Hello World!

William Layton

Department of Mathematics, University of Pittsburgh

Interested in Math Department Research? ClickHere

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 across the spectrum of mathematics to graph theory.
I've worked in all these areas and a number of others too, authoring more than
160-odd refereed publications, 1 undergrad text, 1 grad text and 2 research monographs.
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 approaching 40 PhD students almost all of whom are active and accomplished researchers.
My PhD students now direct PhD students who are themselves doing very well.
I've directed a number of MS students who wrote excellent theses and I've advised undergrad researchers who did terrific work and moved forward to great things.

Outside of mathematics my other interests have been quite varied.
I was chess champion of Georgia in 1976 and 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:
Victor Decaria, Joe Fiordelino, Michael Mclaughlin, Ali Pakzad, Haiyun Zhao.

Some recent PhD students are:
Yong Li (Mellon Fellow, co-advised with Catalin Trenchea),
Yong is an analyst with Goldman-Sachs.

Marina Moraiti (SIAM Student Research Prize Winner),

Nan Jiang
(Mellon Fellow), Nan was a postdoc at FSU.
Now she is a tenure track Asst. Professor at Missouri Univ. of Science and Technology.

Nick Hurl. Nick is at Slippery Rock State Univ.

Sarah Khankan Sarah defended her PhD thesis after only 3 years at Pitt!

Ross Ingram (Bettis),
Nate Mays (WJU but joining EPIC fall 2014),
Hoang Tran (Mellon Fellow),
Hoang has a permenent position as a staff scientist at ORNL, co-advised with Catalin Trenchea),
Aziz Takhirov (Mellon Fellow, Postdoc at Texas A&M),
Michaela Kubacki (tenure track Asst Prof at Middlebury College) and
Xin Xiong (Global Analytics, Walmart eCommerce).

I also have had a number of PhD students visiting from other countries and other advisors:
(a partial list of recent ones) Lars Roehe (2 months), Li Shen, Haibiao Zheng, Yao Rong and Osman Isik (all for the full year).
These visits to our academic family here have been happy and productive!

Scientific Interests: some papers by area.
some older papers.
Turbulent, Multi-Physics flows.

*New Models and Algorithms for Complex Turbulence
, e.g., turbulence not at statistical equilibrium.
This project involves developing models with a rigorous mathematical foundation
as well as numerical analysis of new methods for their solution.

* Ensemble Simulation Models and Algorithms for Turbulence. A joint project with Nan Jiang.
Ensemble simulations are necessary for extending forecast skill,
dealing with errors from unknown data and parameterizations and
estimating sensitivities. However, they lead to the inevitable conflict
of high resolution single realizations and computing ensembles.
We have new methods breaking this deadlock. The new methods lead
to new models as well since ensemble data gives a simple
and calculable way to specify, WITHOUT MODELLING, the TKE.
We have also developed a NEW MIXING LENGTH which seems to
correct the inadequacies of standard mixing length theories.

* Partitioned time stepping methods for coupled, multi domain & physics:

Atmosphere-Ocean coupling in climate models,
multi-physics coupling, e.g., NSE & Maxwells eqns in MHD,
Fully evolutionary Stokes-Darcy-Biot coupling.

*Large Eddy Simulation
- LES reports. Modern LES models in complex applications and legacy codes
including VMS methods, Nonlinear filters, Approximate deconvolution models.

*Uncertainty Quantification. A joint project with Clayton Webster at ORNL and Catalin Trenchea at Pitt.
We are interested in improving current approaches, developing new ones and
and studying the interplay between numerical and modelling errors in turbulence.

*The BigData in turbulent combustion project.
This is a new collaboration including Alexandros Labrinidis, Peyman Givi,
Panos Chrysanthis and Patrick Pisciuneri.

*Algorithms for Flow in complex pebble bed geometries
in which geometric complexity cannot be resolved.

*Long Time Behavior of Numerical Methods. Click for LTB. A long time interest of mine!

* Energy Transfer theories of turbulence and shell models.
I work on these to let undergrads get involved in research on
turbulent flows. There are a lot of interesting conjectures that can be
interrogated via shell and ET Models.

My personal preference is to work on problems
where mathematics can make a difference in extending
the boundary of what is predictable rather than perfecting
a theory where the main outlines are already known.
(There is nothing wrong in the latter. Math is hard enough that
progress will only be possible when we all work on whatever is
closest to our own hearts.)

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, ISBN: 3-540-26316-0
Click Here for More Information.

Introduction to Finite Element Methods for Incompressible, Viscous Flows,
W. Layton, SIAM publications, 2008 / ISBN: 978-0-898716-57-3
Click Here for More Information.

Approximate Deconvolution Models of Turbulence: Analysis, Phenomenology and Numerical Analysis,
W. Layton and L. Rebholz, Springer Lecture Notes in Math, V. 2042, 2012.

Numerical Linear Algebra,
W Layton and M Sussman, Lulu press, ISBN 9781312329850, 2014.

( 20$ printed and free as a pdf file. This is what is within our power to address the cost of college
and the result of 30 years teaching and polishing.

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, Some OLDER Class Home Pages and the Like:

"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.

More of My Fantastic Students:
*A. Labovschii, PhD 2008: MHD Turbulence and Applications
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 was a postdoc at LLNL and is now a tenure track Asst. Prof at U Conn, Avery Point.

*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 now a postdoc at Texas A&M and will be moving to industry soon..

*Nate Mays,
Nate was a tenure track Asst Prof at WJU and will be joining EPIC in Fall 2014.
High accuracy methods for ill-posed
problems and applications to an biomedical parameter identification problem with diagnostic implications.

*Ross Ingram, Numerical analysis of discrete Brinkman models for porous media flow.
Ross' web page. Ross is a leader in an industrial research group at Bettis since fall 2011.

* Aziz Takhirov
Aziz will start a postdoc Fall, 2014 at Texas A&M University.

* Hoang Tran (co-advisor, primary advisor: C Trenchea)
Hoang is a postdoc at ORNL working on UQ for turbulent flows.

* Michaela Kubacki
Asst Prof, Middlebury College

* Xin Xiong Analyst, Walmart eCommerce,

* Nick Hurl
-PhD April 2017 (expected).

* Marina Moraiti -

* Ali Koseaoglu MS 2011, Dynamic relaxation to statistical equilibrium.

*Sara Hritz,
MS 2010, Phenomenology and Computations of a Regularization of the Navier-Stokes equations related to a
Non-Newtonian fluid model, currently: Scientific Analyst, UPMC.

* A. Sunmonu, Ph.D. 1992, "Numerical Analysis of Nonlinear Models of Electrically and Thermally Conducting Materials"
Currently: Full Professor and Dept Chairman, 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"
Associate 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"
Full 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: Tenured Associate 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 Associate 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,
Tenured Associate Professor,
Dept. of Math and CS,
Ankara Yildirim Beyazit University,
Ankara, Turkey

*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, PSU Albington

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

*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
Associate 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
Associate 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
Associate 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: