D. Guirado Instituto de Astrofísica de Andalucía CSIC, Granda, Spain Polarization school, Aussois, 06/06/2013

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1 MONTE CARLO METHODS D. Guirado Instituto de Astrofísica de Andalucía CSIC, Granda, Spain Polarization school, Aussois, 06/06/2013

2 STOCHASTIC ALGORITHMS PROBLEM = GET OUTPUT FROM INPUT THROUGH AN ALGORITHM Deterministic algorithm: Input 1 Output 1 Stochastic algorithm: Input 1 + random decision Output 1 Output 2 Output 3...

MONTE CARLO METHODS Definition: Iterative

Origin: Study of the diffusion of neutrons

3 MONTE CARLO METHODS Definition: Iterative method including random decisions that gives the correct solution of a problem with a certain probability <1. Named after the district of Monte Carlo (Monaco), European gambling capital. Origin: Study of the diffusion of neutrons in a fusion experiment (Los Álamos Laboratory). Monte Carlo Monte Carlo casino

4 TYPICAL MONTE CARLO METHOD Input parameters + i x i N X = 1 N i=1 x i

5 PRACTICE #1: VALUE OF π 2R 10 0 { Area=4R 2 Area=π R 2 0 N (darts inside) N (darts thrown) } π=4 N (darts inside) N (darts thrown) =4 1 N N i=1 x i 10 N =N (darts thrown) x i = { 1 if dart goes inside 0 if dart goes out }

6 STOCHASTIC METHODS WITH COMPUTERS? HOW? Stochastic method = generating random number + performing operation N t Computers make operations much faster. Computer are deterministic, no random numbers. Solution: pseudo-random numbers. PSEUDO-RANDOM NUMBERS Finite (but large) lists of numbers generated by an algorithm. No correlations beteween the numbers. Examples: Numerical Recipes. MKL (optimized for ifort) ran2 (Numerical Recipes) For any Fortran or C compiler. The authors offer $1000 to the first person who finds a statistical test that proves that ran2 fails.

7 CENTRAL LIMIT THEOREM 2 x 1, x 2,..., x N = collection of N values sampled through. x= Any probability density function with average and variance. Then, the probability density distribution X N = 1 N i=1 N N σ 2 = 1 (x N i μ) 2 i=1, which elements are, has a average of and a variance of, where. N X x i σ 2 N LAW OF LARGE NUMBERS N N gaussian.

8 HOW TO CALCULATE THE ERROR IN A MONTE CARLO METHOD N Fit all calculated to a gaussian distribution and obtain. Let us call the correct solution of the problem. Then: with. with. with. x i X [ N, N ] X [ 2 N,2 N ] X [ 3 N,3 N ] p=0.682 p=0.954 p=0.998 If the accuracy is not enough, increase N (the error decreases as ). ~ 1 N RESULTS FROM MONTE CARLO SIMULATIONS HAVE ERRORS!

PRACTICE #2: INTEGRATION S S 0 + - 0 + 0 Integral=S x i ={ N (darts inside positive) N (darts inside negative) =S 1 N N

9 PRACTICE #2: INTEGRATION S S Integral=S x i ={ N (darts inside positive) N (darts inside negative) =S 1 N N (darts thrown) N i=1 N =N (darts thrown) 0 if dart goes outside 1 if dart goes inside positive 1 if dart goes inside negative} x i

10 PRACTICE MONTE #3: CARLO LINEAR MODEL POLARIZATION OF RADIATIVE IN SCATTERING TRANSFER INCLUDING LINEAR AND CIRCULAR POLARIZATION ( I i 0,Q i 0,U i 0,V i 0 ) I,Q,U, V = 1 I 0 N i,q 0 i,u 0 i,v 0 i i =1 N x I,Q,U, V θ φ z y

11 PRACTICE #3: LINEAR POLARIZATION IN SCATTERING

12 PRACTICE #3: LINEAR POLARIZATION IN SCATTERING Particles: - Any size distribution. - Any refractive index (even optically activity). - Any geometry. - Any orientation (aligned or not). Any optical thickness of the coma (single or multiple scattering). Inhomogeneous coma. Packet of photons (W) - Improves statistics. - Reduces computation time. W=1 at the beginning Ends when W<Wmin

13 PRACTICE #3: LINEAR POLARIZATION IN SCATTERING

14 $~top SOME COMPUTATIONAL CONSIDERATIONS

15 PARALLELIZE YOUR CODE Classical method: MPI. New easier alternative: OpenMP (Fortran and C). How it works: Compile: PROGRAM OMP_SUM2 INTEGER NMAX PARAMETER(NMAX=20000) INTEGER I REAL A(NMAX),C(NMAX) DO I = 1,NMAX A(I) = I * 1.0 C(I)=1. ENDDO C$OMP PARALLEL shared(a,b,c,nmax) private(i,j) C$OMP DO DO I = 1,NMAX DO J=1,I C(I)= C(I)+A(J) END DO Compile: ENDDO C$OMP END DO C$OMP END PARALLEL WRITE(*,*)C(NMAX) END $~gfortran omp_sum2.f -o omp_sum2 Sequential omp_sum2 $~gfortran -fopenmp omp_sum2.f -o omp_sum2 Parallel omp_sum2 Execute: $~time./omp_sum2

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Introduction. Microscopic Theories of Nuclear Structure, Dynamics and Electroweak Currents June 12-30, 2017, ECT*, Trento, Italy

Introduction. Microscopic Theories of Nuclear Structure, Dynamics and Electroweak Currents June 12-30, 2017, ECT*, Trento, Italy Introduction Stefano Gandolfi Los Alamos National Laboratory (LANL) Microscopic Theories of Nuclear Structure, Dynamics and Electroweak Currents June 12-30, 2017, ECT*, Trento, Italy Stefano Gandolfi (LANL)