Noise plays an important role in many systems we study, especially in small scale devices. In other systems such as the turbulent flows, the under-resolved small scales can also be modeled as noise. Small noises can creat rare events which over long time affect the behavior of the systems drastically.
Traditionally noise is modeled by Monte Carlo methods or Langevin equations. While being natural and general, these methods often have the disadvantage of being expensive and produce noisy data that are difficult to process.
In this talk we will describe a method based on least action
principles to compute the optimal pathways for systems
driven by noise. We will also describe a method based on
PDEs to compute the probability density of macroscopic
observables. We will discuss the numerical difficulties as
well as applications to bistable systems, magnetic thin
films and turbulent flows. We end by a dissussion of the
application of our method to the issue of predicability.
[C2.002] Computing Optimal Trajectories of Noise-Driven Systems
Robert S. Maier, Daniel L. Stein (University of Arizona)
If a dissipative physical system, either finite-dimensional or spatially extended, resides in an attractor~S but is perturbed by noise of strength~\epsilon, large fluctuations away from~S may occur. But in the \epsilon\to0 limit, they are exponentially rare. An `optimal' trajectory from S to any specified state is singled~out, in the \epsilon\to0 limit, as the most probable fluctuational trajectory. Optimal trajectories also arise in a PDE context, if a ray expansion is used to approximate the solution of the Fokker--Planck equation of the system as~\epsilon\to0 (which is a~singular, weak-diffusivity limit). When the system is low-dimensional, we show how optimal trajectories may efficiently be computed numerically. As an illustration, we treat a periodically driven overdamped particle that moves within a potential well, and is also driven by weak thermal noise. Upon escape, it is reinjected. Our work, involving an integration along optimal trajectories extending from the well bottom, shows that the particle's time-independent PDF, in a boundary layer near the well top, oscillates logarithmically slowly in the weak-noise limit. This result applies to a mesoscopic particle that is confined to a periodically modulated optical trap. We~indicate how our approach to optimal trajectories, and numerics, may be extended to the case of spatially extended systems.
[C2.003] Numerical simulation of stochastic differential equations, with examples
Wesley Petersen (SAM, Mathematik, HG G-52.1, ETH Zurich)
In this talk, I will describe Monte-Carlo methods for simulations of stochastic differential equations of the Itô type,
\[ dx = b(x) dt + \sigma(x) dw(t) \]
where w(t) is a vector of independent Brownian motions.
Their connections with partial differential equations of
mathematical physics will be highlighted and the Dirichlet
problem and exit times discussed. For illustration, some
examples from finance, cosmology, and statistical mechanics
will be given. My intent is to indicate the considerable
progress of the last few years in implicit and explicit
Runge-Kutta methods for SDEs and their implications for
solving 2nd order partial differential equations by
Feynman-Kac representations.
[C2.004] Simulation of Diffusion-Limited Phase Coaresning
Martin E. Glicksman, Ke-Gang Wang, Paula Crawford (Materials Science and Engineering Department, Rensselaer Polytechnic Institute, Troy, NY 12180-3590)
Developing a comprehensive understanding phase coarsening processes during microstructure evolution remains a keystone of materials physics. The mean-field theory of phase coarsening, in the limit of zero volume fraction, was first formulated by Lifshitz and Slyozov, and by Wagner (LSW). Numerous attempts have been made to extend LSW theory toward phase coarsening with nonzero volume fractions. The successes achieved with analytical theories, however, have been severely limited, due primarily to difficulties encountered in characterizing interactions among particles and the matrix. Such theories predict particle size distributions (PSD's) and coarsening kinetics only qualitatively when compared with experimental observations. Three major questions remain open: (1) What happens when the volume fraction of the dispersed phase approaches zero? (2) How are interactions among coarsening particles best characterized? (3) Can we quantitatively understand the kinetics and PSD's in the phase coarsening?
The importance of large-scale simulation of phase coarsening
processes was realized about 20 years ago, and pursued
vigorously with the increased capability of computer
hardware and software. We recently focused our simulation of
the dynamics of phase coarsening to lower volume fractions
to provide better insight into the nature of the interaction
physics among particles. Rate constants and PSD's can be
extracted through these simulations. Gradually, a more
robust understanding of phase coarsening can be built
through a combination of experiment, theory, and computer
simulations. Some examples of recent progress in simulating
diffusion-limited phase coarsening will be discussed.
[C2.005] Self-similar tumor growth with angiogenesis
Vittorio Cristini (Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis MN 55455), John Lowengrub (School of Math, University of Minnesota, Minneapolis MN 55455), Qing Nie (Department of Mathematics, U.C. Irvine, Irvine CA 92697)
We revisit linear analysis of the transient evolution of a perturbed tumor interface both in two and three dimensions, when the tumor core is nonnecrotic and no inhibitor chemical species are present. A new formulation is developed that demonstrates that tumor evolution is qualitatively unaffected by the number of spatial dimensions and is described by a reduced set of two parameters. One parameter describes the rate of cell proliferation. The other the balance between vascularization and apoptosis.
Three regimes of growth are identified. We find nontrivial stationary solutions in the weak-vascularization regime. In the weak-apoptosis regime, tumors grow unbounded and we find that critical conditions exist under which the shape tends to self-similar at long times due to a balance of mass flux and apoptosis. This behavior is peculiar of 3-D tumors. Self-similar evolution can be enforced identically both in 2-D and 3-D by varying the balance between mass flux and apoptosis with the tumor's size. When the mechanisms don't balance, perturbations may either grow unbounded with respect to the underlying unperturbed shape (an infinite cylinder in 2-D and a sphere in 3-D), or decay to zero. In the strong-vascularization regime, unbounded growth may occur for weak apoptosis and the tumor shape tends to the unperturbed shape.
These linear results are then compared to those obtained by
using boundary integral methods in two dimensions that
illustrate the effect of nonlinearity on the existence of
non-trivial stationary states and self-similar evolution.
[C2.006] Level Set Algorithms for Tracking Discontinuities
Tariq Aslam (Los Alamos National Laboratory)
A level set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference method of lines. The zero of a level set function is used to specify the location of the discontinuity(ies). Since a level set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the "real" state, and one corresponding to a "ghost node" state, analogous to the "Ghost Fluid Method". High-order, point-wise convergence is demonstrated for linear and nonlinear scalar conservation laws, even at discontinuities and in multiple dimensions. The solutions are compared to standard high-order shock-capturing schemes.
This presentation will also present work on systems of
conservation laws. In particular, results for the
multi-dimensional Euler equations will be presented.
Examples will include tracking of material interfaces and
tracking of shock waves. Applications ranging from
compressible fluid instability problems (Rayleigh-Taylor,
Richtmyer-Meshkov and Kelvin-Helmholz) to tracking shocks in
reactive fluid flow will be presented. It will be
demonstrated that the method can be used effectively when
very accurate results are required for problems involving
discontinuities. Also, due to the simplicity of the
formulation, it is relatively easy to incorporate such a
computational strategy in an adaptive mesh refinement
framework. Details of one such implementation will be
presented.
[C2.007] Fourth Order Gradient Symplectic Integrator Methods for Solving the Time-Dependent Schrodinger Equation
Siu A. Chin, C. R. Chen (Dept. of Physics, Texas Aamp;M University, College Station, TX 77843)
We show that the method of splitting the operator
e^\epsilon(T+V) to fourth order with purely positive
coefficients produces excellent algorithms for solving the
time-dependent Schrödinger equation. One 4th order
algorithm only requires four Fast Fourier Transformations
per iteration. These algorithms can produce results of
conventional second order split operator method using time
steps 10 times as large. These algorithms not only can solve
quantum dynamical problems directly, but can also be use to
solve large scale eigenvalue problems, such as the Kohn-Sham
equation, through the use of filter diagonalization.
[C2.008] A New Numerical Method for Initial Value Problems with Nonlinear Hamiltonians
Kostrun Marijan, Javanainen Juha (University of Connecticut)
We suggest a novel numerical scheme for solving the particular initial value problems that occur in theoretical exploration of Bose Einstein Condensation in dilute gases. Such phenomena are known to be well described by nonlinear Schrödinger, or more precisely, Gross-Pitaevskii equations. Unlike the standard split-step method for solving these type of problems, our method implements a locally derived (Dirac) interaction picture. For that reason the construction of the linear part of a complete Hamiltonian H_0 and a choice of the time integration scheme are cruical. The method achieves the second order accuracy, while not significantly increasing the computational time. In our talk, we will discuss the derivation of the basic formulae of the method, and show how the method relies on the evolution operator, e^\frac i H_0 \Delta t \hbar . We will consider some simple realizations of this operator in one, two and three spatial dimensions, and all standard coordinate systems. Finally, by comparing the method with a standard split-step method, we show that the subroutines from the latter can be used in the algorithms of the former, albeit arranged differently. At the end (if time allows) we'll present some of our published results obtained by this method for the problem of atom-molecule BEC.