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1998

Volume 58, Issue 6, pp. 1689-2028


Regularization of Ill-Posed Problems Via the Level Set Approach

Eduard Harabetian and Stanley Osher

SIAM J. Appl. Math. 58, pp. 1689-1706 (18 pages) | Cited 10 times

Online Publication Date: July 26, 2006

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We introduce a new formulation for the motion of curves in R2 (easily extendable to the motion of surfaces in R3), when the original motion generally corresponds to an ill-posed problem such as the Cauchy--Riemann equations. This is, in part, a generalization of our earlier work in [6], where we applied similar ideas to compute flows with highly concentrated vorticity, such as vortex sheets or dipoles, for incompressible Euler equations. Our new formulation involves extending the level set method of [12] to problems in which the normal velocity is not intrinsic. We obtain a coupled system of two equations, one of which is a level surface equation. This yields a fixed-grid, Eulerian method which regularizes the ill-posed problem in a topological fashion. We also present an analysis of curvature regularizations and some other theoretical justification. Finally, we present numerical results showing the stability properties of our approach and the novel nature ofthe regularization, including the development of bubbles for curves evolving under Cauchy--Riemann flow.

Phase Segregation Dynamics in Particle Systems with Long Range Interactions II: Interface Motion

Giambattista Giacomin and Joel L. Lebowitz

SIAM J. Appl. Math. 58, pp. 1707-1729 (23 pages) | Cited 21 times

Online Publication Date: July 26, 2006

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We study properties of the solutions of a family of second-order integrodifferential equations, which describe the large scale dynamics of a class of microscopic phase segregation models with particle conserving dynamics. We first establish existence and uniqueness as well as some properties of the instantonic solutions. Then we concentrate on formal asymptotic (sharp interface) limits. We argue that the obtained interface evolution laws (a Stefan-like problem and the Mullins--Sekerka solidification model) coincide with the ones which can be obtained in the analogous limits from the Cahn--Hilliard equation, the fourth-order PDE which is the standard macroscopic model for phase segregation with one conservation law.

Generation of Localization in a Discrete Chain with Periodic Boundary Conditions: Numerical and Analytical Results

Alexander F. Vakakis and Gary Salenger

SIAM J. Appl. Math. 58, pp. 1730-1747 (18 pages)

Online Publication Date: July 26, 2006

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We study the generation of localization in a discrete chain composed of N subsystems and periodic boundary conditions. Strongly localized motions are studied; that is, time-periodic motions where nearly all of the energy is spatially confined to a single subsystem. For varying N, numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in-phase spatially extended solution. However, in the limit as $N \rightarrow \infty$, the bifurcation point tends to infinity, and a smooth transition from localization to nonlocalization occurs. We then present an analytic technique to complement the numerical results. It is based on the matching local asymptotic expansions of a solution branch using Padé approximants. This leads to global analytic representations of the considered solutions, valid over the entire range of the control parameter.

Midgap Defect Modes in Dielectric and Acoustic Media

Abel Klein and Alexander Figotin

SIAM J. Appl. Math. 58, pp. 1748-1773 (26 pages) | Cited 14 times

Online Publication Date: July 26, 2006

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We consider three dimensional lossless periodic dielectric (photonic crystals) and acoustic media having a gap in the spectrum. If such a periodic medium is perturbed by a strong enough defect, defect eigenmodes arise, localized exponentially around the defect, with the corresponding eigenvalues in the gap. We use a modified Birman-Schwinger method to derive equations for these eigenmodes and corresponding eigenvalues in the gap, in terms of the spectral attributes of an auxiliary Hilbert-Schmidt operator. We prove that in three dimensions, under some natural conditions on the periodic background, the number of eigenvalues generated in a gap of the periodic operator is finite, and give an estimate on the number of these midgap eigenvalues. In particular, we show that if the defect is weak there are no midgap eigenvalues.

A Uniqueness Result for Scattering by Infinite Rough Surfaces

Bo Zhang and Simon N. Chandler-Wilde

SIAM J. Appl. Math. 58, pp. 1774-1790 (17 pages) | Cited 19 times

Online Publication Date: July 26, 2006

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Consider the Dirichlet boundary value problem for the Helmholtz equation in a non-locally perturbed half-plane with an unbounded, piecewise Lyapunov boundary. This problem models time-harmonic electromagnetic scattering in transverse magnetic polarization by one-dimensional rough, perfectly conducting surfaces. A radiation condition is introduced for the problem, which is a generalization of the usual one used in the study of diffraction by gratings when the solution is quasi-periodic, and allows a variety of incident fields including an incident plane wave to be included in the results obtained. We show in this paper that the boundary value problem for the scattered field has at most one solution. For the case when the whole boundary is Lyapunov and is a small perturbation of a flat boundary we also prove existence of solution and show a limiting absorption principle.

A Simple Model for Stress Fluctuations in Plasticity with Application to Granular Materials

David G. Schaeffer and Michael Shearer

SIAM J. Appl. Math. 58, pp. 1791-1807 (17 pages)

Online Publication Date: July 26, 2006

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When granular material is modeled as a continuum, plastic constitutive behavior is often assumed. The use of plasticity amounts to replacing a complicated micromechanical system by its average behavior. Recent experiments have shown that, at least for small-scale systems, stress fluctuations may be of the same order, or even much larger, than average stresses.
In this paper a first generation of discrete models for stress fluctuations is discussed. These models consist of many spring-slider elements in parallel. The sliders all obey the same law for frictional resistance, and this resistance varies with the position, but not the velocity, of the slider. The initial positions (and hence the initial frictional resistances) of the sliders are taken to be random. The usual elastoplastic response emerges as the ensemble average over all possible initial positions of the sliders. The stress response resulting from any particular choice of initial conditions exhibits fluctuations similar to those in the experiments. It is shown that the magnitude of fluctuations is governed by two parameters, namely, the system size and the roughness, the latter defined as the ratio of particle contact length to particle size. In numerical simulations, it is observed that the roughness parameter controls the shape of the stress response as a function of applied strain.

The Motion of Superconducting Vortices in Thin Films of Varying Thickness

D. R. Heron and S. J. Chapman

SIAM J. Appl. Math. 58, pp. 1808-1825 (18 pages) | Cited 5 times

Online Publication Date: July 26, 2006

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The interaction of superconducting vortices with superconductor/vacuum interfaces is considered. A vortex is first shown to intersect such an interface normally. Various thin-film models are then formulated, corresponding to different parameter regimes. A local analysis of a vortex is performed, and a law of motion for each vortex deduced. This law of motion implies that the vortex will move to the locally thinnest part of the film, and is consistent with the vortex moving under the curvature induced by being forced to intersect the boundaries of the film normally.

Solidification Fronts and Solute Trapping in a Binary Alloy

Chaim Charach and Paul C. Fife

SIAM J. Appl. Math. 58, pp. 1826-1851 (26 pages) | Cited 1 time

Online Publication Date: July 26, 2006

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A phase-field model with order parameter, concentration, and temperature as field variables is used to study the properties of solidification fronts in a binary alloy. As in previous papers, the model includes dependence of the free energy density not only on these field variables but also on the gradients of the order parameter and concentration. Terms with these gradients represent surface free energy associated with the phase interface and with the jump in concentration. We treat them as conceptually and physically different; in particular, the thicknesses of the two interfaces will generally be different. Based on the smallness of the coefficients of these gradient terms, and the largeness of the ratio of solute diffusivity in the liquid to that of the solid, asymptotic analyses in various parameter regimes are performed which reveal information on such things as the dependence of the discontinuity of concentration at the front on its velocity and on the above-mentioned parameters. More broadly, we investigate the spatial structure of the concentration jump interface in various parameter ranges. Formulations are given to the problem of directional solidification and to free boundary problems for the free motion of a solidification front. Corrections due to curvature of the interface are found.

On the Existence of Nontrivial Solutions for a Scalar "Light-Nematic" System and its Frederiks transition threshold

Xiao-Ping Wang

SIAM J. Appl. Math. 58, pp. 1852-1861 (10 pages)

Online Publication Date: July 26, 2006

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We study the existence of nontrivial solutions for a paraxial model for optical self-focusing in a nematic liquid crystal. We prove that there exists a threshold input intensity called Frederiks transition, above which a nontrivial solution exists.

Asymptotic Theory of Large Deviations for Markov Chains

G. Lerman and Z. Schuss

SIAM J. Appl. Math. 58, pp. 1862-1877 (16 pages) | Cited 2 times

Online Publication Date: July 26, 2006

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A formal asymptotic expansion is constructed for the joint probability density function (pdf) of a stationary ergodic Markov chain, Xn, and its averages, Yn. Since the pair (Xn,Yn)$is Markovian, the joint pdf satisfies a forward Kolmogorov equation whose solution is expanded asymptotically for large n. An algorithm is proposed for the calculation of the full asymptotic series, but only the three leading terms are found explicitly. It is found that for small values of the average, the asymptotic expansion coincides with the appropriate version of the central limit theorem. The ideas and methods are generalized to a large class of averages and to vector valued Markov chains.

Global Attractivity in Delayed Hopfield Neural Network Models

P. van den Driessche and Xingfu Zou

SIAM J. Appl. Math. 58, pp. 1878-1890 (13 pages) | Cited 148 times

Online Publication Date: July 26, 2006

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Two different approaches are employed to investigate the global attractivity of delayed Hopfield neural network models. Without assuming the monotonicity and differentiability of the activation functions, Liapunov functionals and functions (combined with the Razumikhin technique) are constructed and employed to establish sufficient conditions for global asymptotic stability independent of the delays. In the case of monotone and smooth activation functions, the theory of monotone dynamical systems is applied to obtain criteria for global attractivity of the delayed model. Such criteria depend on the magnitude of delays and show that self-inhibitory connections can contribute to the global convergence.

Regularization for Curve Representations: Uniform Convergence for Discontinuous Solutions of Ill-Posed Problems

Andreas Neubauer and Otmar Scherzer

SIAM J. Appl. Math. 58, pp. 1891-1900 (10 pages) | Cited 12 times

Online Publication Date: July 26, 2006

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This paper is concerned with a new approach for regularizing problems with discontinuous solutions: regularization for curve representations. The idea of this approach is to represent a (discontinous) function as a curve with parameterization (a(t),b(t)). A combination with nonlinear Tikhonov regularization then yields uniform convergence of the regularized solutions. The method is applied to deblurring and denoising problems in signal processing. A numerical example for a deblurring problem is presented.

Modulation Equations for Spatially Periodic Systems: Derivation and Solutions

A. Doelman. and R. Schielen

SIAM J. Appl. Math. 58, pp. 1901-1930 (30 pages)

Online Publication Date: July 26, 2006

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We study a class of partial differential equations in one spatial dimension, which can be seen as model equations for the analysis of pattern formation in physical systems defined on unbounded, weakly oscillating domains. We perform a linear and weakly nonlinear stability analysis for solutions that bifurcate from a basic state. The analysis depends strongly on the wavenumber p of the periodic boundary. For specific values of p, which are called resonant, some unexpected phenomena are encountered. The neutral stability curve which can be derived for the unperturbed, straight problem splits in the neighborhood of the minimum into two, which indicates that there are two amplitudes involved in the bifurcating solutions, each one related to one of the minima. The character of the modulation equation, which describes the nonlinear evolution of perturbations of the basic state, depends crucially on the distance of the bifurcation parameter from the lowest, most critical minimum. In a relatively large part of the parameter space, we derive a coupled system of amplitude equations. This can either be reduced to an equation for a real amplitude with cubic and quadratic terms or it can be written as a Ginzburg--Landau equation for a complex amplitude A, with an additional term, proportional to $\ol{A}$. For this latter equation, we study the existence and stability of periodic solutions. We find that the nonsymmetric term $\ol{A}$ decreases the width of the Eckhaus band of stable solutions. Numerical simulations show that complex periodic solutions bifurcate into stable, real solutions for increasing influence of the $\ol{A}$-term.

Acoustic Scattering by an Inhomogeneous Layer on a Rigid Plate

Bo Zhang and Simon N. Chandler-Wilde

SIAM J. Appl. Math. 58, pp. 1931-1950 (20 pages) | Cited 13 times

Online Publication Date: July 26, 2006

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The problem of scattering of time-harmonic acoustic waves by an inhomogeneous fluid layer on a rigid plate in R2 is considered. The density is assumed to be unity in the media: within the layer the sound speed is assumed to be an arbitrary bounded measurable function. The problem is modelled by the reduced wave equation with variable wavenumber in the layer and a Neumann condition on the plate. To formulate the problem and prove uniqueness of solution a radiation condition appropriate for scattering by infinite rough surfaces is introduced, a generalization of the Rayleigh expansion condition for diffraction gratings. With the help of the radiation condition the problem is reformulated as a system of two second kind integral equations over the layer and the plate. Under additional assumptions on the wavenumber in the layer, uniqueness of solution is proved and the nonexistence of guided wave solutions of the homogeneous problem established. General results on the solvability of systems of integral equations on unbounded domains are used to establish existence and continuous dependence in a weighted norm of the solution on the given data.

Minimax Approximation of Optical Profiles

C. A. Hall, T. A. Porsching, and T. L. Bennett

SIAM J. Appl. Math. 58, pp. 1951-1968 (18 pages)

Online Publication Date: July 26, 2006

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The notion of a profile is important in the theory surrounding the finishing of axisymmetric optical surfaces. Mathematically, a profile is a member of the quotient space $C(M)/K$, where $C(M)$ denotes the space of continuous functions defined on a compact subset $M\subset R$, and $K$ is the subspace of constant functions. In this paper we investigate the minimax approximation of a given profile $[f]\in C(M)/K$ by elements of a closed convex cone in $C(M)/K$. We establish the existence of a minimax approximation (uniqueness does not in general hold) and prove two characterization theorems for any such best approximation. One of these theorems is then used as a basis for a ``bisection' algorithm to compute a best approximation corresponding to a particular type of finishing process known as recursive operator controlled finishing.

Asymptotic Transmission of Solitons through Random Media

Josselin Garnier

SIAM J. Appl. Math. 58, pp. 1969-1995 (27 pages) | Cited 13 times

Online Publication Date: July 26, 2006

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This paper contains a study of the transmission of a soliton through a slab of nonlinear and random media. A random nonlinear Schrödinger equation is considered, where the randomness holds in the potential and the nonlinear coefficient. Using the inverse scattering transform, we exhibit several asymptotic behaviors corresponding to the limit when the amplitudes of the random fluctuations go to zero and the size of the slab goes to infinity. The mass of the transmitted soliton may tend to zero exponentially (as a function of the size of the slab) or following a power law, or else the soliton may keep its mass, while its velocity decreases at a logarithmic rate or even more slowly. Numerical simulations are in good agreement with the theoretical results.

Antiplane Shear Flows in Viscoplastic Solids Exhibiting Isotropic and Kinematic Hardening

D. R. Owen and J. M. Greenberg

SIAM J. Appl. Math. 58, pp. 1996-2023 (28 pages)

Online Publication Date: July 26, 2006

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The authors consider antiplane shearing motions of an incompressible viscoplastic solid. The particular constitutive equation employed assumes that the stress tensor has an "elastic" component and a component which can exhibit hysteresis. The model exhibits both "kinematic" and "isotropic" hardening. Our results consist of a set of energy type estimates for the resulting system, L2 contractivity estimates for the solution operator, and finally an analysis of the approach of our system to a "rate independent" model as a distinguished parameter describing our flow rule approaches zero. We also include some computational results for simple piecewise constant data.

Corrigendum: A Finite Element/Spectral Method for Approximating the Time-Harmonic Maxwell System in $\real^3$

Andreas Kirsch and Peter Monk

SIAM J. Appl. Math. 58, pp. 2024-2028 (5 pages) | Cited 6 times

Online Publication Date: July 26, 2006

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In our paper [SIAM J. Appl. Math., 55 (1995), pp. 1234--1344] Theorem 4.2 b) is incorrect. This note shows how to avoid the use of Theorem 4.2 b) in the remainder of the paper. We emphasize that the main results of the paper (Theorems 4.6 and 5.3) are correct but that the proofs must be modified as shown in this note.
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