Princeton University Projective Integration - a sequence of outer integration steps based on inner simulator + estimation (stochastic inference) Accuracy and stability of these methods – NEC/TR 2001 (w/ C. W.Gear, SIAM J.Sci.Comp. 03, J.Comp.Phys. 03, --and coarse projective integration (inner LB) Comp.Chem.Eng time Time Space Value Project forward in time Projective methods in time : -perform detailed simulation for short periods or use existing/legacy codes - and then extrapolate forward over large steps

Princeton University Coarse projective integration: Accelerating things Simulation results at g = 35, 200,000 agents [α 1,…α ns ](k) [α 1,…α ns ](k+1) [α 1,…α ns ](k+2+M) Microscopic Simulator Microscopic Simulator [α 1,…α ns ](k+2) Projection Run for 5x03 t.u. Project for 5x0.3 t.u.

Princeton University THE CONCEPT: What else can I do with an integration code ? Have equation Write Simulation Do Newton on Do Newton Compile Also Estimate matrix-vector product Matrix free iterative linear algebra The World CG, GMRES Newton-Krylov

Princeton University The Bifurcation Diagram Tracing the branch with arc-length continuation

Princeton University STABILIZING UNSTABLE M*****S Feedback controller design We consider the problem of stabilizing an equilibrium x*, p* of a dynamical system of the form where f and hence x* is not perfectly known To do this the dynamic feedback control law is implemented: Where w is a M-dimensional variable that satisfies Choose matrices K, D such that the closed loop system is stable At steady state:and the system is stabilized in it’s “unknown” steady state In the case under study the control variable is the exogenous arrival frequency of “negative” information v ex - and the controlled variables the coefficients of the orthogonal polynomials used for the approximation of the ICDF

Princeton University STABILIZING UNSTABLE M*****S Control variable: the exogenous arrival frequency of “negative” information v ex -

Princeton University So, again, the main points Somebody needs to tell you what the coarse variables are And then you can use this information to bias the atomistic simulations “intelligently” accelerating the extraction of information In effect: use numerical analysis algorithms as protocols for the design of experiments with the atomistic code

Princeton University and now for something completely different: Little stars ! (well…. think fishes)

Princeton University

Fish Schooling Models Initial State Compute Desired Direction Update Direction for Informed Individuals ONLY Zone of Deflection R ij <  Zone of Attraction R ij <  Normalize  INFORMED UNINFORMED Update Positions Position, Direction, Speed Couzin, Krause, Franks & Levin (2005) Nature (433) 513

Princeton University STUCK ~ typically around rxn coordinate value of about 0.5 INFORMED DIRN STICK STATES INFORMED individual close to front of group away from centroid

Princeton University SLIP ~ wider range of rxn coordinate values for slip 0  0.35 INFORMED DIRN SLIP STATES INFORMED individual close to group centroid