Monte Carlo Methods In Particle Physics
Monte Carlo Methods inParticle PhysicsBryan WebberUniversity of CambridgeIMPRS, Munich19-23 November 2007See alsoESW: “QCD and Collider Physics”, C.U.P. bber/QCD 03/Thanks to Mike Seymour, Torbjörn Sjöstrand, Frank Krauss, Peter Richardson,.Monte Carlo Methods 1Bryan Webber
Monte Carlo Event Generation Basic PrinciplesEvent GenerationParton ShowersHadronizationUnderlying EventEvent Generator SurveyMatching to Fixed OrderBeyond Standard ModelMonte Carlo Methods 1Bryan Webber
Lecture1: Basics The Monte Carlo conceptEvent generationExamples: particle production and decayStructure of an LHC eventMonte Carlo implementation of NLO QCDMonte Carlo Methods 1Bryan Webber
Integrals as Averages Basis of all MonteCarlo methods: Draw N values from auniform distribution: Sum invariant underreordering: randomize Central limit theorem:Monte Carlo Methods 1Bryan Webber
Convergence Monte Carlo integrals governed by Central LimitTheorem: errorc.f. trapezium ruleSimpson’s rulebut only if derivatives exist and are finite:Monte Carlo Methods 1Bryan Webber
Importance SamplingConvergence improved by putting more samples in region wherefunction is largest.Corresponds to a Jacobian transformation.Hit-and-miss: accept points with probability ratio (if 1)Monte Carlo Methods 1Bryan Webber
Stratified SamplingDivide up integration region piecemeal andoptimize to minimize total error.Can be done automatically (eg VEGAS).Never as good as Jacobian transformations.N.B. Puts more pointswhere rapidly varying, notnecessarily where larger!Monte Carlo Methods 1Bryan Webber
Multichannel SamplingMonte Carlo Methods 1Bryan Webber
Multi-dimensional Integration Formalism extends trivially to many dimensions Particle physics: very many dimensions,e.g. phase space 3 dimensions per particles,LHC event 250 hadrons. Monte Carlo error remains Trapezium rule Simpson's ruleMonte Carlo Methods 1Bryan Webber
SummaryDisadvantages of Monte Carlo: Slow convergence in few dimensions.Advantages of Monte Carlo: Fast convergence in many dimensions. Arbitrarily complex integration regions (finite discontinuitiesnot a problem). Few points needed to get first estimate (“feasibility limit”). Every additional point improves accuracy (“growth rate”). Easy error estimate.Monte Carlo Methods 1Bryan Webber
Phase SpacePhase space:Two-body easy:Monte Carlo Methods 1Bryan Webber
Other cases by recursive subdivision:Or by ‘democratic’ algorithms: RAMBO, MAMBOCan be better, but matrix elements rarely flat.Monte Carlo Methods 1Bryan Webber
Particle DecaysSimplest examplee.g. top quark decay:pt · p! pb · pνBreit-Wigner peak of W very strong: must beremoved by Jacobian factorMonte Carlo Methods 1Bryan Webber
Associated DistributionsBig advantage of MonteCarlo integration:simply histogram anyassociated quantities.Almost any othertechnique requiresnew integration foreach observable.Can apply arbitrarycuts/smearing.Monte Carlo Methods 1e.g. lepton momentum in top decays:chargedneutralBryan Webber
Cross SectionsAdditional integrations over incoming partondensities:can have strong peaks, eg Z Breit-Wigner:need Jacobian factors.Hard to make process-independent.Monte Carlo Methods 1Bryan Webber
Leading Order Monte Carlo CalculationsNow have everything we need to make leading ordercross section calculations and distributionsCan be largely automated MADGRAPH GRACECOMPHEP AMEGIC ALPGENBut Fixed parton/jet multiplicityNo control of large logs Parton level Need hadron level event generatorsMonte Carlo Methods 1Bryan Webber
Event GeneratorsUp to here, only considered Monte Carlo as a numericalintegration method.If function being integrated is a probability density (positivedefinite), trivial to convert it to a simulation of physicalprocess an event generator.Simple example:Naive approach: ‘events’ with ‘weights’Can generate unweighted events by keeping them withprobability: give them all weightHappen with same frequency as in nature.Efficiency: fraction of generated events kept.Monte Carlo Methods 1Bryan Webber
Structure of LHC Events1. Hard process2. Parton shower3. Hadronization4. Underlying eventWe’ll return to this later.Monte Carlo Methods 1Bryan Webber
Monte Carlo Calculations of NLO QCDTwo separate divergent integrals:Must combine before numerical integration.Jet definition could be arbitrarily complicated.How to combine without knowing?Two solutions:phase space slicing and subtraction method.Monte Carlo Methods 1Bryan Webber
Illustrate with simple one-dim. example:x gluon energy or two-parton invariant mass.Divergences regularized bydimensions.Cross section in d dimensions is:Infrared safety:KLN cancellation theorem:Monte Carlo Methods 1Bryan Webber
Phase space slicingIntroduce arbitrary cutoff:Two separate finite integralsMonte Carlo.Becomes exact forbut numerical errors blow upcompromise (trial and error).Systematized by Giele-Glover-Kosower.JETRAD, DYRAD, EERAD, . . .Monte Carlo Methods 1Bryan Webber
Subtraction methodExact identity:Two separate finite integrals again.Monte Carlo Methods 1Bryan Webber
Subtraction methodExact identity:Two separate finite integrals again.Much harder: subtracted cross section must be valideverywhere in phase space.Systematized inS. Catani and M.H. Seymour, Nucl. Phys. B485 (1997) 291.S. Catani, S. Dittmaier, M.H. Seymour and Z. Trocsanyi, Nucl. Phys. B627 (2002) 189.any observable in any processanalytical integrals done once-and-for-allEVENT2, DISENT, NLOJET , MCFM, Monte Carlo Methods 1Bryan Webber
Summary Monte Carlo is a very convenient numerical integrationmethod. Well-suited to particle physics: difficult integrands, manydimensions. Integrand positive definite event generator. Fully exclusive treat particles exactly like in data. need to understand/model hadronic final state.N.B. NLO QCD programs are not event generators:Not positive definite.But full numerical treatment of arbitrary observables.We’ll discuss later how to combine/match NLO and EGsMonte Carlo Methods 1Bryan Webber
Monte Carlo Methods 1 Bryan Webber Summary Monte Carlo is a very convenient numerical integration method. Well-suited to part
Quasi Monte Carlo has been developed. While the convergence rate of classical Monte Carlo (MC) is O(n¡1 2), the convergence rate of Quasi Monte Carlo (QMC) can be made almost as high as O(n¡1). Correspondingly, the use of Quasi Monte Carlo is increasing, especially in the areas where it most readily can be employed. 1.1 Classical Monte Carlo
The Markov Chain Monte Carlo Revolution Persi Diaconis Abstract The use of simulation for high dimensional intractable computations has revolutionized applied math-ematics. Designing, improving and understanding the new tools leads to (and leans on) fascinating mathematics, from representation theory through micro-local analysis. 1 IntroductionCited by: 343Page Count: 24File Size: 775KBAuthor: Persi DiaconisExplore furtherA simple introduction to Markov Chain Monte–Carlo .link.springer.comHidden Markov Models - Tutorial And Examplewww.tutorialandexample.comA Gentle Introduction to Markov Chain Monte Carlo for .machinelearningmastery.comMarkov Chain Monte Carlo Lecture Noteswww.stat.umn.eduA Zero-Math Introduction to Markov Chain Monte Carlo .towardsdatascience.comRecommended to you b
Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution - to estimate the distribution - to compute max, mean Markov Chain Monte Carlo: sampling using "local" information - Generic "problem solving technique" - decision/optimization/value problems - generic, but not necessarily very efficient Based on - Neal Madras: Lectures on Monte Carlo Methods .
J.S. Liu and R. Chen, Sequential Monte Carlo methods for dynamic systems , JASA, 1998 A. Doucet, Sequential Monte Carlo Methods, Short Course at SAMSI A. Doucet, Sequential Monte Carlo Methods & Particle Filters Resources Pierre Del Moral, Feynman-Kac
Fourier Analysis of Correlated Monte Carlo Importance Sampling Gurprit Singh Kartic Subr David Coeurjolly Victor Ostromoukhov Wojciech Jarosz. 2 Monte Carlo Integration!3 Monte Carlo Integration f( x) I Z 1 0 f( x)d x!4 Monte Carlo Estimator f( x) I N 1 N XN k 1 f( x k) p( x
Monte Carlo for Machine Learning Sara Beery, Natalie Bernat, and Eric Zhan MCMC Motivation Monte Carlo Principle and Sampling Methods MCMC Algorithms Applications History of Monte Carlo methods Enrico Fermi used to calculate incredibly accurate predictions using statistical sampling methods when he had insomnia, in order to impress his friends.
Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and Materials Monte Carlo Methods for PDEs A Little History on Monte Carlo Methods for PDEs Early History of MCMs for PDEs 1.Courant, Friedrichs, and Lewy: Their pivotal 1928 paper has probabilistic interpretations and MC algorithms for linear elliptic
Monte Carlo methods 5.3. MONTE CARLO CONTROL 105 one of the actions from each state. With no returns to average, the Monte Carlo estimates of the other actions will not improve with experience. This is a serious problem because the purpose of learning action values is to help in choosing among the actions available in each state.
