Random numbers and optimization techniques Jorge Andre Swieca School Campos do Jordão, January,2003 second lecture

References The Nature of Mathematical Modeling, N. Gershenfelder, Cambridge, 1999; Numerical Recipes in C, Second Edition, W.H Press et al., Cambridge, 1992; Statistical Data Analysis, G. Cowan, Oxford, 1998 Computational Physics, Dean Karlen (online), 1998

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Random numbers Important task: generate random variables from known probability distributions. random numbers: produced by the computer in a strictly deterministic way – pseudorandom (→ Monte Carlo Method) random number generators: linear congruential generators: ex: c=3, a=5, m=16 0,2,3,13,4,7,6,1,8,11,10,5,12,15,14,9,0,2….

Generators Arguments from number theory: good values for a, c and m. Good generators: longest possible period individual elements within a period should follow each other “randomly” Ex. 1 RANDU a = 65539, m=2 31, c=0

Generators

“Random numbers fall mainly in the planes” Marsaglia, Proc. Acad. Sci. 61, 25, 1968 What is seen in RANDU is present in any multiplicative congruential generator. In 32 bit machines: maximum number of hyperplanes in the space of d-dimensions is: d=32953 d=4 566 d=6 120 d=10 41 RANDU has much less than the maximum!

Generators Ex. 2 Minimal standard generator (Num. Recp. ran0) a = 7 5 =16807, m= RAN1 and RAN2, given in the first edition, are “at best mediocre”

Generators Ex. 3 RANLUX cernlib (mathlib V115, F.James ) A portable high-quality random generator for lattice field theory simulation, M. Lüscher, Comp. Phys. Comm. 79, 100, 1994 period ≥ Recommendations: 1. Do some simple tests. 2. Check the results with other generator Lots of literature about random generators testing. See a set of programs (Die Hard) Functional definition: an algorithm that generates uniform numbers in acceptable if it is not rejected by a set of tests.

Inverse transform p(x) uniform probability distribution x: uniform deviate Generate y according to g(y). 0 1 x y uniform deviate in out analytically or numerically. G(y) g(y)

Inverse transform x F(x) 3 Ex. 4 discrete case f(x=0)=0.3 f(x=1)=0.2 f(x=2)=0.5 u 0.0 ≤ u ≤ 0.3x=0 0.3 ≤ u ≤ 0.5x=1 0.5 ≤ u ≤ 1.0x=2 for 2000 deviates: x= X= X=

Inverse transform Exponential amount of time until a specific event occurs; time between independent events (time until a part fails);

Acceptance-rejection method y max 0 f(x) x min x max if (x,y) under the curve, accept the point, else, discard.

Acceptance-rejection method

Importance sampling g(x): random numbers are easy to generate g(x) f(x) generate random x according to g(x) generate u uniformily between 0 and g(x); if u < f(x), accept x, if not, reject.

Optimization (minimization) Many strategies for minimization: local search, global search, with derivative evaluation or not, etc. Choose the best parameters: iteractive search starting from a initial (guess) value. Objective: find an acceptable solution. It is possible that many solutions are actually good. Other sources of uncertainty larger than differences among the solutions;

Downhill simplex method Nelder-Mead Most functionality for the least amount of code; Simplex (triangle in 2D, tetrahedron in 3D, etc); Random location, evaluate the function at D+1 vertices; Iterative procedure to improve the vertex with the highest value of the function at each step: reflect, reflect and grow, reflect and shrink, shrink, shrink towards the minimum; Minimum: stop when there is no more improvement; Num. Rec.: amoeba.c

Downhill simplex method

Powell’s method simplex in one dimension: a pair of points; find the minimum of a function in a given direction: line minimization: series of line minimization → multi-dimensional search; Powell’s method: updating the directions searched to find a set of directions that don’t interfere with each other;, D line minimizations, : ; good direction to keep for future minimization if advantageous; is added to the set of directions used for minimization (replacing the most similar to it) if gradiente of function available → conjugate gradiente algorithm

Powell’s method

Simulated annealing Growth of a cristal: difficult optimization problem; Liquid instantaneously frozen: atoms trapped in the configuration they had in the liquid state (energy barrier: from glassy to crystalline state ) If liquid slowly cooled: atoms would explore many local arrangements Thermodynamics: the relative probability of system in state with energy E trapped in the lowest energy configuration slower cooling rate, more likely to find the lowest global energy T=0 lowest state T>0 some chance to be in other state

Simulated annealing Metropolis (53) update a simulation Kirkpatrick (80’s) same ideia for other hard problems: Implementation issues: new state randomly selected (E new evaluated) if E new > E, accept the state, else if E new < E, accept the state with prob. Energy → cost function (simulated annealing) High energy: any move is accepted T →0 lowest minima found 1.Selection of trial moves: downhill simplex or conjugate gradiente, but Boltzman factor allows mistakes; 2.Cooling schedule; freeze the system in a bad solution X waste computer time

Simulated annealing

Genetic algorithms Evolution: a very hard optimization problem; Explore many options in parallel rather than concentrating on trying many changes around a single design. Sim. Annealing : one set of search parameters repeatedly updated X G.A: keep an ensemble of sets of parameters. G.A.: state is given by a population: each member a complete set of parameters for function being searched; Population updated in generations

Genetic algorithms Update criteria: Fitness: evaluation of the function for each member of the population; Reproduction; new population (size fixed) selected based on fitness; low fitness parameters may disappear; Crossover: members of the ensemble can share parameters; Mutation: changes in the parameters: random or taking advantage of what is already known to generate good moves;

Genetic algorithms