, and in nonisotropic mixed-dispersion NLS equations which model propagation in fiber arrays.
235 citations
TL;DR: A two-species fractional reaction-diffusion system is introduced to model activator- inhibitor dynamics with anomalous diffusion such as occurs in spatially inhomogeneous media and suggests that Turing instabilities can exist even when the diffusion coefficient of the activator exceeds that of the inhibitor.
Abstract: We introduce a two-species fractional reaction-diffusion system to model activator- inhibitor dynamics with anomalous diffusion such as occurs in spatially inhomogeneous media. Con- ditions are derived for Turing-instability induced pattern formation in these fractional activator- inhibitor systems whereby the homogeneous steady state solution is stable in the absence of diffusion but becomes unstable over a range of wavenumbers when fractional diffusion is present. The condi- tions are applied to a variant of the Gierer-Meinhardt reaction kinetics which has been generalized to incorporate anomalous diffusion in one or both of the activator and inhibitor variables. The anoma- lous diffusion extends the range of diffusion coefficients over which Turing patterns can occur. An intriguing possibility suggested by this analysis, which can arise when the diffusion of the activator is anomalous but the diffusion of the inhibitor is regular, is that Turing instabilities can exist even when the diffusion coefficient of the activator exceeds that of the inhibitor.
195 citations
TL;DR: A mathematical model of an idealized electrostatically actuated MEMS device is constructed and analyzed for the purpose of investigating the effects of the "pull-in" or "snap-down" instability, and variations in this bifurcation diagram for various dielectric profiles are studied, yielding insight into how this technique may be used to increase the stable range of operation.
Abstract: The "pull-in" or "snap-down" instability in electrostatically actuated microelec- tromechanical systems (MEMS) presents a ubiquitous challenge in MEMS technology of great im- portance. In this instability, when applied voltages are increased beyond a critical value, there is no longer a steady-state configuration of the device where mechanical members remain separate. This severely restricts the range of stable operation of many devices. In an attempt to reduce the effects of this instability, researchers have suggested spatially tailoring the dielectric properties of MEMS devices. Here, a mathematical model of an idealized electrostatically actuated MEMS device is constructed and analyzed for the purpose of investigating this possibility. The pull-in instability is characterized in terms of the bifurcation diagram for the mathematical model. Variations in this bifurcation diagram for various dielectric profiles are studied, yielding insight into how this technique may be used to increase the stable range of operation of electrostatically actuated MEMS devices.
167 citations
TL;DR: A new computational method based on a low-frequency asymptotic analysis of Maxwell's equations for solving the inverse problem is introduced and is expected to find applications particularly in magnetoencephalography.
Abstract: Consider an inverse source problem for Maxwell's equations which arises in determining locations of epileptic foci in the living human brain. The human brain is modeled as a heterogeneous medium, where the electric permittivity, magnetic permeability, and conductivity may all be functions. A current dipole is used to model the epilepsy. The inverse source problem in this context is to determine the current dipole from boundary measurements of the fields. In this paper, a new computational method is introduced for solving the inverse problem. The method is based on a low-frequency asymptotic analysis of Maxwell's equations. Our method is constructive and has good convergence properties. A crucial step of the method is to construct special test functions. Uniqueness and stability results for the inverse problem are also established. The method and results are expected to find applications particularly in magnetoencephalography.
TL;DR: In this article, an extension of Aw and Rascle's model is presented, one which accounts for drivers attempting to travel at the maximum allowable speed, a formulation that leads to an effective computational algorithm for solving the resulting system.
Abstract: In a recent paper Aw and Rascle [SIAM J. Appl. Math., 60 (2000), pp. 916--938] introduced a new model of traffic on a uni-directional highway. Here the author studies an extension of this model, one which accounts for drivers attempting to travel at the maximum allowable speed. The author looks at a Lagrangian reformulation of this problem, a formulation that leads to an effective computational algorithm for solving the resulting system.
TL;DR: A two-strain TB model with an arbitrarily distributed delay in the latent stage of individuals infected with the drug-sensitive strain is formulated and the effects of variable periods of latency on the disease dynamics are looked at.
Abstract: Long periods of latency and the emergence of antibiotic resistance due to incomplete treatment are very important features of tuberculosis (TB) dynamics. Previous studies of two-strain TB have been performed by ODE models. In this article, we formulate a two-strain TB model with an arbitrarily distributed delay in the latent stage of individuals infected with the drug-sensitive strain and look at the effects of variable periods of latency on the disease dynamics.
TL;DR: A Petrov--Galerkin finite element method with cubic Hermite elements is developed to solve the eikonal-diffusion equation on a reasonably coarse mesh and the ratio of the Galerkin and supplementary weights is a function of the Peclet number such that the error in the solution is within a small constant factor of the optimal error achievable in the trial space.
Abstract: An efficient finite element method is developed to model the spreading of excitation in ventricular myocardium by treating the thin region of rapidly depolarizing tissue as a propagating wavefront. The model is used to investigate excitation propagation in the full canine ventricular myocardium. An eikonal-curvature equation and an eikonal-diffusion equation for excitation time are compared. A Petrov--Galerkin finite element method with cubic Hermite elements is developed to solve the eikonal-diffusion equation on a reasonably coarse mesh. The oscillatory errors seen when using the Galerkin weighted residual method with high mesh Peclet numbers are avoided by supplementing the Galerkin weights with C0 functions based on derivatives of the interpolation functions. The ratio of the Galerkin and supplementary weights is a function of the Peclet number such that, for one-dimensional propagation, the error in the solution is within a small constant factor of the optimal error achievable in the trial space. An ...
TL;DR: Axisymmetric indentation of an incompressible elastic layer by a frictionless rigid sphere is considered and the force required is inversely proportional to the cube of the layer thickness, rather than to thelayer thickness itself, if the layer is free to slip.
Abstract: Axisymmetric indentation of an incompressible elastic layer by a frictionless rigid sphere is considered. Estimates of the contact radius and the force required to produce a given indentation are given when the thickness of the layer is small compared to the contact radius. The elastic layer is either bonded to or slips along a rigid substrate. The estimates are obtained by asymptotically matching a lubrication-type expansion valid in the contact region to an edge layer expansion studied using the Wiener--Hopf technique. A substantially larger force is required to equally indent a bonded layer compared to a slipping layer. In the former case the force is inversely proportional to the cube of the layer thickness, rather than to the layer thickness itself, if the layer is free to slip.
TL;DR: In this paper, the authors present numerical solutions of a two-dimensional Riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the Mach reflection of weak shock waves.
Abstract: We present numerical solutions of a two-dimensional Riemann problem for the unsteady transonic small disturbance equations that provides an asymptotic description of the Mach reflection of weak shock waves. We develop a new numerical scheme to solve the equations in self-similar coordinates and use local grid refinement to resolve the solution in the reflection region. The solutions contain a remarkably complex structure: there is a sequence of triple points and tiny supersonic patches immediately behind the leading triple point that is formed by the reflection of weak shocks and expansion waves between the sonic line and the Mach shock. An expansion fan originates at each triple point, thus resolving the von Neumann paradox of weak shock reflection. These numerical solutions raise the question of whether there is an infinite sequence of triple points in an inviscid weak shock Mach reflection.
TL;DR: The dynamical behavior of spike-type solutions to a simplified form of the Gierer--Meinhardt activator-inhibitor model in a one-dimensional domain is studied asymptotically and numerically in the limit of small activator diffusivity.
Abstract: The dynamical behavior of spike-type solutions to a simplified form of the Gierer--Meinhardt activator-inhibitor model in a one-dimensional domain is studied asymptotically and numerically in the limit of small activator diffusivity $\varepsilon$. In the limit $\varepsilon \to 0$, a quasi-equilibrium solution for the activator concentration that has n localized peaks, or spikes, is constructed asymptotically using the method of matched asymptotic expansions. For an initial condition of this form, a differential-algebraicsystem of equations describing the evolution of the spike locations is derived. The equilibrium solutions for this system are discussed. The spikes are shown to evolve on a slow time scale $\tau=\varepsilon^2 t$ towards a stable equilibrium, provided that the inhibitor diffusivity D is below some threshold and that a certain stability criterion on the quasi-equilibrium solution is satisfied throughout the slow dynamics. If this stability condition is not satisfied initially or else is no l...
TL;DR: An asymptotic linear stability analysis of the static spike autosolitons in the Gray--Scott model of an autocatalytic chemical reaction found that in one dimension these ASs destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough.
Abstract: We performed an asymptotic linear stability analysis of the static spike autosolitons (ASs)---self-sustained solitary inhomogeneous states---in the Gray--Scott model of an autocatalytic chemical reaction. We found that in one dimension these ASs destabilize with respect to pulsations or the onset of traveling motion when the inhibitor is slow enough. In higher dimensions, the one-dimensional static spike ASs are always unstable with respect to corrugation and wriggling. The higher-dimensional radially symmetric static spike ASs may destabilize with respect to the radially nonsymmetric fluctuations leading to their splitting when the inhibitor is fast or with respect to pulsations when the inhibitor is slow.
TL;DR: It is shown using matched asymptotic methods that the consideredblow-up mechanism is stable, and small perturbations of the initial data produce a small shifting on the blow-up time and the Blow-up point but not modification in the blow -up mechanism.
Abstract: In this paper, the stability of a blow-up mechanism for a particular Keller--Segel model that yields chemotactic aggregation is considered. It is shown using matched asymptotic methods that the considered blow-up mechanism is stable. More precisely, small perturbations of the initial data produce a small shifting on the blow-up time and the blow-up point but not modification in the blow-up mechanism.
TL;DR: Casal's strain gradient elasticity with two material lengths associated with volumetric and surface energies, respectively is considered, and the question of convergence, as $\ell,{\ell}' \to 0$, is studied in detail both analytically and numerically.
Abstract: We consider Casal's strain gradient elasticity with two material lengths $\ell,{\ell}'$ associated with volumetric and surface energies, respectively. For a Mode III finite crack we formulate a hypersingular integrodifferential equation for the crack slope supplemented with the natural crack-tip conditions. The full-field solution is then expressed in terms of the crack profile and the Green function, which is obtained explicitly. For ${\ell}'=0$, we obtain a closed form solution for the crack profile. The case of small ${\ell}'$ is shown to be a regular perturbation. The question of convergence, as $\ell,{\ell}' \to 0$, is studied in detail both analytically and numerically.
TL;DR: A new morphological multiscale method in three-dimensional (3D) image processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems to smooth the level sets of a 3D image while simultaneously preserving geometric features such as edges and corners on thelevel sets.
Abstract: A new morphological multiscale method in three-dimensional (3D) image processing is presented which combines the image processing methodology based on nonlinear diffusion equations and the theory of geometric evolution problems. Its aim is to smooth the level sets of a 3D image while simultaneously preserving geometric features such as edges and corners on the level sets. This is obtained by an anisotropic curvature evolution, where time serves as the multiscale parameter. Thereby the diffusion tensor depends on a regularized shape operator of the evolving level sets. As one suitable regularization, local L2 projection onto quadratic polynomials is considered. The method is compared to a related parametric surface approach, and a geometric interpretation of the evolution and its invariance properties is given. A spatial finite element discretization on hexahedral meshes and a semi-implicit, regularized backward Euler discretization in time are the building blocks of the easy-to-code algorithm. Different a...
TL;DR: A scaling parameter is introduced and it is shown that the solution to the Becker--Doring equations converges to that of the Lifshitz--Slyozov system as the scaling parameter goes to 0.
Abstract: We investigate the connection between two classical models for the study of phase transition phenomena, the Becker--Doring equations, and the Lifshitz--Slyozov system. More precisely, we introduce a scaling parameter and show that the solution to the Becker--Doring equations converges to that of the Lifshitz--Slyozov system as the scaling parameter goes to 0.
TL;DR: This paper shows that for three problems---a transport problem in optical tomography, an elliptic equation governing near-infrared tomographic, and the wave equation in moving media---that R' is the derivative in the strict sense.
Abstract: In many inverse problems a functional of u is given by measurements, where u solves a partial differential equation of the type L(p)u+Su=q. Here q is a known source term, and L(p), S are operators, with p as an unknown parameter of the inverse problem. For the numerical reconstruction of p, the heuristically derived Frechet derivative R' of the mapping $R:p\rightarrow$ "measurementfunctional of u" is often used. We show for three problems---a transport problem in optical tomography, an elliptic equation governing near-infrared tomography, and the wave equation in moving media---that R' is the derivative in the strict sense. Our method is applicable to more general problems than are established methods for similar inverse problems.
TL;DR: In this paper, the authors considered water-drive for recovering oil from a strongly heterogeneous porous column using a two-phase model using Corey relative permeabilities and Brooks-Corey capillary pressure.
Abstract: In this paper we consider water-drive for recovering oil from a strongly heterogeneous porous column. The two-phase model uses Corey relative permeabilities and Brooks--Corey capillary pressure. The heterogeneities are perpendicular to the flow and have a periodic structure. This results in one-dimensional flow and a space periodic absolute permeability, reflecting alternating coarse and fine layers. Assuming many---or thin---layers, we use homogenization techniques to derive the effective transport equations. The form of these equations depends critically on the capillary number. The analysis is confirmed by numerical experiments.
TL;DR: An asymptotic expansion is developed which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound.
Abstract: We study the short time behavior of the early exercise boundary for American style put options in the Black--Scholes theory. We develop an asymptotic expansion which shows that the simple lower bound of Barles et al. is a more accurate approximation to the actual boundary than the more complex upper bound. Our expansion is obtained through iteration using a boundary integral equation. This integral equation is derived from the time derivative of the option value function, which closely resembles the classical Stefan free boundary value problem for melting ice. Our analytical results are supported by numerical computations designed for very short times.
TL;DR: It is shown that equilibrium states can be approximated using a two-phase model for representing the surface energy, and a precise framework is given that guarantees the existence of solutions to this variational problem.
Abstract: We study a model for shape instabilities of heteroepitaxial crystalline films. Lattice misfits between the substrate and the film induce elastic stresses in the film, which adjusts the shape of its free surface to reduce its total energy, sum of an elastic and a surface energy. We give a precise framework that guarantees the existence of solutions to this variational problem. We show that equilibrium states can be approximated using a two-phase model for representing the surface energy. Numerical results, obtained via this approximation, are presented.
TL;DR: In this paper, the authors investigate the problem of two-dimensional, unsteady expansion of an inviscid, polytropic gas, which can be interpreted as the collapse of a wedge-shaped dam containing water initially with a uniform velocity.
Abstract: We investigate the problem of two-dimensional, unsteady expansion of an inviscid, polytropic gas, which can be interpreted as the collapse of a wedge-shaped dam containing water initially with a uniform velocity. We model this problem by isentropic Euler equations. The flow is quasi-stationary, and using hodograph transform, we describe it by a partial differential equation of second order in the state space if it is irrotational initially. Furthermore, this equation is reduced to a linearly degenerate system of three partial differential equations with inhomogeneous source terms. These properties are used to prove that the flow is globally smooth when a wedge of gas expands into a vacuum, and to analyze that shocks may appear in the interaction of four planar rarefaction waves.
TL;DR: In heteroepitaxial growth, the mismatch between the lattice constants in the film and the substrate causes misfit strain in thefilm, making a flat surface unstable to small perturbations.
Abstract: In heteroepitaxial growth, the mismatch between the lattice constants in the film and the substrate causes misfit strain in the film, making a flat surface unstable to small perturbations. This mor...
TL;DR: The motion of a one-spike solution to a simplified form of the Gierer-Meinhardt activator-inhibitor model is studied in both a one and a two-dimensional domain and the pinning effect on the spike motion associated with the presence of spatially varying coefficients in the differential operator is studied.
Abstract: The motion of a one-spike solution to a simplified form of the Gierer-Meinhardt activator-inhibitor model is studied in both a one-dimensional and a two-dimensional domain. The pinning effect on the spike motion associated with the presence of spatially varying coefficients in the differential operator, referred to as precursor gradients, is studied in detail. In the one-dimensional case, we derive a differential equation for the trajectory of the spike in the limit e → 0, where e is the activator diffusivity. A similar differential equation is derived for the two-dimensional problem in the limit for which e � 1 and D � 1, where D is the inhibitor diffusivity. A numerical finite- element method is presented to track the motion of the spike for the full problem in both one and two dimensions. Finally, the numerical results for the spike motion are compared with corresponding asymptotic results for various examples.
TL;DR: It is shown that the long periods in the oscillations occur when the cell generation rate is small, and an asymptotic analysis of the model is provided, which bears a close resemblance to the analysis of relaxation oscillators, except that the slow manifold is infinite dimensional.
Abstract: We study a class of delay differential equations which have been used to model hematological stem cell regulation and dynamics. Under certain circumstances the model exhibits self-sustained oscillations, with periods which can be significantly longer than the basic cell cycle time. We show that the long periods in the oscillations occur when the cell generation rate is small, and we provide an asymptotic analysis of the model in this case. This analysis bears a close resemblance to the analysis of relaxation oscillators (such as the Van der Pol oscillator), except that in our case the slow manifold is infinite dimensional. Despite this, a fairly complete analysis of the problem is possible.