How do we find meaning in an infinite dimensional continuum model of
nature, such as a system of nonlinear partial differential equations?
How can predictions be made based on these models?
How can these models be used to make forecasts that help our life?
One example of research addressing these questions is the modeling of flow and transport in porous media. A porous medium consists of a solid with a large amount of connected or disconnected holes allowing the fluid to pass. Soil and human tissues are considered porous to some extent. Applications of flow and transport include groundwater monitoring for water supply, contaminant remediation and storage of high level nuclear waste, blood flow modeling for medical purposes and production of oil and natural gas. From a humane and economical point of view, these applications are of crucial importance and the use of mathematics can help make a difference (energy can be extracted in an environmental and efficient manner..) Our group is developing state of the art numerical methods for simulating the flow and reactive transport in porous media. The methodologies include finite element methods (continuous, discontinuous, mixed) and various tools of numerical analysis (domain decomposition, adaptivity...)
As another example, understanding turbulent flow is central to many
important problems including environmental and energy related applications
(global change, mixing of fuel and oxidizer in engines and drag reduction),
aerodynamics (maneuvering flight of jet aircraft) and biophysical
applications (blood flow in the heart, especially the left ventricle).
Turbulent flow is composed of coherent patches of swirling fluid called
eddies. These range in size from large storm systems such as hurricanes
to the little swirls of air shed from a butterfly's wings. Large Eddy Simulation
(LES for short) seeks to predict the motion of the largest and most important
eddies uncoupled from the small eddies. This uncoupling is important because
the large eddies are resolvable on a computational mesh (a collection of
chunks of the physical problem) which can be handled by a supercomputer.
Our group's research involves modeling the large
eddies (such as storm fronts, hurricanes and tornadoes in the atmosphere)
in turbulent flow, predicting their motion in computational experiments
and validating mathematically the large eddy models and algorithms developed.
Current approaches to LES seem to be presently confronting some barriers
to resolution, accuracy and predictability. It seems likely that many of
these barriers can be traced to the mathematical foundation of the models
used, the boundary conditions imposed and the algorithms employed for the
simulations. The research undertaken is to develop these mathematical foundations
as a guide for practical computation. This research promises to make
it possible to extend the range of accuracy and reliability of predictions
important to applications, such as those described above, where technological
progress requires confronting turbulence!
An important component of technology progress is the ability to make the best design choices for a device or an experiment. Choosing the shape of an air foil to minimize the fuel consumption of an airplane is one such example. Determining the protein structure that best fits a set of chrystallographic measurements is another example. Mathematically and algorithmically this is accomplished by numerical optimization, whose reach is expanded by advanced algorithms developed in our group.
But optimization is useful sometimes in unexpected places. An area that has been pursued in our group has been the simulation of the collision and contact of many bodies by using time steps that are much larger (resulting in a much faster computation) compared to other algorithms. To achieve that, one has to compute one optimization problem every time the future position and velocity of the bodies needs to be predicted. Simulations of this type include simulations of granular matter, such as cereal, or powders used in drug manufacturing. We have been able to simulate with very large time steps exciting and difficult phenomena such as the Brazil nut effect (link to simulation), where the larger nuts rise to the top of a container that is shaked. But this approach is also useful in the design of interactive video games. Some of our methods have been incorporated by a commercial vendor of a physics engine for video games and is currently incorporated by many of the major commercial game developers Karma .
Progress in science and technological innovation depends on simulations at the leading edge of scientific computing on high performance computers using tools developed by computational mathematicians at Pitt and elsewhere. Simulations of many important problems remain beyond the capabilities of computing capabilities for the foreseeable future. The scientific work of researchers in Computational Mathematics is essential to the United States' industrial competitiveness as well as our leadership in fundamental scientific research.
Computational Mathematics has important things
to say about our world.
Come help with these challenges in our group
at Pitt!
Our department provides a full complement of support for strong
graduate students, including Fellowships, Teaching Assistantships, free
tuition and free health insurance.
Apply Online
to our graduate program
Year 1
During year 1, an initial advisor is assigned. (As your interests change
and evolve, it is normal to move from one advisor to another in our department.)
Please feel free to request an advisor in Computational Mathematics.
At the end of year 1 come the Preliminary Examinations, Syllabus.
The goal of your first year is to prepare for the prelim , pass it and
move on to research as quickly as possible.
Course work:
End of year 1: Preliminary Examination.
This consists of two, 3 hour exams, one on advanced calculus and one
on linear algebra. Currently, these are offered in May and in August. There
is no penalty for failing; thus we encourage all of you to take them at
every opportunity to get a feeling for what the exams are like, including
your first August in Pittsburgh. Old exams are available for study Old
Exams.
Summer Year 1.
There are many options for your summer. These include:
As your studies advance you also take advantage of our 4 month summer
term for foreign research travel, visits to summer research institutes,
and work at internships in Computational Laboratories.
Year 2
Typically year 2 is consumed with taking advanced courses in computational
mathematics and getting ready to read research papers in the area. Our
students also begin participating in our research training seminars in
year 2. At the end of year 2 comes the comprehensive examination.
Comprehensive examination.
This exam is based on plan drawn up by a committee of faculty in computational
mathematics with input by you, comps
. The Computational Mathematics Group's philosophy of this exam is
that it tests if you're really ready to begin reading research
papers in computational mathematics. Our goal is for you to be an independent
and accomplished research scientist in the area. We design our comprehensive
exams with this in mind.
Summer Year 2.
The options of summer year 1 are open in year 2, and every subsequent
summer. In addition, we feel it is very important to began research in
the area of Computational Mathematics at an early date. Summer year 2
is an ideal time to begin working on a group research project.
Year 3.
You typically take an advanced topics class in computational mathematics
and forge ahead in your research.
During year 3 you will typically begin a group research project
within our research training group. You will be ready to read papers the
area and be fully integrated with our faculty and researchers in the area.
By this time you'll be ready to select your Ph.D. thesis advisor from among
the relevant faculty in the area. In computational mathematics students
are extremely lucky. All our faculty are highly active in research. All
have an excellent record in funded research and experience directing Ph.D.
theses.
Pick any faculty member as your thesis advisor in computational mathematics and you will "Hitch the wagon to a star".
Computational mathematics is not an area in which research can be done in isolation. If you work with one faculty member, you will receive the full support of all the faculty members in the group. You'll have an academic advisor (sometimes called an academic father or mother) but also the support of academic aunts and uncles. We work as a fully integrated intellectual family .
Upon selecting your thesis advisor you'll begin to plan and prepare for your Ph.D. thesis. This means forming a Ph.D. thesis committee and preparing a research plan and work plan for accomplishing the research. Once this is prepared you present a Ph.D. thesis "Overview presentation" to the faculty in the area. These are all faculty that you will have been working with and taking courses from for 2+ years now.
Summer year 3.
Our summer program focuses on research, research and
more research. This is supported by international travel,
attending special research workshops at our Pittsburgh Supercomputing
Center and summer research institutes.
Year 4 .
The focus of year 4 is your Ph.D. thesis research. To graduate in year
4, you must be be prepared by November or December to enter the job market.
This means, we will train you in the important skills of a scientist:
At the end of year 4 (or year 5 if required) comes the Ph.D.
thesis defense in which you present your work. By this time you will
be excited to give a talk and proud of the work you have done.
When the time comes for your job search, you will have scientific publications and research talks-accomplishments which document your work in computational mathematics. You will know current funding research trends in your area and you'll be highly skilled at presenting this work to both specialists and non-specialists.
MATH 1080, Numerical Linear Algebra
This course in NLA is for students interested in solving scientific
and engineering problems which involve lots of data and more than one dimension.
Basically, any such problem reduces eventually to one in numerical linear
algebra. The course gives an introduction to the direct and
iterative algorithms for solving linear systems. The course will cover
the development and analysis of these numerical algorithms, to be used
in the resolution of linear systems , the algebraic eigenvalue problem
and least squares problems.
Math 1110, Industrial Mathematics
This course introduces various methods of applied mathematics used
to solve industrial type problems. It addresses the five stages in mathematical
modeling: physical problem,
mathematical model, discrete (numerical) model, computation of a solution,
output data analysis. Central topics are differential equations (continuous
model) and matrix equations
(discrete model).
MATH 1100, Linear Programming
The course M1100 gives an introduction to the basic areas of linear
programming. The course will cover the development and analysis of
algorithms for linear programming, with an emphasis on the simplex
algorithm.
Math 2030 Iterative Methods for Linear and
Nonlinear Systems
The course gives an introduction to the iterative algorithms
for solving linear and nonlinear systems. The course will cover the development
and analysis of these
numerical algorithms, to be used in the resolution of linear and nonlinear
systems. (Offered frequently )
Math 2090: Numerical Solution of Ordinary Differential Equations
This course aims to give an in-depth introduction to the numerical
methods for solving ordinary differential equations. Both initial value
problems and boundary value problems are
considered. Important practical issues such as stability, stiffness,
error estimation and control will be considered for Runge-Kutta methods,
multistep methods and finite difference
methods. If time permits, numerical techniques for differential-algebraic
equations will be also presented.
(Offered frequently )
Math 2480 : COMPUTATIONAL APPROXIMATION THEORY
This course in Computational Approximation Theory is designed for students
interested the mathematical foundations of methods for solving scientific
and engineering problems on computers. The emphasis is on the mathematical
ideas behind approximating a function with one determined by a finite number
of degrees of freedom and applications of this , such as quadrature (numerical
integration). (Offered intermittently according to interests of students
and faculty)
Math 2601 ADVANCED SCIENTIFIC COMPUTING 1
Math 2602 ADVANCED SCIENTIFIC COMPUTING 2
Math 2603 ADVANCED SCIENTIFIC COMPUTING 3
Math 2604 ADVANCED SCIENTIFIC COMPUTING 4
The topics of these four courses rotate according to
interests and current trends in scientific computing. Examples include:
Large Eddy Simulation and Computational Turbulence and Domain Decomposition
for PDEs. (Offered every term)
Math 2960 COMPUTATIONAL FLUID MECHANICS
This course studies the mathematical analysis of finite element methods
for approximating the flow of viscous incompressible fluids. This includes
the oldest area of mathematics up to the most modern. We pick a path through
the subject allowing a penetration of the field that is both intuitive
and rigorous.
TOPICS include:
1. Finite Element Approximation of Scalar PDEs.
2. Vectors ,Tensors and Conservation Laws.
3. Approximating Vector Functions: Mixed Methods.
4. The Equations of Fluid Motion.
5. Equilibrium Laminar Flows.
6. Approximating Equilibrium Laminar Flows.
7. The Time Dependent NSE
8. Approximating the Time Dependent NSE
9. Turbulence.
(Offered frequently according to interests of students and faculty)
Math 3030 MATRIX ITERATIVE ANALYSIS
(Offered intermittently according to interests of students and faculty)
Math 3035 DIFFERENCE METHODS
(Offered intermittently according to interests of students and faculty)
Math 3040 TOPICS IN SCIENTIFIC COMPUTING: High
Performance Computing
(Offered according to interests of students and faculty)
Math 3070 NUMERICAL SOLUTION OF NONLINEAR EQUATIONS
(Offered intermittently according to interests of students and faculty)
Math 3071 , Numerical Solutions of Partial Differential Equations
This course is an introduction to modern numerical methods for solving
partial differential equations. It will cover both finite difference and
finite element methods. Accuracy,
stability, and efficiency of the algorithms are studied from
both theoretical and computational standpoint.
(Offered frequently )
Math 3072: Finite Element Method
This course is an introduction to the theoretical and computational
aspects of the finite element method for the solution of boundary value
problems for partial differential equations.
Emphasis will be on linear elliptic, self-adjoint, second order problems,
although some material on fourth order problems will be presented. Topics
include: variational formulation of
boundary value problems, natural and essential boundary conditions,
Lax Milgram lemma, approximation theory, error estimates, element construction,
continuous and discontinuous
finite element methods, and solution methods for the resulting finite
element systems.
(Offered frequently )
Math 3075 PARALLEL FINITE ELEMENT METHODS
(Offered intermittently according to interests of students and faculty)
Math 3077 COMPUTATIONAL FOURIER ANALYSIS &
APPLICATIONS
(Offered intermittently according to interests of students and
faculty)
Math 3435 COMPUTATIONAL WAVELETTS AND FRACTAL IMAGE
ANALYSIS
(Offered intermittently according to interests of students and
faculty)
Math 3480 SPLINE APPROXIMATION
(Offered intermittently according to interests of students and
faculty)
Math 3910 SEMINAR IN SCIENTIFIC COMPUTING
(Offered intermittently according to interests of students and
faculty)
Math 3965 ADVANCED COMPUTATIONAL FLUID MECHANICS
(Offered intermittently according to interests of students and
faculty)
Numerical Optimization,
Constrained Optimization, Differential Algebraic Equation, Applications
Computational Fluid Dynamics,
Large Eddy Simulation and Turbulence More
on LES , Finite Element Methods for Fluid Flow and Natural Convection
Problems,
Parallel Algorithms for Nonsymmetric Problems, Multi-scale Discretizations
of Flow Problems, Uncertainties in Turbulent Flow Simulations.
Numerical analysis of flow and transport processes in porous media,
Large scale scientific computing with applications to flow in porous
media, massively parallel simulations of multiphase porous media and surface
flows on irregular multiblock domains, Coupling of Stokes and Darcy using
DG, Coupling of DG and MFE for single phase flow, Coupling of Stokes
and Darcy using DG coupled with MFE, Single phase flow: highly discontinuous
permeability, fractures, Miscible displacement: stable and unstable flow,
Two-phase flow: BL, Pressure wave propagation
Finite Element Methods
Discontinuous Galerkin finite element methods, numerical analysis of
partial differential equations, Domain Decomposition, Adaptivity,
Mixed Finite Element (MFE) Methods, FEM's for Hyperbolic Systems and
Elliptic-Hyperbolic Singular Perturbation Problems
design and analysis of accurate discretization techniques (mixed finite
elements, finite volumes, finite differences)
Large Scale Scientific Computing
Efficient nonlinear iterative solvers (domain decomposition, multigrid,
Newton-Krylov methods), Multi-numerics Approach: Theoretical Error Analysis
, Multi-physics Approach: Theoretical Error Analysis, Multi-Level Newton
Methods, Numerical Linear Algebra, Adaptive and parallel algorithms.
Analysis
Acoustic wave equations, Multi-physics/Multi-numerics Approach: Theoretical
Error Analysis, Well-posedness of Nonlinear Models,
Nonlinear Analysis Applied to Ordinary and Partial Differential Equations
Applications,
Modeling and Simulation of Aluminum Reduction Cells, Applications of
Porous Media Modeling and Simulation, Uncertainty Sensitivity and Design
in CFD Applications, Turbulence Modeling, parallel computing.
William Layton wjl@pitt.edu, (412)624-8312, Professor Layton's web page
Beatrice Riviere 412-624-8315 riviere@math.pitt.edu, Professor Riviere's web page
Ivan Yotov (412) 624-8338 , yotov@math.pitt.edu, Professor Yotov's web page
M. Sussman, (Adjunct Professor and research scientist at Bechtel-Bettis),
Dr.
Sussman's web page.
Charles Hall, Professor Emeritus ,
Computational Mechanics, Numerical Analysis, Computational Fluid
Dynamics
D. J. Hebert , Professor Emeritus,
djh+@pitt.edu,
Hebert
Computational Image Processing
Martin J. Marsden Professor Emeritus
Spline functions, approximation theory, numerical analysis.
Thomas A. Porsching Professor Emeritus, tap@pitt.edu
Numerical analysis and scientific computing, computational fluid
dynamics, network theory.
Werner C. Rheinboldt , Professor Emeritus, wcrhein@vms.cis.pitt.edu
Numerical analysis and scientific computing.
Professor V. John, Otto-von-Guericke University , Magdeburg, web
Professor L. Tobiska, Otto-von-Guericke University , Magdeburg, LT
Professor F. Schieweck, Otto-von-Guericke University , Magdeburg
Professor L. Berselli, University of Pisa, Berselli
Professor M. Kaya, Gazi University, Turkey
Professor R. Lewandowski, University Rennes, Lewandowski
G. P. Galdi, Department of Mechanical Engineering, web
P. Givi, Department of Mechanical Engineering, , Givi
A.
Robertson , Department of Mechanical Engineering,
C. Zhu, Department of Geology
Other Universities:
Todd Arbogast, University of Texas at Austin
P. Bastian, Heidelberg University, Germany
Markus Berndt, Los Alamos National Laboratory
L. C. Berselli, University of Pisa
C. Dawson, University of Texas at Austin
Q. Du, Penn State University, Du
V. Ervin, Clemson University, web
V. Girault, Paris VI University, France
M. Gunzburger, FSU, MG
V. John, Otto-von-Guericke University , Magdeburg, VJ
H. K.Lee, Clemson University
R. Lewandowski, University Rennes
Konstantin Lipnikov, Los Alamos National Laboratory
G Matthies, Otto-von-Guericke University , Magdeburg
J Maubach, Eindhoven University of Technology
A J Meir, Auburn University, Meir
J.Peterson , Florida State University
Tom Russell, University of Colorado, Denver
F. Schieweck, Otto-von-Guericke University , Magdeburg, Schieweck
P G Schmidt, Auburn University
Misha Shashkov, Los Alamos National Laboratory
L. Tobiska, Otto-von-Guericke University , Magdeburg, Tobiska
N. Troyani,
M. F. Wheeler, University of Texas at Austin
The main thrust of this project is to investigate accurate and
efficient numerical techniques for simulation of flow and transport
phenomena in porous media which are of major importance in the
environmental and petroleum industries. A novel numerical
methodology
based on multiblock domain decomposition formulations is developed.
Multiblock discretizations allow for locally constructed non-matching
grids and involve the introduction of special approximating spaces
(mortars) on interfaces of adjacent subdomains. The approach is
highly
suitable for coupling of multiple physical processes and is efficiently
implemented on massively parallel computers.
A variety of physical models have been implemented under a common
parallel framework IPARS. Current work includes multiblock coupling
of
two-phase and three-phase flow, as well as fully implicit and
semi-implicit time discretizations. Efficient parallel preconditioners
are being developed for the nonlinear interface problem arising
in
multiblock formulations. Multigrid and dynamically adaptive mortar
methods are also being investigated.
PI: Ivan Yotov, # ORAU 05.700325, June 1, 1999 - May 31, 2001, $10,000
Oak Ridge Associated Universities Ralph E. Powe Junior Faculty Enhancement
Award.
We propose to investigate accurate and efficient numerical techniques
for simulation of flow and transport phenomena in porous media which
are of major importance in the environmental and petroleum industries.
We consider a novel numerical methodology based on multiblock domain
decomposition formulations. The simulation domain is decomposed
into a
series of subdomains (blocks). The underlying equations hold with
their usual meaning on the subdomains, with physically meaningful
boundary conditions imposed on the interfaces. Each block is
independently covered by a local grid. The grids do not have to
match
on the interfaces between blocks, allowing for accurate modeling
of
irregular geometries, multi-scale heterogeneities, and internal
boundaries such as faults and layers. This approach also provides
a
basis for a rigorous coupling of different physical processes and
discretization techniques. Crucial for the success of the method
is
imposing physically meaningful matching conditions on subdomain
interfaces in a numerically stable and accurate way, as well as
designing fast converging algorithms for solving the resulting systems
of nonlinear equations. These are the two main goals of the
project. The first goal is achieved by using specially chosen
interface finite element spaces called mortars to impose continuity
of
normal fluxes across interfaces. The second issue is addressed by
developing parallel nonlinear domain decomposition solvers and
preconditioners. The research will involve both theoretical analyses
as well as computational aspects on distributed heterogeneous
platforms. The results of the research will be incorporated
in state
of the art reservoir and environmental simulators.
* A. Dunca, ardst21@pitt.edu,
web page
Adrian will likely complete his Ph.D. in May 2004: Computational
Turbulence-Simulation and Analysis.
[1] A.DUNCA , V.John, W.Layton and N.Sahin,
Numerical analysis of large eddy simulation,
DNS/LES - Progress and Challenges (Proceedings of Third AFOSR
International Conference on DNS and LES), ed. by C. Liu, L. Sakell
and T.
Beutner, Greyden Press Columbus, 359 - 364, 2001
[2] A.Dunca, V.John and W Layton,
Estimates of the commutation error in the space averaged Navier-Stokes
equations,
to appear in: J. Mathematical Fluid Mechanics, 2003.
[3] A. Dunca, and V. John
Finite Element Error analysis of space averaged flow fields defined
by a differential filter,
submitted, currently under review , Mathematical Models and Methods
in Applied Sciences,2003.
[4] A. Dunca, V. John and W. Layton,
Approximating local averages of fluid velocities: the equilibrium
Navier-Stokes equations,
to appear in: Appl. Numerical Methods, July,2003.
[5]A. Dunca,
Investigation of a shape optimization algorithm for turbulent flows,
report number ANL/MCS-P1101-1003, Argonne National Lab,2002,
http://www-fp.mcs.anl.gov/division/publications/.
[6]A. Dunca, T. Iliescu, M. Kaya and W. Layton,
Numerical Analysis and computational testing of an eddy viscosity
model for turbulent flows involving approximate TKE's,
in preparation, 2003.
[7] A.Dunca ,Y.Epshteyn,V.John
Existence and uniqueness of the Smagorinsky model with model for
the commutation error,
work in progress, 2003.
[8] A.Dunca ,Y.Epshteyn,
On the Stolz-Adams deconvolution models for LES
technical report, October ,2003.
*S. Kaya, sokst20@pitt.edu, web
page
Ph.D. thesis area: Computational Fluid Dynamics,
[1] S. Kaya and W. Layton,
Subgrid scale eddy viscosity methods are Variational Multiscale
Methods,
technical report., TR-MATH 03-05, University of Pittsburgh .
[2]S. Kaya,
Numerical Analysis of a Subgrid Scale Eddy Viscosity method for
higher Reynolds number Flow Problems,
Technical Report, TR-MATH 03-04, University of Pittsburgh, 2003.
[3] S. Kaya and B. Riviere,
A discontinuous subgrid eddy viscosity method for the time dependent
Navier Stokes equations,
submitted to: SIAM J. Numerical Analysis, 2003.
[4] V. John and S. Kaya,
Finite Element Error Analysis and Implementation of a Variational
Multiscale Method for the Navier-Stokes equations,
submitted , currently under review, 2003.
* F. Pahlevani, fap4@pitt.edu
thesis area: Computational Fluid Dynamics
[1] M. Anitescu, W. Layton and F. Pahlevani,
Implicit for local effects and explicit for nonlocal effects is
unconditionally stable,
submitted to ETNA, 2003.
* G. Pencheva, gepst12@pitt.edu,
Ph.D. thesis area: flow in porous media
[1] G. Pencheva and I. Yotov,
Balancing domain decomposition for mortar mixed
finite element methods,
Numer. Linear Algebra with Appl. 2003; Vol. 10,pp.159-180
[2] G. Pencheva and I. Yotov,
Balancing domain decomposition for porous media
flow in multiblock domains,
Contemporary Mathematics, Volume 295,Contemporary
Mathematics,, 2002, pp.409-419
[3] T. Grandine, S. Del Valle, T. Moeller, S. K. Natarajan, G. Pencheva,
J. Sherman, S. Wise,
Designing Airplane Struts Using Minimal Surfaces,
IMA Preprint 1866, July 2002
[4] G. Pencheva,
On some numerical convergence studies of mixed
finite element methods for flow in porous media,
technical report, 2003, submitted.
* G. Hart, gdhart@pitt.edu
Ph.D. Thesis Areas: Optimization, DAEs and Applications.
[1] M. Anitescu, and G. Hart.
Solving nonconvex problems of multibody dynamics with contact and
small friction by sequential convex relaxation.
To appear in: Mechanics of Machines and Structures.2003.
[2] M. Anitescu and G.Hart,
A constraint-stabilized time-stepping approach for rigid multibody
dynamics with joints, contact and friction.
Preprint ANL/MCS-1002-1002. Submitted to International Journal
for Numerical Methods in Engineering.
[3] M. Anitescu and G. Hart.
A Fixed-Point Iteration Approach for Multibody Dynamics with Contact
and Small Friction.
Preprint ANL/MCS-P985-0802. To appear in Mathematical Programming Series
B.,2003.
[4] M. Anitescu, A. Miller and G. Hart.
Constraint stabilization for time-stepping approaches for rigid
multibody dynamics with joints, contact and friction.
Preprint ANL/MCS-P1023-0203. To appear in the Proceedings of
the Annual Conference of the American Society of Mechanical Engineers,
2003.
[5] G. Hart and Mihai Anitescu.
A hard constraint time-stepping approach for multibody dynamics
with contact and friction.
To appear in the Proceedings of the Tapia Conference for Diversity
in Computing, 2003.
* C. Cardoso Manica,
[1] THOMPSON, M., CALDER”N, A. U. Z., MANICA, C. C., BORTOLI, ¡.
L.
Local Error Estimates for a Diffusion-Reaction System ,
In: Semin·rio Brasileiro de An·lise, 2002, NiterÛi-RJ.
Anais do 56* Semin·rio Brasileiro de An·lise. , 2002. p.613
- 627
[2] CARDOSO, C., DE BORTOLI, ¡., THOMPSON, M.
Finite Shadowing for Burgers Equation ,
In: Semin·rio Brasileiro de An·lise,
2001, S„o JosÈ do Rio Preto. Anais 54 Semin·rio Brasileiro
de An·lises 2001. , 2001. p.247 - 261
[3] CARDOSO, C., BORTOLI, ¡. L.
An·lise de eficiÍncia na simulaÁ„o do escoamento
transÙnico sobre o aerofÛlio NACA0012 em diferenÁas
finitas.
In: VII Jornadas de Jovens Pesquisadores-
CiÍncia para a Paz, 1999, Curitiba. VII Jornadas de Jovens Pesquisadores
- Grupo Montevideo-CiÍncia para a Paz. ,
1999. p.174
[4] CARDOSO, C., BORTOLI, ¡. L.
SoluÁ„o das equaÁies de Euler para escoamento compressÌvel
para escoemanto
compressÌvel sobre o NACA0012,
In: X SIC -Sal„o de IniciaÁ„o CientÌfica, 1998,
Porto Alegre. X Sal„o de IniciaÁ„o CientÌfica. , 1998.
p.23
[5] MANICA, C. C. ou CARDOSO, C., BORTOLI, ¡. L.
SoluÁ„o de escoamentos compressÌveis usando as equaÁies
de Euler ,
In: V ERMAC - Encontro Regional de Matem·tica Aplicada e Computacional,
Santa Maria, 1998.
[6] MANICA, C. C. ou CARDOSO, C., BORTOLI, ¡. L.
Modelamento aeroel·stico de asas flexÌveis em computador,
In: IX SIC -Sal„o de IniciaÁ„o CientÌfica, 1997, Porto
Alegre. IX Sal„o de IniciaÁ„o CientÌfica. , 1997. p.28.
* L. Rebholz,
* Y. Epshteyn, yee1@pitt.edu,
Ph.D. thesis area: Computational fluid dynamics.
[1]Y.Epshteyn, V.S.Ryaben'kii,V.Turchaninov
The numerical example of algorithms composition for solution of
the boundary-value problems on compound domain based on difference potential
method,
Preprint No.3,Moscow,Keldysh Institute for Applied Mathematics,Russia
Academy of Sciences,2003.
[2] A.Dunca ,Y.Epshteyn,V.John
Existence and uniqueness of the Smagorinsky model with model for
the commutation error,
work in progress, 2003.
[3] A.Dunca ,Y.Epshteyn,
On the Stolz-Adams deconvolution models for LES
submitted to: SIAM J. Mathematical Analysis, 2003.
* Qi Mi,
* M. Hufford,
* I. Stanciulescu,
* A. Labovsky,
2003-
*N. Heitmann, Assistant Professor of Mathematics, Millersville
University, heitmann@millersville.edu, web
page
Ph.D. December 2003 , "Subgrid stabilization of evolutionary diffusive
transport problems".
* N. Sahin, Assistant Professor, nisst6+@pitt.edu,
web
page
PhD December 2003, "Derivation, Analysis and Testing of New Near Wall
Models for Large Eddy Simulation."
2002-
* H. Al-Attas, Assistant Professor of Mathematics, King Fahd
University of Petroleum and Minerals
Ph.D., "Enhancing reliability of porous media flow through sensitivity
computation".
* A. Caglar, Postdoctoral researcher: University
of British Columbia (in beautiful Vancouver) web
page
Ph.D. "Reliable finite element simulation of boundary driven
turbulence,"
*H. Tawfig, Assistant Professor of Mathematics, King
Fahd University of Petroleum and Minerals
Ph.D., "Numerical Modeling of reaction infiltration instabilities"
2000-
* T. Iliescu, Assistant Professor of Mathematics, Virginia Polytechnic
University, web
page
Ph.D: "Numerical Analysis of Large Eddy Simulation"
*S. Ezekiel, Assistant Professor, CS Dept., Ohio Northern University
Ph.D., "Computational fractal methods for image analysis and data compression"
*M.Fitzgerald,
M.S., thesis: "Large Eddy Motion in Shallow Water and Estuaries,"
1999-
* A. Liakos, Assistant Professor of Mathematics US Naval Academy,
Ph.D. , " Weak Imposition of boundary conditions in the Navier Stokes
equations"
* T. L. Bennett, South Connecticut State University,
Ph.D., "Best approximation in quotient spaces with application to the
finishing of optical surfaces"
* F.Fairag, Assistant Professor of Mathematics, King Fahd University
of Petroleum and Minerals, web
page
Ph.D. , "A two-level discretization method for the stream-function
form of the Navier-Stokes equations"
1996-
* M. Brodzik, Cryptologic Mathematician, U.S. Department
of Defense
Ph.D., "Numerical Approximation of Manifolds and Applications"
* Aiping Wang, Software Designer, CommTech Corporation
Ph.D., "New Flux Difference Splitting Methods for Hyperbolic
Conservation Laws"
* Anna Duerr, Programmer-Analyst, USF&G
MS, "On the Convergence of an Asynchronous Parallel
Finite Element Method ".
* D. A. Fleming , Software Tester, Computer Science
Corporation
MS, "The Performance of a Parallel Finite Element Method
Monitored on the Cray T3D"
1994-
* P. Hu,
Ph.D., "Error estimates for finite volume methods for steady
convection-diffusion equations"
1993-
* X. Yan,
Ph.D., "Numerical methods for differential/algebraic equations"
1992-
* A. Sunmonu, Associate Professor of Mathematics, City
University of New York, York campus, web
page
Ph.D., "Numerical Analysis of Nonlinear Models of Electrically
and Thermally Conducting Materials"
1991-
* B. Hong,
Ph.D., "Symmetry on solution manifolds of parameterized equations"
1990-
* X. Ye, Professor of Mathematics, University of Arkansas, Little
Rock, web
Ph D, "Construction of divergence free spaces for the incompressible
Navier-Stokes equations"
* D.C.O'Neal, Pittsburgh Supercomputing Center, web
page
Certificate in Scientific Computing, 1990, "Optimization of Finite
Element Codes"
1988-
* R.-X. Dai,
Ph.D., "Characterization and computation of fold sets for parameter
dependent equations"
1985-
* S.-H. Chou, Professor of Mathematics and Statistics, Department
of Mathematics and Statistics, Bowling Green State University, web
Ph.D., "A network model for incompressible two-fluid flow and
its numerical solution"
Let us know how you doing! We will list your web page and affiliation here once we hear from you.
computational methods for fluid dynamics,
numerical modeling of flow in porous media,
numerical methods for problems with complementarity constraints,
and
computational neuroscience.
---------------------------------
The Scientific Computing Laboratory equipment includes:
1. Eight-processor shared memory computational server
- ProLiant DL760 Pentium III Xeon 700 MHz
- 8GB SDRAM
- Two 72.8GB Ultra3 SCSI Hard Drives
2. File server
- ML370 G2 Pentium III 1.266 GHz
- 512 MB SDRAM
- Three 72.8GB Ultra3 SCSI Hard Drives
- SDLT 220 GB Tape Drive (for backups)
3. Six workstations
- Evo W6000 Intel Xeon 1.8 GHz Dual Processor
- 512 MB RDRAM
- 40 GB Ultra3 SCSI Hard Drive
4. Firewall computer
- Evo D500 Pentium IV 1.7 GHz
- 512 MB SDRAM
- 40 GB Ultra ATA Hard Drive
5. High-speed switch
- Catalyst 2950 with 24 ports
6. Black and white lazer printer
- hp LaserJet 4100
The lab is supported by our departmental system administrator. Software includes Linux, MPI, AFS, Matlab and others. The visualization package Tecplot, produced by Amtec Engineering, has been purchased from a faculty research grant and is also available in the lab.
All computers in the lab are running Linux. The eight-processor
shared memory computer, the six dual-processor workstations, and the file
server are connected through a high-speed network switch. The resulting
heterogeneous architecture enables users to run distributed memory applications
via Message Passing Interface (MPI) on up to 20 processors. Sequential
codes can be run on the multiprocessor shared memory computer. The workstations
are equipped with monitors and can be used by faculty and graduate students
as stand-alone computers. The lab is fully operational and available 24
hours/day to all faculty and graduate students in computational mathematics.
Fellowships, Fellowships, Fellowships.
Mellon fellowships are awarded in a university wide
competition. In 2002 two students from the mathematics department were
awarded Mellon fellowships including Ms. Gergina Pencheva . Gergina is
working with Professor Yotov on large scale scientific computing and has
written three research papers in her short time in our department. Congratulations
to Gergina!
Givens Associate.
Mr. Adrian Dunca spent two months during summer 2002 as
a prestigious Givens Associate at Argonne National Laboratory to conduct
research on turbulence. His visit was funded by the Department of Energy.
Congratulations Adrian!
The IMA Mathematical Modeling in Industry Workshop
During May 26-June 3, 2002, Gergina Pencheva attended
the IMA Industrial math workshop in Minneapolis, Minnesota. There, she
worked in teams of 6 students under the guidance of Dr. Thomas A.
Grandine from Boeing. They modeled, analyzed and did some computational
experiments associated with a real-world industrial problem in designing
an airplane strut - how to construct a surface of minimal area which attaches
to both wing and engine in a prescribed way. At the end of the 9-day
period Gergina and her group made a public oral presentation and submitted
a written report. Gergina commented, " I had a great time in Minneapolis
and would strongly recommend IMA Industrial workshops for all students
in the Computational Math group."
The IMA 2002 Summer Program for Graduate Students in Scientific Computing,
Gergina Pencheva also attended July 1-26, 2002, Lexington, KY program.
Each week had a different speaker(s) and topic:
week1: Craig Douglas and Jun Zhang talked about Parallel Computing
and Visualization.
week2: Jerome Jaffre and Jean Roberts showed us the basics of
FEM.
week3: Iain Duff talked about three types of direct methods
for sparse matrices; Tony Drummond and Osni Marques presented the Department
of Energy's software ACTS Tool kit for parallel computers.
week4: Toni Kazic, Bioinformatics and Its Relation to Scientific
Computing
DOE Intern.
Mr. Noel Heitmann (now at Millersville University) spent the
summer of 2002 working at the Department of Energy’s Bechtel-Bettis lab.
He was there fully funded by DOE to conduct research in scientific computing.
Congratulations Noel!
International Travel.
Our computational math group has a long-standing
scientific cooperation in computational fluid dynamics with the Institute
of Analysis and Numerics at the University of Magdeburg in Germany. During
summer 2002 Adrian Dunca, Songul Kaya and Husein Merdan traveled to Germany
to conduct collaborative research with the group there. Songul gave a seminar
there: "Subgrid scale eddy viscosity methods are variational multiscale
methods". Their extended visits were funded by an NSF-DAAD grants of
the Computational Math group at Pitt and Professor Tobiska's Computational
Mathematics group at Magdeburg.
The October 2002 Finite Element Circus.
Computational math faculty and grad students attended
the October Finite Element Circus. In fact, the Pitt contingent was the
largest one there and our grad students gave a impressive account of their
scientific research at Pitt. The grad students attending and giving research
talks were S. Kaya (Numerical analysis of a subgrid eddy viscosity methods),
A. Dunca, Y. Epshteyn, N. Heitmann (Subgrid scale Eddy Viscosity for
Convection Dominated Diffusive Transport,), G. Pencheva (Balancing
Domain Decomposition for Mortar Mixed Finite Element Methods), F. Pahlevani
and A. Caglar.
News from 2003.
Fellowships, Fellowships, Fellowships.
Songul Kaya and Adrian Dunca won Mellon predoctoral Fellowships in
a university wide competition for 2003-2004. The excellent entering students
this year included fellowship recipient A. Labovskii who selected Pitt
for grad school because of its strength in computational mathematics. Congratulations
to all!
Conferences
Computational math Ph.D. students Carolina Manica and Faranak Pahlevani
attended the ETNA Anniversary conference and Faranak gave a very well received
talk there.
Songul Kaya attended the (by invitation) conference Perspectives
on incompressible flows, comparison of different computational strategies,
CSCAMM, University of Maryland, April 2003.
Research Talks and Colloquiums.
Our Ph.D. students have given a number of seminar talks and colloquiums
at other universities. Some of them are:
Noel Heitmann: Appalachian State University, Jan. 17, 2003 (Subgrid
scale Stabilization of Evolutionary Convection Dominated Diffusive Transport),
Marist
College Mathematics Seminar, Feb. 7, 2003, Fordham University Colloquium,
Feb. 10, 2003, James Madison University Mathematics and Statistics Colloquia,
Feb. 24, 2003, and Rochester Institute of Technology Colloquium Series,
March 13, 2003.
Our RTG Seminar.
Graduate students continue to be an exciting part of our RTG (Research-Training
Group) seminar. Talks by Ph.D. students in 2003 included:
Adrian Dunca , September, ("On Stolz-Adams deconvolution models"), Songul
Kaya , April , (Numerical analysis of a subgrid eddy viscosity
methods for Navier Stokes equations),
Seminar
Announcements, seminar03-1,
seminar03-2
DGWave: A 2D code for solving the acoustic wave problem using DG.
DGStokes: A 2D code for solving the Stokes problem using DG.
DG2phase: A 3D code for solving the two-phase flow problem using
DG.
Contains gravity, a well model.
Note that all the codes above create output files that can be viewed
by
Tecplot.