Research Activities

Stochastic models in microbiology

A National Science Foundation CAREER grant is supporting several research initiatives in stochastic modeling for microbiology. I have been working with Paul Atzberger (University of California at Santa Barbara) and Charles Peskin (Courant Institute) on the development of an approach for simulating microscopic physiological processes where thermal fluctuations play a significant role.  Our numerical algorithm is based on Peskin's "Immersed Boundary Method," which is a useful approximation for treating fluid systems with flexible immersed structures.  In connection with some physical validation studies of these computations, we have also recently examined how the statistical mechanical framework of osmotic pressure differs in microscopic containers relative to macroscopic containers.Through support from a National Science Foundation ``Computational Science Training in the Mathematical Sciences'' grant, undergraduate student Sam Hughes is applying the computational method for simulating microswimmers.

Another stochastic computational biology project, initiated with Shekhar Garde (Chemical Engineering, Rensselaer) and recent graduate student Adnan Khan, is the development of effective stochastic models for the dynamics of water molecules near protein surfaces for the purpose of accelerating biomolecular simulations. With Grigorios Pavliotis (Imperial College) and graduate student Juan Latorre, we are developing a computational technique based on homogenization theory for the calculation of transport properties of molecular motor models.

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Turbulent transport

Another line of research is in turbulent diffusion, the study of transport of heat and substances through flows which have some sort of disordered structure.  This physical phenomenon plays a crucial role in mixing and combustion systems in engineering, and in the dynamics of the atmosphere and ocean.  Detailed resolution of turbulence is generally impractical in the simulations of such systems, so in practice, one must employ some sort of approximation theory (often called a "closure") which represents the effects of the turbulence on the large-scale (resolvable) system variables through a managable set of parameters characterizing the statistics of the unresolved turbulence.  I am generally interested in the analysis and development of such approximation methods through their exploration in the context of mathematical model flows which represent some important and challenging aspects of the system of interest, but are sufficiently simplified to permit analytical studies.  This line of work is intended to complement and interact with numerical and experimental investigations in addition to less rigorous physical theories. 

I have recently begun a collaboration with Shafer Smith (Center for Atmosphere and Ocean Science, Courant Institute) and graduate student Banu Baydil, through the auspices of a National Science Foundation ``Collaborations in Mathematical Geosciences'' grant, with the purpose of applying mathematical developments in the theory of turbulent diffusion to the practical parameterization of transport by unresolved mesoscale turbulence and structures in computational ocean circulation models. One methodology feeding into our parameterization approach is that of homogenization theory, on which I worked with Grigorios Pavliotis (Imperial College) and recent PhD student Adnan Khan. We extended previous results to develop rigorous eddy-diffusivity representations of transport by a flow consisting of both a spatiotemporal mean flow and turbulent subgrid scale fluctuations. We will also draw from the Lagrangian stochastic modeling approach, which was investigated through their application to mathematical shear flow models in the recent PhD thesis of Emilio Castronovo.

Multiscale Random Field Simulation

During a fortuitous overlap of our visits to Berlin in 2004, Karl Sabelfeld (Institute of Computational Mathematics and Mathematical Geophysics; Weierstrass Institute for Applied Analysis and Stochastics), Orazgeldi Kurbanmuradov (Turkmenistan State University), and I revisited two leading methods for the efficient simulation of multiscale random fields (the Randomization Method and the Fourier-Wavelet Method) and are investigating the quality of more detailed statistical properties, such as ergodicity, of the random fields simulated by these methods.

Nonstandard Multiple Scale Asymptotics

Multiple scale asymptotics run through much of my research, usually in the context of the derivation of simplified effective equations to describe multiscale stochastic systems (such as the turbulent transport problem described above).  I have been working with Christof Schuette (Freie Universitaet Berlin) and Jessika Walter (Ecole Polytechnique Federale de Lausanne) on such asymptotic mode reduction projects for biomolecular modeling, accounting particularly for metastable behavior.  In this application, as well as in some of my other projects, the multiple scale asymptotics naturally involve three time scales.  Together with Gregor Kovacic (Rensselaer), Robert E. Lee DeVille (Courant Institute) and Rensselaer graduate students Yunfeng Li, Adnan Khan, and Phil Stathos, I am working on clarifying some subtle aspects that arise in these kinds of asymptotic calculations in ordinary differential equation models as well as partial differential equations.

Laser Propagation through Disordered Media

The statistical properties of Maxwell-Bloch solitons propagating through  media with spatially disordered (random) preparation are being investigated in joint work with Gregor Kovacic (Rensselaer), graduate student Katie Newhall (Rensselaer), and graduate student Ethan Atkins (Courant Institute).

Weak turbulence theory for discrete lattice models

Joseph Biello (University of California, Davis), Yuri L'vov (Rensselaer), and I are applying the weak turbulence methodology to some simple nonlinear physical lattice models.  The weak turbulence theory produces approximate equations to describe the transfer of energy in systems with many weakly interacting modes.  Our aims are to provide another perspective on the relaxation to thermal equilibrium in these lattice models, and to provide a more rigorous understanding of the circumstances in which the weak turbulence theory is applicable.  Current and former graduate students Emilio Castronovo, Warren Towne, and Matthew Ferrara have been working with us, partially supported by an NSF Research Training Grant, on more effective ways of numerically simulating weakly nonlinear lattice models.

Stochastic modeling in ecology

Brad Lister (Biology, Anderson Center for Innovation in Undergraduate Education) and I have begun supervising undergraduate projects involving stochastic modeling approaches in ecosystem models, with the support of a ``Computational Science Training in the Mathematical Sciences'' grant from the National Science Foundation. Tom Wentworth is currently working with us in developing more realistic coevolutionary network models for the purpose of exploring how various features of ecological interactions affect the food web structure that emerges. Recent projects with other students have looked at foraging strategies of ant colonies (senior thesis of Chris Rainey), stochastic modeling of environmental influences on population dynamics, and the effects of social network structure on the spread of epidemics.

Some of the above projects are supported by NSF grants DMS-0449717 (CAREER), OCE-0620956 (CMG), DMS-0636358 (RTG), DUE-0639321 (CSUMS). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

References:

P. R. Kramer, A. Khan, P. Stathos, and R. E. L. DeVille, "Method of Multiple Scales with Three Time Scales," submitted to Proceedings in Applied Mathematics and Mechanics. (preprint version)

J. C. Latorre, P. R. Kramer, and G. A. Pavliotis, "Effective Transport Properties for Flashing Ratchets Using Homogenization Theory," Proceedings in Applied Mathematics and Mechanics 7(1), 2008: 1080501-1080502 . (preprint version)

B. Baydil, P. R. Kramer, and K. S. Smith, "Parameterization for Mesoscale Ocean Transport through Random Flow Models," submitted to Proceedings in Applied Mathematics and Mechanics. (preprint version)

P. J. Atzberger and P. R. Kramer, "Error analysis of a stochastic immersed boundary method incorporating thermal fluctuations," Mathematics and Computers in Simulation, in press (preprint version)

P. R. Kramer, C. S. Peskin, and P. J. Atzberger, "On the foundations of the stochastic immersed boundary method," Computer Methods in Applied Mechanics and Engineering197(25-28), 2008: 2232-2249 (preprint version)

P. J. Atzberger and P. R. Kramer, "A Microscopic Theoretical Framework for Osmotic Phenomena," Phys. Rev. E 75, 2007, 061125. (preprint version)

P. R. Kramer, O. Kurbanmuradov, and K. Sabelfeld,  "Comparative Analysis of Multiscale Gaussian Random Field Simulation Algorithms," Journal of Computational Physics226 (1), 2007: 897--924. (preprint version) 

P. J. Atzberger, P. R. Kramer, and C. S. Peskin, "A stochastic immersed boundary method for biological fluid dynamics at microscopic length scales," Journal of Computational Physics 224 (2), 2007: 1255-1292. (preprint version)

P. R. Kramer,  "Brownian Motion,"  Encyclopedia of Nonlinear Science,  ed., Alwyn Scott.  New York and London:  Routledge, 2005. (preprint version)

P. R. Kramer,  "Fokker-Planck Equation,"  Encyclopedia of Nonlinear Science,  ed., Alwyn Scott.  New York and London:  Routledge, 2005. (preprint version)

E. Castronovo and P. R. Kramer, "Subdiffusion and Superdiffusion in Lagrangian Stochastic Models of Oceanic Transport," Monte Carlo Methods and Applications, 10 (3-4), 2004:  245-256. (preprint version)

L. J. Borucki, T. Witelski, C. Please, P. R. Kramer, and D. Schwendeman, "A theory of pad conditioning for chemical-mechanical polishing,"  Journal of Engineering Mathematics, 50 (1), 2004:  1--24. (preprint version)

P. R. Kramer and C. S. Peskin, "Incorporating Thermal Fluctuations into the Immersed Boundary Method," Proceedings of the Second MIT Conference on Computational Fluid and Solid Mechanics, K. J. Bathe, ed., Elsevier, 2, 1755--1758. (preprint version)

P. R. Kramer, J. A. Biello, and Y. Lvov, ``Application of weak turbulence theory to FPU model,'' Discrete and Continuous Dynamical Systems, Expanded Volume for the Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, 2003:  482--491. (preprint version)

J. A. Biello, P. R. Kramer, and Yuri Lvov, ``Stages of energy transfer in the FPU model,'' Discrete and Continuous Dynamical Systems, Expanded Volume for the Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, 2003:  113--122. (preprint version)

P. R. Kramer and A. J. Majda, "Stochastic mode reduction for immersed boundary method," SIAM J. Appl. Math. 64 (2), 2003:  369--400. (PDF)

P. R. Kramer and A. J. Majda, "Stochastic mode reduction for particle-based simulation methods for complex microfluid systems," SIAM J. Appl. Math. 64 (2), 2003:  401--422. (PDF)

P. R. Kramer, A. J. Majda, and E. Vanden-Eijnden, "Closure approximations for passive scalar turbulence:  A comparative study on an exactly solvable model with complex features," J. Stat. Phys., 111 (3/4), 2003:  565--679. (PDF)

P. R. Kramer, "Two different rapid decorrelation in time limits for turbulent diffusion,"  J. Stat. Phys., 110 (1/2), 2003: 87--136.  (PDF)

A. J. Majda and P. R. Kramer, "Simplified models for turbulent diffusion:  Theory, numerical modelling and physical phenomena," Physics Reports, 314 (4-5), 1999: 237-574. (PS)

P. R. Kramer, "A review of some Monte Carlo simulation methods for turbulent systems,'' Monte Carlo Methods and Applications, 7 (3--4) 2001: 229--244. (PDF)