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Table of Contents
News
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Technical Reports - 2005
SMU Math Report 2005-01
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Computational methods and results for structured multiscale models of
tumor invasion
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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.
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| Bruce P. Ayati, Glenn F. Webb and Alexander R.A. Anderson |
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Get paper |
SMU Math Report 2005-02
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Negative-coupling resonances in pump-coupled lasers
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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.
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| Thomas W. Carr, Michael L. Taylor and Ira B. Schwartz |
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Get paper |
SMU Math Report 2005-03
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Modeling the Role of the Cell Cycle in Regulating Proteus mirabilis Swarm-Colony Development
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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.
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| Bruce P. Ayati |
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Get paper |
SMU Math Report 2005-04
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Numerical computation of local vapor-liquid interface shape and
heat transfer near steady contact line on heated surface
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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.
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| H.R.Quach and V.S.Ajaev |
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Get paper |
SMU Math Report 2005-05
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A Theory of Inertial Range Similarity in Isotropic Turbulence
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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.
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| Mogens V. Melander and Bruce R. Fabijonas |
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Get paper |
Technical Reports - 2004
SMU Math Report 2004-01
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Effects of Basis Selection and H-Refinement on Error Estimator Reliability
and Solution Efficiency for High-Order Methods in Three Space Dimensions
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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.
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| Get paper:
P. Moore |
SMU Math Report 2004-02
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Localized patterns in homogeneous networks of diffusively coupled reactors
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| 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.
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| Get paper:
P. Moore and Werner Horsthemke |
SMU Math Report 2004-03
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Viscous flow of a volatile liquid on an inclined heated surface
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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.
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| Vladimir Ajaev |
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Get pdf file. |
SMU Math Report 2004-04
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A Structured-Population Model of Proteus mirabilis Swarm-Colony Development
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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.
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| Bruce Ayati |
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Get pdf file. |
Technical Reports - 2003
SMU Math Report 2003-01
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Quadrature Formulas for ``Moments" of B-spline Wavelets
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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.
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| Bentley T. Garrett |
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Get pdf file. |
SMU Math Report 2003-02
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A Spectral Integration Method for Linear Two-Point Boundary
Value Problems
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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.
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| Bentley T. Garrett |
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Get pdf file.
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Technical Reports - 2002
SMU Math Report 2002-01
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A Variable Order Wavelet Method for
the Sparse Representation of Layer Potentials in the Non-Standard
Form
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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.
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| Get paper: J. Tausch |
SMU Math Report 2002-02
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Period Locking due to Delayed Feedback in a Laser with Saturable Absorber
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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.
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| Thomas W. Carr |
| Get pdf file |
SMU Math Report 2002-03
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An Incomplete Assembly with Thresholding Algorithm for Systems of
Reaction-Diffusion Equations in Three Space Dimensions
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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.
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| Get paper:
Peter K. Moore |
SMU Math Report 2002-04
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An Implicit Interpolation Error-Based Error Estimation Strategy for
HP-Adaptivity
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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.
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| Get paper:
Peter K. Moore |
SMU Math Report 2002-05
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Implicit Interpolation Error-Based Error Estimation for Semilinear
Elliptic and Parabolic Equations in Two Space Dimensions
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| 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.
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| Get paper:
Peter K. Moore |
Technical Reports - 2001
SMU Math Report 2001-01
|
Efficient Analysis of Periodic Dielectric Waveguides using
Dirichlet-to-Neumann Maps.
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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.
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J. Tausch and J. Butler
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(get compressed ps file) |
SMU Math Report 2001-02
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Improved Integral Formulations for Fast 3-D Method-of-Moments Solvers.
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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.
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J. Tausch, J. Wang
and J. White
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(get compressed ps file) |
SMU Math Report 2001-03
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Solving Bordered Almost Block Diagonal Linear Systems
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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.
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B. Garrett and I. Gladwell
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(get compressed ps file) |
SMU Math Report 2001-04
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Applications of Lobatto Polynomials to an Adaptive Finite
Element Method: Estimating Solution Derivatives and Grid-to-Grid Interpolation
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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.
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Peter K. Moore
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(get compressed ps file) |
SMU Math Report 2001-05
|
Onset of instabilities in self-pulsing semiconductor lasers with
delayed feedback
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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.
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Thomas W. Carr
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(get pdf file) |
SMU Math Report 2001-06
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Laplace's Method on a computer algebra system with an
application to the real valued modified Bessel functions
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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.
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Bruce R. Fabijonas
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(get pdf file) |
SMU Math Report 2001-07
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Interpolation Error-Based A Posteriori Error Estimation
for P-Refinement Using First and Second Derivative Jumps
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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
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|
(get pdf file) |
SMU Math Report 2001-08
|
Interpolation Error-Based A Posteriori Error Estimation in
Three Dimensions: A First Step
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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.
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Peter K. Moore
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(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
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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.
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Johannes Tausch and
Jacob White
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(get compressed ps file) |
SMU Math Report 2000-02
|
On the Relationship of Various Discontinuous Finite Element Methods
for Second-Order Elliptic Equations
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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.
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Zhangxin Chen
|
|
(get compressed ps file) |
SMU Math Report 2000-03
|
Error Analysis for Characteristics-Based
Methods for Degenerate Parabolic Problems
|
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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
|
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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
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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) |
|
