superdiffusion and lévy flights a particle transport monte...

Centro de FísicaEscola de Ciências

Universidade do Minho

Superdiffusion and Lévy Flights

A Particle Transport Monte Carlo Simulation Code

Eduardo J. Nunes-Pereira

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ANOMALOUS TRANSPORTDefinitions and Examples

SUPERDIFFUSION AND LÉVY FLIGHTSSerial code and opportunities for parallelization

DYNAMICS OF SUPERDIFFUSIONNumerical algorithms

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Brownian motionCharacteristic scale

Mean Free PathDiffusion Equation

Diffusion Coefficient

Anomalous TransportNo characteristic scale (fractal self-similar)

No second moment (sometimes no mean free path)Diffusion Equation not valid

No Diffusion Coefficient

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Anomalous TransportNo characteristic scale

Jump size distributionIs a power law

(all scales; self similar fractal)

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THEORY

VERIFIED EXPERIMENTALLY

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SUPERDIFFUSIVE RADIATIVE TRANSPORTWITH POWER LAWS (LÉVY FLIGHTS)

PARTICLE (LIGHT) TRANSPORT MONTE CARLO SIMULATION

“LE CAMEMBERT” EXPERIMENT (with R.KAISER, NICE)

Transmission profile under superdiffusion …

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SUPERDIFFUSIVE RADIATIVE TRANSPORTWITH POWER LAWS (LÉVY FLIGHTS)

PARTICLE (LIGHT) TRANSPORT MONTE CARLO SIMULATION

PARTICLE (LIGHT) TRANSPORTMONTE CARLO SIMULATION

GENERATE TRAJECTORIES ANDKEEP TRACK AND ACCUMULATE

MONTE CARLOMASSIVE USE OF RANDOM NUMBERS

SLOW CONVERGENCE STATISTICS (MONTE CARLO)

ANDPOWER LAWS (EVENTS IN THE TAILOF THE DISTRIBUTION DOMINANT)

HIGH NUMBER OF TRAJECTORIES

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SERIAL RANDOM NUMBER GENERATORSPORTABLE, PSEUDO-RANDOM NUMBERS GENERATORS

STEP 1: UNIFORM DEVIATES IN [0,1[

STEP 2 (EXAMPLE): EXPONENTIAL DEVIATES

x is distributed uniformly in [0,1[

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STEP 3: GENERAL TRANSFORMATION METHOD(FOR ANY STATISTICAL DISTRIBUTION)

Solve … with

SOLVE ANALYTICALLY OR NUMERICALLY

STEP 2 (EXAMPLE): EXPONENTIAL DEVIATES ANALYTICAL

Indefinite integral of desired distribution

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STEP 3: GENERAL TRANSFORMATION METHOD(FOR ANY STATISTICAL DISTRIBUTION)

SOLVE ANALYTICALLY OR NUMERICALLY

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RANDOM NUMBER GENERATORSIN SERIAL CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT

STEP 1: UNIFORM DEVIATES IN [0,1[

x is distributed uniformly in [0,1[

BASIC ALGORITHM LINEAR CONGRENTIAL GENERATOR

INTEGER ARITHMETIC

/ IS THE DESIRED REAL RANDOM NUMBER

SEED + “SET OF MAGIC NUMBERS”, FOR MAXIMUM REPETITION PERIOD

INTEGER ARITHMETIC

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2 G

ENER

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RS P

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ADD

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PRO

CED

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ENTI

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ELAT

ION

)

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ANALYTICAL GENERAL TRANSFORMATION METHODIN MONTE CARLO TRAJECTORIES

SPHERICAL COORDINATESTHE DIRECTION OF THE NEXT STEP

IS ANALYTICAL


… HOWEVER, THE STEP LENGTHIS NOT ANALYTICAL

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NUMERICAL GENERAL TRANSFORMATION METHODIN MONTE CARLO TRAJECTORIES

THE SINGLE STEP F(y) IS COMPUTED NUMERICALLY ONCE IN A SET OF DISCRETE DATA POINTS


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… THEN A NATURAL CUBIC SPLINE INTERPOLATION IMPLEMENTS THE TRANSFORMATION METHOD

CAS

E I

DO

PPLE

R

RANDOMDEVIATESRANDOMDEVIATES


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CAS

E II

VO

IGT

RANDOMDEVIATES


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THE CUBIC SPLINE INTERPOLATION IS A 2 STEP ALGORITHM: STEP 1: SOLVE LINEAR TRIDIAGONAL SYSTEM (ONCE; NO TIME COST) STEP 2: INTERPOLATE BY TABLE LOOKUP AND BISSECTION (REPEAT …)

THE MONTE CARLO CODE IS EXPECTED TO SPENT MOST OF THE TIME:

GENERATING UNIFORM RANDOM NUMBERS IN [0,1[

IMPLEMENTING NUMERICALLY THE TRANSFORMATION METHOD (BISSECTION)

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OPPORTUNITIES FOR PARALLELIZATION (PROVIDE FEEDBACK)

MONTE CARLO PARTICLE TRANSPORT CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT:• SERIAL CODE IN FORTRAN 90• BOTTLENECK RANDOM NUMBERS (UNIFORM DEVIATES & TRANSFORMATION METHOD)• EACH TIME A RANDOM NUMBER IS NEEDED, IS GENERATED• FIGURE OF MERIT TO STOP SIMULATION TOTAL NUMBER TRAJECTORIES

MONTE CARLO PARTICLE TRANSPORT CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT ALGOTITHM:•GENERATE TRAJECTORIES•ACUMULATE POSITIONS•REPEAT UNTIL TOTAL NUMBER OF TRAJECTORIES

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OPPORTUNITY #1 MODEL MPI

•MASTER/SLAVE MODEL WITH DYNAMIC LOAD BALANCE IN HETEROGENEOUS SYSTEMS

•TIME IS LINEAR IN # TRAJECTORIES => LAUNCH REPLICAS OF SMALL MONTE CARLO SIMULATIONS IN EACH COMPUTATIONAL NODE AND, BASED ON FEEDBACK, PARTITION TOTAL NUMBER OF TRAJECTORIES TO OBTAIN OVERALL LOAD BALANCE

•DISADVANTAGE: EACH NODE IS INDEPENDENT AND DOES NOT SHARES RANDOM NUMBER SEQUENCES

•TESTED IN PROTOTYPE CODE


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OPPORTUNITY #2 PARALLEL RANDOM NUMBER GENERATORS

•DATA SHARING THE UNIFORM RANDOM NUMBER SEQUENCE IS GLOBALLY ACCESSIBLE FROM ALL COMPUTING NODES

•POSSIBLY COMBINE WITH OPPORTUNITY #1; MASTER/SLAVE MODEL WITH DYNAMIC LOAD BALANCE IN HETEROGENEOUS SYSTEMS

•NO EXPERIENCE WITH PARALLEL RANDOM NUMBER GENERATORS


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OPPORTUNITY #3 HYBRID CPU/GPU MODELS

•LEVEL #1: LINEAR CONGRUENTIAL GENERATOR USES INTEGER ARITHMETIC (IS THIS STILL RELEVANT TO EXPLORE, BASED IN LATEST HARDWARE?)

•LEVEL #2: RANDOM NUMBER GENERATOR BY NUMERICAL INTERPOLATION IS A “LOOKUP” ALGORITHM IN AN ORDERED TABLE; COULD BE MORE EFFICIENTLY RECODED IN GPU ?


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GOAL ALGORITHM/NUMERICAL CODEDYNAMICS OF ANOMALOUS TRANSPORT

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SUPE

RDIF

FUSI

ON

(PO

WER

LAW

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… APPLICABLE FOR OTHER POWER LAW CASES

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MAS

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DES

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YNAM

ICS

3. LINEAR/NON LINEAR REGIME AND INTEGRATION MASTER EQUATION IN TIME AND SPACE

2. LINEAR REGIME AND EXPANSION IN EIGENVALUES/EIGENVECTORS

1. LINEAR REGIME AND EXPANSION IN “# OF JUMPS”(MONTE CARLO CODE)

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NU

MER

ICAL

ALG

ORI

THM

S1. LINEAR EXPANSION IN “# OF JUMPS”

+ MONTE CARLO SIMULATION

MONTE CARLO SIMULATION OF TRAJECTORIES (ONLY SPACE)BOTTLENECK RANDOM NUMBER GENERATION/MANIPULATION

SPATIAL DISTRIBUTIONS IN # JUMPS(MONTE CARLO)

TEMPORAL DISTRIBUTIONS ARE ANALYTICAL(DECOUPLED FROM SPACE)

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2. LINEAR REGIME AND EIGENVALUE/EIGENVECTOR DECOMPOSITION+ MONTE CARLO SIMULATION (FOR RANDOM CONFIGURATIONS)

“CORE” COMPUTACIONAL LINEAR ALGEBRA (EIGENVALUES/EIGENVECTORS) + RANDOM NUMBERS

CPU or CPU/GPU HYBRID LIBRARIES

TRANSITION MATRIX FOR EACH CONFIGURATION

(EIGENVALUES/EIGENVECTORS)+

CONFIGURATIONAL ENSEMBLES(MONTE CARLO REPLICAS OF

DIFFERENT CONFIGURATIONS)

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3. LINEAR/NON LINEAR REGIME USING FRACTIONAL DERIVATIVES MODELS FOR SPACE AND TIME DISCRETIZATION

THE MASTER EQUATION CAN BE WRITTE AS A GENERALIZATION OF THE DIFFUSION EQUATION, BUT NOW USING (FRACTIONAL) DERIVATIVE ORDER

FRACTIONAL CALCULUS / FRACTIONAL DIFFUSION / FRACTIONAL KINETICS

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“CORE” COMPUTATIONAL LINEAR ALGEBRA“HOT TOPIC” AND matlab CODE

@ TIME(SUBDIFFUSION)

@ SPACE(SUPERDIFFUSION)


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superdiffusion and lévy flights a particle transport monte...

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