Habib Ammari's Home Page

Habib Ammari

Research Scientist at CNRS (tenured, 1997-)
Head of the Electromagnetic Modeling and Inversion Group

Center of Applied Mathematics
CNRS UMR 7641 & Ecole Polytechnique
91128 Palaiseau Cedex, France

Current Research Interests

* Potential theory
* Medical imaging
* Modeling in electromagnetics


Selected Publications

* Selected Reprints
* Selected Preprints
* Selected Talks

Books

* (with H. Kang) Reconstruction of Small Inhomogeneities from Boundary Measurements,
* Lecture Notes in Mathematics, Volume 1846, Springer-Verlag, Berlin 2004, 238 pages. (with H. Kang) Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory.
* To appear, 2005, 306 pages. (edited with H. Kang) Inverse Problems, Multi-Scale Analysis, and Homogenization. Proceedings of a Workshop in Seoul.
Contemporary Mathematics Volume, American Mathematical Society. To appear, 2006.

Former PhD Students

* A. Khelifi, Electromagnetic Scattering From Small Dielectric Inhomogeneities, PhD Thesis, Ecole Polytechnique, February 2002
* F. Triki, Asymptotics of Resonant Frequencies in Electromagnetics, PhD Thesis, Ecole Polytechnique, December 2002
* K. Touibi, Imaging Microstructures, PhD Thesis, Ecole Polytechnique, April 2004
* S. Soussi , Mathematical Modeling in Optics, PhD Thesis, Ecole Polytechnique, September 2004
* E. Iakovleva (supervised with D. Lesselier), Inverse Scattering from Small Inclusions, PhD Thesis, Ecole Polytechnique, November 2004

Current PhD Students

A. Dossevi (supervised with L. Garnero), A. Kozhemyak, K. Louati, H. Zribi.

Current Post-Docs

M. Lim, E. Kim, H. Lee.

Inverse Problems Seminar
Workshop on Inverse Problems, Multi-scale Analysis and Homogenization, Seoul, June 22-24, 2005
Ecole d'été, Méthodologies de l'Inversion des Ondes et Modèles Directs, 5-9 septembre, 2005, SupElec, Gif-Sur-Yvette

Moving-grid methods

Moving-grid methods

Paul Zegeling

---

What are moving-grid methods?

Moving-grid methods are solution-adaptive methods for time-dependent partial differential equations (PDEs). These methods, also characterized by the terms r-refinement (*), continuously deforming grids, or dynamically moving grids, move the spatial grid continuously in time while the discretization of the PDE and the grid selection procedure are intrinsically coupled.

(*) h-refinement methods ('local refinement') adapt the grid only at discrete time levels, whereas p-refinement methods (mostly of finite-element type) increase or decrease the order of approximation where, when and if necessary. In advanced applications also hybrid methods exist: h-p-, h-r-, or r-p-refinement.

Moving-grid methods use a fixed number of spatial grid points, without need of interpolation, and let them move with "whatever fronts are present". Using these methods, if properly applied, can save up to a factor of 10 (in 1D) or 100 (in 2D) spatial grid points in the case of steep moving layers stemming from a convection-diffusion or reaction-diffusion system.

Comparison of a fixed grid with a moving grid

The following four pictures illustrate the difference between fixed grid and moving grid methods for the one-dimensional Burger's equation.

[4 pictures, 21 kB in total]

The following movie shows a uniform non-moving central-difference scheme for a two-dimensional Burgers' type equation :

[mpeg-movie, 351 kB]

Grid degeneration

An intrinsic difficulty of topological nature exists: so-called `grid distortion' or `grid degeneration'. In terms of transformation of variables (non-uniform in (x,y,t) -> uniform ()) this means that for some instances the Jacobian may become zero or almost zero.

The following pictures show three possible cases of grid degeneration.

[3 pictures, 27 kB in total]

Applications

An interesting application of a moving-grid method in 1D can be found in a tidally-averaged advection-dispersion model. Application of the same method to a `lava-dome' model can be found in lava-dome model. Recent work by the author dealing with moving grids concerns the application to the Gray-Scott reaction-diffusion model, which describes two irreversible chemical reactions. The PDE-system reads :

where k is the dimensionless rate constant of the second reaction and F is the dimensionless feed rate. It is known from experiments that many different complex patterns (regular, stationary, periodic, chaotic) can be found for different choices of the parameters.

Even in one space dimension interesting solution behaviour may occur for which moving-grid methods could be benificial.

Self replicating patterns in 1D

For the 1D-case a moving-finite-difference method based on equidistribution (with spatial and temporal smoothing) is used to detect the splitting of the pulses.

Self-replicating patterns recognized by moving grids :

[gif, 76 kB]

Irregular solution behaviour created by changing the parameters :

[gif, 72 kB]

Self replicating patterns in 2D

For the 2D-case a moving-finite-element method based on minimizing the PDE-residual (with regularization) gives:

The following movie shows the U-component of another pattern in two dimensions (a "moving vulcano"):

[mpeg-movie, 1.5 MB]

Moving Finite Differences in 2D

Below you find some results obtained by a moving finite difference method in two space dimensions, which is based on smoothed equidistribution. It is applied to a 2D Burgers' equation.

> Moving-finite-differences (N=21) for Burgers' equation at t=0, t=0.1 :

Many references on this subject can be found [HERE]

Other interesting pages:

Gradient-weighted moving-finite-elements: Neil Carlson, Purdue University (also 3D!).

Moving-finite-elements: Mike Baines, University of Reading.

Moving-finite-volumes: John Mackenzie, Strathclyde University.

Moving-mesh-PDEs: Weizhang Huang, University of Kansas; and Bob Russell, Simon Fraser University.

The method-of-lines and adaptive grids: Bill Schiesser, Lehigh University.

Moving grids with the deformation method: Guojun Liao, Univ. of Texas at Arlington.

Application of moving grids to PDE models with Boussinesq convection (also movies): Hector Ceniceros, Univ. of California (Santa Barbara).

Moving grids and harmonic maps: Tang Tao, Hong Kong Baptist University.

Books on adaptive grids and grid generation

A collection of references on High-Resolution Non-Oscillatory Central Schemes

-- CENTRAL STATION --
A collection of references on
High-Resolution Non-Oscillatory Central Schemes
Second-order central schemes in one-space dimension
Third-and higher-order central schemes in one-space dimension
Non oscillatory central schemes in several space dimensions
Non oscillatory central schemes on unstructured and overlapping grids
Non oscillatory central schemes for incompressible flows
Non oscillatory central schemes for Hamilton-Jacobi equations
Applications of non-oscillatory central schemes for semi-conductors
Applications to Sedimentation, Flocculations and related models
Non oscillatory central schemes -- applications to...
Multi-component problems
Relaxation problems and stiff source terms
Extended thermodynamics
Balance laws: Shallow-water/Saint-Venant, Solar atmosphere, ...
Saturating dissipation
Homogenization and multiscale
Discrete kinetic models
MHD
The p-system
Riemann problems
Granular Avalanches
Second order central schemes in one space dimension
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H. Nessyahu & E. Tadmor (1990) [pdf file]Non-oscillatory central differencing for hyperbolic conservation lawsJournal of Computational Physics 87 (1990) 408-463.
D. Levy & E. Tadmor (1997) [postscript file]Non-oscillatory boundary treatment for staggered central schemes
G.-S. Jiang, D. Levy, C.-T. Lin, S. Osher & E. Tadmor (1998) [pdf file]High-resolution non-oscillatory central schemes with non-staggered grids for hyperbolic conservation lawsSIAM Journal on Numerical Analysis 35 (1998) 2147-2168.
A. Kurganov & E. Tadmor (2000) [pdf file]New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equationsJournal of Computational Physics 160 (2000) 214-282.
A. Kurganov & G. Petrova (2000) [pdf file]Central schemes and contact discontinuitiesMathematical Modelling and Numerical Analysis 34 (2000) 1259-1275.
K.-A. Lie & S. Noelle (2003) [pdf file]On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation lawsSIAM J. Scientific Computation 24 (2003) 1157-1174.
Third and higher order central schemes in one space dimension
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X-D. Liu & E. Tadmor (1998) [pdf file]Third order nonoscillatory central scheme for hyperbolic conservation lawsNumerische Mathematik 79 (1998) 397-425.
F. Bianco, G. Puppo & G. Russo (1998) [pdf file]High order central schemes for hyperblic systems of conservation laws"Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 7th international conference held in Zurich, Feb. 1998 (M. Fey and R. Jeltsch, eds.), Int'l series in Numer. Math. Vol 129, Birkhauser, 1999, 55-64.
F. Bianco, G. Puppo & G. Russo (1999) [pdf file]High order central schemes for hyperbolic systems of conservation lawsSIAM Journal on Scientific Computing 21 (1999) 294-322.
D. Levy, G. Puppo & G. Russo (1999) [pdf file]Central WENO schemes for hyperbolic systems of conservation lawsMathematical Modelling and Numerical Analysis 33 (1999) 547-571.
D. Levy, G. Puppo & G. Russo (2000) [pdf file]On the behavior of the total variation in CWENO methods for conservation lawsApplied Numerical Mathematics 33 (2000) 407-414.
A. Kurganov & D. Levy (2000) [pdf file]A third-order semi-discrete central scheme for conservation laws and convection-diffusion equationSIAM Journal on Scientific Computing 22 (2000) 1461-1488.
G. Puppo (2002) [pdf file]Numerical entropy production on shocks and smooth transitionsJournal of Scientific Computing 17 (2002) 263-271.
J. Qiu & C.-W. Shu (2002) [pdf file]On the construction, comparison, and local characteristic decompositions for high order central WENO schemesJournal of Computational Physics 183 (2002) 187-209.
E. Tadmor & J. Tanner (2003) [pdf file]An adaptive order Godunov type central scheme"Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 9th international conference held in CalTech, Mar. 2002, (T. Hou & E. Tadmor eds.) Springer 2003, pp. 871-880.
Central schemes in several space dimensions
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G.-S. Jiang & E. Tadmor (1998) [pdf file]Non-oscillatory central schemes for multidimensional hyperbolic conservation lawsSIAM Journal on Scientific Computing 19 (1998) 1892-1917.
D. Levy (1998) [gzipped file]Third-order 2D Central Schemes for Hyperbolic Conservation LawsINRIA School on Hyperbolic Systems Vol. I (1998) 489-504.
T. Katsaounis & D. Levy (1999) [pdf file]A modified structured central scheme for 2D hyperbolic conservation lawsApplied Mathematics Letters 12 (1999) 89-96.
D. Levy, G. Puppo & G. Russo (2000) [pdf file]A third order central WENO scheme for 2D conservation lawsApplied Numerical Mathematics 33 (2000) 415-421.
D. Levy, G. Puppo & G, Russo (2000) [pdf file]Compact central WENO schemes for multidimensional conservation lawsSIAM Journal on Scientific Computing 22 (2000) 656-672.
A. Kurganov & G. Petrova (2001) [pdf file]A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problemsNumerische Mathematik 88 (2001) 683-729.
A. Kurganov, S. Noelle & G. Petrova (2000) [pdf file]Semi-discrete central-upwind schemes for hyperbolic conservation laws and Hamilton-Jacobi equationsSIAM Journal on Scientific Computing 23 (2001) 707-740.
W. Rosenbaum, M. Rumpf & S. Noelle (2000) [gzipped file]An adaptive staggered scheme for conservation laws"Hyperbolic Problems: Theory, Numerics, Applications"Proceedings of the 8th international conference held in Magdeburg, Feb. 2000
K.-A. Lie & S. Noelle (2003) [pdf file]An improved quadrature rule for the flux computation in staggered central difference schemes in multi-dimensionsJournal of Scientific Computing 18 (2003) 69-81.
K.-A. Lie & S. Noelle (2000) [gzipped file]A naive implementation of ACM in nonoscillatory central difference schemes for 2D Euler equationsProceedings of the 11th ECMI conference.
R. Liska & B. Wendroff (2002) [pdf file]Comparison of several difference schemes on 1D and 2D test problems for the Euler equationsPreprint
D. Levy, G. Puppo & G. Russo (2002) [pdf file]A forth order central WENO scheme for multi-dimensional hyperbolic systems of conservation lawsSIAM J. Scientific Computing 24 (2002) 480-506.
K.-A Lie, S. Noelle & Rosenabaum (2002) [pdf file]On the resolution and stability of central difference schemesProceedings of the Third International Symposium on Finite Volumes for Complex ApplicationsPorquerolles, France, Hermes Penton Ltd, London (2002) 793-800.
X.-D. Liu & P. D. Lax (2003) [pdf file]Positive schemes for solving multi-dimensional hyperbolic systems of conservation laws IIJ. Computational Physics 187 (2003) 428-440.
P. Arminjon & A. St-Cyr (2003) [pdf file]Nessyahu朤admor-type central finite volume methods without爌redictor for 3D Cartesian and unstructured tetrahedral爂ridsApplied Numer. Math. 46 (2) (2003) 135-155.
L. Pareschi, G. Puppo & G. Russo (2004) [pdf file]Central Runge-Kutta for conservation lawsPreprint
A. Kurganov & G. Petrova (2005) [pdf file]Central-upwind schemes on triangular grids for hyperbolic systems of conservation lawsNumerical Methods for Partial Differential Equations 21 (2005), 536-552.
Central schemes on unstructured and overlapping grids
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P. Arminjon, M.-C. Viallon & A. Madarne (1997) [html file]A finite volume extension of the Lax-Friedrichs and Nessyahu-Tadmor schemes for conservation laws on unstructured gridsInt'l J. Computational Fluid Dynamics 9(1) (1997), 1-22.
P. Arminjon & M.-C. Viallon (1999) [pdf file]Convergence of a finite volume extension of the Nessyahu-Tadmor scheme on unstructured grids for a two-dimensional linear hyperbolic equations SIAM Journal on Numerical Analysis 36 (1999) 738-771.
B. Haasdonk, B. Kroner & D. Rohde (2001) [pdf file]Convergence of a staggered Lax-Friedrichs scheme for nonlinear conservation laws on unstructured two-dimensional gridsNumerische Mathematik 88 (2001) 459-484.
M. Kuther (2001) [pdf file]Error estimates for the staggered Lax-Friedrichs scheme on unstructured gridsSIAM Journal on Numerical Analysis 39 (2001) 1269-1301.
S. Karni, A. Kurganov & G. Petrova (2002) [pdf file]A smoothness indicator for adaptive algorithms for hyperbolic systemsJ. of Computational Physics 178 (2003) 323-341.
M. Kuther and M. Ohlberger (2003) [pdf file]Adaptive second-order central schemes on unstructured staggered gridsProceedings of the 9th international conference on "Hyperbolic Problems: Theory, Numerics, Applications"CalTech, Mar. 2002, (T. Hou & E. Tadmor eds.) Springer 2003, pp. 675-684.
P. Arminjon & A. St-Cyr (2003) [pdf file]New space staggered and time interleaved 2nd order finite volume methodsProceedings of the 9th international conference on "Hyperbolic Problems: Theory, Numerics, Applications"CalTech, Mar. 2002, (T. Hou & E. Tadmor eds.) Springer 2003, pp. 295-304.
Yingjie Liu (2004) [pdf file]Central Schemes on Overlapping CellsPreprint
Non-oscillatory central schemes for incompressible flows
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R. Kupferman & E. Tadmor (1997) [pdf file]A fast high-resolution second-order central scheme for incompressible flowsProceedings of the National Academy of Sciences 94 (1997) 4848-4852.
D. Levy & E. Tadmor (1997) [gzipped file]Non-oscillatory central schemes for the incompressible 2-D Euler equationsMathematics Research Letters 4 (1997) 1-20.
R. Kupferman (1998) [pdf file]Simulation of viscoelastic fluids: Couette-Taylor FlowJournal of Computational Physics 147 (1998) 22-59.
R. Kupferman (1998) [pdf file]A numerical study of the axisymmetric Couette-Taylor problem using a fast high-resolution second-order central schemeSIAM Journal on Scientific Computing 20 (1998), 858-877.
R. Kupferman & M. Denn (1999) [pdf file]Simulation of the evolution of concentrated shear layers in a Maxwell fluid with a fast high-resolution finite-difference schemeJournal of Non-Newtonian Fluid Mechanics 84 (1999) 275-287.
R. Kupferman (2001) [pdf file]A central-difference scheme for a pure streamfunction formulation of incompressible viscous flowSIAM Journal on Scientific Computing 23 (1) (2001) 1-18.
V. Naulin & A. Nielsen (2003) [pdf file]Accuracy of spectral and finite difference schemes in 2D advection problemsSIAM Journal of Scientific Computing 25 (2003) 104-126.
R. Grauer & F. Spanier (2003) [pdf file]A note on the use of central schemes for the incompressible Navier-Stokes flowsJournal of Computational Physics 192 (2003) 727-731.
Non-oscillatory central schemes -- Hamilton-Jacobi equations
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C.-T. Lin & E. Tadmor (1998) [pdf file]L1-stability and error estimates for approximate Hamilton-Jacobi solutionsNumerische Mathematik 87 (2001) 701-735.
C.-T. Lin & E. Tadmor (2000) [pdf file]High-resolution non-oscillatory central scheme for Hamilton-Jacobi equationsSIAM Journal on Scientific Computation 21 (2000) 2163-2186.
A. Kurganov & E. Tadmor (2000) [pdf file]New high-resolution semi-discrete central schemes for Hamilton-Jacobi equationsJournal of Computational Physics 160 (2000) 720-742.
S. Bryson & D. Levy (2003) [pdf file]High-order semi-discrete central-upwind schemes for multi-dimensional Hamilton-Jacobi equationsJournal of Computational Physics 189 (2003) 63-87.
S. Bryson & D. Levy (2003) [pdf file]High-order schemes for multi-dimensional Hamilton-Jacobi equations"Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 9th international conference held in CalTech, Mar. 2002, (T. Hou & E. Tadmor eds.) Springer 2003, pp. 387-396..
S. Bryson & D. Levy (2003) [pdf file]Central schemes for multi-dimensional Hamilton-Jacobi equationsSIAM Journal on Scientific Computing 25, 2003, 769-791.
S. Bryson & D. Levy (2003) [pdf file]High-order central WENO schemes for multi-dimensional Hamilton-Jacobi equationsSIAM Journal of Numerical Analysis 41, 2003, 1339-1369.
S. Bryson, A. Kurganov, D. Levy & G. Petrova (2004) [pdf file]Semi-discrete central-upwind schemes with reduced dissipation for Hamilton-Jacobi equationsIMA J. Numerical Analysis 25, 2005, 113-138.
Applications of non-oscillatory central schemes to semi-conductors
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M. Trovato & P. Falsaperla (1998) [pdf file]Full nonlinear closure for a hydrodynamical model of transport in siliconPhysical Review B-Condensed Matter 57 (1998) 4456-4471.
V. Romano & G. Russo (2000) [gzipped file]Numerical solution for hydrodynamical models of semiconductorsMathematical Models & Methods in Applied Sciences 10 (2000) 1099-1120.
A. M. Anile, N. Nikiforakis & R. M. Pidatella (2000) [pdf file] Assessment of a high resolution centered scheme for the solution of hydrodynamical semiconductor equationsSIAM Journal of Scientific Computing 22 (2000) 1533-1548.
A. M. Anile, V. Romano & G. Russo (2000) [pdf file]Extended hydrodynamical model of carrier transport in semiconductorsSIAM Journal Applied Mathematics 61 (2000) 74-101.
A. M. Anile & V. Romano (2000)Hydrodynamical Modeling of Charge Carrier Transport in SemiconductorsMeccanica 35 (2000) 249-296.
V. Romano (2001) [pdf file]2D simulation of a silicon MESFET with a nonparabolic hydrodynamical model based on the maximum entropy principleJournal of Computational Physics 176 (2001) 70-92.
V. Romano (2001) [pdf file]Non-parabolic band hydrodynamical model of silicon semiconductors and simulation of electron devicesMathematical Methods in eth Applied Sciences 24 (2001) 439-471.
C. Gardner, A. Gelb & J. Hernandez (2002) [gzipped file]A comparison of modern hyperbolic methods for semiconductor device simulation: NTK central schemes vs. CLAWPACKVLSI Design 15 (2002) 721-728.
Applications to Sedimentation, Flocculations and related models
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R. Burger & F. Concha (1998) [pdf file]Mathematical model and numerical simulation of the setting of flocculated suspensionsInternational Journal of Multiphase Flow 24 (1998) 1005-1023.
E. B. Pitman (1998) [pdf file]Forces on bins: The effect of random frictionPhysical Review E 57 (1998) 3170-3175.
R. Burger, F. Concha, K. K. Fjelde, & K. H. Karlsen (2000) [pdf file]Numerical simulation of the setlling of polydisprese suspensions of spheresPowder Technology 113 (2000) 30-54.
R. B黵ger, S. Evje, K. H. Karlsen & K.-A. Lie (2000) [pdf file]Numerical methods for the simulation of the settling of flocculated suspensionsChemical Engineering Journal 80 (2000) 91-104.
R. Burger, K. -K Fjelde, K. Hofler, & K. H. Karlsen (2000) [pdf file]Central difference solutions of the kinematic model of settling of polydispersesuspensions and three-dimensional particle-scale simulationsJournal of Engineering Mathematics 41 (2001) 167-187.
B. Xue & Y. Sun (2003) [pdf file]Modeling of sedimentation of polydisperese spherical beads with a broad size distributionChemical Engineering Science 58 (2003) 1531-1543.
S. Berres & R. Bürger (2003) [pdf file]On gravity and centrifugal settling of polydisperse suspensions forming compressible sedimentsInt'l J. Solids & Structures 40 (2003), 4965-4987.
S. Berres, R. Bürger, K. H. Karlsen & E. M. Tory (2003) [pdf file]Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compressionSIAM J. Applied Mathematics 64 (2003), 41-80.
S. Berres, R. Bürger & K. H. Karlsen (2004) [pdf file]Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensionsJ. Comp. Appl. Math. 164-165 (2004) 53-80
S. Berres, R. Bürger & E. M. Tory (2004) [pdf file]Mathematical model and numerical simulation of the liquid fluidization of polydisperse solid particle mixturesComput. Visual Sci. 6 (2004) 67-74.
Non-oscillatory central schemes -- applications to > Multi-component problems
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B. Engquist & O. Runborg (1996) [pdf file]Multiphase computations in geometrical opticsJournal of Computational and Applied Mathematics 74 (1996) 175-192.
R. Fazio & G. Russo (2000) [gzipped file]A Lagrangian central scheme for multi-fluid flows"Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 8th Int'l conference held in Magdeburg, Feb. 2000
L. Gosse (2002) [gzipped file]Using K-branch entropy solutions for multivalued geometric optics computationsPreprint
S. Karni, E. Kirr, A. Kurganov & G. Petrova (2003) [pdf file]Compressible two-phase flows by central and upwind schemesMathematical Modeling and Numerical Analysis 38 (3) (2004) 477-494.
Non-oscillatory central schemes -- applications to > Relaxation problems and stiff source terms
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F. Bereux & L. Sainsaulieu (1997) [pdf file]A Roe-type Riemann solver for hyperbolic systems with relaxation based on time-dependent wave-decompositionNumerische Mathematik 77 (1997) 143-185.
S. F. Liotta, V. Romano & G. Russo (2000) [pdf file]Central schemes for balance laws of relaxation typeSIAM Journal on Numerical Analysis 38 (2000) 1337-1356.
L. Pareschi (2001) [pdf file]Central differencing based numerical schemes for hyperbolic conservation laws with relaxation termsSIAM Journal Numerical Analysis 39 (2001) 1395-1417.
C. Arvanitis, T. Katsaounis & C. Makridakis (2001) [pdf file]Adaptive finite element relaxation schemes for hyperbolic conservation lawsMathematical Modeling and Numerical Analysis 35 (1) (2001) 17-33.
S. Jin, L. Pareschi & M. Slemrod (2002) [gzipped file]A relaxation scheme for solving the Boltzmann equation based on the Chapman-Enskog expansionActa Mathematicas Applicatae Sinica (English Series) 18, No.1, (2002).
A. Kurganov (2002) [pdf file]An accurate deterministic projection method for hyerbolic systems with stif source term"Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 9th international conference held in CalTech, Mar. 2002, (T. Hou & E. Tadmor eds.) Springer 2003, pp. 665-674.
Non-oscillatory central schemes -- applications to > Extended thermodynamics
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M. Torrilhon (2000) [pdf file]Characteristic waves and dissipation in the 13-moment-caseContinuum Mech. Thermodynamics 12 (2000) 289-301.
Non-oscillatory central schemes -- applications to > Balance laws
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S. F. Liotta, V. Romano & G. Russo (1998) [gzipped file]Central schemes for systems of balance laws"Hyperbolic Problems: Theory, Numerics, Applications", Proceedings of the 7th Int'l conference held in Zurich, Feb. 1998 (M. Fey and R. Jeltsch, eds.), Int'l Series on Numerical Mathematics, Birkhauser, Vol. 130, 1999, 651-660
G. Russo (2002) [pdf file]Central schemes and systems of balance lawsin "Hyperbolic Partial Differential Equations, Theory, Numerics and Applications", (A. Meister and I. Struckmeier,eds.) Vieweg, Wiesbaden (D), 2002.
A. Kurganov & D. Levy (2002) [pdf file]Central-Upwind Schemes for the Saint-Venant SystemMathematical Modeling and Numerical Analysis 36 (2002) 397-425.
N. Crnjaric-Zic, S. Vukovic & L. Sopta (2004) [pdf file]Balanced finite volume WENO and central WENO schemes for the shallow water and the open-channel flow equationsJournal of Computational Physics 200 (2004) 512-548.
S. Bryson, A. Kosovichev & D. Levy (2005) [pdf file]High-order shock-capturing methods for modeling dynamics of the solar atmosphereNonlinearity 201 (2005) 1-26.
A. Chertok & A. Kurganov (2004) [pdf file]On a Hybrid Finite-Volume-Particle MethodMathematical Modeling and Numerical Analysis 38(6) (2004) 1071-1091.
Non-oscillatory central schemes -- applications to > Saturating dissipation
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A. Kurganov & P. Rosenau (1997) [pdf file]Effects of a saturating dissipation in Burgers-type equationsCommunications on Pure and Applied Math. L (1997) 753-771.
A. Kurganov, D. Levy & P. Rosenau (1998) [pdf file]On Burgers-type equations with nonmonotonic dissipative fluxesCommunications on Pure & Applied Mathematics LI (1998) 443-473.
J. Goodman, A. Kurganov & P. Rosenau (1999) [pdf file]Breakdown in Burgers-type equations with saturating dissipation fluxesNonlinearity 12 (1999) 247-268.
J. Otero, L. A. Dontcheva, H. Johnston, R. A. Worthing, A. Kurganov, G. Petrova & C. Doering (2004) [pdf file]High Raleigh Number Convection in a Fluid Saturated Porous LayerJournal of Fluid Mechanics 500 (2004), 263-281.
Non-oscillatory central schemes -- applications to > Homogenization and Multiscale
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E. Tadmor & T. Tassa (1997) [pdf file]On the homogenization of oscillatory solutions to nonlinear convection-diffusion equationsAdvances in Mathematical Sciences and Applications 7(1) (1997) 93-117.
X. Li & W. E (2004) [pdf file]Multiscale modeling of the dynamics of solids at finite temperatureJ. Mech. Phys. solids(53) 2004, 1650-1685.
Non-oscillatory central schemes -- applications to > Discrete kinetic models
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E. Gabetta, L. Pareschi & M. Ronconi (2000) [gzipped file]Central schemes for hydrodynamical limits of discrete-velocity kinetic modelsTransport Theory and Statistical Physics 29 (2000) 465-477.
A. Kurganov (2002) [pdf file]Semi-discrete central schemes for balance laws. Application to the Broadwell modelProceedings of the Third International Symposium on Finite Volumes for Complex Applications
Non-oscillatory central schemes -- applications to > MHD
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C.C. Wu & T. Chang (2001) [pdf file]Further study of the dynamics of two-dimensional MHD coherent structures -- a large-scale simulationJournal of Atmospheric and Solar-Terrestrial Physics 63 (2001) 1447-1453.
K. Germaschewski, A. Bhattacharjee, T. Linde, R. Rosner, A. Siegel, D. Keyes & F. Dobrian (2003) [pdf file]The magnetic reconnection code: framework and applicationSciDAC-TOPS, CMRS poster.
M. Torrilhon (2003) [pdf file]Non-uniform convergence of finite volume schemes for Riemann problems of ideal magnetohydrodynamicsJournal of Computational Physics 192(1) (2003) 73-94.
J. Kleimann, A. Kopp, H. Fichtner, R. Grauer, & K. Germaschewski (2004) [pdf file]Thre-dimensional MHD high-resolution computations with CWENO employing adaptive mesh refinmentComput. Physics Communication 158 (2004) 47-56.
J. Balbas, E. Tadmor, & C.-C. Wu?2004) [pdf file]Non-oscillatory central schemes for one- and two-dimensional MHD equationsJournal of Computational Physics 201 (2004) 261-285.
P. Arminjon & R. Touma?2004) [pdf file]Central finite volume methods with constrained transport divergence treatment for ideal MHDJournal of Computational Physics (2004)
J. Balbas & E. Tadmor (2005) [pdf file]Non-oscillatory central schemes for one- and two-dimensional MHD equations. II: High-order semi-discrete schemesPreprint (2005)
Non-oscillatory central schemes -- applications to > The p-system
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F. Hoch & M. Rascle (1999) [pdf file]A numerical study of a pathological example of p-systemSIAM Journal of Numerical Analysis 36 (1999) 1588-1603.
Non-oscillatory central schemes -- applications to > Riemann problems
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A. Kurganov & E. Tadmor (2000) [pdf file]Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solversNumerical Methods for Partial Differential Equations 18 (2002) 548-608.
Non-oscillatory central schemes -- applications to > Granular Avalanches
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Y.C. Tai, S. Noelle, J.M. N.T. Gray & K. Hutter (2001) [pdf file]Shock-capturing and front-tracking methods for granular avalanchesJournal of Computational Physics 175 (2001) 269-301.

Analytical and Stochastic Fluid Dynamics

Analytical and Stochastic Fluid DynamicsOrganized by: Craig Evans, Susan Friedlander, Boris Rozovsky, Daniel Tataru and David A. EllwoodOctober 10, 2005 to October 14, 2005
Lecture Schedule & Talk Abstracts
Download schedule in PDF format
Monday, October 10
9:30-10:00 Mohammed Ziane, “Remarks on the normal form of the Navier-Stokes equations”Abstract: We introduce a construction of regular solutions to the Navier-Stokes which is specifically designed for the study of their asymptotic expansions. Using this construction, we give sufficient conditions for the convergence of those expansions. We also construct suitable normed spaces in which they converge. Moreover, in thesespaces, the normal form of the Navier-Stokes equations associated with the terms of the asymptotic expansions is a well-behaved infinite system of differential equations.This is a joint work with Foias, Hoang and Olson.
10:00-10:45 Peter Constantin, “Nonlinear Fokker-Planck Navier-Stokes Systems”Abstract: We will describe regularity results concerning Navier-Stokes equations coupled with nonlinear Fokker-Planck equations describing the probability distribution of microscopic inclusions.
10:45-11:15 Morning Tea (6th Floor)
11:15-12:00 Edriss Titi, “Global Regularity for the Three-dimensional Primitive Equations of Ocean and Atmosphere Dynamics”Abstract: The basic problem faced in geophysical fluid dynamics is that a mathematical description based only on fundamental physical principles, which are called the ``Primitive Equations'', is often prohibitively expensive computationally, and hard to study analytically. In this talk we will survey the main obstacles in proving the global regularity for the three-dimensional Navier—Stokes equations and their geophysical counterparts. Even though the Primitive Equations look as if they are more difficult to study than thethree-dimensional Navier--Stokes equations we will show in this talk that they have globally (in time) unique regular solution for all initial data. This is a joint work with Chongshen Cao.
12:00-2:45 Lunch/Discussion
2:45-3:30 Sergei B. Kuksin, “Asymptotic properties of some SPDE with small dissipation”Abstract: I will discuss recent progress in the study of asymptotic properties of a class of randomly forced PDE, where the force is proportional to the square root of the dissipation. The class contains, for example, the randomly forced 2D Navier-Stokes equation and the forced-driven KdV equation. The latter equation will be the main object of my talk.
3:30-4:00 Afternoon Tea (6th Floor)
4:00-4:45 Franco Flandoli, “Markov selections and their regularity for 3D stochastic Navier-Stokes equations”Abstract: Existence of weak solutions to the 3D stochastic Navier-Stokes equations in the martingale sense is known, but their uniqueness is open as in the deterministic case. We prove the existence of selections with a weak form of Markov property. Under special assumptions on the noise, we prove the full Markov property and a form of continuous dependence on initial conditions, similar to the Strong Feller property. The methods are strongly inspired to an approach of Da Prato and Debussche.
4:45-5:15 Roman Shvydkoy, “Spectral problem for the Euler and Navier-Stokes equations”Abstract: In this talk we give an overview of recent results on the essential spectrum of the Euler equation, related instability issues, and the vanishing viscosity limit for the linearized problem.
Tuesday, October 11
9:30-10:00 Eric Vanden-EijndenAbstract: We will present a class of simple and exactly solvable linear shell models which are formally conservative yet may display anomalous dissipation via cascade of energy. As a result, these models can be forced and yet have a unique invariant measure.
10:00-10:45 Jonathan Mattingly, “Exponential Mixing for the Degenerately forced Navier Stokes Equations”Abstract: I will describe how to prove exponential convergence to the unique equilibrium of the stochastically forced 2D Navier-Stokes equations when the forcing is minimal. The techniques are quite general and make use of infinite dimensional Malliavin calculus and coupling in the Wasserstein Metric on a function space.
10:45-11:15 Morning Tea (6th Floor)
11:15-12:00 Vladimir Sverak, “Regularity of L^{3,\infty} solutions of the Navier-Stokes equations"Abstract: This is a joint work with Luis Escauriaza and Gregory Seregin showing that solutions of the three-dimensional incompressible Navier-Stokes equations with bounded spatial L^3 norm are regular. The proof uses a blow-up technique, together with a backward uniqueness result for the heat equation.
12:00-2:45 Lunch/Discussion
2:45-3:30 Charlie Fefferman, “The surface QG-alpha equation”Abstract: Will explain recent work of Cordoba et al on a 2D fluid equation intermediate between surface QG and 2D Euler.
3:30-4:00 Afternoon Tea (6th Floor)
4:00-4:45 Tom Hou, “The Interplay between Local Geometric Properties and the Global Regularity for 3D Incompressible Flows”Abstract: Whether the 3D incompressible Euler equation can develop a finite time singularity from smooth initial data has been an outstanding open problem. It has been believed that a finite singularity of the 3D Euler equation could be the onset of turbulence. Here we review some existing computational and theoretical work on possible finite blow-up of the 3D Euler equation. Further, we show that there is a sharp relationship between the geometric properties of the vortex filament and the maximum vortex stretching. By exploring this local geometric property of the vorticity field, we have obtained a global existence of the 3D incompressible Euler equations provided that the unit vorticity vector and the velocity field have certain mild regularity property in a very localized region containing the maximum vorticity. Our assumption on the local geometric regularity of the vorticity field and the velocity field seems consistent with recent numerical experiments. Further, we discuss how viscosity may help preventing singularity formation for the 3D Navier-Stokes equations, and prove the global existence of the 2D dissipative Boussinesq equations. We will also present some recent global existence results for a model of the 3D Navier-Stokes equations and the 3D alpha-averaged Euler equations.
4:45-5:15 Jim Kelliher, “Bounded domain limit for Navier-Stokes and Euler equations”Abstract: Suppose we have well-posedness for some PDE both in all of R^d and in a bounded domain U of R^d. Imagine we take the initial data for the whole space and restrict it to U, modifying it slightly to satisfy any required boundary conditions. If we scale U by r, does the solution on rU approach the solution in R^d as r goes to infinity in some appropriate sense? We answer this question for weak solutions to the Navier-Stokes and Euler equations in two dimensions, showing that strong convergence occurs in some norms and weak convergence in other norms of interest. Critical to this is showing that the initial velocity can be modified to match the appropriate boundary conditions while changing its norms by a vanishingly small amount as r goes to infinity. A secondary goal is to establish, for initial velocity that decays at infinity as slowly as possible, the same existence, uniqueness, and regularity results as for classical (that is, finite energy) solutions. The original motivation for this work was to establish the existence and uniqueness of statistical solutions to the Navier-Stokes and Euler equations in the entire plane with infinite energy and, for the Euler equations, unbounded initial vorticities, by taking advantage of the well developed theory of statistical solutions in a bounded domain of the plane.
Wednesday, October 12
9:30-10:00 Igor Kukavica, “One direction and one component regularity for the Navier-Stokes equations”Abstract: We consider sufficient conditions for regularity of Leray-Hopf solutions of the Navier-Stokes equation. By a result of Neustupa and Panel, the Leray-Hopf weak solutions are regular provided a single component of the velocity is bounded. In this talk we will survey existing and present new results on one component and one direction regularity. This is a joint work with M. Ziane
10:00-10:45 Andrea Bertozzi, “Electrowetting in a Hele-Shaw geometry”Abstract: Electrowetting has recently been explored as a mechanism for moving small amounts of fluid in confined spaces. We propose a diffuse interface model for droplet motion, due to electrowetting, in a Hele-Shaw geometry. In the limit of small interface thickness, asymptotic analysis shows the model is equivalent to Hele-Shaw flow with a voltage-modified Young-Laplace boundary condition on the free surface. We show that details of the contact angle significantly affect the timescale of motion in the model. We measure receding and advancing contact angles in the experiments and derive their influences through a reduced order model. These measurements suggest a range timescales in the Hele-Shaw model which include those observed in the experiment. The shape dynamics and topology changes in the model agree well wtih the experiment, down to the length scale of the diffuse interface thickness. This is joint work with Hsiang-Wei Lu, Karl Glasner, and C J Kim.
10:45-11:15 Morning Tea (6th Floor)
11:15-12:00 Giovanni Gallavotti, “Chaotic motions and developed turbulence: heuristic ideas”Abstract: An interpretation of Ruelle's proposal to regard turbulence as due to strange attractors can be interpreted as the statement that, for practical purposes, the attractor of a chaotic evolution can be regarded as an Anosov system. The problem is how to extract information that can be tested experimentally or theoretically from this point of view. I shall discuss some recent attempts in this direction.
12:00-2:45 Lunch/Discussion
2:45-3:30 Claude Bardos, “Analytic stability and singularities for Kelvin Helmholtz, Rayleigh Taylor, Problems Comparison with the stability of water waves problems”Abstract: Rayleigh Taylor and Kelvin Helmholtz are equations describing interfaces in fluids. Such interfaces are known by experiments and numerical simulations to be highly unstable. On the other hand a series of recent of results (Sijie Wu, Gilles Lebeau and Vladimir Kamotski) show that these solutions whenever they exist with a minimal regularity are solutions of nonlinear elliptic equations. Therefore they are analytic. This shows for instance that any solution with small regularity (say in C1+∞ but not in C∞ ) at some time will evolve in very singular systems. In the present lecture I will discuss these issues and compare the situation corresponding to Kelvin Helmholtz, Rayleigh Taylor and water waves equations. Conclusions are as follow The equation for water waves describes the stability of the phenomena as long as the surface of the water (which may be a multivalued function) do not self intersect. From this point of view Rayleigh Taylor and Kelvin Helmholtz problem are very similar but very different from water waves. When a singularity appears say a cusp in vorticity as predicted by the numerical computations of Moore and Orzag and proven on one example by Caflisch and Orellana the solution can in no way be extended by a curve of the same regularity what will appear will be a very weak solution in the sense of Delors or a curve which will not be arc-chord in the sense of Guy David and may be in agreement with the numerical simulations of Krasny. Therefore mathematical proof of the existence of weak enough solutions concentrated on such curves is a challenging open problem.
3:30-4:00 Afternoon Tea (6th Floor)
4:00-4:45 Herbert Koch, “Regularity for a free boundary problem and a conjecture of De Giorgi”Abstract: In this joint work with G. Leoni and M. Morini we transform free boundary problems to systems of elliptic boundary value problems. Examples are minimizers to the Mumford-Shah functional and stationary velocity fields of fluids with surface tension. Koch: Regularity for a free boundary problem and a conjecture of De Giorgi.
4:45-5:15 Natasa Pavlovic, “Long time behavior of solutions to the 3D Navier-Stokes”Abstract: In this talk we shall discuss long time behavior of solutions to the 3D Navier-Stokes equations that evolve from initial data in the space BMO^{-1} intersected with certain Morrey space. This is a joint work with James Colliander, Carlos Kenig and Gigliola Staffilani. Local in time existence of solutions to the Navier-Stokes equations in BMO^{-1} and global in time existence of solutions corresponding to small initial data in BMO^{-1} were proved by Koch and Tataru in 2001. We consider apriori global solution in the space BMO^{-1} corresponding to initial data that are in BMO^{-1} intersected with certain Morrey space. A persistence of Morrey norm combined with microlocal analysis of solutions help us identify a time decay of solutions.
Thursday, October 13
9:30-10:00 Anna L. Mazzucato, “On the decay of the energy spectrum for weak solutions to the Navier-Stokes equations”Abstract: We observe that known regularity results imply weak solutions of the 3D Navier-Stokes equations are regular if their energy spectrum decays as k^s, s<-3, for large wavenumbers k. We allow for weak solutions having infinite total energy by localizing the equations. We then consider certain modified Leray backwards self-similar solutions and show that their energy spectrum decays exactly at the critical rate. Therefore, this rate of decay is consistent with the appearance of a self-similar singularity.
10:00-10:45 Anatoli Babin, “Linear superposition of nonlinear waves”Abstract: Nonlinear waves are described by nonlinear differential equations. Their solutions are determined by initial data, which are functions of the spatial variables. When the equation is linear, if the initial function equals the sum of two or several functions, the solution equals the sum of corresponding solutions. For nonlinear equations this linear superposition principle is not valid. Nevertheless, there are important and physically relevant systems and classes of initial data for which the solution equals the sum of corresponding solutions with a small error. The examples include: Fermi-Pasta-Ulam system, nonlinear wave equation, nonlinear Schrodinger equation, Navier-Stokes and Euler systems in a rotating frame, Boussinesq system with a strong rotation or stratification. The described approximate linear superposition of nonlinear waves is explained by a destructive wave interference between different wavepackets in the process of their time evolution, this interference drastically reduces nonlinear interactions between the wavepackets.
10:45-11:15 Morning Tea (6th Floor)
11:15-12:00 Susan Friedlander, “Nonlinear Instability for the Navier Stokes Equations”Abstract: It is proved that linear instability implies nonlinear instability for the Navier Stokes equations in L^p, p > 1. The result holds in all spatial dimensions and both finite domains and R^n. The method of proof uses a bootstrap argument. This is joint work with Roman Shvydkoy and Natasa Pavlovic. Friedlander: Nonlinear Instability for the Navier Stokes Equations
12:00-2:45 Lunch/Discussion
2:45-3:30 Poster Session
3:30-4:00 Afternoon Tea (6th Floor)
4:00-4:45 Poster Session
4:45-5:15 Discussion Workshop
Friday, October 14
9:30-10:00 Boris Rozovsky, “Passive Scalar Equation in a Turbulent Gaussian Velocity Field”Abstract: Time evolution of a passive scalar is considered in a turbulent homogeneous incompressible Gaussian flow. The turbulent nature of the flow results in non-smooth coefficients in the corresponding evolution equation. A strong, in the probabilistic sense, solution of the equation is constructed using Wiener Chaos expansion, and the properties of the solution are studied. Among the results obtained are a certain L_{p}-regularity of the solution and Feynman-Kac-type, or Lagrangian, representation formula. The results apply to both viscous and conservative flows. Rozovsky: Passive Scalar Equation in a Turbulent Gaussian Velocity Field.
10:00-10:45 Alexandre Chorin, “Scaling laws in turbulence”Abstract: I will present a mathematical model for the scaling ofwall-bounded turbulent flows, as well as recent results on the scaling of homogeneous istropic turbulence. (joint work with G.I. Barenblatt).
10:45-11:15 Morning Tea (6th Floor)
11:15-12:00 Remigijus Mikulevicius, “On stochastic Euler equation”Abstract: The existence of a martingale problem solution to a degenerated Navier-Stokes equation is proved. It is a weak limit of solutions to Navier-Stokes equation.
12:00-2:45 Lunch/Discussion
2:45-3:30 Xiaoming Wang, “The Emergence of Large Scale Coherent Structure under Small Scale Random Bombardments”Abstract: We provide mathematical justification of the emergence of large scale coherent structure in a two dimensional fluid system under small scale random bombardments with small forcing and appropriate scaling assumptions. The analysis shows that the large scale structure emerging out of the small scale random forcing is not the one predicted by equilibrium statistical mechanics. But the error is very small which explains earlier successful prediction of the large scale structure based on equilibrium statistical mechanics. This is a joint work with Andrew Majda of New York University.
3:30-4:00 Afternoon Tea (6th Floor)

TMSCSCS: Topological Methods in Scientific Computing, Statistics and Computer Science

http://math.stanford.edu/comptop/#People
References
Papers

[ABB] O. Alter, P.O. Brown, D. Botstein, “Singular value decomposition for genome-wide expression data processing and modeling,” Proceedings of the National Academy of Sciences, 97 (18), August 2000, pp. 10101–10106. PDF

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[ABET] N. Amenta, M. Bern, D. Eppstein, S.-H. Teng, “Regression depth and center points,” Discrete and Computational Geometry, 23 (3), April 2000, pp. 305–323. PDF

[BIH] C.A.L. Bailer-Jones, M. Irwin, T. von Hippel, “Automated classification of stellar spectra II: two-dimensional classification with neural networks and principal components analysis,” Monthly Notices of the Royal Astronomical Society, 298 (2), August 1998, pp. 361–377. PDF

[BW] G.E.P. Box, “Sampling and Bayes’ inference in scientific modelling and robustness,” Journal of the Royal Statistical Society, Series A, 143 (4), 1980, pp. 383–430. PDF

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[DW] R. Durbin, D. Willshaw, “An analogue approach to the travelling salesman problem using an elastic net method,” Nature, 326 (6114), 22 April 1987, pp. 689–691. PDF

[Ed] H. Edelsbrunner, “Shape reconstruction with Delaunay complex,” pp. 119–132 in [LNCS1380]. PDF

[EdM] H. Edelsbrunner, E.P. Mücke, “Three-dimensional alpha shapes,” ACM Transactions on Graphics, 13 (1), January 1994, pp. 43–72. PDF

[EdS] H. Edelsbrunner, N.R. Shah, “Triangulating topological spaces,” Tenth Annual ACM Symposium on Computational Geometry, International Journal of Computational Geometry and Applications, 7 (4), August 1997, pp. 365–378. PDF

[Ep] D. Eppstein, “Clustering for faster network simplex pivots,” Networks, 35 (3), 2000, pp. 173–180. PDF

[F] J. Friedman, “Computing Betti numbers via combinatorial Laplacians,” Algorithmica, 21 (4), 1998, pp. 331–346. PDF

[FR] J.H. Friedman, L.C. Rafsky, “Graph theoretic measures of multivariate association and prediction,” Annals of Statistics, 11 (2), 1983, pp. 377–391. PDF

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[KPP] W. Klein, R. Plomp, L.C.W. Pols, “Vowel spectra, vowel spaces, and vowel identification,” Journal of the Acoustical Society of America, 48 (4), 1970, pp. 999–1009. PDF

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[Ma] G. Marsaglia, “Random numbers fall mainly in the planes,” Proceedings of the National Academy of Sciences, 61 (1), September 1968, pp. 25–28. PDF

[MS1] T. Martinetz, K. Schulten, “Topology Representing Networks,” Neural Networks, 7 (3), 1994, pp. 507–522. PDF

[MS2] R.B. Morgan, D.S. Scott, “Preconditioning the Lanczos algorithm for sparse symmetric eigenvalue problems,” SIAM Journal on Scientific Computing, 14 (3), May 1993, pp. 585–593. PDF

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[SLD] P. Simard, Y. Le Cun, J. Denker, “Efficient pattern recognition using a new transformation distance,” pp. 50–58 in [ANIPS5]. PDF

[TSM] P. Tamayo, D. Slonim, J. Mesirov, Q. Zhu, S. Kitareewan, E. Dmitrovsky, E.S. Lander, T.R. Golub, “Interpreting patterns of gene expression with self-organizing maps: Methods and application to hematopoietic differentiation,” Proceedings of the National Academy of Sciences, 96 (6), March 1999, pp. 2907–2912. PDF

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Conference Proceedings

[BS3] J.-M. Bernardo, M.H. DeGroot, D.V. Lindley, A.F.M. Smith (Eds.), Bayesian Statistics, 3, Proceedings of the Third Valencia International Meeting, June 1–5, 1987, Oxford University Press, New York, NY, 1988. (ISBN: 0-19-852220-7)

[ANIPS5] C. Giles, S. Hanson, J. Cowan (Eds.), Advances in Neural Information Processing Systems, 5, Morgan Kaufmann, San Francisco, CA, 1993. (ISBN: 1-55860-274-7)

[ANIPS7] G. Tesauro, D.S. Touretzky, T.K. Leen (Eds.), Advances in Neural Information Processing Systems, 7, MIT Press, Cambridge, MA, 1995. (ISBN: 0-262-20104-6)

[ANIPS10] M.I. Jordan, M.J. Kearns and S.A. Solla (Eds.), Advances in Neural Information Processing Systems, 10, MIT Press, Cambridge, MA, 1997. (ISBN: 0-262-10076-2)

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[Ko] T. Kohonen, Self Organisation and Associative Memory, Springer Series in Information Sciences, 8, 3nd Ed., Springer-Verlag, Berlin, 1989. (ISBN: 0-387-51387-6)

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[Mi] J. Milnor, Singular Points of Complex Hypersurfaces, Annals of Mathematics Studies, 61, Princeton University Press, Princeton, NJ, 1968. (ISBN: 0-691-08065-8)

Measure valued solutions to conservation laws

R. DiPerna, Arch.Rat.Mech.Anal 88 1985, 223-270

J. Carrillo, Entropy Solutions for nonlinear degenerate problems, Arch.Rat.Mech.Anal 147 1999, 269-361

Elliptic operator in the sense of ADN

3. Zbl 0922.35042 Agmon, Shmuel
Representation theorem for solutions of the Helmholtz equation on $\bbfR^n$. (English)
Buslaev, V. (ed.) et al., Differential operators and spectral theory. M. Sh. Birman's 70th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 189(41), 27-43 (1999). MSC 2000: *35J05 35C15, Reviewer: P.K.Mahanti

MR0162050 (28 #5252) Agmon, S.; Douglis, A.; Nirenberg, L. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math. 17 1964 35--92. (Reviewer: G. Geymonat) 35.46

MR0125307 (23 #A2610)
Agmon, S.; Douglis, A.; Nirenberg, L.
Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I.
Comm. Pure Appl. Math. 12 1959 623--727.
35.43

Andrew D. Lewis

http://penelope.mast.queensu.ca/~andrew/

Andrew D. Lewis's Notes
2001a
Title: A brief on controllability of nonlinear systemsDetail: Mathematics & Statistics Control Seminar, Queen's University, Fall 2001
1997a
Title: Reduction of simple mechanical systemsDetail: Mechanics and symmetry seminars, University of Warwick, Fall 1997
1995c
Title: Some two-dimensional manifolds with affine connectionsDetail: Geometry reading group, California Institute of Technology, Fall 1995 - Winter 1996
1995b
Title: Notes on linear frame bundlesDetail: Geometry reading group, California Institute of Technology, Fall 1995 - Winter 1996
1993b
Title: Notes on mechanics, symmetry, reduction, holonomy, and controlDetail: Mechanical systems reading group, California Institute of Technology, Summer 1993
1993a
Title: Notes on principal fibre bundlesDetail: Mechanical systems reading group, California Institute of Technology, Summer 1993

HLNUM.org

This page is about "Hochleistungsnumerik" or, for the english speaking visitors: "high performance numerics". It contains information about various projects I am/was involved in which are more or less related to this topic:
Numerical Mathematics
H-matrices,
FEM/BEM applications,
Convection-Diffusion-Equations,
Parallel Computations
Beowulf Systems,
ThreadPool,
libBSP,
Memory-Management
rmalloc,
Tools
cpuflags,