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2007

Volume 68, Issue 3, pp. 599-905


A Phase Field Method for Joint Denoising, Edge Detection, and Motion Estimation in Image Sequence Processing

T. Preusser, M. Droske, C. S. Garbe, A. Telea, and M. Rumpf

SIAM J. Appl. Math. 68, pp. 599-618 (20 pages)

Online Publication Date: December 07, 2007

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The estimation of optical flow fields from image sequences is incorporated in a Mumford–Shah approach for image denoising and edge detection. Possibly noisy image sequences are considered as input and a piecewise smooth image intensity, a piecewise smooth motion field, and a joint discontinuity set are obtained as minimizers of the functional. The method simultaneously detects image edges and motion field discontinuities in a rigorous and robust way. It is able to handle information on motion that is concentrated on edges. Inherent to it is a natural multiscale approximation that is closely related to the phase field approximation for edge detection by Ambrosio and Tortorelli. We present an implementation for two-dimensional image sequences with finite elements in space and time. This leads to three linear systems of equations, which have to be solved in a suitable iterative minimization procedure. Numerical results and different applications underline the robustness of the approach presented.

Homogeneous Branched-Chain Explosions

Luis L. Bonilla, Manuel Carretero, and J. B. Keller

SIAM J. Appl. Math. 68, pp. 619-628 (10 pages)

Online Publication Date: December 07, 2007

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A model of homogeneous explosions with competing branching and recombination processes due to Kapila is analyzed by singular perturbation methods. In this model, the concentration of radicals is very low during a long induction period that ends with a rapid radical-growth stage in which all the reactants are consumed as the radicals reach their peak concentrations. The sudden jump in radical concentration is then followed by a long period of chain termination. Based on an exact relation between the fuel concentration and a slowly varying combination of fuel and radicals, we find a composite of two matched asymptotic expansions providing very good agreement with the numerical solution. This approximation is compared to another composite obtained by the method of multiple self-adjusting scales. Both approximations seem to be similarly accurate provided the induction time is calculated beyond leading order.

Spectral Theory For an Elastic Thin Plate Floating on Water of Finite Depth

Christophe Hazard and Michael H. Meylan

SIAM J. Appl. Math. 68, pp. 629-647 (19 pages) | Cited 1 time

Online Publication Date: December 12, 2007

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The spectral theory for a two-dimensional elastic plate floating on water of finite depth is developed (this reduces to a floating rigid body or a fixed body under certain limits). Two spectral theories are presented based on the first-order and second-order formulations of the problem. The first-order theory is valid only for a massless plate, while the second-order theory applies for a plate with mass. The spectral theory is based on an inner product (different for the first- and second-order formulations) in which the evolution operator is self-adjoint. This allows the time-dependent solution to be expanded in the eigenfunctions of the self-adjoint operator which are nothing more than the single frequency solutions. We present results which show that the solution is the same as those found previously when the water depth is shallow, and show the effect of increasing the water depth and the plate mass.

Hydrodynamic Limit of a Fokker–Planck Equation Describing Fiber Lay-Down Processes

L. L. Bonilla, T. Götz, A. Klar, N. Marheineke, and R. Wegener

SIAM J. Appl. Math. 68, pp. 648-665 (18 pages) | Cited 2 times

Online Publication Date: December 12, 2007

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In this paper, a stochastic model for the turbulent fiber lay-down in the industrial production of nonwoven materials is extended by including a moving conveyor belt. In the hydrodynamic limit corresponding to large noise values, the transient and stationary joint probability distributions are determined using the method of multiple scales and the Chapman–Enskog method. Moreover, exponential convergence towards the stationary solution is proven for the reduced problem. For special choices of the industrial parameters, the stochastic limit process is an Ornstein–Uhlenbeck process. It is a good approximation of the fiber motion even for moderate noise values. Moreover, as shown by Monte-Carlo simulations, the limiting process can be used to assess the quality of nonwoven materials in the industrial application by determining distributions of functionals of the process.

Inverse Bounds of Two-Component Composites

Christian Engström

SIAM J. Appl. Math. 68, pp. 666-679 (14 pages)

Online Publication Date: December 13, 2007

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A method is presented for estimating microstructural parameters from permittivity measurements of two-component composites. This structural information is described by a particular positive measure in the Stieltjes integral representation of the effective permittivity. The dependence on the geometrical structure can be reduced to the problem of calculating the moments of the measure. We present a method that uses measurement data at a set of distinct frequencies or temperatures to calculate bounds on several moments. These inverse bounds are improved when the volume fraction is known or the material is isotropic. Composites with known geometrical structure illustrate the method.

Traveling Waves in a Bioremediation Model

Shangbing Ai

SIAM J. Appl. Math. 68, pp. 680-693 (14 pages)

Online Publication Date: December 19, 2007

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We study a bioremediation model that arises in restoring ground water and soil contaminated with organic pollutants. It describes an in situ bioredimedation scenario in which a sorbing substrate of contaminated soil is degraded by indigenous microorganisms in the presence of an injected nonsorbing electron acceptor. The model relates to the coupling of the advection, dispersion, and biological reaction simultaneously for the substrate, electron acceptor, and the total biomass by two advection-reaction-diffusion equations and an ODE. We establish the existence of traveling waves for the model with wider classes of kinetic functions. Our result generalizes previous results for this model which were established only for multiplicative Monod kinetics. In addition, the proof of our result, which is based on a dynamical systems approach, is simpler.

Cucker–Smale Flocking under Hierarchical Leadership

Jackie (Jianhong) Shen

SIAM J. Appl. Math. 68, pp. 694-719 (26 pages) | Cited 5 times

Online Publication Date: December 19, 2007

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A mathematical theory on flocking serves the foundation for several ubiquitous multiagent phenomena in biology, ecology, sensor networks, and economics, as well as social behavior like language emergence and evolution. Directly inspired by the recent fundamental works of Cucker and Smale on the construction and analysis of a generic flocking model, we study the emergent behavior of Cucker–Smale flocking under hierarchical leadership. The rates of convergence towards asymptotically coherent group patterns in different scenarios are established. The consistent convergence towards coherent patterns may well reveal the advantages and necessities of having leaders and leadership in a complex (biological, technological, economic, or social) system with sufficient intelligence.

Neural Timing in Highly Convergent Systems

Colleen Mitchell and Michael Reed

SIAM J. Appl. Math. 68, pp. 720-737 (18 pages)

Online Publication Date: December 20, 2007

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In order to study how the convergence of many variable-response neurons on a single target can sharpen timing information, we investigate the limit as the number of input neurons and the number of incoming spikes required to fire the target both get large with the ratio fixed. We prove that the standard deviation of the firing time of the target cell goes to zero in this limit, and we derive the asymptotic forms of the density and the standard deviation near the limit. We use the theorems to understand the behavior of octopus cells in the mammalian cochlear nucleus.

Stability of Traffic Flow Behavior with Distributed Delays Modeling the Memory Effects of the Drivers

Rifat Sipahi, Fatihcan M. Atay, and Silviu-Iulian Niculescu

SIAM J. Appl. Math. 68, pp. 738-759 (22 pages) | Cited 5 times

Online Publication Date: December 20, 2007

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Stability analysis of a single-lane microscopic car-following model is studied analytically from the perspective of delayed reactions of human drivers. In the literature, the delayed reactions of the drivers are modeled with discrete delays, which assume that drivers make their control decisions based on the stimuli they receive from a point of time in the history. We improve this model by introducing a distribution of delays, which assumes that the control actions are based on information distributed over an interval of time in history. Such an assumption is more realistic, as it takes into consideration the memory capabilities of the drivers and the inevitable heterogeneity of their delay times. We calculate exact stability regions in the parameter space of some realistic delay distributions. Case studies are provided demonstrating the application of the results.

Shock Solutions for Particle-Laden Thin Films

Benjamin P. Cook, Andrea L. Bertozzi, and A. E. Hosoi

SIAM J. Appl. Math. 68, pp. 760-783 (24 pages) | Cited 3 times

Online Publication Date: January 04, 2008

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We derive a lubrication model describing gravity-driven thin film flow of a suspension of heavy particles in viscous fluid. The main features of this continuum model are an effective mixture viscosity and a particle settling velocity, both depending on particle concentration. The resulting equations form a $2 \times 2$ system of conservation laws in the film thickness $h(x,t)$ and in $\phi h$, where $\phi(x,t)$ is the particle volume fraction. We study flows in one dimension under the constant flux boundary condition, which corresponds to the classical Riemann problem, and we find the system can have either double-shock or singular shock solutions. We present the details of both solutions and examine the effects of the particle settling model and of the microscopic length scale $b$ at the contact line.

Identification and Characterization of a Mobile Source in a General Parabolic Differential Equation with Constant Coefficients

Steven Kusiak and John Weatherwax

SIAM J. Appl. Math. 68, pp. 784-805 (22 pages)

Online Publication Date: January 04, 2008

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We discuss an inverse source problem for a general parabolic differential equation in $\mathbb{R}^n \times \mathbb{R}_+$ with constant coefficients and a source whose strength and support may vary with time. We demonstrate that a knowledge of the solution on any bounded open set $\mathcal{M}$ in $\mathbb{R}^n$ located away from the source for any fixed time $T \geq 0$ determines the so-called carrier support (originally defined in the article “Notions of support for far fields” [J. Sylvester, Inverse Problems, 22 (2006), pp. 1273–1288] as a nontrivial subset of the support of the true source) at that coincident time. Additionally, we provide a reconstruction algorithm which can locate the time-varying position of the carrier support of the assumed unknown source with extremely few discrete (possibly nonuniform) measurements taken on such an open set over a wide range of regularity classes of the source. Finally, we provide a few numerical examples which illustrate the efficacy and robustness of this location and tracking method.

Diffeomorphic Surface Flows: A Novel Method of Surface Evolution

Sirong Zhang, Laurent Younes, John Zweck, and J. Tilak Ratnanather

SIAM J. Appl. Math. 68, pp. 806-824 (19 pages) | Cited 1 time

Online Publication Date: January 09, 2008

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We describe a new class of surface flows, diffeomorphic surface flows, induced by restricting diffeomorphic flows of the ambient Euclidean space to a surface. Different from classical surface PDE flows such as mean curvature flow, diffeomorphic surface flows are solutions of integro-differential equations in a group of diffeomorphisms. They have the potential advantage of being both topology-invariant and singularity free, which can be useful in computational anatomy and computer graphics. We first derive the Euler–Lagrange equation of the elastic energy for general diffeomorphic surface flows, which can be regarded as a smoothed version of the corresponding classical surface flows. Then we focus on diffeomorphic mean curvature flow. We prove the short-time existence and uniqueness of the flow, and study the long-time existence of the flow for surfaces of revolution. We present numerical experiments on synthetic and cortical surfaces from neuroimaging studies in schizophrenia and auditory disorders. Finally we discuss unresolved issues and potential applications.

The Nonlinear Critical Layer for Kelvin Modes on a Vortex with a Continuous Velocity Profile

S. A. Maslowe and N. Nigam

SIAM J. Appl. Math. 68, pp. 825-843 (19 pages) | Cited 1 time

Online Publication Date: January 16, 2008

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We consider in this paper the propagation of neutral modes along a vortex with velocity profile $\bar{V}(r)$, $r$ being the radial coordinate. In the linear inviscid stability theory for swirling flows, modes that are singular at some value of $r$ denoted $r_c$, the critical point, are particularly significant. The singularity can be dealt with by adding viscous and/or nonlinear effects within a thin critical layer centered on the critical point. At high Reynolds numbers, the case of most interest in applications such as aeronautics and geophysical fluid dynamics, nonlinearity is the appropriate choice, although viscosity may still play a subtle role. We determine here the scaling and equations that govern the nonlinear critical layer. The method of characteristics is then employed to obtain an exact solution of the governing inviscid system composed of four coupled PDEs, of which two are nonlinear and two are linear. Finally, assuming zero phase change across the critical layer, solutions are obtained for the outer eigenvalue problem demonstrating the existence of modes not possible in a linear theory. This result may have important implications for the short wave cooperative instability mechanism that has received so much attention in the context of aircraft trailing vortices.

Partially Reflected Diffusion

A. Singer, Z. Schuss, A. Osipov, and D. Holcman

SIAM J. Appl. Math. 68, pp. 844-868 (25 pages) | Cited 2 times

Online Publication Date: January 16, 2008

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The radiation (reactive or Robin) boundary condition for the diffusion equation is widely used in chemical and biological applications to express reactive boundaries. The underlying trajectories of the diffusing particles are believed to be partially absorbed and partially reflected at the reactive boundary; however, the relation between the reaction constant in the Robin boundary condition and the reflection probability is not well defined. In this paper we define the partially reflected process as a limit of the Markovian jump process generated by the Euler scheme for the underlying Itô dynamics with partial boundary reflection. Trajectories that cross the boundary are terminated with probability $P\sqrt{\Delta t}$ and otherwise are reflected in a normal or oblique direction. We use boundary layer analysis of the corresponding master equation to resolve the nonuniform convergence of the probability density function of the numerical scheme to the solution of the Fokker–Planck equation in a half-space, with the Robin constant $\kappa$. The boundary layer equation is of the Wiener–Hopf type. We show that the Robin boundary condition is recovered if and only if trajectories are reflected in the conormal direction $\boldsymbol{\sigma}\boldsymbol{n}$, where $\boldsymbol{\sigma}$ is the (possibly anisotropic) constant diffusion matrix and $\boldsymbol{n}$ is the unit normal to the boundary. Otherwise, the density satisfies an oblique derivative boundary condition. The constant $\kappa$ is related to $P$ by $\kappa= rP\sqrt{\sigma_n}$, where $r=1/\sqrt{\pi}$ and $\sigma_n=\boldsymbol{n}^T\boldsymbol{\sigma}\n$. The reflection law and the relation are new for diffusion in higher dimensions.

A Mathematical Model for the Steady Activation of a Skeletal Muscle

J.-P. Gabriel, L. M. Studer, D. G. Rüegg, and M.-A. Schnetzer

SIAM J. Appl. Math. 68, pp. 869-889 (21 pages)

Online Publication Date: January 25, 2008

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A skeletal muscle is composed of motor units, each consisting of a motoneuron and the muscle fibers it innervates. The input to the motor units is formed of electrical signals coming from higher motor centers and propagated to the motoneurons along a network of nerve fibers. Because of its complexity, this network still escapes actual direct observations. The present model describes the steady state activation of a muscle, i.e., of its motor units. It incorporates the network as an unknown quantity and, given the latter, predicts the input-force relation (activation curve) of the muscle. Conversely, given a suitable activation curve, our model enables the recovery of the network. This step is performed by using experimental data about the activation curve, and the whole activation process of a muscle can then be theoretically investigated. In this way, this approach provides a link between the macroscopic (activation curve) and microscopic (network) levels. From a mathematical viewpoint, solving the preceding inverse problem is equivalent to solving an integral equation of a new type.

Random Covering of Multiple One-Dimensional Domains with an Application to DNA Sequencing

Michael C. Wendl

SIAM J. Appl. Math. 68, pp. 890-905 (16 pages)

Online Publication Date: January 25, 2008

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Classical results for randomly covering a one-dimensional domain are generalized to multiple domains. The density function for the number of gaps is derived in the context of Bell's polynomials. Limiting forms are determined as well. The multiple domain configuration is a good model for DNA sequencing scenarios in which the target is fragmented, e.g., filtered DNA libraries and macronuclear genomes. Large-scale sequencing efforts are now starting to focus on such projects. Fragmentation effects are most prominent for small targets but vanish for very large targets. Here, the current model converges with classical theory. Pyrosequencing has been suggested as a viable, much cheaper alternative for large filtered projects. However, our model indicates that a recently demonstrated microscale Sanger reaction will likely be far more effective.
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