FOAM: general purpose Monte Carlo Cellular Algorithm S. Jadach

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FOAM: general purpose

Monte Carlo Cellular Algorithm S. Jadach

Institute of Nuclear Physics, Krak ´ ow, Poland Presented with help of Thorsten Ohl

Outline:

• Introduction and motivation

• Cellular algorithm of FOAM

• Examples of numerical results

• Conclusions

These and related slides on http://home.cern.ch/jadach

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Introduction: What is general purpose?

For the problem of function minimalization one takes MINUIT or some other program and applies it to arbitrary user-function. One may find also general

purpose programs for integration of “arbitrary” integrand. General-purpose means that these tools work, in principle, for a very wide range of user-functions.

For multi-dimensional Monte Carlo simulation problem, that is for the problem of generation randomly points according to a given n -dimensional distribution, there is precious little examples of the General Purpose Monte Carlo Simulators (GPMCS), that is programs which work (in principle) for arbitrary integrand.

Two essential reasons for scarcity of GPMCS’s:

(a) lack of ideas about the efficient algorithm,

(b) need of much CPU power and memory – only recently available/affordable.

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General features of General Purpose Monte Carlo Simulators (GPMCS)

Inevitably the GPMCS has to work in 2 stages: exploration and generation.

During exploration GPMCS is “digesting” the entire shape of the n -dimensional distribution ρ(x ₁ , x ₂ , ...x _n ) to be generated and memorize it as efficiently as possible using all CPU power and memory available.

Obviously, for the memorized ρ ⁰ (x ₁ , x ₂ , ...x _n ) a method of the MC generation of the points ~x ^exactly according to ρ ⁰ (~x) , has to be available.

The quality of the distribution of the weight w = ρ/ρ ⁰ for events in the generation (small variance, good ration of maximum to average, etc.) is determined by the algorithm of the exploration. In other words, the “target weight distribution” in the generation is determining the algorithm of exploration.

The GPMCS programs will be always limited to “small dimensions”.

With presently available computers “small” means in practice n < 10

(up to n < 15 for certain function). This is already not so bad!!!

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Cellular exploration of the distribution

The most obvious method to minimize the variance (or maximum weight) of the target weight distribution in generation (proposed already some 40 years ago) is to split integration domain into many cells, such that ρ(~x) is approximated by constant

ρ ⁰ (~x) within each cell. This is a cellular class of general purpose MC algorithms.

(I think that “stratified sampling”, used in the literature, has a narrower meaning.)

FOAM: shape of cells, how to cover space with cells?

Three shapes of cells are used pure simplices, pure hypercubes and Cartesian products of them. They can be rather easily and efficiently parametrized.

The system of cells can be created all at once (like in Vegas) or in a more evolutionary way, by the “split process”. In the Foam algorithm we rely on the

binary split of cells. )Choice of a cell to be split driven by target weight distribution.) The binary split assures automatically full coverage of the space,

simply because the primary “root cell” is the entire integration domain.

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Variance reduction versus maximum weight reduction

In construction of the FOAM algorithm I have put most effort on the minimization of the ratio of the maximum weight to the average weight w _max /hwi ^.

This parameter is essential, if we want to transform w -ted events into w = 1

events, at the latter stage of the MC generation.

Minimizing maximum weight is not the same as minimizing variance

σ = phw ² i − hwi ² . Usually minimizing w _max is more difficult.

In FOAM minimizing variance is also implemented and optionally available.

It can be useful if case them w -ted events are acceptable.

Next slide shows two examples of the weight distribution evolution in the Foam,

when adding more and more cells.

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Generation weight distribution: minimization of variance

Number of cells: 200, 2k, 20k

Generation weight distribution: minimization of maximum weight

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Number of cells: 200, 2k, 20k

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Cell split algorithm: covers both (a) w _max and (b) variance minimization

We define two auxiliary distributions ρ ⁰ (x) ^and ρ loss (x) related to integrand ρ(x) ^.

Both are constructed together with the foam of cells, in the exploration process.

(1) EXPLORATION of ρ(x) and BUILD-UP of Foam of cells :

ρ _loss (x) ^and ρ ⁰ (x) are evolving in the process of the division of cells.

R loss = R ρ loss d ⁿ x is minimized in the same process.

(2) MC event generation:

Events are generated according to ρ ⁰ (x) ^. R ⁰ = R ρ ⁰ d ⁿ x is known exactly.

R = R ⁰ hwi ⁰ ^where w = ρ/ρ ⁰ ^. The average h...i ⁰ is over events generated according to ρ ⁰ ^.

(a) Minimization of w _max

ρ ⁰ (x) ≡ max _y∈Cell _i ρ(y) ^{, for} x ∈ Cell i , the “ceiling function”.

R _loss = R d ⁿ x [ρ ⁰ (x) − ρ(x)] = R d ⁿ x ρ _loss (x) ^,

Note that rejection rate in final MC run = R _loss /R ^.

(b) Minimization of of variance

ρ ⁰ (x) ≡ phρ ² i _i ^, ^for x ∈ Cell _i ^. The average h...i i is over i − th Cell assuming flat distribution.

ρ _loss (x) ≡ phρ ² i _i − hρi _i ^, ^for x ∈ Cell _i . Final MC variance is just ' R _loss ^.

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Two Rules governing binary split of a cell

Each split of a Cell: ω → ω ⁰ + ω ⁰⁰ should decrease total R _loss ^. R ^ω _loss ⁰ + R ^ω _loss ⁰⁰ << R _loss ^ω

How to get the best total decrease ∆R _loss ^?

[1] For each next cell split we choose a cell with the biggest R _loss ^.

[2] Position/direction of a plane dividing a parent cell into two daughter cells is chosen to get the smallest total R _loss ^.

How do we split a cell into two daughter cells?

General method relies on the small MC exercise on which events are generated with flat distribution, weighted with ρ and projected onto n (simplical case) or

n(n + 1)/2 (h-cubical case) of the cells.

Resulting histograms are analysed and the best “division geometry” found, for

which the estimate of ∆R loss is calculated. See next slides...

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Geometry of n -dim. Simplical Cell division, 3-Dim. case

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Pair of vertices x i and x j is chosen and a new vertex Y is put somewhere on the line in between: Y = λx i + (1 − λ)x j , 0 < λ < 1

Two daughter simplices are defined with the list of vertices:

(x 1 , x 2 , ..., x i−1 , Y, x i+1 , ..., x j−1 , x j , x j+1 , ..., x n , x n+1 ),

(x 1 , x 2 , ..., x _i−1 , x i , x i+1 , ..., x _j−1 , Y, x j+1 , ..., x n , x n+1 ).

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Geometry of n -dim. Simplical Cell division, 3-Dim. case

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How do we choose (i, j) pair and the value of λ ^?

Short sample of the MC events (100-1000) is generated ∈ ^cell.

Each MC point projected X → Y onto a given edge (i, j), i 6= j ^: Y = λ ij x i + (1 − λ ij )x j , λ ij (X) = _|Det ^|Det ⁱ ^|

i |+|Det _j | , Det i = Det(r 1 , ..., r i−1 , r i+1 , ...r n , r n+1 ),

Det j = Det(r ¹ , ..., r _j−1 , r j+1 , ...r n , r n+1 ), r k = x k − X ^,

where Det(x ¹ , x 2 , ..., x n ) determinant.

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Choice of the best division edge, Simplical 3-Dim. case

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(2,3) (2,4)

How do we select (i, j) ^? ^{out of} ⁿ⁽ⁿ⁻¹⁾ ₂ possibilities.

For each (i, j) ^the dN/dλ is histogrammed, and its LOSS functional R _loss ^is

estimated. The edge (i, j) with the biggest LOSS is selected! For the cell division

λ is read from the histogram. λ is always a rational number, n/N bin !

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Binary split 2-dim. example

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• Integrand covers narrow strip along edges.

• We intend to split upper triangle.

• 1000 w-ted events are generated and projected onto 3 sides of the parent triangle.

• 3 Projections are analyzed.

• Choosen is the cell with the smallest R loss (middle plot).

• Two resulting daughter triangles are shown at leftmost plot.

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2-dimensional example of binary split: projection on one of edges

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• Projected integrand ρ(x) ^.

• ^Old ρ ⁰ for parent cell (majorizing ρ(x) ^).

• ^New ρ ⁰ for two daughter cells.

• ^OLD R _loss all area above red line, for the parent cell.

• ^New R _loss between red line and New ρ ⁰ , for 2 daughters.

• ^Obviously I _{N ew} ⁰ < I _Old ⁰ , the division

point ? is chosen to MINIMIZE THE

LOSS functional/integral R _loss ^!

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Number of cells= 10, 70,

250, 2500.

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FOAM: general purpose Monte Carlo Cellular Algorithm S. Jadach