News

## Technical Reports - 2005

SMU Math Report 2005-01

Computational methods and results for structured multiscale models of tumor invasion
We present multiscale models of cancer tumor invasion with components at the molecular, cellular, and tissue levels. We provide biological justifications for the model components, present computational results from the model, and discuss the scientific-computing methodology used to solve the model equations. The models and methodology presented in this paper form the basis for developing and treating increasingly complex, mechanistic models of tumor invasion that will be more predictive and less phenomenological. Because many of the features of the cancer models, such as taxis, aging and growth, are seen in other biological systems, the models and methods discussed here also provide a template for handling a broader range of biological problems.
Bruce P. Ayati, Glenn F. Webb and Alexander R.A. Anderson
Get paper

SMU Math Report 2005-02

Negative-coupling resonances in pump-coupled lasers
We consider coupled lasers, where the intensity deviations from the steady state, modulate the pump of the other lasers. Most of our results are for two lasers where the coupling constants are of opposite sign. This leads to a Hopf bifurcation to periodic output for weak coupling. As the magnitude of the coupling constants is increased (negatively) we observe novel amplitude effects such as a weak coupling resonance peak and, strong coupling subharmonic resonances and chaos. In the weak coupling regime the output is predicted by a set of slow evolution amplitude equations. Pulsating solutions in the strong coupling limit are described by discrete map derived from the original model.
Thomas W. Carr, Michael L. Taylor and Ira B. Schwartz
Get paper

SMU Math Report 2005-03

Modeling the Role of the Cell Cycle in Regulating Proteus mirabilis Swarm-Colony Development
We present models and computational results which indicate that the spatial and temporal regularity seen in Proteus mirabilis swarm-colony development is largely an expression of a sharp age of dedifferentiation in the cell cycle from motile swarmer cells to immotile dividing cells (also called swimmer or vegetative cells.) This contrasts strongly with reaction-diffusion models of Proteus behavior that ignore or average out the age structure of the cell population and instead use only density-dependent mechanisms. We argue the necessity of retaining the explicit age structure, and suggest experiments that may help determine the underlying mechanisms empirically. Consequently, we advocate Proteus as a model organism for a multiscale understanding of how and to what extent the life cycle of individual cells affects the macroscopic behavior of a biological system.
Bruce P. Ayati
Get paper

SMU Math Report 2005-04

Numerical computation of local vapor-liquid interface shape and heat transfer near steady contact line on heated surface
We develop an efficient and robust numerical procedure for computing the local interface shape and heat flux near a steady contact line on a heated surface using Matlab. The approach is based on reduction of the full system of coupled equations for liquid flow and heat transfer to a single ordinary differential equation for interface position. Results are obtained for a number of values of non-dimensional parameters. Common features of the solutions include high values of the evaporative mass flux and curvature close to the apparent contact line, as well as strong capillary pressure gradients that drive the local flow.
H.R.Quach and V.S.Ajaev
Get paper

SMU Math Report 2005-05

A Theory of Inertial Range Similarity in Isotropic Turbulence
We consider equilibrium statistics for high Reynolds number isotropic turbulence in an incompressible flow driven by steady forcing at the largest scale. Motivated by shell model observations, we develop a similarity theory for the inertial range from clearly stated assumptions. In the right variables, the theory is scaling invariant, but in traditional variables it shows anomalous scaling. We obtain the underlying probability density function, the scaling exponents, and the coefficients for the structure functions. An inertial range length scale also emerges.
Mogens V. Melander and Bruce R. Fabijonas
Get paper

## Technical Reports - 2004

SMU Math Report 2004-01

Effects of Basis Selection and H-Refinement on Error Estimator Reliability and Solution Efficiency for High-Order Methods in Three Space Dimensions
Designing effective high-order adaptive methods for solving stationary reaction-diffusion equations in three dimensions requires the selection of a finite element basis, {\it a posteriori} error estimator and refinement strategy. Estimator accuracy may depend on the basis chosen, which in turn, may lead to unreliability or inefficiency via under- or over-refinement, respectively. The basis may also have an impact on the size and condition of the matrices that arise from discretization, and thus, on algorithm effectiveness. Herein, the interaction between these three components is studied in the context of an h-refinement procedure. The effects of these choices on the robustness and efficiency of the algorithm are examined for several linear and nonlinear problems.
Get paper: P. Moore

SMU Math Report 2004-02

Localized patterns in homogeneous networks of diffusively coupled reactors
We study the influence of network topology on instabilities of the homogeneous steady state of diffusively coupled, monostable nonlinear cells. A particular focus are diffusion-induced instabilities, i.e., Turing instabilities. We present various theorems that make it possible to determine analytically the stability properties of networks with arbitrary topologies and general monostable dynamics of the individual cells. This work aims in particular to determine those topologies that will give rise to localized stationary patterns. Specific examples focus on well-stirred chemical reactors. The reactors are coupled by diffusion-like mass transfer, and the kinetics is given by the Lengyel-Epstein model, a two-variable scheme for the chlorine dioxide-iodine-malonic acid reaction.
Get paper: P. Moore and Werner Horsthemke

SMU Math Report 2004-03

Viscous flow of a volatile liquid on an inclined heated surface
We investigate the effects of evaporation on a gravity driven flow of a viscous liquid on a heated solid. Vapor molecules are adsorbed on the dry areas of the solid and form a microscopic adsorbed film. The thickness of this film is calculated from the formulas for disjoining pressure and the principles of equilibrium thermodynamics. A lubrication-type approach is used to derive an evolution equation capable of describing both the macroscopic shape of the vapor-liquid interface and the adsorbed film on the vapor-solid interface. Under the conditions of negligible evaporation, the numerical solution of the evolution equation predicts translational motion and formation of capillary ridge, in agreement with previous investigations. Moderate evaporation is shown to slow down the flow and decrease the height of the capillary ridge, which implies stabilizing effect of evaporation on the well-known instability observed in gravity driven thin film flows. We also study combined effects of evaporation and thermocapillary stresses and show that the latter act to reduce the velocity of the downward motion, but increase the height of the capillary ridge. Apparent contact angles are found from the solution and shown to increase with evaporation and contact line speed. For strong evaporation, steady state solutions are found such that evaporation balances the downward motion of the interface under the action of gravity.
Get pdf file.

SMU Math Report 2004-04

A Structured-Population Model of Proteus mirabilis Swarm-Colony Development
In this paper we present continuous age- and space-structured models and numerical computations of Proteus mirabilis swarm-colony development. We base the mathematical representation of the cell-cycle dynamics of Proteus mirabilis on those developed by Esipov and Shapiro, which are the best understood aspects of the system, and we make minimum assumptions about less-understood mechanisms, such as precise forms of the spatial diffusion. The models in this paper have explicit age-structure and, when solved numerically, display both the temporal and spatial regularity seen in experiments, whereas the Esipov and Shapiro model, when solved accurately, shows only the temporal regularity. The composite hyperbolic-parabolic partial differential equations used to model Proteus mirabilis swarm-colony development are relevant to other biological systems where the spatial dynamics depend on local physiological structure. We use computational methods designed for such systems, with known convergence properties, to obtain the numerical results presented in this paper.
Bruce Ayati
Get pdf file.

## Technical Reports - 2003

SMU Math Report 2003-01

Quadrature Formulas for Moments" of B-spline Wavelets
We derive exact quadrature formulas for integrals involving integrands that are the products of linear functions and wavelets based on the cardinal B-spline. More specifically, the formulas evaluate the first and second moments" of the B-spline wavelet with arbitrary shift and scale, and the double integral of the B-spline wavelet of arbitrary shift and scale. In the process, we derive formulas that are efficient in terms of the number of B-spline evaluations. These quadratures are useful in solving certain integral equations discretized with a wavelet expansion of the solution and collocation.
Bentley T. Garrett
Get pdf file.

SMU Math Report 2003-02

A Spectral Integration Method for Linear Two-Point Boundary Value Problems
In recent years, a robust method based on spectral integration with Chebyshev expansions has been applied to stiff linear two-point boundary value problems. Furthermore, the complexity of the solution process has been reduced by solving local integral equations over subintervals, resulting in $O(Np^2)$ instead of $O(N^3)$ arithmetic operations, where $N$ is the number of nodes, and $p$ is the order of the Chebyshev approximation on each subinterval. An adaptive version has also been devised. In this paper, a spectral analysis is used to demonstrate how the order affects the accuracy. Also the specialized method for the constant coefficient case, the general nonadaptive method, and the general adaptive method are compared against one another in terms of complexity. The complexity of all three methods are then compared to that of COLNEW, a collocation code that is currently one of the best established, robust codes for solving general boundary value problems.
Bentley T. Garrett
Get pdf file.

## Technical Reports - 2002

SMU Math Report 2002-01

A Variable Order Wavelet Method for the Sparse Representation of Layer Potentials in the Non-Standard Form
We discuss a variable order wavelet method for boundary integral formulations of elliptic boundary value problems. The wavelets are restrictions of piecewise polynomial functions in three variables on the boundary manifold. This construction is especially suited to obtain sparse approximate representations of integral operators on complicated geometries. For integral equations of the second kind we will show that the non-standard form can be compressed to contain only $O(N)$ non-vanishing entries while retaining the asymptotic converge of the full Galerkin scheme. Here, $N$ is the number of degrees of freedom in the discretization. The leading terms of the complexity estimates for the overhead are of the form $C_1 N \log^3 N + C_2 N$, where $C_1$ is much smaller than $C_2$. The constants in the complexity estimates are independent of the geometry of the boundary manifold.
Get paper: J. Tausch

SMU Math Report 2002-02

Period Locking due to Delayed Feedback in a Laser with Saturable Absorber
We consider laser with saturable absorber operating in the pulsating regime that is subject to delayed optical feedback. Alone, both the saturable absorber and delayed feedback cause the CW output to become unstable to periodic output via Hopf bifurcations. The delay feedback causes the laser pulse period to lock to an integer fraction of the feedback time. We derive a map from the original model to describe the periodic pulsations of the laser. Equations for the period of the laser predict the occurrence of the different locking states as well as the value of the pump when there is a switch between the locked states.
Thomas W. Carr
Get pdf file

SMU Math Report 2002-03

An Incomplete Assembly with Thresholding Algorithm for Systems of Reaction-Diffusion Equations in Three Space Dimensions
Solving systems of reaction-diffusion equations in three space dimensions can be prohibitively expensive both in terms of storage and cpu time. Herein I present a new incomplete assembly procedure that is designed to reduce storage requirements. Incomplete assembly is analogous to incomplete factorization in that only a fixed number of nonzero entries are stored per row and a drop tolerance is used to discard small values. The algorithm is incorporated in a finite element method-of-lines code and tested on a set of reaction-diffusion systems. The effect of incomplete assembly on cpu time and storage and on the performance of the temporal integrator DASPK, algebraic solver GMRES and preconditioner ILUT is studied.
Get paper: Peter K. Moore
SMU Math Report 2002-04

An Implicit Interpolation Error-Based Error Estimation Strategy for HP-Adaptivity
Hp-adaptive finite element methods require error estimates of the solution at the current order and one order higher. In [Moore,94] it was proved that a p-refinement (hierarchical) error estimation strategy was asymptotically exact for nonlinear parabolic equations. An extension of this strategy was proposed for computing higher-order estimates [Flaherty, Moore95]. Recently a new approach, interpolation error based (IEB) error estimation, for constructing a posteriori error estimates at both orders has been developed. I show that: i) IEB error estimation can be applied to semilinear two-point boundary value problems and parabolic equations in one space dimension; ii) the hierarchical estimator is an implicit IEB method and thus, works for semilinear two-point boundary value problems; iii) the hierarchical extension for computing higher-order error estimates is asymptotically exact. Computational results illustrating the theory and comparing the implicit (hierarchical) strategy with the earlier explicit IEB methods are presented.
Get paper: Peter K. Moore
SMU Math Report 2002-05

Implicit Interpolation Error-Based Error Estimation for Semilinear Elliptic and Parabolic Equations in Two Space Dimensions
Several authors have proposed an error estimation strategy for the finite element method applied to linear elliptic and parabolic equations in two space dimensions based on an odd/even order dichotomy principle. For odd-order approximations error estimates are computed directly from the finite element solution via jumps in the first derivatives across element boundaries. With even-order approximations an error estimate is obtained by computing a second solution on each element. Although both estimators are asymptotically exact the even-order estimators are typically more robust than the odd-order ones. In this paper the even-order method is extended to a family of methods for all orders greater than one, thereby recovering robustness for odd-orders. Proofs of asymptotic exactness are extended to semilinear elliptic and parabolic equations. Computational results demonstrating the effectiveness of the approach and comparing different members of the family are presented.
Get paper: Peter K. Moore

## Technical Reports - 2001

SMU Math Report 2001-01

Efficient Analysis of Periodic Dielectric Waveguides using Dirichlet-to-Neumann Maps.
We present a numerical scheme for the analysis of periodic dielectric waveguides using Floquet-Bloch theory. The problem of finding the fundamental propagation modes is reduced to a nonlinear eigenvalue problem involving Dirichlet-to-Neumann maps. This approach leads to much smaller matrix problems than the ones that have appeared previously. We discuss an eigensolver and extend the conventional rule to choose the branches of the transverse wave numbers. This ensures analytic dependence on the Floquet multiplier and convergence of the nonlinear solver. This methology allows arbitrary precision by increasing the discretization fineness. We will demonstrate that even for a complicated multilayer waveguide structure, the propagation factors can be calculated within seconds to several digits of accuracy.
J. Tausch and J. Butler
(get compressed ps file)
SMU Math Report 2001-02

Improved Integral Formulations for Fast 3-D Method-of-Moments Solvers.
This paper introduces a new integral formulation to calculate charge densities of conductor systems that may include multiple dielectric materials. We show that the conditioning of our formulation is much better than that of the standard equivalent charge formulation. When combined with a nonstandard discretization scheme, results can be obtained with higher accuracy at reduced numerical cost. We present a multipole accelerated implementation of our formulation. The results demonstrate that the new approach can cut the iteration count by a factor between two and four. Moreover, we will demonstrate that in the presence of sparsification errors and multiple dielectric materials second-kind formulations are much more accurate than the standard first-kind formulations.
J. Tausch, J. Wang and J. White
(get compressed ps file)
SMU Math Report 2001-03

Solving Bordered Almost Block Diagonal Linear Systems
Almost block diagonal (ABD) linear systems arise in a variety of contexts, specifically in the numerical solution of systems of ordinary differential equation two-point boundary value problems with separated boundary conditions. The stable, efficient solution of ABDs has received much attention recently. Bordered almost block diagonal (BABD) linear systems arise when the boundary conditions are non-separated. After an introduction to the problem, we describe a variety of direct approaches to solving BABD systems, some of them extensions of ABD techniques. We pay particular attention to questions of efficiency and stability.
(get compressed ps file)
SMU Math Report 2001-04

Applications of Lobatto Polynomials to an Adaptive Finite Element Method: Estimating Solution Derivatives and Grid-to-Grid Interpolation
Hp-adaptive finite element methods require algorithms for estimating the error in the solution for different discretizations and for interpolating solutions between grids. The first often involves estimating high-order derivatives of the solution. The second typically leads to the solution of large linear systems. The Lobatto interpolant, which possesses a variety of superconvergence properties for two-point boundary value problems and parabolic equations provides one approach to developing these algorithms. I derive a Taylor-like'' series for the pointwise error in the Lobatto interpolant. Differentiating this series leads to high-order derivative approximations using the interpolant. These estimates are extended to the finite element solution using the weak form of the equations. Explicit formulas for the inverses of the Lobatto interpolation matrices are given. Computational results illustrate the theory.
Peter K. Moore
(get compressed ps file)
SMU Math Report 2001-05

Onset of instabilities in self-pulsing semiconductor lasers with delayed feedback
We consider the deterministic dynamics of a semiconductor laser with saturable absorber that is subject to delayed optical feedback. Alone, both the saturable absorber and delayed feedback cause the CW output to become unstable to periodic output via Hopf bifurcations. We examine the combined effects of these two destabilizing mechanisms to determine new conditions for the Hopf bifurcations. We also describe the transient as the unstable CW output evolves to the oscillatory state. A main result is that the presence of a saturable absorber can increase the sensitivity of the laser to delayed feedback.
Thomas W. Carr
(get pdf file)
SMU Math Report 2001-06

Laplace's Method on a computer algebra system with an application to the real valued modified Bessel functions
We examine a Maple implementation of two distinct approaches to Laplace's Method used to obtain asymptotic expansions of Laplace type integrals. One algorithm uses power series reversion, whereas the other expands all quantities in Taylor or Puiseux series. These algorithms are used to derive asymptotic expansions for the real valued modified Bessel functions of pure imaginary order and real argument that mimic the well-known corresponding expansions for the unmodified Bessel functions.
Bruce R. Fabijonas
(get pdf file)
SMU Math Report 2001-07

Interpolation Error-Based A Posteriori Error Estimation for P-Refinement Using First and Second Derivative Jumps
Hp-adaptive finite element methods require estimates of the error in the solution at the current order and one order higher. Interpolation-error based a posteriori error estimates offer one solution to this problem. I show how such estimates at the current order can be obtained in one dimension for odd (even) order bases by using jumps in the first (second) derivative of the finite element solution at element boundaries. Additionally the jumps in the second (first) derivative for odd (even) order elements allow error estimates of the finite element solution one order higher than the current order to be computed. These estimates are compared with interpolation-error based methods that use high-order derivative approximations of the finite element solution. Computational results illustrate the theory and the impact of the estimation strategy on the refinement algorithm is discussed.
Peter K. Moore
(get pdf file)
SMU Math Report 2001-08

Interpolation Error-Based A Posteriori Error Estimation in Three Dimensions: A First Step
Interpolation error-based a posteriori error estimation for elliptic and parabolic equations requires finding an interpolant that is asymptotically equivalent to the finite element solution. In three dimensions such an interpolant is obtained by taking the tensor product of the one-dimensional Lobatto interpolant. Formulas for the error in L^2 and H^1 of this interpolant are derived. These formulas involve high-order derivatives of the solution of the partial differential equations. Approximations of these derivatives from jumps in first and second derivatives of the interpolant across element boundaries and from differences of high-order derivatives of the interpolant are presented. Projections of the interpolant onto finite element spaces constructed from hierarchical and modified-hierarchical bases are examined and the effect of using these smaller bases on the error estimation strategy is considered.
Peter K. Moore
(get compressed ps file)

## Technical Reports - 2000

SMU Math Report 2000-01

Multiscale bases for the sparse representation of boundary integral operators on complex geometry
A multilevel transform is introduced to represent discretizations of integral operators from potential theory by nearly sparse matrices. The new feature presented here is to construct the basis in a hierarchical decomposition of the three-space and not, as in previous approaches, in a parameter space of the boundary manifold. This construction leads to sparse representations of the operator even for geometrically complicated, multiply connected domains. We will demonstrate that the numerical cost to apply a vector to the operator using the non-standard form is essentially equal to performing the same operation with the Fast Multipole Method. With a second compression scheme the multiscale approach can be further optimized. Moreover, the diagonal blocks of the transformed matrix can be used as an efficient and inexpensive preconditioner.
Johannes Tausch and Jacob White
(get compressed ps file)
SMU Math Report 2000-02

On the Relationship of Various Discontinuous Finite Element Methods for Second-Order Elliptic Equations
In this paper we introduce a family of discontinuous finite element methods for fairly general second-order elliptic equations with variable coefficients. Lower-order terms are included in these equations, so the analysis and results apply also to time-dependent equations. We first write this family of methods in a mixed formulation, and then establish their equivalent versions in a nonmixed formulation by incorporating some projection operators. Within this framework, we can recover all existing discontinuous finite element methods by changing appropriate parameters. Stability and convergence properties are studied for these discontinuous methods; stability results and sharp error estimates are established for general boundary conditions and with reasonable assumptions. We show that when discontinuous finite element methods are defined in mixed form, they not only preserve good features of these methods, also have some advantages over classical Galerkin discontinuous methods such as they are more stable in this form.
Zhangxin Chen
(get compressed ps file)
SMU Math Report 2000-03

Error Analysis for Characteristics-Based Methods for Degenerate Parabolic Problems
We consider characteristics-based finite element methods for solving nonlinear, degenerate, advection-diffusion equations. These equations have applications in simulation of petroleum reservoirs and groundwater aquifers and in modeling of free boundary problems. Standard finite element Galerkin methods have been studied for these equations. In this paper we analyze the characteristics-based finite element methods for them. The main difficulty in the analysis is that they are degenerate and the solution lacks in regularity. Here we develop a technique that respects the degeneracy and the known minimal regularity. This technique is based on the Green operator for standard elliptic equations and is developed directly for the degenerate advection-diffusion equations. We concentrate our analysis on the modified method of characteristics (MMOC) and one of its variants, the modified method of characteristics with adjusted advection (MMOCAA), which preserves mass. We derive error estimates in various norms. The extension to other variants is discussed. The present technique is also applied to nondegenerate problems; error estimates previously obtained for the MMOC are derived under much weaker regularity assumptions on the solution, and the error estimates for the MMOCAA appear new even in the nondegenerate case. Finally, numerical results are presented to show the sharpness of the error estimates derived.
Zhangxin Chen, Richard E. Ewing, Ellen Q. Jiang, and Anna M. Spagnuolo
(get compressed ps file)
SMU Math Report 2000-04

Characteristics-Discontinuous Finite Element Methods in Mixed Form for Advection-Dominated Diffusion Problems
In this paper three characteristics-discontinuous finite element methods in mixed form are introduced for time dependent advection-dominated diffusion problems. Namely, the diffusion problems are discretized using discontinuous finite elements in mixed form, and the temporal differentiation and advection terms in these problems are treated by a characteristic tracking scheme. The first method is based on the standard modified method of characteristics. It is simple to set up and analyze, but fails to preserve an integral identity satisfied by the solution of the differential problems. The second method is formulated using the modified method of characteristics with adjusted advection and preserves the integral identity globally. The third method is defined in terms of a local Eulerian-Lagrangian technique and preserves the identity locally. These three methods not only preserve the conceptual and computational merits of both characteristics-based procedures and discontinuous finite element schemes, also possess some new features. Stability and convergence properties are studied for all three methods; unconditionally stable results and sharp error estimates are established. Their relationships to standard characteristics-based methods such as MMOC, MMOCAA, ELLAM, and CMFEM are described in detail.
Zhangxin Chen
(get compressed ps file)
SMU Math Report 2000-05

Convergence and Stability of Two Families of Discontinuous Finite Element Methods for Second-Order Problems
In this paper, we study two families of discontinuous finite element methods for second-order problems. For the first family for a model elliptic problem, we obtain optimal error estimates in both the energy norm and the $L^2(\O)$-norm. We prove a stability result for this family when the coefficient in the lower-order term of the elliptic problem is positive; without this lower-order term, we show a similar stability result for finite element spaces with piecewise polynomials of degree greater than one. For the second family, we derive an optimal estimate in the energy norm and a sub-optimal estimate in the $L^2(\O)$-norm. We establish the stability result for the second family with or without the lower-order term of the elliptic problem. The stiffness matrix arising from the first family is symmetric and positive definite, while it is positive definite but nonsymmetric from the second one. We extend the convergence and stability results to parabolic and convection-dominated problems as well.
Zhangxin Chen
(get compressed ps file)
SMU Math Report 2000-06

Distributional modes for scaler field quantization
We propose a mode-sum formalism for the quantization of the scalar field based on distributional modes, which are naturally associated with a slight modification of the standard plane-wave modes. We show that this formalism leads to the standard Rindler temperature result, and that these modes can be canonically defined on any Cauchy surface.
Alfonso F. Agnew and Tevian Dray
(get compressed ps file)