Superdiffusion and Lévy Flights. A Particle Transport Monte Carlo Simulation Code
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1 Superdiffusion and Lévy Flights A Particle Transport Monte Carlo Simulation Code Eduardo J. Nunes-Pereira Centro de Física Escola de Ciências Universidade do Minho Page 1 of 49
2 ANOMALOUS TRANSPORT Definitions and Examples SUPERDIFFUSION AND LÉVY FLIGHTS Serial code and opportunities for parallelization DYNAMICS OF SUPERDIFFUSION Numerical algorithms Page 2 of 49
3 Page 3 of 49
4 BROWNIAN MOTION ONE OF PARADIGMS OF XX CENTURY Page 4 of 49
5 BROWNIAN MOTION DIFFUSION EQUATION Page 5 of 49
6 ANOMALOUS TRANSPORT WHY ANOMALOUS? NOT VALID DIFFUSION EQ. Page 6 of 49
7 Brownian motion Characteristic scale Mean Free Path Diffusion Equation Diffusion Coefficient Anomalous Transport No characteristic scale (fractal self-similar) No second moment (sometimes no mean free path) Diffusion Equation not valid No Diffusion Coefficient Page 7 of 49
8 Anomalous Transport No characteristic scale Jump size distribution Is a power law (all scales; self similar fractal) Page 8 of 49
9 ANOMALOUS TRANSPORT (SUPERDIFFUSION) PATHLENGTH DISTRIBUTION IS A POWER LAW LÉVY FLIGHT FLUORESCENT LAMPS Page 9 of 49
10 ANOMALOUS TRANSPORT (SUPERDIFFUSION) SUPERDIFUSION IS FASTER (SUBDIFFUSION IS SLOWER; NOT SHOWN) Page 10 of 49
11 POWER LAWS (ARE EVERYWHERE) Page 11 of 49
12 POWER LAWS (ARE EVERYWHERE) Page 12 of 49
13 ANOMALOUS TRANSPORT EXAMPLE 1 FORAGING BEHAVIOR IN SCARCE RESOURCES ENVIRONMENTS Page 13 of 49
14 ANOMALOUS TRANSPORT EXAMPLE 1 FORAGING BEHAVIOR IN SCARCE RESOURCES ENVIRONMENTS Page 14 of 49
15 ANOMALOUS TRANSPORT EXAMPLE 1 FORAGING BEHAVIOR IN SCARCE RESOURCES ENVIRONMENTS Page 15 of 49
16 ANOMALOUS TRANSPORT EXAMPLE 1 ANIMAL TRACKING Page 16 of 49
17 ANOMALOUS TRANSPORT EXAMPLE 2 POPULATION DYNAMICS AND EPIDEMICS Page 17 of 49
18 ANOMALOUS TRANSPORT EXAMPLE 2 POPULATION DYNAMICS AND EPIDEMICS Page 18 of 49
19 ANOMALOUS TRANSPORT EXAMPLE 2 POPULATION DYNAMICS AND EPIDEMICS Page 19 of 49
20 ANOMALOUS TRANSPORT EXAMPLE 3 EXTREME STATISTICS AND NATURAL HAZARDS Page 20 of 49
21 ANOMALOUS TRANSPORT EXAMPLE 4 ANOMALOUS TRASPORT FOR LIGHT SUPERDIFFUSIVE RADIATIVE TRANSPORT Page 21 of 49
22 ANOMALOUS TRANSPORT EXAMPLE 4 ANOMALOUS TRASPORT FOR LIGHT SUPERDIFFUSIVE RADIATIVE TRANSPORT Page 22 of 49
23 ANOMALOUS TRANSPORT Definitions and Examples SUPERDIFFUSION AND LÉVY FLIGHTS Serial code and opportunities for parallelization DYNAMICS OF SUPERDIFFUSION Numerical algorithms Page 23 of 49
24 THEORY VERIFIED EXPERIMENTALLY Page 24 of 49 SUPERDIFFUSIVE RADIATIVE TRANSPORT
25 LE CAMEMBERT EXPERIMENT (with R.KAISER, NICE) Transmission profile under superdiffusion SUPERDIFFUSIVE RADIATIVE TRANSPORT WITH POWER LAWS (LÉVY FLIGHTS) PARTICLE (LIGHT) TRANSPORT MONTE CARLO SIMULATION Page 25 of 49
26 SUPERDIFFUSIVE RADIATIVE TRANSPORT WITH POWER LAWS (LÉVY FLIGHTS) PARTICLE (LIGHT) TRANSPORT MONTE CARLO SIMULATION PARTICLE (LIGHT) TRANSPORT MONTE CARLO SIMULATION GENERATE TRAJECTORIES AND KEEP TRACK AND ACCUMULATE MONTE CARLO MASSIVE USE OF RANDOM NUMBERS SLOW CONVERGENCE STATISTICS (MONTE CARLO) AND POWER LAWS (EVENTS IN THE TAIL OF THE DISTRIBUTION DOMINANT) HIGH NUMBER OF TRAJECTORIES Page 26 of 49
27 SERIAL RANDOM NUMBER GENERATORS PORTABLE, PSEUDO-RANDOM NUMBERS GENERATORS STEP 1: UNIFORM DEVIATES IN [0,1[ x is distributed uniformly in [0,1[ STEP 2 (EXAMPLE): EXPONENTIAL DEVIATES Page 27 of 49
28 SERIAL RANDOM NUMBER GENERATORS PORTABLE, PSEUDO-RANDOM NUMBERS GENERATORS STEP 3: GENERAL TRANSFORMATION METHOD (FOR ANY STATISTICAL DISTRIBUTION) Solve with Indefinite integral of desired distribution SOLVE ANALYTICALLY OR NUMERICALLY STEP 2 (EXAMPLE): EXPONENTIAL DEVIATES ANALYTICAL Page 28 of 49
29 SERIAL RANDOM NUMBER GENERATORS PORTABLE, PSEUDO-RANDOM NUMBERS GENERATORS STEP 3: GENERAL TRANSFORMATION METHOD (FOR ANY STATISTICAL DISTRIBUTION) SOLVE ANALYTICALLY OR NUMERICALLY Page 29 of 49
30 RANDOM NUMBER GENERATORS IN 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 Page 30 of 49
31 RANDOM NUMBER GENERATORS IN SERIAL CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT Page 31 of 49 ran2 NUMERICAL RECIPES 2 GENERATORS PLUS ADDITIONAL SHUFFLING PROCEDURE (BREAK SEQUENTIAL CORRELATION)
32 RANDOM NUMBER GENERATORS IN SERIAL CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT ANALYTICAL GENERAL TRANSFORMATION METHOD IN MONTE CARLO TRAJECTORIES SPHERICAL COORDINATES THE DIRECTION OF THE NEXT STEP IS ANALYTICAL HOWEVER, THE STEP LENGTH IS NOT ANALYTICAL Page 32 of 49
33 RANDOM NUMBER GENERATORS IN SERIAL CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT NUMERICAL GENERAL TRANSFORMATION METHOD IN MONTE CARLO TRAJECTORIES THE SINGLE STEP F(y) IS COMPUTED NUMERICALLY ONCE IN A SET OF DISCRETE DATA POINTS Page 33 of 49
34 RANDOM NUMBER GENERATORS IN SERIAL CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT CASE I DOPPLER RANDOM DEVIATES THEN A NATURAL CUBIC SPLINE INTERPOLATION IMPLEMENTS THE TRANSFORMATION METHOD Page 34 of 49
35 RANDOM NUMBER GENERATORS IN SERIAL CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT CASE II VOIGT RANDOM DEVIATES THEN A NATURAL CUBIC SPLINE INTERPOLATION IMPLEMENTS THE TRANSFORMATION METHOD Page 35 of 49
36 RANDOM NUMBER GENERATORS IN SERIAL CODE FOR SUPERDIFFUSIVE RADIATIVE TRANSPORT THEN A NATURAL CUBIC SPLINE INTERPOLATION IMPLEMENTS THE TRANSFORMATION METHOD 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) Page 36 of 49
37 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 Page 37 of 49
38 OPPORTUNITIES FOR PARALLELIZATION (PROVIDE FEEDBACK) 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 Page 38 of 49
39 OPPORTUNITIES FOR PARALLELIZATION (PROVIDE FEEDBACK) 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 Page 39 of 49
40 OPPORTUNITIES FOR PARALLELIZATION (PROVIDE FEEDBACK) 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? Page 40 of 49
41 ANOMALOUS TRANSPORT Definitions and Examples SUPERDIFFUSION AND LÉVY FLIGHTS Serial code and opportunities for parallelization DYNAMICS OF SUPERDIFFUSION Numerical algorithms Page 41 of 49
42 GOAL ALGORITHM/NUMERICAL CODE DYNAMICS OF ANOMALOUS TRANSPORT Page 42 of 49
43 APPLICABLE FOR OTHER POWER LAW CASES Page 43 of 49 SUPERDIFFUSION (POWER LAWS)
44 MASTER EQUATION INTEGRO-DIFFERENTIAL EQUATION DESCRIBES DYNAMICS 1. LINEAR REGIME AND EXPANSION IN # OF JUMPS (MONTE CARLO CODE) 2. LINEAR REGIME AND EXPANSION IN EIGENVALUES/EIGENVECTORS 3. LINEAR/NON LINEAR REGIME AND INTEGRATION MASTER EQUATION IN TIME AND SPACE Page 44 of 49
45 1. LINEAR EXPANSION IN # OF JUMPS + MONTE CARLO SIMULATION NUMERICAL ALGORITHMS SPATIAL DISTRIBUTIONS IN # JUMPS (MONTE CARLO) TEMPORAL DISTRIBUTIONS ARE ANALYTICAL (DECOUPLED FROM SPACE) MONTE CARLO SIMULATION OF TRAJECTORIES (ONLY SPACE) BOTTLENECK RANDOM NUMBER GENERATION/MANIPULATION Page 45 of 49
46 2. LINEAR REGIME AND EIGENVALUE/EIGENVECTOR DECOMPOSITION + MONTE CARLO SIMULATION (FOR RANDOM CONFIGURATIONS) TRANSITION MATRIX FOR EACH CONFIGURATION (EIGENVALUES/EIGENVECTORS) + CONFIGURATIONAL ENSEMBLES (MONTE CARLO REPLICAS OF DIFFERENT CONFIGURATIONS) CORE COMPUTACIONAL LINEAR ALGEBRA (EIGENVALUES/EIGENVECTORS) + RANDOM NUMBERS CPU or CPU/GPU HYBRID LIBRARIES Page 46 of 49
47 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 Page 47 of 49
48 3. LINEAR/NON LINEAR REGIME USING FRACTIONAL DERIVATIVES MODELS FOR SPACE AND TIME TIME SPACE (SUPERDIFFUSION) CORE COMPUTATIONAL LINEAR ALGEBRA HOT TOPIC AND matlab CODE Page 48 of 49
49 3. LINEAR/NON LINEAR REGIME USING FRACTIONAL DERIVATIVES MODELS FOR SPACE AND TIME DISCRETIZATION Page 49 of 49
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