Paul J. Atzberger, Assistant Professor
PhD Mathematics · Courant Institute, NYU · 2003
Research Interests
Stochastic Analysis, Numerical Analysis, Scientific Computing. Applications in Physics, Engineering, and Biology.
Publications:
1. A Stochastic Immersed Boundary Method for Fluid-Structure Dynamics at Microscopic Length Scales, Atzberger, P.J., Kramer, P.R., and Peskin, C.S., J. Comp. Phys., Vol. 224, Iss. 2, (2007). [PDF]
2. Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations (brief introduction), Atzberger, P.J., Kramer, P.R., and Peskin, C.S., (proceedings of ICIAM 2007). [PDF]
3. Theoretical Framework for Microscopic Osmotic Phenomena, Atzberger, P.J. and Kramer, P.R. , Phys. Rev. E, 75, 1, (2007). [PDF]
4. Spatially Adaptive Stochastic Numerical Methods for Intrinsic Fluctuations in Reaction-Diffusion Systems, Atzberger, P.J., (2008), (preprint). [PDF]
5. A Microfluidic Pumping Mechanism Driven by Non-equilibrium Osmotic Effects, Atzberger, P.J., Isaacson, S.A., and Peskin, C.S., (2008), (preprint). [PDF]
6. On the Foundations of the Stochastic Immersed Boundary Method, Kramer, P.R., Peskin, C.S., and Atzberger, P.J., Comp. Meth. in Appl. Mech. and Eng., Vol. 197, Iss. 25-28, 15 April, pp. 2232-2249, (2008). [PDF]
7. Error Analysis of a Stochastic Immersed Boundary Method Incorporating Thermal Fluctuations , Atzberger, P.J. and Kramer, P.R. (to appear in Mathematics and Computers in Simulation). [PDF]
8. A Note on the Correspondence of the Immersed Boundary Method with Thermal Fluctuations To Stokesian-Brownian Dynamics, Atzberger, P.J., Physica D, Vol. 226, Iss. 2, 15, pg. 144-150, (2007). [PDF]
9. Velocity Correlations of a Thermally Fluctuating Brownian Particle: A Novel Model of the Hydrodynamic Coupling , Atzberger, P.J., Phys. Lett. A, Vol. 351, Iss. 4-5, 6, March, pp. 225-230, (2006). [PDF]
10. A Brownian Dynamics Model of Kinesin in Three Dimensions Incorporating the Force-Extension Profile of the Coiled-Coil Cargo Tether, Atzberger, P.J. and Peskin, C.S., Bull. Math. Biol., vol. 68, no. 1, pp. 131-160, (2006). [PDF]
We gratefully acknowledge the following source of support:
Grant DMS-0635535*.
* This material is based upon work supported by the National Science Foundation under Grant No. DMS-0635535. Any opinions, findings and conclusions or recomendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).
Numerical Codes:
Class Websites and Course Notes:
Mathematical Finance Majors:
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Introduction to Stochastic Analysis and Applications
(Fall 2008)
-
TR 11:00am - 12:15pm.
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Numerical Analysis I
(Fall 2008)
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TR 8:00am - 9:15pm.
-
Introduction to Mathematical Biology
(Winter 2008)
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TR 8:00am - 9:15pm, Old Little Theatre, Rm. 160B.
-
Finite Element Methods
(Fall 2007)
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TR 11:00am - 12:15pm, South Hall Rm. 6635.
-
Introduction to Complex Variables
(Fall 2007)
-
TR 8:00am - 9:15pm, GIRV Rm. 1116.
Additional Class Websites:
Advising:
Postdoctoral Researchers:
-
Chetan Pahlajani
Graduate Students:
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Lacey Heubel
-
David Valdman
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Per Danzl (co-advisor with Jeff Moehlis, IGERT)
Undergraduate Students:
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Joe Rudzinski
Administration:
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Mathematical Finance Majors (Undergraduate Faculty Adviser)
Research Summaries:
Below we give some brief summaries concerning a few projects on which we are currently working.
Stochastic Immersed Boundary Methods /
Computational Fluid Dynamics:
The Stochastic Immersed Boundary Method (SIB) is a numerical approach for studying the mechanics of elastic structures which interact with a fluid in the presence of thermal fluctuations. The hydrodynamic interactions of the composite system are handled by an approximate treatment of the fluid-structure stresses in which a Lagrangian representation of the immersed structures is coupled to an Eulerian representation of the fluid. Thermal fluctuations are accounted for in the system by an appropriate stochastic forcing of the fluid-structure equations in accordance with the principles of statistical mechanics. The formalism is cast in terms of a system of Stochastic Partial Differential Equations (SPDE's). Fast time scales introduced into the fluid dynamics by the thermal fluctuations pose a challenge for conventional approaches to numerical approximation. Using results from stochastic calculus, we are developing efficient stochastic numerical methods for the formalism. Additional work includes development of stochastic numerical methods for non-periodic and adaptive multilevel meshes, see papers.
Immersed structures in SIB can be used to represent the mechanics
of a variety of microscopic hydrodynamic systems, for example,
in a complex fluid the structures could represent solute
particles, polymers, or membrane structures. In the figure
some simulations demonstrating the methodology for a few
basic physical systems are shown. Click on the
images to play the associated movies. From top to bottom are:
(i) polymer knot simulations showing SIB preservation of knot
topology without the need for excluded volume interactions,
(ii) simulations demonstrating a tethered membrane model
using SIB for the hydrodynamic coupling, (iii) simulations
showing how the methodology may be used to simulate
more complex mechanical systems subject to thermal fluctuations,
in this case a basic model from biology of a motor protein
transporting a cargo vesicle under an imposed hydrodynamic load.
In collaboration with the
Brown Group, Department of Chemistry,
dynamic coarse-grained models of lipid bilayer
membranes are being developed using the SIB formalism
to account for molecular level interactions, lipid-lipid and
lipid-solvent hydrodynamic coupling, and thermal fluctuations.
For a more in-depth discussion see the
publications section.
Osmotic Phenomena:
Osmosis is a general phenomenon in mechanical systems where pressures and forces are generated in a fluid or gas solvent by solute particles which are confined by semi-permeable membrane boundaries or electric fields. Such forces and pressures play a fundamental role in many biological systems and technological devices. Some examples in biology include mechanisms responsible for exchange of nutrients and wastes in microscopic capillaries in tissues and means of propulsion in cell motility. In technological applications, microscopic devices have been designed which utilize osmotic effects to pump fluids, actuate forces via swelling, or deliver controlled drug doses. The classical theories for osmotic pressure, such as van't Hoff's relation, become inaccurate for microscopic systems where solute particles interact on length scales comparable to the size of the confining chamber. Classical theories of osmosis also typically assume systems are in thermodynamic equilibrium.
To study systems in regimes applicable to biological systems and microscopic devices we are developing theory to describe microscopic osmotic phenomena both in equilibrium and non-equlibrium settings. Some recent work includes studying how osmotic pressures generated by polymer solutes depend on polymer topology, stiffness, and excluded volume. In related work, a pumping mechanism for a microfluidic device has been proposed which uses reversible chemical reactions to drive fluid flows, see the above papers.
Molecular Biology / Applications:
Technological advances such as optical traps and molecular tagging are yielding a wealth of quantitative information about the mechanics and localization of processes in biological systems at the single molecule level. In cell biology an interesting class of molecules are motor proteins which act essentially as microscopic machines within cells to form cellular structures, transport materials, or generate forces. We are using stochastic modeling, analysis, and simulation to study the basic mechanisms underlying how molecular motor proteins operate with an effort toward making connections with available experimental data. In previous work we have studied the kinesin motor protein and developed a coarse-grained mechanical model consistent with available optical trap experimental data, see papers.
Aptamers are short segments of DNA or RNA which bind to proteins. In applications, specific aptamers (nucleic acid sequences) are sought which bind to a desired target protein with high affinity and specificity. Applications include extraction methods for proteins, biosensor devices, and pharmaceutical drug development. Obtaining strong binding aptamers poses a significant challenge in practice. In collaboration with the H.T. Soh Laboratory, Department of Mechanical Engineering along with the undergraduate Joe Rudzinski (CCS Program, UCSB) mathematical approaches are being used to study experimental protocols for the selection of DNA-Aptamers from a library of random sequences.
Spatially Adaptive Numerical Methods for Stochastic Partial Differential Equations:
Results for a stochastic reaction-diffusion system with intrinsic concentration fluctuations:
* Growth of a noise-induced spatial pattern resolved with adaptive meshing [AVI Movie].
For a more in-depth discussion of this work and information about additional projects please see the publications section.
