foam multi-purpose s. jadach

Presentation compact version F-1

FOAMMulti-Purpose

Celular Monte Carlo Generator

S. JadachInstitute of Nuclear Physics, Krakow, Poland

Outline:

• General introduction.

• Foam algorithm and new MCell (Mega-Cell MC).

• Numerical tests.

• Conclusions.

Slides, program sources: http://home.cern.ch/jadach

S. Jadach Krakow, September, 2001

Introduction F-2

Just to give an idea...

Modern PC desktop can create within 15minCPU

nC =5k-50k cells and subsequently generate N =10M

events for relatively strongly peaked n-dimesional

integrand, n < 12.

With σ<W>

= 0.3 that provides 3-digit value of the

integral, δII

= σ√N

' 10−3,

or alternatively 3M events with W = 1,

assuming Wmax

<W>= 0.3.

Not bad!! What are the key limitations?

Memory: ∼ 16 × n × nC Bytes. <50MB.

CPU time: ∼ nC /1k Minutes CPU. <1h.

Efficient algorithm of build-up of cells:

maximizing <W>Wmax

and/or minimizing σ<W>

.

Algorithms based on BINARY SPLIT of cells seems to fulfill

the constraints. Parent cells kept in record →→ hierarchical data organization (tree-like linked list).


Picture Gallery F-3

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Number of cells= 10, 70, 250, 2500.


Picture Gallery F-4

Evolution of hyper-cubic. foam

Number of cells= 10, 70, 250, 2500.


Picture Gallery F-5

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Foam Algorithm F-6

Terminology&Notation

Integration domain Ω = [0, 1]n is split into Cells ω1...ωk .

Two AUXILIARY distributions ρ′(x) and ρloss(x) related

to integrand ρ(x), which are CONSTANT over each cell

are constructed. How? See next slides.

TWO STEP SCENARIO:

(1) Foam of cells BUILD-UP:

ρloss(x) is evolving with the foam, it DRIVES the division of

cells, Iloss =∫

ρlossdnx is minimized in the process.

(2) MC generation:

Events are generated according to ρ′(x).

I ′ =∫

ρ′dnx is known exactly.

I = I ′ < W >′ where W = ρ/ρ′,

The average < .. >′ means over events generated according to ρ′

OUR AIMS:

Minimize rejection (loss) rate L = 1 − <W>Wmax

, OR

Minimize variance σ =√

< W 2 > − < W >2.


Foam Algorithm F-7

binary split of a cell

How do we minimize L = 1 − <W>

Wmax

?

DEF: ρ′(x) ≡ maxy∈Celli ρ(y), for x ∈ Celli.

is a maximum of ρ(x) within each cell:

Maximum over cell found experimentally using ∼1000 MC events.

We MINIMIZE during the foam construction:

Iloss =∫

dnx [ρ′(x) − ρ(x)] ≡∫

dnx ρloss(x)

Each SPLIT of a Cell: ω → ω′ + ω

′′

should obey: Iω′

loss+ I

ω′′

loss<< I

ωloss

.

RULES to follow:• For next split we choose a CELL with the biggest Iloss.

• Position/direction of a plane dividing a parent CELL into

two daughter CELLs is chosen to minimize total Iloss.


Foam Algorithm F-8

Binary split 1-dim. example

! "$# %'&)(+*,*

-/.02143658797

-:.0

;

ρ(x) is integrand

Old ρ′ for parent cell (majorizing ρ(x))

New ρ′ for two daughter cells

OLD Iloss all area above red line, for the parent cell

New Iloss between red line and New ρ′, for 2 daughters.

Obviously I ′New < I ′

Old, the division point ? is chosen to

MINIMIZE THE LOSS functional/integral Iloss!


Foam Algorithm F-9


The two-maximum case is a little bit more complicated.

We follow simple prescription:

Determine the blue line MAXIMIZING Iloss, see picture,

and apply the division procedure twice (obtaining 3 cells).

Division points are ? and


Foam Algorithm F-10


! " # ! # ! $ # % ! !

! " #

! #

! $ #

% !

&'(

)

Integrand covers narrow strip along edges. Intend to split upper triangle.

Generate 1000 w-ted events and project them on 3 sides of the parent triangle.

3 Projections are analyzed.

We choose the one with the smallest LOSS functional Iloss (middle plot).

Two resulting daughter triangles are shown.

The DIVISION line EXPELLED from the “empty” area in the middle!!!


Foam Algorithm F-11

Minimizing variance

How to Minimize σ2

<W>2 = <W 2>

<W>2 − 1 ?

DEF: ρ′(x) ≡ Vi ×√

< ρ2 >i, for x ∈ Celli.

The < .. >i is for uniformly distributed points ∈ Celli.

Cells with bigger contribution to total variance, “promoted” in MC generation.

We MINIMIZE during the foam build-up:

Iloss =∫

dnx ρloss(x) where

ρloss(x) ≡ Vi ×√

< ρ2 >i − < ρ >2i , for x ∈ Celli.

The algorithm avoids dividing cells in the areas where the integrand is flat,

the dispersion of the function over the cell is small.

All the algorithm of cell division remain the same.

We did less effort to optimize this mode of the program.


Foam Algorithm F-12

Weight distribution: minimization of variance

Number of cells: 200, 2k, 20k

Weight distribution: minimization of rejection rate

! " " ! # $" %

# $" %

& $" %

' $" %

( $" %

)* +) ,* )) ,* +) -* )))* )).,)/

)* +).,)/

,* )).,) /

,* +).,)/

01 20 31 00 31 20 41 0001 005306

01 205306

31 00530 6

31 205306

41 005306

41 20530 6

Number of cells: 200, 2k, 20k


New development F-13

Why hypercubes as cells? Why not?

Consideration in favor of simplexes:MEMORY: In n-dimensions, simplical foam (binary division)

requires only one vertex vector ~v for each new cell!

That means: 8n Bytes/Cell.

Every new hypercubic at n=15 has 32k vertices :-(

However, it can be parameterized using only 2 vectors: ~v for

“lower left corner” vertex position, and ~d defining n lengths

along each direction :-) Hence only 16n Bytes/Cell.

(The interior of the hypercube is xi = vi + λidi, 0 < λi < 1.)

Consideration in favor of hypercubes:Simplexes limited to n < 6 and NCell <5k because:

(1) Already for n=10 initial h-cube → n! ∼ 106 simplexes,

(2) geometry become CPU consuming (determinants).

(2) memory limitation,

However, memory consumption for h-cubes can be limited to

below 50Bytes/Cell indep. of n, albeit with some tolerable CPU overhead,

and no determinants :-))

BOTTOM LINE:

With hypercubes we can reach 1Mega Cells at n ∼ 15.


Picture Gallery F-14

Hyper-cubics = rectangles in 2-dimen.

The “Camel” Double-Gauss functionof Peter Lepage.


New development F-15

Miracle of ∼50Bytes/Cell

Two essential ingredients:

• ~v and ~x are not stored but (re)calculated using the

division parameter λ and the index i of the edge along

which the division occurred, all over ∼ ln2 NCell

steps, up to first Cell.

• Furthermore parameter λ = j/nbin can be stored as

a short inter j (2Bytes sufficient).

Each cell has several other attributes like I ′, Iloss (16B),

pointers to parent and daughter cells (12B), status (8B)

etc., altogether ∼ 50B/Cell.

Can be easily reduced to 32B/Cell if necessary.

BOTTOM LINE:

The way to 1Mega Cells is open.

We hit CPU barrier instead. See next slide.


Foam Algorithm F-16

CPU barrier: one quick fix is found

Playing with hypercubical code MCell for Nc ∼100k Cells

at n ∼ 10 dimensions shows that the obtained efficiency<W>Wmax

and variance σ is improved ESSENTIALLY by the

INCREASING of Nc.

However, another important factor is the number of MC

events used to explore each newly created Cell Nsamp.

A necessary value of Nsamp increases for higher n.

→ Dramatic increase of total CPU time for the foam

build-up, which is T ∼ n × Nc × Nsamp.

The trick limiting the above effect:

During MC exploration of a new cell continuously monitor

the no. of accumulated effective W = 1 events:

Neff = (∑

wi)2

∑

w2i

and stop when Neff/nbin > 25.

nbin is the number of bins in the histogram used to

estimate the best division direction/edge and parameter.

The increase of Nsamp is not wasted for cells in which integrand

is already not varying too much. We are now able to push input

Nsamp to high values, with little increase of CPU time.


Foam F-17

Program usage very easy (f77)

*--------------------------------------------------------

DOUBLE PRECISION Density

EXTERNAL Density

CALL MCellA_SetkDim( 3) ! number of dimensions

CALL MCellA_SetnBuf( 2000) ! no. of cells

CALL MCellA_SetnSampl( 500) ! no. MC ev./cell (init.)

CALL MCellA_Initialize(Density) ! initialization

DO loop = 1, 200000

CALL MCellA_MakeEvent(Density) ! generate MC event

CALL MCellA_GetMCvector(MCvector) ! get MC event vector

CALL MCellA_GetMCwt(MCwt) ! get MC event weight

CALL GLK_Fil1(1000, MCwt,1d0) ! histogramming

ENDDO

CALL MCellA_Finalize(MCresult,MCerror) !get integr.+err.

*--------------------------------------------------------

User has to set only the number of dimensions kDim.

The other input variables are already preset for the user,

thus calling setters for them is optional.

User needs to provide his own integrand function,

which is here Density.


Foam in C++ F-18

Program usage in C++ also easy

//-------------------------------------------------------

Foam FoamX; // Create Simulator

FunTest Density1(FunType); // Create integrand fun.

FoamX.SetnDim( nDim); // Dimension simplical

FoamX.SetkDim( kDim); // Dimension hyp-cubical

FoamX.SetnCells( nCells); // No. of cells

FoamX.Initialize( &Density1 ); // Initialize Foam cells

double *MCvect =new double[nDim+kDim];

for(long loop=0; loop<NevTot; loop++)

FoamX.MakeEvent(); // Generate MC event

FoamX.GetMCvect( MCvect); // Get MC event

MCwt=FoamX.GetMCwt(); // Get MC weight

double MCresult, MCerror;

FoamX.Finalize( &MCresult, &MCerror); // get integral

//------------------------------------------------------

User has to set only the number of dimensions kDim, nDim.

User needs to provide his own integrand function,

which is here Density. Its type FunTest has to inherit from

abstract class FIntegrand.


MCell and Foam f77 codes F-19

Programs

MCell v2.02, f77:with up to 1Mega cells is specialized for higher dimensions,

n ≤ 20, hypercubical cells only.

Foam v2.02, f77 :with Nc ≤ 10000 cells, is aimed for up to six-dimensions.

It is upgraded and improved:

both simplices (n ≤ 5) and hypercubes (n ≤ 10) available.

Foam v2.02, c++ :unlimited number of cells Nc and dimension nDim+kDim.

Both simplices and hypercubes available.

Includes hypercubical algorithm MCell with low memory

consumtion as default option.

Foam v2.02, c++: bbased on ROOTpackage (R. Brun et.el.)

Other improvements:

• Cells can be simultaneously simplices in n-dimensions

and hypercubics in k-dimensions.

• It is possibile to start FOAM from SINGLE simplex instead

of unit hyper-cubic.


Numerical tests F-20

Tests of MCell/Foam at low dimensions

Three 2-dimensional testing functions:

fa = 1 − Θ(0.5 − |x1 − 0.5| − γ) Θ(0.5 − |x1 − 0.5 − γ|), γ = 0.05,

fb =1

4πR2

γ

π[(R −√

(x1 − 0.25)2 + (x2 − 0.40)2)2 + γ2)],

γ = 0.02, R = 0.35

fc =γ

π[(x1 + x2)2 + γ2)], γ = 0.02.

Functions, 2-dimens. Foam MCell VEGAS

fa(x1, x2) (diagonal ridge) 0.94 0.74 0.03

fb(x1, x2) (circular ridge) 0.80 0.80 0.16

fc(x1, x2) (edge of square) 1.00 1.00 0.53

Results from Foam/MCell are for 5000 cells (2500 active cells)

and cell exploration based on 200 MC events/cell.

Efficiencies are < W > /W εmax with ε=0.0005.

Functions, 3-dimens. Foam MCell VEGAS

fa (thin diagonal) 0.56 0.62 0.002

fb (thin sphere) 0.27 0.50 0.11

fc (surface of cube) 0.66 1.00 0.30

The main point is that efficiency of MCell can be trivially

increased to 100% by increasing no of cells, while for VEGAS

we can NEVER do better.


Numerical tests F-21

Tests of MCell at higher dimensions

Double-hump CAMEL test-function of P. Lepage. "! !$#&% # ('$'&% #$' $$ #$! ) +* %$, *$* $$$ % $$ # +* ! * # $$ % $$ ' !$! ! * , *- $$ .& $$ # /$ "! * ,$% . 0%$, ('$'$'$!$' $$ , ! /$ ) * &* 0%$% $$ ! $$ , . /$ ! * %$, 0% $$ $$ , ! /$ * # 0% # $$&*$* $$ ,$,' /$ ) !$! 0% . ('$' * #&, $+* !' /$ ! . !&, (# . ('$'&% , $ ! ' /$ .& % (#$' ('$'$#&% $ !$!+* /$ ) $ !&% ! ! (# .& ' $&*$*-+* /$ ! $ !$# !(!$! . #$'&, $&* !&,+* /$ $ ! . . # ('$'&, +* $ ! %+* /$ $ $ ! * ,- * $$$ ! $ ! #+* 1$ $ ,$, !(#$' ('$'&%$% ' $&* %$,

Efficiency depends mainly on number of cells MCell.

Number of MC trials per Cell cannot be too small.


Conclusions F-22

Summary

• New f77 program MCell for 1 Mega cells and n<20 is now available, in

addition to upgraded f77 Foam.

• C++ version of the Foam with dynamic memory allocation now also

available! Also version with ROOT.

• MCell/Foam are are general purpose Monte Carlo simulators/integrators based

on a self-adapting FOAM OF CELLS.

• Cells are simplices and/or hyper-cubics in Foam (f77 and c++), only

hyper-cubics in MCell (f77).

• f77 MCell (and c++ Foam) is able to cope with CPU and MEMORY barriers at

higher dimensions, as numerical tests demonstrate.

• Looking forward for a KillerApp!

Postscript of these slides: http://home.cern.ch/jadach/public/MCell-talk.ps.gz


foam multi-purpose s. jadach

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