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Using the computer to select the right variables

Rationale:

Lake Carnegie, Princeton, NJ

Straight Line Distance

Actual transition difficulty represented

by curved path

Curved Transition Distance

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Using the computer to choose the right variables

Rationale:

Lake Carnegie, Princeton, NJLake Carnegie, Princeton, NJ

Euclidean Distance

Selected Datapoint

XY

Z3D Dataset with 2D manifold

Euclidean distance in input spacemay be weak indicator

of INTRINSIC similarity of datapoints

Geodesic distance is good for this dataset

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Dataset in x, y, z Dataset Diffusion Map

N datapoints N datapoints

eigencomputation

, , , 1,ii i ix y z i N x 2 3, , 1,i i i i N

Diffusion Maps

R. Coifman, S. Lafon, A. Lee, M. Maggioni, B. Nadler, F. Warner, and S. Zucker,Geometric diffusions as a tool for harmonic analysis and structure definition of data: Diffusion maps.PNAS 102 (2005).

B. Nadler, S. Lafon, R. Coifman, and I. G. Kevrekidis,Diffusion maps, spectral clustering and reaction coordinates

of dynamical systems.Appl. Comput. Harmon. Anal. 21 (2006).

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Diffusion Map (2, 3)

2

3

2

ABSOLUTE Coordinates SIGNED Coordinates

Report absolute distanceof all uninformed individuals

to informed individual to DMAP routine

Report (signed) distanceof all uninformed individuals

to informed individual to DMAP routine

STICK

SLIPSTICK

SLIP

Reaction Coordinate

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So, again, the same simple theme

• If there is some reason to believe that there exist slow, effective dynamics in some smart collective variables

• Then this can be used to accelerate some features of the computation

• Tools for data-based detection of coarse variables…

• BUT– You can start COMPUTATIONS wherever you want

– You CANNOT (not easily!) start experiments wherever you want

– TALK to experiments with spatiotemporal resolution

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Effective simplicity

• Construct predictive models (deterministic, Markovian)

• Get information from them: CALCULUS, Taylor series– Derivatives in time to jump in time

– Derivatives in parameter space for sensitivity /optimization

– Derivatives in phase space for contraction mappings

– Derivatives in physical space for PDE discretizations

In complex systems --- no derivatives at the level we need them

sometimes no variables ---- no calculus

If we know what the right variables are, we can

PERFORM differential operations

on the right variables – A Calculus for Complex Systems

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Coming full circle1. No equations ? Isn’t that a little medieval ? Equations =“Understanding”

AGAIN matrix free iterative linear algebra

A x = b

PRECONDITIONING, B A x = B b

B approximate inverse of A

Use “the best equation you have”

to precondition equation-free computations.

2. With enough initialization authority: equation free laboratory experiments

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Computer-Aided Analysisof Nonlinear Problems in Transport Phenomena

Robert A. Brown, L. E. Scriven and William J. Silliman

in HOLMES, P.J., New Approaches to Nonlinear Problems in Dynamics, 1980

ABSTRACT The nonlinear partial differential equations of mass, momentum, energy, Species and charge transport…. can be solved in terms of functions of limited differentiability,no more than the physics warrants, rather than the analytic functions of classical analysis…….. basis sets consisting of low-order polynomials. …. systematically generating andanalyzing solutions by fast computers employing modern matrix techniques.

….. nonlinear algebraic equations by the Newton-Raphson method. … The Newton-Raphsontechnique is greatly preferred because the Jacobian of the solution is a treasure trove, not onlyfor continuation, but also for analysing stability of solutions, for detecting bifurcations ofsolution families, and for computing asymptotic estimates of the effects, on any solution, ofsmall changes in parameters, boundary conditions, and boundary shape……

In what we do, not only the analysis, but the equations themselves are obtained on thecomputer, from short experiments with an alternative, microscopic description.