Welcome to the Computational Mathematics Research-Training Group!

What is Computational Mathematics?

Computational Mathematics addresss questions like:

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!

The Mathematics Ph.D. Program  for Computational Mathematics.


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

Our Research Training Program: Why study Computational Mathematics?

    Computational Mathematics is an exciting and challenging area. The advantages of accepting the challenge with our group include:
  Here is an example of the typical trajectory to the Ph.D. in Computational Mathematics to give you an  overview of what our program is all about. Please see the mathematics department  Graduate Handbook  for the exact rules and details of each step or contact the Graduate Chair or Graduate Secretary.

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.

Our Courses.

Undergraduate Courses Open to Graduate Students

MATH 1070, Numerical Analysis
This course in numerical mathematics is designed for students interested  in solving scientific and engineering problems on computers and is  intended to expose students to a wide range of up to date numerical methods. The emphasis is on algorithms, the mathematical ideas behind them  and their use in obtaining numerical solutions. Our goal will be to understand how and when the methods work. The concept of numerical error will be used to quantify the accuracy of approximation. We will also study the stability and the efficiency of algorithms.

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.
 

Graduate Courses in Computational Mathematics

Our gateway course for students who have no experience with computing or numerical analysis is:
Math 2070: Numerical Methods in Scientific Computing I, and
Math 2071: Numerical Methods in Scientific Computing II
The sequence M2070-M2071 gives an in-depth introduction to the basic areas of numerical analysis.  The courses will cover the development and mathematical analysis of practical algorithms for the basic areas of numerical analysis . The course M2071 does not assume a knowledge of M2070; and material from M2070 that is needed in M2071 will be reviewed as necessary .
These courses also include a novel  Computational Laboratory (Offered every year)

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)
 

Research activity of our group

Research

The diversity of this group is reflected in its research interests, which range over such areas as approximation theory, numerical analysis of partial differential equations, adaptive methods for scientific computing, computational methods of fluid dynamics and elasticity, numerical solution of nonlinear problems, numerical optimization, algorithm design, high level  programming languages, simplicial grid computations, automatic image and pattern analysis,  and simulation of stochastic reaction diffusion systems. There are weekly seminars, as well  as lectures and workshops at the Pittsburgh Supercomputing Center on current trends in scientific supercomputing.
 
 

Topics of Current Research Interest

Our interests continue to evolve and expand. Here is a selection of some of our current interests:

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.
 


 

Associated faculty.

Current faculty

Mihai Anitescu  412-624-8352   anitescu@math.pitt.edu, Professor Anitescu's web page

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.
 

Emeritus faculty

Charles G. Cullen   Professor Emeritus ,
Numerical Linear Algebra

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.
 

Some of our recent longer term research visitors.

Professor V. Ervin, Clemson University,  web

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
 

Other Recent Research Collaborators

Pitt Faculty:

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

Earlier Computational Math Faculty Members

J. Boland,
S. Choudhury,
J. Maubach,  JM
M. Melander,
E. Overman,  EO
J. Peterson, JPJP2
N. Zabusky,  Zabusky

Recent Funded projects

PI: Ivan Yotov, U.S. Department of Energy, Grant #DE-FG03-99ER25371, Dec. 1, 1998- Nov. 30, 2001.
Joint with the University of Texas at Austin, subcontract # UTA99-0210, $100,356.

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.
 

Journals edited

SIAM Journal on Numerical Analysis (WJL),  Numerical Linear Algebra and its Applications (WJL),
Guest Editor, Computational Geosciences, Special Issue on "Locally Conservative Numerical Methods for Flow in Porous Media" (IY)

History of our group and ICMA: the Institute for Computational Mathematics and Applications

under development.

Research accomplishments of our current Ph.D. students.


* 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., CALDERN, 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, So 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 simulaÁo 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.
SoluÁo das equaÁies de Euler para escoamento compressÌvel para escoemanto
compressÌvel sobre o NACA0012,
In: X SIC -Salo de IniciaÁo CientÌfica, 1998,
Porto Alegre. X Salo de IniciaÁo CientÌfica. , 1998. p.23

[5] MANICA, C. C. ou CARDOSO, C., BORTOLI, ¡. L.
SoluÁo 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 -Salo de IniciaÁo CientÌfica, 1997, Porto Alegre.  IX Salo de IniciaÁo 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,

Some of our alumni!

This list is incomplete. We'd really like to hear from all our alumni, find out how you're doing and add your name, affiliation and web page to our list. Drop us a line!

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.

Our report server, see the faculty and student web pages for more, reports

The computational mathematics seminar series.

We have a fun, active and intellectually stimulating seminar each week. There are a variety of speakers: faculty, grad students and visitors. Seminar Announcements
Previous years seminars,
 seminar03-1
 seminar03-2

Our NSF-SCREMS Scientific Computing Laboratory

Our NSF-SCREMS  Scientific Computing Laboratory  is a large-scale environment for scientific computing research in the mathematical sciences. The computing equipment includes an Intel eight-processor shared memory computational server, a six-node Intel cluster, and an Intel file server, all running Linux and connected through a high speed network.  The  SCREMS Lab supports several research projects, including in particular:

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.
 

News from the Computational Mathematics Program.

News from 2002.
Culver-Teplitz Awards.
The Culver-Teplitz awards are given to select graduate students who have set standards for excellence in all their academic endeavors. They are given in recognition of teaching excellence and research accomplishment. In 2002 our department's  winners included 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 !

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 Energys 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

Software

DG2D framework:  An hp adaptive code for two-dimensional problems for
discontinuous Galerkin formulations.  This code solves single phase flow
problem, coupled flow and transport problem such as miscible displacement

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.

A gallery

Under development!