Mathematical Modeling and Analysis
Mathematical Modeling and Analysis
@incollection{KLipnikov_JDMoulton_DSvyatskiy_2009b,
author = {K. Lipnikov and J. D. Moulton and D. Svyatskiy},
title = {A Multilevel Multiscale Mimetic ({M}$^3$) Method
for an Anisotropic Infiltration Problem},
booktitle = {Computational Science--ICCS 2009,
9$^{th}$ International Conference Baton Rouge, LA,
May 25-27, 2009. Proceedings, Part I},
pages = {685--694},
series = {Lecture Notes in Computer Science},
volume = 5544,
doi = {10.1007/978-3-642-01970-8_68},
publisher = {Springer-Verlag},
year = 2009,
editor = {G. Allen and J. Nabrzyski and E. Seidel and G. D. van Albada
and J. J. Dongarra and P. M. A. Sloot},
abstract = {Modeling of multiphase flow and transport in highly
heterogeneous porous media must capture a broad
range of coupled spatial and temporal scales.
Recently, a hierarchical approach dubbed the
Multilevel Multiscale Mimetic (M$^3$) method, was
developed to simulate two-phase flows in porous
media. The M$^3$ method is locally mass conserving
at all levels in its hierarchy, it supports
unstructured polyhedral grids and full tensor
permeabilities, and it can achieve large coarsening
factors. In this work we consider infiltration of
water into a two-dimensional layered medium. The
grid is aligned with the layers but not the
coordinate axes. We demonstrate that with an
efficient temporal updating strategy for the
coarsening parameters, fine-scale accuracy of
prominent features in the flow is maintained by the
M$^3$ method.}
}
@techreport{JEDendy_JDMoulton_2009a,
author = {J. E. Dendy and J. D. Moulton},
title = {Black Box Multigrid with Coarsening by a Factor of Three},
institution = {Applied Mathematics and Plasma Physics Group},
year = 2009,
number = {{LA-UR} 09-03783},
address = {Los Alamos National Laboratory},
note = {\emph{submitted to} Numerical Linear Algebra
with Applications},
abstract = {Black Box Multigrid (BoxMG) is a robust variational
multigrid solver for diffusion equations on
logically structured grids. BoxMG standardly uses
coarsening by a factor of two. It handles
cell-centered discretizations on logically
rectangular grids by treating the cell-centers as
the unknowns to be coarsened. Such a strategy does
not preserve the cell structure. That is,
coarse-grid cells are not the union of fine grid
cells. In some applications, such as local grid
refinement, it is desirable that the cell structure
be preserved. In this paper we develop a method
that employs coarsening by a factor of three. It is
a natural generalization of standard BoxMG, using
operator-induced interpolation (which approximately
preserves the continuity of the normal flux),
restriction as the transpose of interpolation, and
Galerkin coarsening. We present numerical results
that demonstrate its robustness with respect to
discontinuous diffusion coefficients.}
}
@techreport{DHigdon_etal_2008a,
author = {D. Higdon and C. S. Reese and J. D. Moulton and
J. A. Vrugt and C. Fox},
title = {Posterior exploration for computationally
intensive forward models},
institution = {Statistical Sciences Group},
year = 2009,
number = {{LA-UR} 08-05905},
address = {Los Alamos National Laboratory},
note = {\emph{to appear in} The Handbook of Markov Chain Monte Carlo,
Eds. X.-L. Meng, A. Gelman, and G.Jones, CRC press.}
}
@article{KLipnikov_JDMoulton_DSvyatskiy_2008a,
author = {K. Lipnikov and J. D. Moulton and D. Svyatskiy},
title = {A {M}ultilevel {M}ultiscale {M}imetic ({M}$^3$) Method for
Two-Phase Flows in Porous Media},
journal = journalofcomputationalphysics,
year = 2008,
volume = 227,
number = 14,
pages = {6727--6753},
doi = {10.1016/j.jcp.2008.03.029},
abstract = {We describe a multilevel multiscale mimetic (M$^3$)
method for solving two-phase flow (water and oil) in
a heterogeneous reservoir. The governing equations
are the elliptic equation for the reservoir pressure
and the hyperbolic equation for the water saturation.
On each time step, we first solve the pressure
equation and then use the computed flux in an
explicit upwind finite volume method to update the
saturation. To reduce the computational cost, the
pressure equation is solved on a much coarser grid
than the saturation equation. The coarse-grid
pressure discretization captures the influence of
multiple scales via the subgrid modeling technique
for single-phase flow recently proposed in
\cite{YAKuznetsov_2006a,Gvozdev:2007,Kuznetsov:08}. We
extend significantly the applicability of this
technique by developing a new robust and efficient
method for estimating the flux coarsening parameters.
Specifically, with this advance the M$^3$ method can
handle full permeability tensors and general
coarsening strategies, which may generate polygonal
meshes on the coarse grid. These problem dependent
coarsening parameters also play a critical role in
the interpolation of the flux, and hence, in the
advection of saturation for two-phase flow.
Numerical experiments for two-phase flow in highly
heterogeneous permeability fields, including layer 68
of the SPE Tenth Comparative Solution Project,
demonstrate that the M$^3$ method retains good
accuracy for high coarsening factors in both
directions, up to 64 for the considered models.
Moreover, we demonstrate that with a simple and
efficient temporal updating strategy for the
coarsening parameters, we achieve accuracy comparable
to the fine-scale solution, but at a fraction of the
cost.},
pdf = {http://math.lanl.gov/~moulton/papers/upscaling/M3.pdf}
}
@article{MBerndt_GHansen_JDMoulton_2008a,
author = {M. Berndt and G. Hansen and J. D. Moulton},
title = {Efficient Nonlinear Solvers for
{L}aplace-{B}eltrami Smoothing
of Three-Dimensional Unstructured Grids},
journal = computersandmathematics,
year = 2008,
volume = 55,
number = 12,
pages = {2791--2806},
doi = {10.1016/j.camwa.2007.10.029},
abstract = {The Laplace-Beltrami system of nonlinear, elliptic,
partial differential equations has utility in the
generation of computational grids on complex and
highly curved geometry. Discretization of this
system with Finite Elements readily accommodates
unstructured grids, but generates a large, sparse,
ill-conditioned system of nonlinear discrete
equations. The extensive use of the
Laplace-Beltrami approach, particularly in
large-scale applications, has been limited by the
scalability and efficiency of solvers. This paper
addresses this limitation by developing two
nonlinear solvers based on the Jacobian-Free
Newton-Krylov (JFNK) methodology. A key feature of
these methods is that the Jacobian is not formed
explicitly for use by the underlying linear
solver. Iterative linear solvers such as the
Generalized Minimal RESidual (GMRES) method do not
technically require the stand-alone Jacobian;
instead its action on a vector is approximated
through two nonlinear function evaluations. The
preconditioning required by GMRES is also discussed;
two different preconditioners are developed, both of
which are readily treated with existing Algebraic
Multigrid (AMG) methods. Further, the most efficient
preconditioner overall for the problems considered
is based on a Picard linearization. Numerical
examples demonstrate that these new solvers are
significantly faster than a standard Newton-Krylov
approach; a speedup factor of approximately 26 was
obtained for the Picard preconditioner on the
largest grids studied here. In addition, these JFNK
solvers exhibit good algorithmic scaling with
increasing grid size.},
pdf = {http://math.lanl.gov/~berndt/Papers/setup-paper.pdf}
}
@article{JDMoulton_CFox_DSvyatskiy_2008a,
author = {J. D. Moulton and C. Fox and D. Svyatskiy},
title = {Multilevel Approximations in Sample-Based Inversion from
the {D}irichlet-to-{N}eumann Map},
journal = journalofphysicsconferenceseries,
volume = {124},
pages = {012035 (10pp)},
url = {http://stacks.iop.org/1742-6596/124/012035},
doi = {10.1088/1742-6596/124/1/012035},
year = {2008},
abstract = {In 2005, Christen and Fox introduced a delayed
acceptance Metropolis-Hastings (DAMH) algorithm that
improved computational efficiency in sample-based
imaging of electrical conductivity (EIT). That work
used a linear approximation to the forward map in
the first step of the algorithm. In this paper, we
develop an alternative approximation for use in
DAMH, namely a multilevel approximation developed
from the hierarchy of coarse-scale models obtained
by variational coarsening. This approach builds on
two important strengths of robust multigrid
solvers. First, the cost of a fine-scale solution of
the forward map scales linearly with the degrees of
freedom, and hence, it is provides better efficiency
for algorithms performing sample-based
inference. Second, the homogenization implicit in
robust variational multigrid methods gives better
solutions at coarse scales than homogenization by
averaging of coefficients. We report results from a
stylized example in electrical impedance imaging
where data is a noisy and incomplete measurement of
the Dirichlet-to-Neumann map.}
}
@techreport{JDMoulton_CFox_DSvyatskiy_2007a,
author = {J. D. Moulton and C. Fox and D. Svyatskiy},
title = {Multilevel Approximations in Sample-Based Inversion from
the Dirichlet-to-Neumann Map},
institution = {Mathematical Modeling and Analysis Group},
year = 2007,
number = {{LA-UR} 07-7958},
address = {Los Alamos National Laboratory},
note = {\emph{submitted to the} Proceedings of the
First International Congress of IPIA
Conference on Applied Inverse Problems 2007:
Theoretical and Computational Aspects,
June 25--29, 2007, Vancouver, Canada.},
abstract = {In 2005, Christen and Fox introduced a delayed
acceptance Metropolis-Hastings (DAMH) algorithm that
improved computational efficiency in sample-based
imaging of electrical conductivity (EIT). That work
used a linear approximation to the forward map in
the first step of the algorithm. In this paper, we
develop an alternative approximation for use in
DAMH, namely a multilevel approximation developed
from the hierarchy of coarse-scale models obtained
by variational coarsening. This approach builds on
two important strengths of robust multigrid solvers.
First, the cost of a fine-scale solution of the
forward map scales linearly with the degrees of
freedom, and hence, it is provides better efficiency
for algorithms performing sample-based inference.
Second, the homogenization implicit in robust
variational multigrid methods gives better solutions
at coarse scales than homogenization by averaging of
coefficients. We report results from a stylized
example in electrical impedance imaging where data
is a noisy and incomplete measurement of the
Dirichlet-to-Neumann map.}
}
@techreport{JDMoulton_TMAustin_MShashkov_JEMorel_DSvyatskiy_2007a,
author = {J. D. Moulton and T. M. Austin and M. Shashkov
and J. E. Morel and D. Svyatskiy},
title = {A Mimetic Preconditioner for Mixed Discretizations
of the Diffusion Equation},
institution = {Mathematical Modeling and Analysis Group},
year = 2007,
number = {{LA-UR} 07-8396},
address = {Los Alamos National Laboratory}
}
@techreport{JDMoulton_KLipnkov_JFung_SRunnels_2006a,
author = {D. Moulton and K. Lipnikov. and J. Fung and S. Runnels},
title = {Discretization Schemes on Polygonal and
Polyhedral Grids for Diffusion Problems},
institution = {Mathematical Modeling and Analysis Group},
address = {Los Alamos National Laboratory},
year = 2007,
number = {{LA-UR} 07-1588},
note = {\emph{to appear in the} Proceedings of NECDC 2006,
October 23--27, Los Alamos National Laboratory,
Los Alamos, NM}
}
@article{SPMacLachlan_JDMoulton_2006a,
author = {S. P. MacLachlan and J. D. Moulton},
title = {Multilevel Upscaling through Variational Coarsening},
journal = waterresourcesresearch,
year = 2006,
volume = 42,
eid = {W02418},
doi = {10.1029/2005WR003940},
abstract = {A new efficient multilevel upscaling procedure for
single-phase saturated flow in porous media is
presented. While traditional approaches to this
problem have focused on the computation of an
upscaled hydraulic conductivity, here the
coarse-scale model is created explicitly from the
fine-scale model through the application of
operator-induced variational coarsening. This
technique, which originated with robust multigrid
solvers, has been shown to accurately capture the
influence of fine-scale heterogeneous structure over
the complete hierarchy of coarse-scale models that
it generates. Moreover, implicit in this hierarchy
is the construction of interpolation operators that
provide a natural and complete multiscale basis for
the fine-scale problem. Thus, this new multilevel
upscaling methodology is similar to the Multiscale
Finite Element Method (MSFEM) and, indeed, it
attains similar accuracy in computations of the
fine-scale hydraulic head and coarse-scale normal
flux on a variety of problems; yet it is an order of
magnitude faster on the examples considered here.},
pdf = {http://math.lanl.gov/~moulton/papers/upscaling/2005WR003940-af.pdf}
}
@techreport{TMAustin_MBerndt_JDMoulton_2004a,
author = {T. M. Austin and M. Berndt and J. D. Moulton},
title = {A memory efficient parallel tridiagonal solver},
institution = {Mathematical Modeling and Analysis Group},
year = 2004,
number = {{LA-UR} 03-4149},
address = {Los Alamos National Laboratory, Los Alamos, NM},
abstract = {We present a memory efficient parallel algorithm for the
solution of tridiagonal linear systems of equations
that are diagonally dominant on a very large number
of processors. Our algorithm can be viewed as a
parallel partitioning algorithm. We illustrate its
performance using some examples. Based on this
partitioning algorithm, we introduce a recursive
version that has logarithmic communication
complexity.},
pdf = {http://math.lanl.gov/~moulton/papers/multigrid/parallel-line-solver.pdf}
}
@article{MBerndt_KLipnikov_JDMoulton_MJ_Shashkov_2001a,
author = {M. Berndt and K. Lipnikov and J. D. Moulton
and M. J. Shashkov},
title = {Convergence of Mimetic Finite Difference
Discretizations of the Diffusion Equation},
journal = {East--West Journal of Numerical Mathematics},
year = 2001,
volume = 9,
pages = {253--316},
abstract = {The main goal of this paper is to establish the
convergence of mimetic discretizations of the
first-order system that describes linear
diffusion. Specifically, mimetic discretizations based
on the support-operators methodology (SO) have been
applied successfully in a number of application areas,
including diffusion and electromagnetics. These
discretizations have demonstrated excellent
robustness, however, a rigorous convergence proof has
been lacking. In this research, we prove convergence
of the SO discretization for linear diffusion by first
developing a connection of this mimetic discretization
with Mixed Finite Element (MFE) methods. This
connection facilitates the application of existing
tools and error estimates from the finite element
literature to establish convergence for the SO
discretization. The convergence properties of the SO
discretization are verified with numerical examples.
}
}
@article{DMTartakovsky_JDMoulton_VAZlotnik_2000a,
author = {D. M. Tartakovsky and J. D. Moulton and V. A. Zlotnik},
title = {Kinematic Structure of Minipermeameter Flow},
journal = waterresourcesresearch,
year = 2000,
volume = 36,
pages = {2433--2442},
abstract = {Minipermeameters are rapidly becoming a popular tool for
collecting localized measurements of permeability in
both laboratory and field studies. While one of the
main advantages of minipermeameters is their ability
to collect data on various support volumes, there have
been only limited attempts to analyze their size and
geometry. We define the support volume of
minipermeameter measurements as a region containing
$90\%$ of the total gas flow, i.e., a region bounded
by the $10\%$ streamline. Using our new
semi-analytical solutions for the Stokes' stream
function we demonstrate that the support volume has
the shape of a semi-toroid adjacent to the sample
surface. Hence there is a blind spot directly below
the minipermeameter, which is not probed by the
measurement. We demonstrate that the support volume
of the minipermeameter measurements decreases with the
tip-seal's ratio (a ratio of the inner tip-seal radius
to the outer tip-seal radius), while the size of the
corresponding blind spot increases.
},
pdf = {http://math.lanl.gov/~moulton/papers/permeameter/permeameter.pdf}
}
@inproceedings{JEDendy_JDMoulton_1999a,
author = {J. E. Dendy and J. D. Moulton},
title = {Some Aspects of Multigrid for Mixed Discretizations},
booktitle = {Multigrid VI, Proceedings of the Sixth European
Multigrid Conference, held in Gent, Belgium,
September 27-30, 1999},
pages = {80--86},
year = 2000,
editor = {E Dick and K. Riemslagh and J. Vierendeels},
volume = 14,
series = {Lecture Notes in Computational Science and Engineering},
publisher = {Springer--Verlag},
abstract = {A broad class of discretizations of the diffusion
operator is based on its first order form, allowing
the rigorous enforcement of many desirable physical
properties of the continuous model. In this research
we investigate the development of multilevel solvers
for the \emph{local} or \emph{hybrid} forms of these
discretizations on logically rectangular
quadrilateral meshes. In this case, the local
elimination of flux leads to a system that contains
both cell- and edge-based scalar unknowns. Based on
this natural partitioning of the system we develop
approximate reduced systems that reside on a single
logically rectangular grid. Each such approximate
reduced system, formed as an approximate Schur
complement or as a variational product, are used as
the first coarse-grid in a multigrid hierarchy or as
a preconditioner for Krylov based methods.
},
pdf = {http://math.lanl.gov/~moulton/papers/preconditioners/EMG-mixed-1999.pdf}
}
@techreport{JDMoulton_SKnapek_JEDendy_1999a,
author = {J. David. Moulton and Stephan Knapek and Joel E. Dendy},
title = {Multilevel Upscaling in Heterogeneous Porous Media},
institution = {Center for Nonlinear Studies},
year = 1999,
type = {Research Highlights},
number = {{LA-UR} 99-4754},
address = {Los Alamos National Laboratory, Los Alamos, NM},
month = {January},
abstract = {The multiscale structure of heterogeneous porous media
prevents a straightforward numerical treatment of the
underlying mathematical flow models. In particular,
fully resolved flow simulations are intractible and
yet the fine-scale structure of a porous medium may
significantly influence the coarse-scale properties
of the solution (e.g., average flow rates).
Consequently, homogenization or upscaling procedures
are required to define approximate coarse-scale
models suitable for efficient computation.
Unfortunately, inherent in such a procedure is a
compromise between its computational cost and the
accuracy of the resulting coarse-scale solution. In
general, most popular upscaling methods do not
balance these competing demands. In this paper we
highlight a new efficient, numerical method, which
combines our recent work on multigrid homogenization
(MGH) with the work of Dvo\v{r}\'{a}k (1994) to compute
bounded estimates of the homogenized permeability for
single phase saturated flows. Our approach is
motivated by the observation that the coarse-scale
influence of multiscale structures are captured
automatically by robust variationally defined
multigrid methods. The effectiveness of this new
algorithm is demonstrated with numerical examples.
},
pdf = {http://math.lanl.gov/~moulton/papers/homogenization/cnls-1999-01.pdf}
}
@unpublished{warren-1999-avalon,
author = {Michael S. Warren and Aric Hagberg and J. David Moulton
and David Neal and John K. Salmon},
title = {{A}valon: Champagne computing on a beer budget},
journal = {},
year = {1999},
volume = {},
pages = {},
abstract = {Avalon is a 140 processor Alpha/Linux Beowulf cluster
constructed entirely from commodity personal computer
technology and freely available
software. Computational Physics simulations performed
on Avalon resulted in the award of a 1998 Gordon Bell
price/performance prize for significant achievement
in parallel processing. Avalon ranked as the 113th
fastest computer in the world on the November 1998
TOP500 list, obtaining a result of 47.8 Gigaflops on
the parallel Linpack benchmark.
Avalon currently provides over 15,000 node-hours of
production computing time per week, split among about
10 production users. Obtaining an equivalent amount
of computing through Los Alamos institutional sources
would cost a minimum of \$45,000 per week. The
machine also supports code development for another 50
users. The largest single simulation was performed in
April and May 1998 using the SPaSM molecular dynamics
code, which computed a total of $1.12 \times 10^{16}$
floating point operations. This simulation is among
the few scientific simulations to have ever involved
more than 10 Petaflops of computation.
The price of hardware and final assembly labor for
Avalon totalled \$313,000 dollars. The monetary cost
of the development and OS software used for the
applications mentioned above was \$0. Perhaps most
extraordinary, all of the hardware and software
maintenance on the machine is performed in the spare
time of four people, averaging less than 10 man-hours
of labor per week overall.
},
url = {http://cnls.lanl.gov/avalon/},
note = {Extended abstract}
}
@article{JDMoulton_JEDendy_JMHyman_1998a,
author = {J. D. Moulton and J. E. Dendy and J. M. Hyman},
title = {The Black Box Multigrid Numerical Homogenization Algorithm},
journal = journalofcomputationalphysics,
year = 1998,
volume = 141,
pages = {1--29},
abstract = {In mathematical models of flow through porous media, the
coefficients typically exhibit severe variations in
two or more significantly different length scales.
Consequently, the numerical treatment of these
problems relies on a \emph{homogenization} or
\emph{upscaling} procedure to define an approximate
coarse-scale problem that adequately captures the
influence of the fine-scale structure. Inherent in
such a procedure is a compromise between its
computational cost and the accuracy of the resulting
coarse-scale solution. Although techniques that
balance the conflicting demands of accuracy and
efficiency exist for a few specific classes of
fine-scale structure (e.g., fine-scale periodic),
this is not the case in general.
In this paper we propose a new, efficient, numerical
approach for the \emph{homogenization} of the
permeability in models of single-phase saturated
flow. Our approach is motivated by the observation
that multiple length scales are captured
automatically by robust multilevel iterative solvers,
such as Dendy's \emph{black box multigrid}. In
particular, the operator-induced variational
coarsening in black box multigrid produces
coarse-grid operators that capture the essential
coarse-scale influence of the medium's fine-scale
structure. We derive an explicit local, cell-based,
approximate expression for the symmetric, $2 \times
2$ homogenized permeability tensor that is defined
implicitly by the black box coarse-grid operator.
The effectiveness of this black box multigrid
numerical homogenization method is demonstrated
through numerical examples.
},
pdf = {http://math.lanl.gov/~moulton/papers/homogenization/JCP-mgh-1998.pdf}
}
@article{JDMoulton_JEMorel_UMAscher_1998a,
author = {J. D. Moulton and J. E. Morel and U. M. Ascher},
title = {Approximate Schur Complement Preconditioning of the
Lowest-Order Nodal Discretizations},
journal = siamjournalsc,
year = 1998,
volume = 19,
number = 1,
pages = {185--205},
month = {Jan},
abstract = {Certain classes of nodal methods and mixed-hybrid finite
element methods lead to equivalent, robust, and
accurate discretizations of second-order elliptic
PDEs. However, widespread popularity of these
discretizations has been hindered by the awkward
linear systems which result. The present work
overcomes this awkwardness and develops
preconditioners which yield solution algorithms for
these discretizations with an efficiency comparable to
that of the multigrid method for standard
discretizations. Our approach exploits the natural
partitioning of the linear system obtained by the
mixed-hybrid finite element method. By eliminating
different subsets of unknowns, two Schur complements
are obtained with known structure. Replacing key
matrices in this structure by lumped approximations,
we define three optimal preconditioners. Central to
the optimal performance of these preconditioners is
their sparsity structure which is compatible with
standard finite difference discretizations and hence
treated adequately with only a single multigrid
cycle. In this paper we restrict the discussion to the
two-dimensional case; these techniques are readily
extended to three dimensions.
},
pdf = {http://math.lanl.gov/~moulton/papers/preconditioners/SIAM-nodal-1998.pdf}
}
@phdthesis{JDMoulton_1996a,
author = {J. D. Moulton},
title = {Nodal Methods: Analysis, Performance and Fast Iterative Solvers},
school = {University of British Columbia},
year = 1996,
address = {Institute of Applied Mathematics},
month = {November},
abstract = {\emph{Nodal Methods} have long been one of the most
popular discretization techniques employed within the
reactor physics community, while remaining
conspicuously absent from the mainstream numerical
analysis literature. A fundamental reason for this
anomaly is that the physical arguments which were used
to develop and enhance these methods seemed at odds
with more rigorous discretization techniques. To
facilitate communication between these distinct
communities, a detailed chronological study of the
lowest-order nodal methods for linear second order
elliptic problems is presented. The presentation
highlights the underlying motivation of these methods
and formalizes many of their renowned properties. In
addition, various equivalence relations within this
family of discretizations are demonstrated, and
equivalences with specific non-conforming and
mixed-hybrid finite element methods (FEMs) are
established. Rigorous error bounds and stability
properties follow immediately from these latter
equivalence relations, corroborating the results of a
more rudimentary truncation error analysis.
An inherent difficulty of reactor simulation is that
the coefficients in the mathematical model exhibit
severe variations on two significantly different
length scales. As in many other applications this is
treated by defining an appropriate homogenization
procedure which yields a simplified model with
piecewise constant coefficients on a coarse scale
suitable for efficient computation. Significant
enhancements in accuracy are possible if the processes
of homogenization and discretization are unified. We
review the popular techniques that are based on this
premise and rely on certain properties of the nodal
discretization. In addition, we address the factors
that contribute to their success in reactor modelling
and deter their generalization outside of the reactor
physics community. As an alternative to these highly
specialized methods, we introduce a new multi-level
homogenization technique which is readily applicable
in a general setting, and is shown to have many
important attributes.
Widespread acceptance of nodal methods has also been
hindered by their use of nonstandard unknowns, as this
results in stencils that appear awkward and
incompatible with sophisticated iterative solution
techniques. Specifically, equivalence with certain
mixed-hybrid FEMs reveals that the nodal
discretizations result in an indefinite system, which
in two dimensions contains both cell-based and
edge-based unknowns. Yet, inherent in this structure
is a natural partitioning of the system which may be
exploited to define a hierarchy of reduced systems
(i.e. Schur complements) that are symmetric positive
definite. Unfortunately, the reduced systems that
involve unknowns of only one type, and hence seem most
compatible with sophisticated iterative methods,
suffer a loss of sparsity. However, the structure
inherent in this hierarchy may be utilized in the
construction of sparse approximate Schur complements
for these systems. It is demonstrated that any one of
these approximate operators, which are of either the
standard 5 or 9-point family, may be utilized as an
excellent preconditioner for conjugate gradient
iterations. The efficiency of this approach is fully
realized when the preconditioner is
\emph{approximately} inverted using only a single V-
or W-cycle of a robust \emph{Black Box} multigrid solver.
}
}
@article{MSPatterson_JDMoulton_BCWilson_1991a,
author = {M. S. Patterson and J. D. Moulton and B. C. Wilson
and K. W. Berndt and J. R. Lakowicz},
title = {Frequency-Domain Reflectance for the Determination of the
Scattering and Absorption Properties of Tissue},
journal = {Appl. Opt.},
year = 1991,
volume = 30,
number = 31,
pages = {4474--4476},
abstract = {Measurements of the phase and modulation of
amplitude-modulated light diffusely reflected by
turbid media can be used to deduce absorption and
scattering coefficients.}
}
@inproceedings{MSPatterson_SJMadsen_JDMoulton_BCWilson_1991a,
author = {M. S. Patterson and S. J. Madsen and J. D. Moulton
and B. C. Wilson},
title = {Diffusion Equation Representation of Photon Migration
in Tissue},
booktitle = {International Microwave Symposium},
pages = {905--908},
year = 1991,
editor = {G. L. Hieter},
volume = 2,
organization = {{IEEE} {MTT-S}},
abstract = {A time dependent diffusion model of photon migration
in tissue is used to develop analytic expressions for
the diffusely reflected pulse detected some distance
from a delta function input. Particular attention is
paid to the nature of the boundary between the tissue
and the surrounding non-scattering medium, and it is
shown that the pulse shape is relatively insensitive
to the nature of this boundary. Monte Carlo
simulation and experimental results are presented
which confirm the accuracy of the diffusion model.
}
}
@inproceedings{SJMadsen_MSPatterson_BCWilson_etal_1991a,
author = {S. J. Madsen and M. S. Patterson and B. C. Wilson
and Y. D. Park and J. D. Moulton and S. L. Jacques
and Y. Hefetz},
title = {Time Resolved Diffuse Reflectance and Transmittance Studies
in Tissue Simulating Phantoms: A comparison between Theory
and Experiment},
booktitle = {Time Resolved Spectroscopy and Imaging of Tissues},
pages = {42--51},
year = 1991,
editor = {Britton Chance},
volume = 1431,
organization = {SPIE},
abstract = {When a picosecond light pulse is incident on an
optically turbid medium such as tissue, the temporal
distribution of diffusely reflected and transmitted
photons depends on the optical absorption and
scattering properties of the medium. From
diffusion theory it is possible to derive analytic
expression for the pulse shape in terms of optical
interaction coefficients of a homogeneous
semi-infinite medium. Experimental tests of this
model in tissue-simulating liquid phantoms of
different geometries are presented here.
}
}
@mastersthesis{JDMoulton_1990a,
author = {J. D. Moulton},
title = {Diffusion Modelling of Picosecond Pulse Propagation
in Turbid Media},
school = {McMaster University},
year = 1990,
address = {Hamilton, Ontario},
month = {August},
abstract = {The increasing use of visible and near infrared light
in therapeutic and diagnostic techniques has created a
need to model its propagation in tissue. One of the
fundamental objectives of such a model is the
noninvasive evaluation of the optical properties of
tissue. The focus of this thesis was the development
of the \emph{diffusion approximation} in the
semi-infinite, slab, cylindrical and spherical
geometries. This development required the derivation
of approximate boundary conditions which included the
zero, extrapolated and partial current boundary
conditions. Calculations of the fluence and its
related quantities arising from the extrapolated
boundary condition were found to be in excellent
agreement with the results of the more rigorous
partial current boundary condition.
A preliminary evaluation of the validity of diffusion
theory was performed by comparing its predictions to
exact analytical calculations of the fluence in an
infinite medium as well as Monte Carlo simulations of
the reflectance and transmittance in 1-dimensional
planar geometries. In all cases the agreement at
late times was excellent. A practical test of the
diffusion model was accomplished with the analysis of
the reflectance data from a phantom of known optical
properties in both the semi-infinite and slab
geometries. The model perfomed well at low
concentrations of added absorber, but a considerable
discrepancy was found at the highest concentration.
A systematic examination of the accuracy of the
diffusion model as a function of the fundamental
parameters is required to resolve this
inconsistency.
Approximate expressions describing the equivalent
information in the frequency domain were also
developed for a semi-infinite medium. These
expressions were then used to analyze the phase and
modulation obtained from phantoms of known optical
properties. Once again reasonable results were
obtained at low concentrations of added absorber
while a significant discrepancy arose at the highest
concentration. The resolution of these descrepancies
requires further investigation.
}
}
@inproceedings{MSPatterson_JDMoulton_BCWilson_BChance_1990a,
author = {M. S. Patterson and J. D. Moulton
and B. C. Wilson and B. Chance},
title = {Applications of Time-Resolved Light Scattering
Measurements to Photodynamic Therapy Dosimetry},
booktitle = {Photodynamic Therapy Mechanisms II},
pages = {62--75},
year = 1990,
editor = {T. J. Dougherty},
volume = 1203,
organization = {SPIE},
abstract = {Since biological response to photodynamic therapy (PDT)
depends on the light fluence distribution and
photosensitizer concentration in the tissue, these
two variables should ideally be measured
noninvasively in individual cases. This can be
reduced to determining the optical absorption and
transport scattering coefficients of the tissue
because, if these two parameters are known, the
fluence distribution can be calculated and the
photosensitizer concentration can be deduced from tis
characteristic contribution to the absorption
spectrum. The temporal spreading of a picosecond
laser pulse as it propagates through tissue carries
information about both the scattering and absorption
properties of the tissue. A mathematical model is
presented which allows derivation of the interaction
coefficients from the pulse shape, and preliminary
experiments are reported which demonstrate the
potential of these techniques in PDT dosimetry.
Equivalent information can be obtained in the
frequency domain by using modulated light sources and
measuring the phase and modulation of the detected
light. Analytical expressions are developed for
these observable quantities in terms of the optical
interaction coefficients. Particular attention is
drawn to the potential of low (less than 200MHz)
frequency measurements as these can be made with
relatively simple instrumentation.
}
}
@inproceedings{BCWilson_MSPatterson_STFlock_JDMoulton_1988a,
author = {B. C. Wilson and M. S. Patterson and S. T. Flock
and J. D. Moulton},
title = {The Optical Absorption and Scattering Properties
of Tissues in the Visible and Near-Infrared
Wavelength Range},
booktitle = {Light in Biology and Medicine},
pages = {45--52},
year = 1988,
editor = {R. H. Douglas and J. Moan and F Dall'Acqua},
volume = 1,
publisher = {Plenum Publishing Corp.},
abstract = {The development of diagnostic and therapeutic
photomedicine has generated a need to determine the
optical properties of tissues in the U.V., visible
and intrared regions of the spectrum. In this paper
we will review the experimental techniques and
resulting data on the optical prperties of mamalian
tissues. These will include recent results from this
laboratory as well as a summary of the work of other
groups. Measurements are most abundant at around
630nm, the wavelength of greatest current interest
for clinical photodynamic therapy. We will use these
data as the reference values for examaning the
wavelength-dependence of the optical properties.
}
}
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