Hello World!

William Layton

Department of Mathematics, University of Pittsburgh
Pittsburgh, PA 15260 , U.S.A.

My email address is: wjl@pitt.edu
phone: (412)624-8312/Math Office:(412)624-8375

Who am I?
I am a full professor in Pitt's math department. My research is mainly in numerical analysis.
Numerical analysis is one of the broadest areas of mathematics as it includes convergence
questions of analysis, including new convergence issues related to the rate of convergence, problems
of mechanics, nonlinear PDEs and applications
all the way across the spectrum of mathematics
to graph theory.
I've worked in all these areas and a number of others too, authoring more than
140 refereed publications and 3 research level books. Through all this variety I have had an
abiding fascination with the deep mathematics of fluid motion, which, paraphrasing, is an area of
mathematics in which " a gnat may bathe and an elephant may drown."
I've had 24 PhD students almost all of whom are active and accomplished researchers (my best accomplishment).
Outside of mathematics my other interests have been quite varied.
I was chess champion of Georgia in 1976, played various sports over time. I am currently an avid mid-level
whitewater kayaker. I get many mathematical ideas from observations of turbulent flows in nature!

My current and fantastic PhD students in progress are:
Ross Ingram, Nate Mays, Hoang Tran (co-advised with Catalin Trenchea), Aziz Takhirov,
Nick Hurl, Marina Moraiti, Michaela Kubacki and Xin Xiong.

This year I had the great pleasure of doing research with two terrific undergrads:
Keith Wyss and Becky Negley. Remember their names; we will all hear more from them!
This year I also have 4 PhD students visiting from other countries and other advisors:
Lars Roehe (2 months), Li Shen, Haibiao Zheng and Osman Isik (all for the full year).
These visits to our academic family here have been happy and productive!

Scientific Interests: Click for some papers by area.
My Current Interests are on Turbulent, Multi-Physics flows on Complex Domains:
*Variational Methods ( FEM, Spectral...)
-especially for flow problems
* Partitioned (non iterative domain decomposition) time stepping methods for multi domain / multi physics coupled problems:
Atmosphere-Ocean coupling in climate models,
multi-physics coupling such as NSE & Maxwells eqns in MHD problems,
Fully evolutionary Stokes-Darcy coupling.
*Large Eddy Simulation - LES reports. Implementation of modern LES models in complex applications and legacy codes
including VMS methods, Nonlinear filters, Approximate deconvolution models.
*Adaptive and parallel algorithms.
Flow in complex pebble bed geometries in which
geometric complexity cannot be resolved, but must be resolved!
*Long Time Behavior of Numerical Methods. Click for LTB. Current projects include accelerating "spin-up" of climate models, initialization with statistically steady flow data, and
long time error behavior ( slow flows + large domains means that several through flow times = long time for geophysical flows) of
partitioned methods for groundwater - surface water coupling.
* Transfer theories of turbulence and a connection to shell models. This is a new interest.

In all these areas, one common theme is:
Mathematical analysis as a guide to practical computation.

Mathematics of Large Eddy Simulation of Turbulent Flows

Springer Verlag: Series: Scientific Computation
L. C. Berselli, T. Iliescu, W. Layton
2005, Approx. 345 p., Hardcover
ISBN: 3-540-26316-0
Click Here for More Information.

My second book is out:
Introduction to Finite Element Methods for Incompressible, Viscous Flows,
W. Layton,
SIAM publications, 2008 / xx + 213 pages / Softcover / ISBN: 978-0-898716-57-3 Click Here for More Information.

My third book is completed and under consideration:
Approximate Deconvolution Models of Turbulence: Analysis, Phenomenology and Numerical Analysis, W Layton and L Rebholz,
under consideration for publication, 2010.

Editorial Boards: Journal Mathematical Analysis and Applications (JMAA), Numerical Linear Algebra and Applications (NLAA), inception to 2009, SIAM Journal on Numerical Analysis (SINUM), Advances in Numerical Analysis (ANA).

Research Training Group in Computational Mathematics Read this if you are interested in grad school!!.

Teaching Resources, Class Home Pages and the Like:
Here's information about our NEW SCIENTIFIC COMPUTING major!
"How to give a 20 minute talk"
Resources for Math 1080-Numerical Linear Algebra.
Resources for Math 1070-Numerical Analysis.
Resources for Math 1360-Mathematical Modeling.
Resources for Math 2070-the Gateway course in Computational Mathematics.
Resources for Math 1470-the undergraduate PDE course.

My Fantastic Students:
IN PROGRESS and currently postdocs in exciting places:

*A. Labovschii, PhD 2008: MHD Turbulence and Applications
Alex just acccepted a tenure track assistant professor position at Michigan Technological University.
Michigan Tech is a mid-sized math department with a strong PhD program in applied math.

*Jeff Connor, PhD 2010, Uncoupling Ocean--Atmosphere models
Jeff completed his PhD in 4 years and wrote one of the most consequential PhD theses of our department.
Jeff is a postdoc at LLNL Jeff's web page.

*Alex Lozovskiy, PhD 2010, Alex opened a new area of computational science up to rigorous numerical analysis:
Prediction and modelling of the noise generated by turbulence
Alex is beginning a postdoc in France in computational mechanics.

*Nate Mays, High accuracy methods for ill-posed
problems and applications to an biomedical parameter identification problem with diagnostic implications.
web page: http://www.pitt.edu/~nhm3/
Nate's Papers (so far):
1. "A Superconvergence Theory for a Family of Iterated Regularization Methods" - R. Ingram, W. Layton, N. Mays (under revision)

2. "GRACE - An Algorithm for Finding Regression Coefficients" - N. Mays, C. Rosano, H. Aizenstein, R. Boudreau, L. Marai, K. Wong, S.L. Yilmaz. W. Layton (in progress)

3. "Stability and Convergence Analysis: Leray-Iterated-Tikhonov Navier Stokes Equation with Time Relaxation" - R. Ingram, C. Manica, I. Stanculescu (in progress)

4. "Analysis of the Modified Iterated Tikhonov Method of Deconvolution" - N. Mays and M. Zhong (in progress)

*Ross Ingram, Numerical analysis of discrete Brinkman models for porous media flow.
Ross' web page. Ross will work in an industrial research group at Bettis starting fall 2011.

* Aziz Takhirov

* Hoang Tran (co-advisor, primary advisor: C Trenchea)

* Michaela Kubacki

* Xin Xiong

* Nick Hurl

* Marina Moraiti

* Ali Koseaoglu MS 2010-2011.
Project: Dynamic relaxation to statistical equilibrium.

* A. Sunmonu, Ph.D. 1992, "Numerical Analysis of Nonlinear Models of Electrically and Thermally Conducting Materials"
Currently: Tenured Associate Professor, Department of Mathematics and Computer Science,
City University of New York- York
A.Sunmonu's web page.

* F.Fairag, Ph.D., 1998, "A two-level discretization method for the stream-function form of the Navier-Stokes equations"
Tenure Track Assistant Professor,
Department of Mathematics, King Fahd University of Petroulem and Minerals
F.Fairag's web page.

*Dave O'Neal, Certificate in Scientific Computing, 1990,
Certificate project: "Optimization of Finite Element Codes"
currently: Senior Scientific Specialist, Application Development and Technical Support
Pittsburgh Supercomputing Center
Dave O'Neal's web page.

* A. Liakos, Ph.D., 1999: Weak Imposition of boundary conditions in the Navier Stokes equations
Tenured Associate Professor, Department of Mathematics,
U.S. Naval Academy, Annapolis
Second place in 1998 SIAM Student Research Paper Competition
Postdoc: Center for Fibers and Films, Clemson University
E-mail: liakos@usna.edu
Tas Liakos' web page.

* Traian Iliescu, Ph.D., 2000: "Numerical Analysis of Large Eddy Simulation"
Tenured Associate Professor, Mathematics Department,
Virginia Polytechnic University.
Culver Prize, Department of Mathematics, University of Pittsburgh
Mellon Predoctoral Fellow, University of Pittsburgh
First prize: 1999 SIAM Student Research Paper Competition
2000 Wilkinson Fellow, Argonne National Labs,
T. Iliescu's web page.
E-mail: iliescu@math.vt.edu

*Melanie Fitzgerald, M.S. 2000, thesis: Large Eddy Motion in Shallow Water and Estuaries,
M. Fitzgerald's web page.

* Huasin Al-Attas, Ph.D. 2002: Reliability of Porous Media Flow and Parallel Sensitivity Computation,
Tenure Track Assistant Professor of Mathematics, King Fahd University of Petroleum and Minerals
E-mail: huast1+@pitt.edu

* Atife Caglar,
Ph.D. 2002: "Reliable finite element simulation of boundary driven turbulence",
Honorable Mention: 2002 SIAM Student Research Paper Competition
Mathematics of Information Technology and Complex Systems Postdoctoral Research Fellowship
Department of Mathematics, University of British Columbia
Currently: Tenure Track Assistant Professor,
Department of Mathematics, University of Wisconsin-Green Bay
E-mail: caglara@uwgb.edu
A. Caglar's web page.

*Noel Heitmann, Ph.D. 2003: " Subgrid Stabilization of Evolutionary Diffusive Transport Problems" ,
Currently: Tenured Assistant Professor, Millersville State University
E-mail: email Noel Heitmann's web page.

* Niyazi Sahin, Ph.D. 2003: "Derivation, Analysis and Testing of New Near Wall Models for Large Eddy Simulation",
Honorable Mention: 2000 SIAM Student Reserarch Paper Competition,
Tenure Track Assistant Professor,
Mathematics Department, Mugla University,

*Faranak Pahlevani, (co-advisor: M. Anitescu) PhD 2004: "Sensitivity Analysis of Eddy Viscosity Models"
Postdoctoral Researcher,
Montana State University, Department of Mathematical Sciences and currently:
tenure track assistant professor, CUNY

* Adrian Dunca, PhD 2004: Space Averaged Navier Stokes Equations in the Presence of Walls
Postdoc, Department of Mathematics, University of Michigan, 2004-2006.

*Songul Kaya-Merdan, (co-advisor: B. Riviere) PhD 2004: "Numerical Analysis of a Variational Multiscale Method for Turbulence",
Currently: Tenured Associate Professor
Middle East Technical University
Winner of the 2004 HALES DISTINGUISHED RESEARCH AWARD for the Best Doctoral Dissertation.
2003-2004 Andrew Mellon PreDoctoral Fellow

Songul's web page.

*Carolina Cardoso Manica, PhD August 2006: Numerical Methods in Turbulence
Assistant Professor,
Federal University of Rio Grande do Sul,
Porto Alegre-RS- Brazil
First Place: 2005 SIAM Student Research Paper Competition
Mellon predoctoral fellow, 2005-2006 and Culver Prize, 2006
Carolina's web page.
E-mail: cac15@pitt.edu

*Patricia Butler,
MS in Mathematics August 2006
MS thesis: Calculating functionals of solutions of large sparse systems
Currently: Algebra specialist, Davis schools , California

*Leo Rebholz, PhD December 2006: Helicity and physical fidelity in models and algorithms
Tenure Track Assistant Professor,
Department of Mathematical Sciences, Clemson University.
Culver Prize, 2006
Leo's webpage.

*Sara Marashinia, MS 2007, Analysis of Bernoulli's equation in LES turbulence models

*Gary Hart,(co-advisor, primary advisor: Mihai Anitescu)
PhD 2007: A Constrained-Stabilized Time-Stepping Approach for Piecewise Smooth Multibody Dynamics
Currently: Assistant Professor, University of Pittsburgh at Greensburg
Some papers:
[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.

*Monika Neda, PhD 2007: Numerical analysis of homogeneous isotropic turbulence generated by higher order models of turbulence
Tenure track Assistant Professor, University of Nevada Las Vegas

Another Research Award:
Monika Neda's 30 minute research talk: "Time relaxation for turbulent flow problems"
won one of 6 awards (which included a cash prize) for:
The Most Outstanding Research Presentation by a PhD student
at the Joint MAA & SIAM Southeast Atlantic Section Meeting, 2006.
Mellon predoctoral fellow, 2006-2007
. Monika's web page.

*I. Stanculescu, PhD 2008: Optimal Approximate De-Convolution Models of turbulence.
Iuliana is an assistant professor at Nova Southeastern University in Ft Lauderdale Fl.
Iuliana's web page.

* Kristen Harnett, M.S. 2008, thesis: On duality and the bi-conjugate gradient method,

Updated 12/08.

Other Scientific Intrests:
I've worked in several related areas and I still have research interests
in these areas. In no particular order, some of these other areas
in which I've written a few papers include:

*Parallel Algorithms for Highly Nonsymmetric Problems
-we have developed massively data parallel algorithms for solving linear and nonlinear systems
arising from convection dominated phenomena. We have in particular
developed an element wise data parallel solution method which is proven
to converge uniformly in the Peclet number.

Here is a page which lists a few reports on data parallel domain decomposition algorithms.

*Finite Element Methods for Natural Convection Problems
Here is a page which lists a few reports in natural convection problems.

* Nonlinear Analysis Applied to Ordinary and Partial Differential Equations

Here is a page which lists a few reports in nonlinear analysis.

* Finite Difference Methods and FEM's for Hyperbolic Systems and Elliptic-Hyperbolic
Singular Perturbation Problems

Here is a page which lists some reports on FEM's and FDM's for this topic.

* Multi-Level Newton Methods
I have done some work developing these ideas for fluid flow problems.

* Modelling and Simulation of Aluminum Reduction Cells

*Numerical Linear Algebra

More pictures.

I like turbulence!

Miscellaneous Math Stuff:

Here are some math related links.
Pitt's Math Department
Quotes to see some quotes about mathematics.
"How to give a 20 minute talk"

Let me know if you found something of interest here so I'll know to work more on these pages.
E-mail me: