method of particles as a universal solver witold dzwinel agh - department of computer science...

Method of Particles as a Universal Solver

Witold Dzwinel

AGH - Department of Computer Science

Dzwinel W, Alda W, Kitowski J, Yuen DA, Molecular Simulation, 20/6, 361-384 2000Dzwinel W, Future Generation Computer Systems, 12, 371-389, 1997 Dzwinel W, Yuen DA, Boryczko K, Chemical Engineering Sci., 61, 2169-2185, 2006.

Universal solver

= automata or a formalism having universal computational capabilities (equivalent to TM, or lambda calculus or 110 Wolfram rule)

= paradigm, which can be a common platform of an offspring of algorithms designated for solving a broad class of seemingly unrelated problems from e.g. modeling and simulation (PM, CA, ANN, MA …) optimization (GA, SA, ANN, PM, MA …) learning theory and systems (ANN, GA …) etc.

Method of particles – in simulation and modeling

The algorithms employing moving and interacting particles as primitives.

Taxonomy due to definition of particle (quark, atom, molecule, granule, cluster,

chunk of something, item, many items, galaxies etc) definition of interactions (hard, soft: pair, manybody, multipole) moving scheme (deterministic, stochastic) granularity (?) of space and time

continuous/continuous (MD, DPD, FPM, SC-DPD, SPH) continuous/discrete (hard spheres, DSMC) discrete/continuous (lattice dynamical systems) discrete/discrete (LG, LBG, percolation, DLA etc.)

http://www.amara.com/papers/nbody.html#p3m


Boryczko K, Dzwinel W, Yuen DA, J Mol. Modeling,8,33-45,2002


OOOFFFFFF--- GGGRRRIIIDDD PPPAAARRRTTTIIICCCLLLEEE MMMEEETTTHHHOOODDDSSS

conservative interactions

MD

DDPPDD + dissipative and Brownian forces

FFPPMM + particles rotation, non-central forces

SSPPHH Regularized tensor interactions

AT

OM

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UID

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MOVING MES H NODES

+ variable mass and particle

volume non-isothermal model TTCC-- DDPPDD F

LU

ID P

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Dzwinel W, Boryczko K, Yuen DA, Finely Dispersed Particles: Micro-, Nano-, and Atto-Engineering A.M. Spasic & J.P. Hsu eds., Taylor&Francis, CRC Press, 715-778, 2006

Fluid Particles

colloidal bead

dissipative particle

BOTTOM-UP APPROACH

MD – particles creating Voronoy clusters

Colloidal bead

Dissipative particle

TOP-DOWN APPROACH

Finite Volume - contiuum description.

Flekkoy and Coveney, 1999

Serrano and Espanol, 2002


Boryczko K, Dzwinel W, Yuen DA, J Mol. Modeling,9,16-33,2003 Dzwinel W, Boryczko K, Yuen DA, J Colloid Int Sci, 258/1, 163-173, 2003

Method of particles – in graphics

www.graphics.stanford.edu/.../vortex_particle-sig05/

Method of particles – crowding

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),(

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ijji

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Helbing D., Farkas I., Vicsek T., Nature, Vol. 407, pp. 487-490. 2000


Two groups of people running from opposite directions

Optimization - L-J cluster

http://www.uniovi.es/qcg/d-MolSym/LJ1/

Global minimum searchFunkcja m ultim odalna

http://www.mat.univie.ac.at/~neum/glopt.html

Global minimum search - particles

GA vs. MD - global minimum of N-D function (N~10)

Many interacting solutions (particles) the lowest can attract stronger the higher others

Clustering around wells Bad derivatives approach (mimics

annealing process)

Bad derivative method

U d U KT r r r r r0 0 ,

erUerUe

rUrF Te 0000 2

1

U(r)

Ue1(r0)

Ue2(r0)

Ie1(r,r0)) Ie2(r,r0))

GA vs. MD - global minimum of N-D function

parents

domain searched

final cluster of particles in minimum well

Jasińska-Suwada, A., Dzwinel, W., Rozmus, K., Sołtysiak, J., Computer Science, 2, 13-51, 2000

… more dimensions??

More advanced version of coordinate decent scheme Problems when irregular and xi have very different domains

Needs regularization and normalization procedures Additional difficulties with gradient calculations

Best fit: is the sum of simple functions, like in MD, the total force acting on a single particle

ii

kkki

NN

xxx

xxxy

rF

r

rrr

),,(

;,...,,,...,,

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Multi-dimensional scaling – not a trivial example

The MDS mapping from D-dimensional space to d-dimensional (D>>d) consists in minimization of the quadratic loss function, called “the stress function”:

where CN and wij are free parameters, which depend on the MDS goals. Smaller values of the “stress function“ mean better correspondence between source and target data structures.

jiij

N

i

N

ij

kij

kijij

NN

r

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1,...,,

Multi-dimensional scaling – not a trivial example

N-D

i

j

Dij

Static N-D structure.

2-D

i

j

Initially dynamic structure evolves to static -“frozen”- system of particles representing minimum of the

“stress function”

FT

rij fij

1. N-particles interact via the fij = -grad((Dij-rij)m. Dij

mw) two-body forces. 2. The total force acting on a single particle is FTi=ijfij. 3. The system evolves in time due to the Newtonian equations of motion. 4. Friction removes energy from the system. 5. Eventually, the particles stop reaching minimal potential energy = minimum of mapping

criterion

MAPPING N-D into 2-D

1.Dzwinel W, Blasiak J, Future Generation Computers Systems, 15, 365-379, 1999

Examples – periodic boundary conditions

Arodz, Boryczko, Dzwinel, Kurdziel, Yuen: IEEE Visualization 2005: 90

Examples - mammograms

Examples - earthquakes

Yuen, W. Dzwinel, Yehuda Ben-Zion, B.Kadlec, Encyclopedia of Complexity and System Science, Springer Verlag, 2007

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