Homework 4 will be posted tonight, due Wednesday, May 8.

Transcription

1 Multiscale Computing and Escape Problems Friday, April 26, :01 PM Homework 4 will be posted tonight, due Wednesday, May 8. Multiscale computation is a way to use the ideas of asymptotic reductions of stochastic systems in situations where some of the analytical steps cannot be solved exactly. We'll focus on the case of method of averaging, which is one of the most common settings. The most difficult aspect in general in implementing the method of averaging is solving for the conditional distribution of the fast variables given the slow variables. Need: This satisfies the stationary Fokker-Planck equation, which may not be explicitly solvable in a system of interest. But even if you can't solve it, maybe you could estimate this conditional probability density through simulation. And this is the basis of multiscale computation. References: Abdulle, E, Engquist, and Vanden-Eijnden, "The heterogeneous multiscale method" E, Liu, and Vanden-Eijnden, "Analysis of Multiscale Methods for Stochastic Differential Equations" Two key numerical challenges in fast-slow systems, if attacked directly: The number of fast variables, which are typically of subsidiary interest, is AppSDE13 Page 1

2 much larger than the number of slow variables. The time step needed to resolve the fast variables is very small, but one wants to simulate for a large time in order to see the slow variables do something interesting. (Stiff system) Multiscale computation proceeds through the following cycle: 1. Freeze the slow variables X=x and simulate the Y variables for a time long enough for them to reach their (conditional) stationary distribution. Run multiple realizations (perhaps in parallel). 2. From the simulated trajectories of the Y variables, estimate the effective drift coefficient for the slow X variables by the following Monte Carlo procedure: (frozen x) And the samples should be taken from the stationary distribution of this SDE, meaning after waiting "long enough" along the trajectories. 3. Take O(1) time step (appropriate for the slow X variables) with this estimated averaged drift coefficient: 4. Reinitialize the fast variables Y to be consistent with/appropriate for the new value of the slow variable X. This can be a subtle part of the method, particularly if the Y variables must satisfy some constraints based on the X variables. If there aren't hard constraints, then one reasonable thing to do is simply recycle the data from the end of the last cycle. And E, Ren, Vanden- Eijnden, "A general strategy for designing seamless multiscale methods" formulated an approach that justifies this approach in a wide range of contexts. When is this efficient relative to direct simulation? The point of the method is that each variable has time step suitable for it, so this removes nominally the numerical stiffness. That is, we are not using the small time step needed for the Y variables to simulate the X variables. But we only will save time if the amount of time we take to simulate the Y trajectories is much smaller than the length of a coarse time step for the X variables. Since we need to simulate the Y trajectories on a time scale long enough for them to reach their stationary distribution, this means that multiscale computation in the form I've written requires that the time scale on which Y variables reach their stationary distribution should be short compared to the coarse time step for the X variables. One can do some numerical analysis to AppSDE13 Page 2

3 encode this idea (see E, Liu, and Vanden-Eijnden, "Analysis of Multiscale Methods for Stochastic Differential Equations") Comment: In atmosphere-ocean science, multiscale computation is called superparameterization. See for example: Xing, Majda, and Grabowski, "New Efficient Sparse Space-Time Algorithms for Superparameterization on Mesoscales" Extensions: Can also do multiscale in space. Sorin Mitran: extend the multiscale method to situations where the statistics of the fast variables actually don't relax fast relative to the slow variables. Introduced a third level (mesoscale) tracking the PDF of the fast variables, like a kinetic equation. Time parallelization (parareal computing) using GPU parallel computing. Tp-CKM: time-parallel continuous-kinetic-molecular. Mitran, "Time parallel kinetic-molecular interaction algorithm for CPU/GPU computers" Young & Mitran, "A numerical model of cellular blebbing: A volume-conserving, fluid-structure interaction model of the entire cell" Escape/Exit Time Problems Abstractly, an exit problem is of the following type: Given SDE: AppSDE13 Page 3

4 Two key exit questions: 1) What are the statistics of the first exit time: 2) What is the probability distribution for where the system exits the specified domain? The formulas for addressing these follow from applications of Dynkin's formula. To get Dynkin's formula, we start with just applying the Ito lemma to an arbitrary nice function f. the infinitesmal generator or backward Kolmogorov operator AppSDE13 Page 4

5 Note that Is the forward Kolmogorov or fokker-planck operator Suppose we take the equation we derived from Ito's lemma and integrate it and then take a conditional average, given some initial condition: Actually let's evaluate this at the random exit time T 1 Take conditional average with respect to a given initial condition. What happens to the last term: If T 1 were deterministic, we know it would be zero. AppSDE13 Page 5

6 For general random T 1 we cannot make that conclusion. But if T 1 is a Markov time, like a first exit time, then the argument carries over and this term can be shown to vanish. AppSDE13 Page 6