VAMP, Version 1.0: Vegas AMPlified

400 Pages·2016·1.65 MB·English

VAMP, Version 1.0: Vegas AMPlified:
Anisotropy, Multi-channel sampling and
Parallelization
Thorsten Ohl12
Darmstadt University of Technology
Schloßgartenstr. 9
D-64289 Darmstadt
Germany
IKDA 98/??
hep-ph/yymmnnn
October 1999
DRAFT: November 3, 2022
1e-mail: [email protected]
2Supported by Bundesministerium fu¨r Bildung, Wissenschaft, Forschung und
Technologie, Germany.
Abstract
WepresentannewimplementationoftheclassicVegasalgorithmforadaptive
multi-dimensional Monte Carlo integration in Fortran95. This implementa-
tion improves the performance for a large class of integrands, supporting
stratified sampling in higher dimensions through automatic identification of
the directions of largest variation. This implementation also supports multi
channel sampling with individual adaptive grids. Sampling can be performed
in parallel on workstation clusters and other parallel hardware. Note that for
maintenance of the code, and especially its usage within the event generator
WHIZARD, some features of Fortran2003 have been added.
Revision Control
i
Contents
1 Introduction 1
2 Algorithms 3
2.1 Importance Sampling . . . . . . . . . . . . . . . . . . . . . 4
2.2 Stratified Sampling . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Vegas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Vegas’ Inflexibility . . . . . . . . . . . . . . . . . . . 6
2.3.2 Vegas’ Dark Side . . . . . . . . . . . . . . . . . . . . 7
2.4 Multi Channel Sampling . . . . . . . . . . . . . . . . . . . . 7
2.5 Revolving . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.6.1 Multilinear Structure of the Sampling Algorithm . . . 8
2.6.2 State and Message Passing . . . . . . . . . . . . . . 13
2.6.3 Random Numbers . . . . . . . . . . . . . . . . . . . 13
2.6.4 Practice . . . . . . . . . . . . . . . . . . . . . . . . 14
3 Design Trade Offs 19
3.1 Programming Language . . . . . . . . . . . . . . . . . . . . 20
4 Usage 21
4.1 Basic Usage . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.1.1 Basic Example . . . . . . . . . . . . . . . . . . . . . 22
4.2 Advanced Usage . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.1 Types . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.2 Shared Arguments . . . . . . . . . . . . . . . . . . . 25
4.2.3 Single Channel Procedures . . . . . . . . . . . . . . . 27
4.2.4 Inout/Output and Marshling . . . . . . . . . . . . . 28
4.2.5 Multi Channel Procedures . . . . . . . . . . . . . . . 31
4.2.6 Event Generation . . . . . . . . . . . . . . . . . . . 34
4.2.7 Parallelization . . . . . . . . . . . . . . . . . . . . . 34
4.2.8 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . 35
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4.2.9 Other Procedures . . . . . . . . . . . . . . . . . . . . 36
4.2.10 (Currently) Undocumented Procedures . . . . . . . . 36
5 Implementation 37
5.1 The Abstract Datatype division . . . . . . . . . . . . . . . 37
5.1.1 Creation, Manipulation & Injection . . . . . . . . . . 38
5.1.2 Grid Refinement . . . . . . . . . . . . . . . . . . . . 44
5.1.3 Probability Density . . . . . . . . . . . . . . . . . . . 48
5.1.4 Quadrupole . . . . . . . . . . . . . . . . . . . . . . . 49
5.1.5 Forking and Joining . . . . . . . . . . . . . . . . . . 49
5.1.6 Inquiry . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1.7 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . 57
5.1.8 I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.1.9 Marshaling . . . . . . . . . . . . . . . . . . . . . . . 66
5.1.10 Boring Copying and Deleting of Objects . . . . . . . 69
5.2 The Abstract Datatype vamp grid . . . . . . . . . . . . . . . 70
5.2.1 Container for application data . . . . . . . . . . . . 77
5.2.2 Initialization . . . . . . . . . . . . . . . . . . . . . . 77
5.2.3 Sampling . . . . . . . . . . . . . . . . . . . . . . . . 85
5.2.4 Forking and Joining . . . . . . . . . . . . . . . . . . 96
5.2.5 Parallel Execution . . . . . . . . . . . . . . . . . . . 103
5.2.6 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.7 Multi Channel . . . . . . . . . . . . . . . . . . . . . 112
5.2.8 Mapping . . . . . . . . . . . . . . . . . . . . . . . . 126
5.2.9 Event Generation . . . . . . . . . . . . . . . . . . . 134
5.2.10 Convenience Routines . . . . . . . . . . . . . . . . . 140
5.2.11 I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.2.12 Marshaling . . . . . . . . . . . . . . . . . . . . . . . 157
5.2.13 Boring Copying and Deleting of Objects . . . . . . . 163
5.3 Interface to MPI . . . . . . . . . . . . . . . . . . . . . . . . 166
5.3.1 Parallel Execution . . . . . . . . . . . . . . . . . . . 168
5.3.2 Event Generation . . . . . . . . . . . . . . . . . . . 181
5.3.3 I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.3.4 Communicating Grids . . . . . . . . . . . . . . . . . 186
6 Self Test 192
6.1 No Mapping Mode . . . . . . . . . . . . . . . . . . . . . . . 192
6.1.1 Serial Test . . . . . . . . . . . . . . . . . . . . . . . 192
6.1.2 Parallel Test . . . . . . . . . . . . . . . . . . . . . . 202
6.1.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . 204
6.2 Mapped Mode . . . . . . . . . . . . . . . . . . . . . . . . . 204
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6.2.1 Serial Test . . . . . . . . . . . . . . . . . . . . . . . 204
6.2.2 Parallel Test . . . . . . . . . . . . . . . . . . . . . . 219
6.2.3 Output . . . . . . . . . . . . . . . . . . . . . . . . . 221
7 Application 222
7.1 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . 222
A Constants 245
A.1 Kinds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
A.2 Mathematical and Physical Constants . . . . . . . . . . . . . 245
B Errors and Exceptions 247
C The Art Of Random Numbers 251
C.1 Application Program Interface . . . . . . . . . . . . . . . . . 251
C.2 Low Level Routines . . . . . . . . . . . . . . . . . . . . . . 254
C.2.1 Generation of 30-bit Random Numbers . . . . . . . . 254
C.2.2 Initialization of 30-bit Random Numbers . . . . . . . 256
C.2.3 Generation of 52-bit Random Numbers . . . . . . . . 259
C.2.4 Initialization of 52-bit Random Numbers . . . . . . . 260
C.3 The State . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
C.3.1 Creation . . . . . . . . . . . . . . . . . . . . . . . . 262
C.3.2 Destruction . . . . . . . . . . . . . . . . . . . . . . . 263
C.3.3 Copying . . . . . . . . . . . . . . . . . . . . . . . . 264
C.3.4 Flushing . . . . . . . . . . . . . . . . . . . . . . . . 265
C.3.5 Input and Output . . . . . . . . . . . . . . . . . . . 265
C.3.6 Marshaling and Unmarshaling . . . . . . . . . . . . . 270
C.4 High Level Routines . . . . . . . . . . . . . . . . . . . . . . 273
C.4.1 Single Random Numbers . . . . . . . . . . . . . . . . 275
C.4.2 Arrays of Random Numbers . . . . . . . . . . . . . . 276
C.4.3 Procedures With Explicit tao random state . . . . . 278
C.4.4 Static Procedures . . . . . . . . . . . . . . . . . . . . 279
C.4.5 Generic Procedures . . . . . . . . . . . . . . . . . . 280
C.4.6 Luxury . . . . . . . . . . . . . . . . . . . . . . . . . 281
C.5 Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
C.5.1 30-bit . . . . . . . . . . . . . . . . . . . . . . . . . . 283
C.5.2 52-bit . . . . . . . . . . . . . . . . . . . . . . . . . . 285
C.5.3 Test Program . . . . . . . . . . . . . . . . . . . . . . 286
D Special Functions 287
D.1 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
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E Statistics 291
F Histogramming 294
G Miscellaneous Utilities 304
G.1 Memory Management . . . . . . . . . . . . . . . . . . . . . 304
G.2 Sorting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
G.3 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . 310
G.4 I/O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
H Linear Algebra 313
H.1 LU Decomposition . . . . . . . . . . . . . . . . . . . . . . . 313
H.2 Determinant . . . . . . . . . . . . . . . . . . . . . . . . . . 315
H.3 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . 316
H.4 Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
I Products 323
J Kinematics 324
J.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . 324
J.2 Massive Phase Space . . . . . . . . . . . . . . . . . . . . . . 326
J.3 Massive 3-Particle Phase Space Revisited . . . . . . . . . . . 329
J.4 Massless n-Particle Phase Space: RAMBO . . . . . . . . . . . 332
J.5 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
K Coordinates 336
K.1 Angular Spherical Coordinates . . . . . . . . . . . . . . . . . 336
K.2 Trigonometric Spherical Coordinates . . . . . . . . . . . . . 340
K.3 Surface of a Sphere . . . . . . . . . . . . . . . . . . . . . . 343
L Idiomatic Fortran90 Interface for MPI 344
L.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
L.2 Point to Point . . . . . . . . . . . . . . . . . . . . . . . . . 347
L.3 Collective Communication . . . . . . . . . . . . . . . . . . . 353
M Ideas 358
M.1 Toolbox for Interactive Optimization . . . . . . . . . . . . . 358
M.2 Partially Non-Factorized Importance Sampling . . . . . . . . 358
M.3 Correlated Importance Sampling (?) . . . . . . . . . . . . . 358
M.4 Align Coordinate System (i.e. the grid) with Singularities (or
the hot region) . . . . . . . . . . . . . . . . . . . . . . . . . 359
M.5 Automagic Multi Channel . . . . . . . . . . . . . . . . . . . 359
v
N Cross References 360
N.1 Identifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
N.2 Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . 379
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Program Summary:
(cid:136)
Title of program: VAMP, Version 1.0 (October 1999)
(cid:136)
Program obtainable by anonymous ftp from the host
crunch.ikp.physik.th-darmstadt.deinthedirectorypub/ohl/vamp.
(cid:136)
Licensing provisions: Free software under the GNU General Public
License.
(cid:136)
Programming language used: From version 2.2.0 of the program:
Fortran2003 [8] Until version 2.1.x of the program: Fortran95 [9] (For-
tran90 [7] and F [14] versions available as well)
(cid:136)
Number of program lines in distributed program, including
test data, etc.: ≈ 4300 (excluding comments)
(cid:136)
Computer/Operating System: AnywithaFortran95(orFortran90
or F) programming environment.
(cid:136)
Memory required to execute with typical data: Negligible on
the scale of typical applications calling the library.
(cid:136)
Typical running time: A small fraction (typically a few percent) of
the running time of applications calling the library.
(cid:136)
Purpose of program:
(cid:136)
Nature of physical problem:
(cid:136)
Method of solution:
(cid:136)
Keywords: adaptive integration, event generation, parallel processing
vii
—1—
Introduction
We present a reimplementation of the classic Vegas [1, 2] algorithm for adap-
tive multi-dimensional integration in Fortran95 [9, 13]1 (Note that for the
maintenance of the program and especially its usage within the event gen-
erator WHIZARD parts of the program have been adapted to Fortran2003).
The purpose of this reimplementation is two-fold: for pedagogical reasons
it is useful to employ Fortran95 features (in particular the array language)
together with literate programming [4] for expressing the algorithm more
concisely and more transparently. On the other hand we use a Fortran95
abstract type to separate the state from the functions. This allows multiple
instances of Vegas with different adaptions to run in parallel and in paves
the road for a more parallelizable implementation.
The variable names are more in line with [1] than with [2] or with [17,
18, 19], which is almost identical to [2].
Copyleft
Mention the GNU General Public License (maybe we can switch to the GNU
Library General Public License)
1 ⟨Copyleft notice 1⟩≡ (37a70b167a192a202a204b219e222a245247a273d274a287a289b291a294a304a313a321b323324a336344a) 2▷
! Copyright (C) 1998 by Thorsten Ohl <[email protected]>
!
! VAMP is free software; you can redistribute it and/or modify it
! under the terms of the GNU General Public License as published by
! the Free Software Foundation; either version 2, or (at your option)
! any later version.
!
1Fully functional versions conforming to preceeding Fortran standard [7], High Perfor-
mance Fortran (HPF) [10, 11, 15], and to the Fortran90 subset F [14] are available as
well. A translation to the obsolete FORTRAN77 standard [6] is possible in principle, but
extremely tedious and error prone if the full functionality shall be preserved.
1