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2005

Volume 65, Issue 4, pp. 1101-1462


Modeling of Wave Resonances in Low-Contrast Photonic Crystals

Dmitri Agueev and Dmitry Pelinovsky

SIAM J. Appl. Math. 65, pp. 1101-1129 (29 pages) | Cited 9 times

Online Publication Date: July 31, 2006

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Coupled-mode equations are derived from Maxwell equations for modeling of low-contrast cubic-lattice photonic crystals in three spatial dimensions. Coupled-mode equations describe resonantly interacting Bloch waves in stop bands of the photonic crystal. We study the linear boundary-value problem for stationary transmission of four counter-propagating and two oblique waves on the plane. Well-posedness of the boundary-value problem is proved by using the method of separation of variables and generalized Fourier series. For applications in photonic optics, we compute integral invariants for transmission, reflection, and diffraction of resonant waves.

A Universal Procedure for Normalizing n-Degree-of-Freedom Polynomial Hamiltonian Systems

Susana Gutiérrez-Romero, Jesús F. Palacián, and Patricia Yanguas

SIAM J. Appl. Math. 65, pp. 1130-1152 (23 pages) | Cited 2 times

Online Publication Date: July 31, 2006

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We depart from an n-degree-of-freedom Hamiltonian formed by the sum of homogeneous polynomials in $n$ coordinates and n momenta with arbitrary coefficients. By extending formally an integral of the principal part of the system to the full Hamiltonian and truncating higher-order terms, we obtain a simplified Hamiltonian. This "normalization" procedure can be used to extract qualitative features of the departure system. In this paper we present the symbolic routines needed to achieve the normalization. The power and generality of the algorithm are exhibited through two examples.

Spatially Discrete FitzHugh--Nagumo Equations

Christopher E. Elmer and Erik S. Van Vleck

SIAM J. Appl. Math. 65, pp. 1153-1174 (22 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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We consider pulse and front solutions to a spatially discrete FitzHugh--Nagumo equation that contains terms to represent both depolarization and hyperpolarization of the nerve axon. We demonstrate a technique for deriving candidate solutions for the McKean nonlinearity and present and apply solvability conditions necessary for existence. Our equation contains both spatially continuous and discrete diffusion terms.

Combustion Stabilization by Forced Oscillations in a Duct

Abram Dorfman

SIAM J. Appl. Math. 65, pp. 1175-1199 (25 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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The feasibility of stabilizing premixed combustion by forced oscillation is analytically demonstrated through the simulation of an active control input-output mechanism. The developed model is used for analysis of the interactions between an autonomous oscillation in a duct, a loudspeaker's input, and the unsteady heat release. We assume that the autonomous oscillations (at frequency $\omega_0$) exist in a duct containing a flame with a loudspeaker at the input. At $t = 0$, the loudspeaker starts to generate oscillations at a different frequency $\omega$. To find the resulting oscillations (the output), a mathematical technique is needed that takes into account (1) the pressure and velocity fields in the duct when the loudspeaker starts; (2) the variable amplitudes of the resulting oscillations, which depend on time and location; and (3) coupling of the fresh and burnt gas flows at the flame. Such a technique differs significantly from that used by previous authors for studying single oscillation/flame interactions. The mathematical development leads to an exact solution that gives a stability criterion in the form of a system of two integro-differential equations. Analysis shows that the stability domains of the time lag depend mainly on the flame location and the fresh/burnt gases temperature ratio. Numerical results are obtained for a centrally located flame and for the temperature ratio 1500 K/300 K.

$\omega$-Harmonic Functions and Inverse Conductivity Problems on Networks

Soon-Yeong Chung and Carlos A. Berenstein

SIAM J. Appl. Math. 65, pp. 1200-1226 (27 pages) | Cited 5 times

Online Publication Date: July 31, 2006

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In this paper, we discuss the inverse problem of identifying the connectivity and the conductivity of the links between adjacent pair of nodes in a network, in terms of an input-output map. To do this we deal with the weighted Laplacian $\Delta_{\omega}$ and an $\omega$-harmonic function on the graph, with its physical interpretation as a diffusion equation on the graph, which models an electric network. After deriving the basic properties of $\omega$-harmonic functions, we prove the solvability of (direct) problems such as the Dirichlet and Neumann BVPs. Our main result is the global uniqueness of the inverse conductivity problem for a network under a suitable monotonicity condition.

Distance Functions and Geodesics on Submanifolds of $\R^d$ and Point Clouds

Facundo Mémoli and Guillermo Sapiro

SIAM J. Appl. Math. 65, pp. 1227-1260 (34 pages) | Cited 6 times

Online Publication Date: July 31, 2006

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A theoretical and computational framework for computing intrinsic distance functions and geodesics on submanifolds of ${\Bbb {R}}^d$ given by point clouds is introduced and developed in this paper. The basic idea is that, as shown here, intrinsic distance functions and geodesics on general co-dimension submanifolds of ${\Bbb {R}}^d$ can be accurately approximated by extrinsic Euclidean ones computed inside a thin offset band surrounding the manifold. This permits the use of computationally optimal algorithms for computing distance functions in Cartesian grids. We use these algorithms, modified to deal with spaces with boundaries, and obtain a computationally optimal approach also for the case of intrinsic distance functions on submanifolds of ${\Bbb{R}}^d$. For point clouds, the offset band is constructed without the need to explicitly find the underlying manifold, thereby computing intrinsic distance functions and geodesics on point clouds while skipping the manifold reconstruction step. The case of point clouds representing noisy samples of a submanifold of Euclidean space is studied as well. All the underlying theoretical results are presented along with experimental examples for diverse applications and comparisons to graph-based distance algorithms.

A Mixture Theory for the Genesis of Residual Stresses in Growing Tissues I: A General Formulation

Robyn P. Araujo and D. L. Sean McElwain

SIAM J. Appl. Math. 65, pp. 1261-1284 (24 pages) | Cited 18 times

Online Publication Date: July 31, 2006

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In this paper a theoretical framework for the study of residual stresses in growing tissues is presented using the theory of mixtures. Such a formulation must necessarily be a solid-multiphase model, comprising at least one phase with solid characteristics, owing to the fundamental role played by the incompatibility of strains in generating residual stresses. Since biological growth involves mass exchange between cellular and extracellular phases, field equations are presented for individual phases and for the mixture as a whole which incorporate this phenomenon. Appropriate constitutive equations are then deduced from first principles, appealing to the second law of thermodynamics.
The analysis shows that the distinguishing feature of multiphase models involving mass exchange is the necessity to propose an additional constitutive postulate between the variables in the mass-balance equation in order to close the model. In particular, the defining characteristic of a solid-multiphase model which describes biological growth is a constitutive postulate which relates the process of interphase mass exchange (cell proliferation/cell death) with the expansion or contraction of the solid phase. Thus, the framework presented here represents a new class of mathematical models which extends the concepts of poroelasticity to accommodate continuous volumetric growth. A set of modelling equations is then proposed for the simplest case of a solid-multiphase model, being a biphasic mixture of a linear-elastic solid and an inviscid fluid.

Modeling, Design, and Optimization of a Solid State Electron Spin Qubit

R. E. Caflisch, Mark F. Gyure, Hans D. Robinson, and Eli Yablonovitch

SIAM J. Appl. Math. 65, pp. 1285-1304 (20 pages) | Cited 3 times

Online Publication Date: July 31, 2006

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This paper describes a solid state system in which a qubit is realized as the spin of a single trapped electron in a quantum dot and read functionality is via an adjacent quantum wire with a single or a small number of conductive states. Because of the limited design window for this system, simulation is an important guide to an experimental search for successful designs. We use a semianalytic approximation that is accurate enough to provide meaningful results and computationally simple enough to allow high throughput, as needed for design and optimization. In particular, we find designs that achieve double pinchoff (i.e., a single trapped electron in the dot and a single conductive state in the wire). After relaxing the design requirements to allow for a small number of conductive states in the wire, we find successful designs that are optimally robust, in the sense that their success is unlikely to be affected by fabrication errors.

The Effect of Dispersal Patterns on Stream Populations

Frithjof Lutscher, Elizaveta Pachepsky, and Mark A. Lewis

SIAM J. Appl. Math. 65, pp. 1305-1327 (23 pages) | Cited 19 times

Online Publication Date: July 31, 2006

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Individuals in streams are constantly subject to predominantly unidirectional flow. The question of how these populations can persist in upper stream reaches is known as the "drift paradox." We employ a general mechanistic movement-model framework and derive dispersal kernels for this situation. We derive thin- as well as fat-tailed kernels. We then introduce population dynamics and analyze the resulting integrodifferential equation. In particular, we study how the critical domain size and the invasion speed depend on the velocity of the stream flow. We give exact conditions under which a population can persist in a finite domain in the presence of stream flow, as well as conditions under which a population can spread against the direction of the flow. We find a critical stream velocity above which a population cannot persist in an arbitrarily large domain. At exactly the same stream velocity, the invasion speed against the flow becomes zero; for larger velocities, the population retreats with the flow.

A Mathematical Study of the Hematopoiesis Process with Applications to Chronic Myelogenous Leukemia

Mostafa Adimy, Fabien Crauste, and Shigui Ruan

SIAM J. Appl. Math. 65, pp. 1328-1352 (25 pages) | Cited 22 times

Online Publication Date: July 31, 2006

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This paper is devoted to the analysis of a mathematical model of blood cell production in the bone marrow (hematopoiesis). The model is a system of two age-structured partial differential equations. Integrating these equations over the age, we obtain a system of two nonlinear differential equations with distributed time delay corresponding to the cell cycle duration. This system describes the evolution of the total cell populations. By constructing a Lyapunov functional, it is shown that the trivial equilibrium is globally asymptotically stable if it is the only equilibrium. It is also shown that the nontrivial equilibrium, the most biologically meaningful one, can become unstable via a Hopf bifurcation. Numerical simulations are carried out to illustrate the analytical results. The study may be helpful in understanding the connection between the relatively short cell cycle durations and the relatively long periods of peripheral cell oscillations in some periodic hematological diseases.

On the Sound in Unbounded and Ducted Vortex Flows

L. M. B. C. Campos and P. G. T. A. Serrão

SIAM J. Appl. Math. 65, pp. 1353-1368 (16 pages) | Cited 1 time

Online Publication Date: July 31, 2006

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The propagation of sound is considered in a potential cylindrical vortex, with superimposed axial flow, by means of explicit analytical solutions. The sound waves are sinusoidal in time and in the axial and azimuthal directions; the convected wave equation leads to a radial dependence specified by an ordinary second-order differential equation, with two singularities, at the origin and at infinity. Both singularities are irregular, implying that the acoustic fields have an essential singularity. In the neighborhood of the vortex axis, the essential singularity of the acoustic field is specified by an exponential of the integrated Doppler shift; using the latter as a factor, the acoustic fields are specified by asymptotic expansions in ascending powers of the radius. In the neighborhood of the point at infinity, where the tangential mean flow velocity vanishes, the leading terms are outward or inward propagating cylindrical waves; these factors multiply asymptotic expansions in descending powers of the radius. The two pairs of solutions, around the vortex axis and the point at infinity, are valid in all space or overlapping regions, as far as the asymptotic expansions can be calculated. The case of an annular nozzle, with uniform axial flow, and potential swirl is used as an example; the eigenvalues are obtained for rigid wall boundary conditions and the corresponding eigenfunctions are plotted.

On a Regularization Scheme for Linear Operators in Distribution Spaces with an Application to the Spherical Radon Transform

Thomas Schuster and Eric Todd Quinto

SIAM J. Appl. Math. 65, pp. 1369-1387 (19 pages) | Cited 8 times

Online Publication Date: July 31, 2006

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This article provides a framework to regularize operator equations of the first kind where the underlying operator is linear and continuous between distribution spaces, the dual spaces of smooth functions. To regularize such a problem, the authors extend Louis' method of approximate inverse from Hilbert spaces to distribution spaces. The idea is to approximate the exact solution in the weak topology by a smooth function, where the smooth function is generated by a mollifier. The resulting regularization scheme consists of the evaluation of the given data at so-called reconstruction kernels which solve the dual operator equation with the mollifier as right-hand side. A nontrivial example of such an operator is given by the spherical Radon transform which maps a function to its mean values over spheres centered on a line or plane. This transform is one of the mathematical models in sonar and radar. After establishing the theory of the approximate inverse for distributions, we apply it to the spherical Radon transform. The article also contains numerical results.

Modeling of Seismic Data in the Downward Continuation Approach

Christiaan C. Stolk and Maarten V. de Hoop

SIAM J. Appl. Math. 65, pp. 1388-1406 (19 pages) | Cited 20 times

Online Publication Date: July 31, 2006

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Seismic data are commonly modeled by a high-frequency single scattering approximation. This amounts to a linearization in the medium coefficient about a smooth background. The discontinuities are contained in the medium perturbation. The high-frequency part of the wavefield in the background medium is described by a geometrical optics representation. It can also be described by a one-way wave equation. Based on this we derive a downward continuation operator for seismic data. This operator solves a pseudodifferential evolution equation in depth, the so-called double-square-root equation. We consider the modeling operator based on this equation. If the rays in the background that are associated with the reflections due to the perturbation are nowhere horizontal, the singular part of the data is described by the solution to an inhomogeneous double-square-root equation.

Some Properties of the Capacity Value Function

B. A. Chiera, A. E. Krzesinski, and P. G. Taylor

SIAM J. Appl. Math. 65, pp. 1407-1419 (13 pages)

Online Publication Date: July 31, 2006

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In a previous paper [B. A. Chiera and P. G. Taylor, Probab. Engrg. Inform. Sci., 16 (2002), pp. 513--522], two of the authors developed a method for ascribing a value to an extra unit of capacity on a telecommunications link. Specifically, they expressed the value of an extra unit of capacity as a function of current capacity, current occupancy, and a planning horizon. The intention was to use this function as an ingredient in a bandwidth reallocation scheme for ensuring efficient operation of a telecommunications network.
Unfortunately, direct evaluation of the function requires numerical inversion of a Laplace transform expressed in terms of Charlier polynomials, a task that is beyond the processing capabilities of typical switches in today's telecommunications networks. Because of this, it is desirable to have more easily computable methods of either calculating or approximating the capacity value function. We develop two approaches to this problem: the first is a recursive method of computing the Laplace transform of the capacity value function, and the second is a linear approximation to the capacity value function itself.

Diffusive and Chemotactic Cellular Migration: Smooth and Discontinuous Traveling Wave Solutions

K. A. Landman, M. J. Simpson, J. L. Slater, and D. F. Newgreen

SIAM J. Appl. Math. 65, pp. 1420-1442 (23 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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A mathematical model describing cell migration by diffusion and chemotaxis is considered. The model is examined using phase plane, numerical, and perturbation techniques. For a proliferative cell population, travelingwave solutions are observed regardless of whether the migration is driven by diffusion, chemotaxis, or a combination of the two mechanisms. For pure chemotactic migration, both smooth and discontinuous solutions with shocks are shown to exist using phase plane analysis involving a curve of singularities, and identical results are obtained numerically. Alternatively, pure diffusive migration and combinations of diffusive and chemotactic migration yield smooth solutions only. For all cases the wave speed depends on the exponential decay rate of the initial cell density, and it is bounded by a minimum value which is numerically observed whenever the initial cell distribution has compact support. The minimum wave speed $c_{min}$ is proportional to $\sqrt{\chi}$ or $\sqrt {D}$ for pure chemotaxis and pure diffusion cases, respectively. The value of $c_{min}$ for combined diffusion and chemotactic migration is examined numerically. The rate at which the mixed migration system approaches either a diffusion-dominated or chemotaxis-dominated system is investigated as a function of a dimensionless parameter involving $D/\chi$. Finally, a perturbation analysis provides details of the steep critical layer when $D/\chi \ll1$, and these are confirmed with numerical solutions. This analysis provides a deeper qualitative and quantitative understanding of the interplay between diffusion and chemotaxis for invading cell populations.

Asymptotic Theory of Electroseismic Prospecting

Benjamin S. White

SIAM J. Appl. Math. 65, pp. 1443-1462 (20 pages) | Cited 7 times

Online Publication Date: July 31, 2006

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In a porous medium such as the earth's subsurface, electromagnetic (EM) waves and mechanical waves are coupled through the phenomenon of electrokinetics, for which a complete set of partial differential equations was derived by S. Pride. In this paper, we derive from Pride's equations an asymptotic theory that enables forward modeling of the seismic response to an EM source in fully three-dimensional geometries on a scale that is relevant to exploration. For simplicity, we consider piecewise homogeneous media separated by interfaces which are curved surfaces in three dimensions. The following physical picture emerges: An EM source excites an EM wave which propagates into the earth, stirring up local mechanical movement. At an interface, EM energy is converted to seismic waves, which may be described by ray theory. Instantly, on the seismic time scale, every interface becomes a wavefront for both compressional and shear waves; that is, seismic P- and S-waves explode from both sides of each interface, at every point on it. The rays for these waves leave the interface in the orthogonal direction and propagate up and down into the homogeneous media on both sides of the surface. We derive formulas for the initial amplitudes of these waves. Conventional seismic ray theory then describes propagation of the P- and S-waves, including reflection, transmission, and mode conversion at any other interfaces that they may encounter. Thus, three-dimensional electroseismic modeling may be accomplished with conventional EM and conventional seismic modeling tools, using the present theory to provide the link between them.
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