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Projective Integration - a sequence of outer integration steps

Value. Time. Project forward in time. time. Space. 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,

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Projective Integration - a sequence of outer integration steps

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  1. Value Time Project forward in time time Space 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. 2002 • Projective methods in time: • perform detailed simulation for short periods • or use existing/legacy codes • - and then extrapolate forward over large steps

  2. Microscopic Simulator Microscopic Simulator Projection [α1,…αns](k+1) [α1,…αns](k+2) [α1,…αns](k+2+M) [α1,…αns](k) Coarse projective integration: Accelerating things Simulation results at g = 35, 200,000 agents Run for 5x03 t.u. Project for 5x0.3 t.u.

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

  4. The Bifurcation Diagram Tracing the branch with arc-length continuation

  5. 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 andhence 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 matricesK, 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 vex- and the controlled variables the coefficients of the orthogonal polynomials used for the approximation of the ICDF

  6. STABILIZING UNSTABLE M*****S Control variable: the exogenous arrival frequency of “negative” information vex-

  7. 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

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

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

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

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

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