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2007

Volume 67, Issue 5, pp. 1213-1521


Multiphase Image Segmentation via Modica–Mortola Phase Transition

Yoon Mo Jung, Sung Ha Kang, and Jianhong Shen

SIAM J. Appl. Math. 67, pp. 1213-1232 (20 pages) | Cited 12 times

Online Publication Date: June 15, 2007

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We propose a novel multiphase segmentation model built upon the celebrated phase transition model of Modica and Mortola in material sciences and a properly synchronized fitting term that complements it. The proposed sine-sinc model outputs a single multiphase distribution from which each individual segment or phase can be easily extracted. Theoretical analysis is developed for the $\Gamma$-convergence behavior of the proposed model and the existence of its minimizers. Since the model is not quadratic nor convex, for computation we adopted the convex-concave procedure (CCCP) that has been developed in the literatures of both computational nonlinear PDEs and neural computation. Numerical details and experiments on both synthetic and natural images are presented.

Scattering by a Semi-Infinite Periodic Array and the Excitation of Surface Waves

C. M. Linton, R. Porter, and I. Thompson

SIAM J. Appl. Math. 67, pp. 1233-1258 (26 pages) | Cited 4 times

Online Publication Date: June 15, 2007

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The two-dimensional problem of acoustic scattering of an incident plane wave by a semi-infinite array of either rigid or soft circular scatterers is solved. Solutions to the corresponding infinite array problems are used, together with a novel filtering approach, to enable accurate solutions to be computed efficiently. Particular attention is focused on the determination of the amplitude of the Rayleigh–Bloch waves that can be excited along the array. In general, the far field away from the array consists of the sum of a finite number of plane waves propagating in different directions (the number depending on the observation angle) and a circular wave emanating from the edge of the array. In certain resonant cases (characterized by one of the scattered plane waves propagating parallel to the array), a different far field pattern occurs, involving contributions that are neither circular waves nor plane waves. Uniform asymptotic expansions that vary continuously across all of the shadow boundaries that exist are given for both cases.

On the Convergence of the Harmonic $B_z$ Algorithm in Magnetic Resonance Electrical Impedance Tomography

J. J. Liu, J. K. Seo, M. Sini, and E. J. Woo

SIAM J. Appl. Math. 67, pp. 1259-1282 (24 pages) | Cited 9 times

Online Publication Date: June 15, 2007

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Magnetic resonance electrical impedance tomography (MREIT) is a new medical imaging technique that aims to provide electrical conductivity images with sufficiently high spatial resolution and accuracy. A new MREIT image reconstruction method called the harmonic $B_z$ algorithm was proposed in 2002, and it is based on the measurement of $B_z$ that is a single component of an induced magnetic flux density $\mathbf{B}=(B_x,B_y,B_z)$ subject to an injection current. Since then, MREIT imaging techniques have made significant progress, and recent published numerical simulations and phantom experiments show that we can produce high-quality conductivity images when the conductivity contrast is not very high. Though numerical simulations can explain why we could successfully distinguish different tissues with small conductivity differences, a rigorous mathematical analysis is required to better understand the underlying physical and mathematical principle. The purpose of this paper is to provide such a mathematical analysis of those numerical simulations and experimental results. By using a uniform a priori estimate for the solution of the elliptic equation in the divergent form and an induction argument, we show that, for a relatively small contrast of the target conductivity, the iterative harmonic $B_z$ algorithm with a good initial guess is stable and exponentially convergent in the continuous norm. Both two- and three-dimensional versions of the algorithm are considered, and the difference in the convergence property of these two cases is analyzed. Some numerical results are also given to show the expected exponential convergence behavior.

Asymptotic Profiles of the Steady States for an SIS Epidemic Patch Model

L. J. S. Allen, B. M. Bolker, Y. Lou, and A. L. Nevai

SIAM J. Appl. Math. 67, pp. 1283-1309 (27 pages) | Cited 7 times

Online Publication Date: June 19, 2007

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Spatial heterogeneity, habitat connectivity, and rates of movement can have large impacts on the persistence and extinction of infectious diseases. These factors are shown to determine the asymptotic profile of the steady states in a frequency-dependent SIS (susceptible-infected-susceptible) epidemic model with $n$ patches in which susceptible and infected individuals can both move between patches. Patch differences in local disease transmission and recovery rates characterize whether patches are low-risk or high-risk, and these differences collectively determine whether the spatial domain, or habitat, is low-risk or high-risk. The basic reproduction number ${\cal R}_0$ for the model is determined. It is then shown that when the disease-free equilibrium is stable (${\cal R}_0 < 1$) it is globally asymptotically stable, and that when the disease-free equilibrium is unstable (${\cal R}_0 > 1$) there exists a unique endemic equilibrium. Two main theorems link spatial heterogeneity, habitat connectivity, and rates of movement to disease persistence and extinction. The first theorem relates the basic reproduction number to the heterogeneity of the spatial domain. For low-risk domains, the disease-free equilibrium is stable (${\cal R}_0 < 1$) if and only if the mobility of infected individuals lies above a threshold value, but for high-risk domains, the disease-free equilibrium is always unstable (${\cal R}_0 > 1$). The second theorem states that when the endemic equilibrium exists, it tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. This limiting disease-free equilibrium has a positive number of susceptible individuals on all low-risk patches and can also have a positive number of susceptible individuals on some, but not all, high-risk patches. Sufficient conditions for whether high-risk patches in the limiting disease-free equilibrium have susceptible individuals or not are given in terms of habitat connectivity, and these conditions are illustrated using numerical examples. These results have important implications for disease control.

Nonlinear Dynamics of Electrified Thin Liquid Films

Dmitri Tseluiko and Demetrios T. Papageorgiou

SIAM J. Appl. Math. 67, pp. 1310-1329 (20 pages) | Cited 9 times

Online Publication Date: June 21, 2007

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We study a nonlinear nonlocal evolution equation describing the hydrodynamics of thin films in the presence of normal electric fields. The liquid film is assumed to be perfectly conducting and to completely wet the upper or lower surface of a horizontal flat plate. The flat plate is held at constant voltage, and a vertical electric field is generated by a second parallel electrode kept at a different constant voltage and placed at a large vertical distance from the bottom plate. The fluid is viscous, and gravity and surface tension act. The equation is derived using lubrication theory and contains an additional nonlinear nonlocal term representing the electric field. The electric field is linearly destabilizing and is particularly important in producing nontrivial dynamics in the case when the film rests on the upper side of the plate. We give rigorous results on the global boundedness of positive periodic smooth solutions, using an appropriate energy functional. We also implement a fully implicit numerical scheme and perform extensive numerical experiments. Through a combination of analysis and numerical experiments we present evidence for the global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena, which are also observed in hanging films when electric fields are absent.

Elastic Scatterer Reconstruction via the Adjoint Sampling Method

S. Nintcheu Fata and B. B. Guzina

SIAM J. Appl. Math. 67, pp. 1330-1352 (23 pages) | Cited 8 times

Online Publication Date: July 11, 2007

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An inverse problem dealing with the reconstruction of voids in a uniform semi-infinite solid from near-field elastodynamic waveforms is investigated via the linear sampling method. To cater to active imaging applications that are characterized by a limited density of illuminating sources, existing formulation of the linear sampling method is advanced in terms of its adjoint statement that features integration over the receiver surface rather than its source counterpart. To deal with an ill-posedness of the integral equation that is used to reconstruct the obstacle, the problem is solved by alternative means of Tikhonov regularization and a preconditioned conjugate gradient method. Through a set of numerical examples, it is shown (i) that the adjoint statement elevates the performance of the linear sampling method when dealing with scarce illuminating sources, and (ii) that a combined use of the existing formulation together with its adjoint counterpart represents an effective tool for exposing an undersampling of the experimental input, e.g., in terms of the density of source points used to illuminate the obstacle.

Inverse Source Problem in Nonhomogeneous Background Media

Anthony J. Devaney, Edwin A. Marengo, and Mei Li

SIAM J. Appl. Math. 67, pp. 1353-1378 (26 pages) | Cited 7 times

Online Publication Date: July 11, 2007

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The scalar wave inverse source problem (ISP) of determining an unknown radiating source from knowledge of the field it generates outside its region of localization is investigated for the case in which the source is embedded in a nonhomogeneous medium with known index of refraction profile $n(\mathbf{r})$. It is shown that the solution to the ISP having minimum energy (the so-called minimum energy source) can be obtained via a simple method of constrained optimization. This method is applied to the special case when the nonhomogeneous background is spherically symmetric ($n(\mathbf{r})=n(r)$), and it yields the minimum energy source in terms of a series of spherical harmonics and radial wave functions that are solutions to a Sturm–Liouville problem. The special case of a source embedded in a spherical region of constant index is treated in detail, and results from computer simulations are presented for this case.

Global Dynamics of a Predator-Prey Model with Stage Structure for the Predator

Paul Georgescu and Ying-Hen Hsieh

SIAM J. Appl. Math. 67, pp. 1379-1395 (17 pages) | Cited 4 times

Online Publication Date: July 18, 2007

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The global properties of a predator-prey model with nonlinear functional response and stage structure for the predator are studied using Lyapunov functions and LaSalle's invariance principle. It is found that, under hypotheses which ensure the uniform persistence of the system and the existence of a unique positive steady state, a feasible a priori lower bound condition on the abundance of the prey population ensures the global asymptotic stability of the positive steady state. A condition which leads to the extinction of the predators is indicated. We also obtain results on the existence and stability of periodic solutions. In particular, when (4.2) fails to hold and the unique positive steady state $E^*$ becomes unstable, the coexistence of prey and predator populations is ensured for initial populations not on the one-dimensional stable manifold of $E^*$, albeit with fluctuating population sizes.

Spatiotemporal Symmetries in the Disynaptic Canal-Neck Projection

Martin Golubitsky, LieJune Shiau, and Ian Stewart

SIAM J. Appl. Math. 67, pp. 1396-1417 (22 pages) | Cited 2 times

Online Publication Date: July 20, 2007

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The vestibular system in almost all vertebrates, and in particular in humans, controls balance by employing a set of six semicircular canals, three in each inner ear, to detect angular accelerations of the head in three mutually orthogonal coordinate planes. Signals from the canals are transmitted to eight (groups of) neck motoneurons, which activate the eight corresponding muscle groups. These signals may be either excitatory or inhibitory, depending on the direction of head acceleration. McCollum and Boyle have observed that in the cat the relevant network of neurons possesses octahedral symmetry, a structure that they deduce from the known innervation patterns (connections) from canals to muscles. We rederive the octahedral symmetry from mathematical features of the probable network architecture, and model the movement of the head in response to the activation patterns of the muscles concerned. We assume that connections between neck muscles can be modeled by a “coupled cell network,” a system of coupled ODEs whose variables correspond to the eight muscles, and that this network also has octahedral symmetry. The network and its symmetries imply that these ODEs must be equivariant under a suitable action of the octahedral group. It is observed that muscle motoneurons form natural “push-pull pairs" in which, for given movements of the head, one neuron produces an excitatory signal, whereas the other produces an inhibitory signal. By incorporating this feature into the mathematics in a natural way, we are led to a model in which the octahedral group acts by signed permutations on muscle motoneurons. We show that with the appropriate group actions, there are six possible spatiotemporal patterns of time-periodic states that can arise by Hopf bifurcation from an equilibrium representing an immobile head. Here we use results of Ashwin and Podvigina. Counting conjugate states, whose physiological interpretations can have significantly different features, there are 15 patterns of periodic oscillation, not counting left-right reflections or time-reversals as being different. We interpret these patterns as motions of the head, and note that all six types of pattern appear to correspond to natural head motions.

A Method to Compute Statistics of Large, Noise-Induced Perturbations of Nonlinear Schrödinger Solitons

R. O. Moore, G. Biondini, and W. L. Kath

SIAM J. Appl. Math. 67, pp. 1418-1439 (22 pages) | Cited 8 times

Online Publication Date: July 20, 2007

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We demonstrate in detail the application of importance sampling to the numerical simulation of large noise-induced perturbations in soliton-based optical transmission systems governed by the nonlinear Schrödinger equation. The method allows one to concentrate numerical Monte Carlo simulations around the noise realizations that are most likely to produce the large pulse deformations connected with errors, and yields computational speedups of several orders of magnitude over standard Monte Carlo simulations. We demonstrate the method by using it to calculate the probability density functions associated with pulse amplitude, frequency, and timing fluctuations in a prototypical soliton-based communication system.

The Inverse Conductivity Problem with an Imperfectly Known Boundary in Three Dimensions

Ville Kolehmainen, Matti Lassas, and Petri Ola

SIAM J. Appl. Math. 67, pp. 1440-1452 (13 pages) | Cited 5 times

Online Publication Date: July 20, 2007

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We consider the inverse conductivity problem in a strictly convex domain whose boundary is not known. Usually the numerical reconstruction from the measured current and voltage data is done assuming that the domain has a known fixed geometry. However, in practical applications the geometry of the domain is usually not known. This introduces an error, and effectively changes the problem into an anisotropic one. The main result of this paper is a uniqueness result characterizing the isotropic conductivities on convex domains in terms of measurements done on a different domain, which we call the model domain, up to an affine isometry. As data for the inverse problem, we assume the Robin-to-Neumann map and the contact impedance function on the boundary of the model domain to be given. Also, we present a minimization algorithm based on the use of Cotton–York tensor, which finds the push forward of the isotropic conductivity to our model domain and also finds the boundary of the original domain up to an affine isometry. This algorithm works also in dimensions higher than three, but then the Cotton–York tensor has to replaced with the Weyl tensor.

Heteroclinic Bifurcation in the Michaelis–Menten-Type Ratio-Dependent Predator-Prey System

Bingtuan Li and Yang Kuang

SIAM J. Appl. Math. 67, pp. 1453-1464 (12 pages) | Cited 6 times

Online Publication Date: July 20, 2007

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The existence of a heteroclinic bifurcation for the Michaelis–Menten-type ratio-dependent predator-prey system is rigorously established. Limit cycles related to the heteroclinic bifurcation are also discussed. It is shown that the heteroclinic bifurcation is characterized by the collision of a stable limit cycle with the origin, and the bifurcation triggers a catastrophic shift from the state of large oscillations of predator and prey populations to the state of extinction of both populations. It is also shown that the limit cycles related to the heteroclinic bifurcation originally bifurcate from the Hopf bifurcation.

Lift on Slender Bodies with Elliptical Cross section Evaluated by Using an Oseen Flow Model

Edmund Chadwick and Nina Fishwick

SIAM J. Appl. Math. 67, pp. 1465-1478 (14 pages) | Cited 4 times

Online Publication Date: August 22, 2007

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Consider uniform, incompressible flow past a slender body with an elliptical cross section such that the major axis of the body is inclined slightly to the flow direction. Assume that the flow is inviscid everywhere except in a thin boundary layer region and in the vortex core of trailing line vortices that emanate from the body into the vortex wake. Hence, the flow is quasi-inviscid, and so the slip (impermeability) boundary condition is applied. Further assume that outside the boundary layer the velocity is to first order the uniform stream velocity. Then the Oseen approximation can be applied. The resulting solution, up to the slender body approximation, is given, and the lift over the slender body is determined. This solution is then compared with the theoretical and experimental results for flow past a delta wing, the viscous cross-flow method and experimental results for flow past a body with a circular cross section, and Newtonian impact theory and experimental results for flow past a body with an elliptical cross section.

Stationary Pattern of a Ratio-Dependent Food Chain Model with Diffusion

Rui Peng, Junping Shi, and Mingxin Wang

SIAM J. Appl. Math. 67, pp. 1479-1503 (25 pages) | Cited 12 times

Online Publication Date: August 24, 2007

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In the paper, we investigate a three-species food chain model with diffusion and ratio-dependent predation functional response. We mainly focus on the coexistence of the three species. For this coupled reaction-diffusion system, we study the persistent property of the solution, the stability of the constant positive steady state solution, and the existence and nonexistence of nonconstant positive steady state solutions. Both the general stationary pattern and Turing pattern are observed as a result of diffusion. Our results also exhibit some interesting effects of diffusion and functional responses on pattern formation. (An erratum to this article has been appended at the end of the pdf file.)

A Simple Illustration of a Weak Spectral Cascade

David J. Muraki

SIAM J. Appl. Math. 67, pp. 1504-1521 (18 pages) | Cited 1 time

Online Publication Date: August 24, 2007

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The textbook first encounter with nonlinearity in a partial differential equation (PDE) is the first-order wave equation: $u_t + u u_x = 0$. Often referred to as the inviscid Burgers equation, this equation is familiar to many in the theoretical contexts of characteristics, wavebreaking, or shock propagation. Another canonical behavior contained within this simplest of PDEs is the spectral cascade. Surprisingly, buried in a little-known 1964 article by G.W. Platzman is an elegant example of an exact Fourier series solution associated with a purely sinusoidal initial condition. This Fourier representation, valid prior to wavebreaking, is generalized to arbitrary continuous initial conditions on both the periodic and infinite domains. For the specific example of Platzman's original problem, the Fourier coefficients decay exponentially with increasing wavenumber, and the decay rate flattens to zero precisely at the time of wavebreaking. It is demonstrated that two simplified descriptions, a downscale truncation and a linearization from initial conditions, also produce an exponential spectral cascade uniformly to large wavenumbers. This weak cascade is responsible for the initial generation of Fourier harmonics in the viscous Burgers equation.
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