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2009-10 Program on Stochastic Dynamics

Research Foci
Description of Activities

Further Information

Introduction

This 12 month SAMSI program is centered around the broad topic of Stochastic Dynamics with particular focus on analysis, computational methods, and applications of systems governed by stochastic differential equations. Two application areas will be emphasized: problems in biological sciences and dynamics of networks.

Researchers from statistics, applied mathematics, mathematical biology, and engineering will be involved, and the program will promote the opportunity for interdisciplinary, methodological and theoretical research.

Organizing Committee:

Program Leaders: Hongyun Wang (UC Santa Cruz), Alejandro Garcia (San Jose State), Cindy Greenwood (ASU)

Local Scientific Coordinators: Alan Karr (NISS), Jonathan Mattingly (Duke), Peter Mucha (UNC)

Directorate Liaison: Michael Minion (SAMSI)

National Advisory Committee Liaison: Rick Durrett (Cornell University)

Additional leaders will be appointed from each of the theme areas mentioned below, from among those who will be long-term visitors.

Research Foci

Stochastic Analysis and Numerical Methods

In recent years it has become increasingly clear that to effectively understand complex stochastic systems, a combination of modern numerical analysis, estimation and sampling techniques, and rigorous analysis of stochastic dynamics is required. Whether one speaks of path sampling techniques, estimation in complex non-linear dynamics, or simulation of rare-events it is important to bring both sophisticated analytic tools and an understanding of what one can compute efficiently.

A working group in stochastic analysis and numerical methods is partially inspired by a recent workshop, sponsored by AIM and the NSF, concerning approaches for the numerical integration of stochastic systems which span many temporal-scales. This subject would fit well with other potential working group topics of multi-scale computing, biological applications, and dynamics networks. Important issues such as the erogodicity of numerical methods for SDEs, the construction of higher order methods for SDEs and SPDEs, the role of holonomic constraints and how to enforce them in numerical methods, or ways to efficiently compute quantities like free energies in chemical kinetic simulations would provide very fertile ground for productive collaboration between mathematicians, statisticians, and computational scientists under the stochastic dynamics banner.

Multi-scale and multi-physics computing

The classical continuum equations arising in fluid flow, elasticity, or electromagnetic propagation in materials require constitutive laws to derive a closed-form system. The constitutive laws appropriate for a given set of equations can be derived in two ways: First, from phenomenological considerations such as the linear behavior underlying elastic deformation or Newtonian fluid flow. Second, from averaging of kinetic theory results describing basic molecular dynamics. This has been possible for situations in which the microscopic behavior has been close to thermodynamic equilibrium. In such cases, Gaussian statistics are well verified at the microscopic level and the moments of Gaussian distributions can be computed analytically.

However, many physical processes exhibit significant localized departure from Gaussian statistics. When a solid breaks, the motion of the atoms in the crystalline lattice along the crack propagation path is no longer governed by a Maxwell-Boltzmann distribution. When a material undergoes a phase change, large scale correlations among atoms are formed (or destroyed) which modify the typically Gaussian statistics of the equilibrium phases. Protein folding can be seen as a large-scale modulation imposed by polymer links of the Gaussian statistics of the component atoms. A common characteristic of these situations is that macroscopic features impose the departure from local thermodynamic equilibrium and macroscopic quantities of are practical interest. Crack propagation is initiated by a force acting on a solid and we wish to know how far the solid deforms before it breaks. In solidification, a heat flux evacuates energy from the melt at some rate and we wish to characterize the type of order arising in the material.

In all such cases, a basic problem is how to extract the statistical distributions of physical quantities when the system is away from thermodynamic equilibrium. Knowledge of the distribution would allow local constitutive laws to be formulated. Direct numerical simulation is prohibitively expensive. Continuum level simulation is incomplete due to lack of constitutive laws. Furthermore, while it is clear that higher-order moments characterizing the microscopic statistical distribution are required, it is not known how many of these moments are needed and what their persistence time might be. The microscopic dynamics are stochastic but subject to multiple macroscopic constraints. One major statistical challenge is how to characterize the microscopic motion in a manner which can be used to derive a constitutive law. A basic computational question is how to advance the system in time efficiently at both the microscopic and macroscopic level. An analysis challenge is how to combine this knowledge and form (e.g. through asymptotic expansions) particular constitutive laws. Thus progress in this area will depend on experts in numerical and stochastic analysis, statistics, and engineering modeling to join forces and methodologies.

Stochastic Modeling and Computation in Biology

The explosion of interest in mathematical and statistical modeling and computation in the biological and medical sciences, where stochasticity is present at nearly every scale, has been one of the most exciting trends in the biosciences in the last ten years. Math biology has grown from a niche area to a major research group in many US math departments, and graduate programs in mathematics are scrambling to cope with a wave of students seeking to do graduate work in interdisciplinary research areas. Programs in bio-statistics, bio-informatics, and bio-medical engineering are also seeing increased growth.

In examples as diverse as bio-chemical networks, diffusion and noise in cellular transport, modeling molecular motors by a stochastic ratchet, the study of epidemics, or the modeling or analysis of neuronal dynamics, stochastic modeling, analysis, and computation permeate the biological sciences. A working group centered in applications of stochastic dynamics in biology and medicine will both allow experts in analysis and computation to be exposed to interesting applications and allow researchers in bio-statistics, math-biology and bio-medical engineering to work together with experts in stochastic analysis and computation.

Dynamics of Social Networks

Social network data are distinguished by the inherent dependencies among units. These dependencies, usually represented by (binary or more general) links, are a primary focus of many analyses.

To date, however, both models of, and techniques for inference for, social networks, have focused on static networks. Virtually no extant models address the appearance or disappearance of nodes, the evolution of link existence or strength, or the characteristics of nodes. One SAMSI-generated exception is Banks, et al. (Q. Appl. Math. 66(2) 233-247, 2008), which uses stochastic differential equations to model joint evolution of the edge set and node characteristics, with a focus on characterizing the role of stochastic variability.

Indeed, at an NSF workshop in October 2007 on "Discovery in Complex or Massive Data: Common Statistical Themes," there was consensus about urgent need for models of the dynamics of networks and associated tools for inference.

Among the central issues are:

  1. Models for inference for dynamic social networks are usually descriptive rather than generative. Moreover, many of the existing descriptive tools treat dynamic networks only through the amalgamation of a sequence of static snapshots. More modeling work is needed on both fronts, both for adequate description but also to attempt to explain the "physics" of the network dynamics.
  2. Sampling: Do data represent the entire network or are they based on only a subnetwork or subgraph? This problem can be considered from both a sample designed-based or a model-based perspective. It is understood poorly for static networks, where it strongly impacts the study of stochastic processes between statically-connected nodes, and it is essentially not understood at all for dynamic networks.
  3. Embeddability: Underlying existing dynamic network models use a continuous time stochastic process even though the data used to study the models and their implications may come in the form of repeated snapshots at discrete time points--a form of time sampling as opposed to node sampling--or cumulative network links. Can we represent and estimate the continuous-time parameters in the actual data realizations used to fit models?
  4. In dynamic social network settings, data generated over time present a series of forecasting problems. How should we evaluate alternative predictions from different models?

Description of Activities

Workshops: The Opening Workshop will be held August 30-September 2, 2009 at SAMSI. This workshop will aim to engage as large a part of the statistics, mathematics, and relevant scientific communities as possible, with representative sessions from all of the main program topics. The Transition Workshop at the end of the program will disseminate program results and chart a path for future research in the area. Other workshops being considered include:

  • A workshop centered around the analysis and computation of multi-scale systems with small scale stochastic forcing. This is a critical topic in areas such as bio-fluid dynamics, meteorology, combustion, and materials science. Engaging mathematical analysts, statisticians, computational scientists, and application stake-holders interested in this topic could lead to fundamental breakthroughs in this emerging field.
  • A workshop on the dynamics of networks
  • A workshop on stochastic modeling in the bio-sciences.

Working Groups: Working groups will meet throughout the program to pursue particular research topics identified in the kickoff workshop (or subsequently chosen by the working group participants). The working groups consist of SAMSI visitors, postdoctoral fellows, graduate students, and local faculty and scientists.

Further Information

Additional information about the program and opportunities to participate in it is available:

 
 

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