Chapter 5 Function-based Monte Carlo

Transcription

1 Chapter 5 Functon-based Monte Carlo 5.1 Four technques or estmatng ntegrals Our net set o mathematcal tools that we wll develop nvolve Monte Carlo ntegraton. In the grand scheme o thngs, our study so ar o samplng rom dstrbutons has provded us wth the tools we need to run a smple Monte Carlo smulaton; what the study o ntegraton o unctons wll gve us relates to keepng score n a smulaton. In addton, we wll use ths lesson to ntroduce n the smplest contet I can thnk o the rst technques o varance reducton. Ths lesson s dvded nto man parts: 1. Our basc mathematcal ramework or perormng the ntegraton; and. Development o our partcular methods usng the ramework. Mathematcal ramework or ntegraton We wll be consderng the use o Monte Carlo methods to sample the value o a basc ntegral o the orm: I b ( d (5-1 a We are gong to use ths as a very smple eample o a complete Monte Carlo smulaton. (There are much better ways o solvng ths ntegral. Our basc mathematcal ramework s a two step procedure whereby we: Sample (somehow between a and b, gvng us an ˆ ; and then Score a guess or the ntegral soluton, I, based on that ˆ. Derent methods or solvng ths ntegral are possble and can be characterzed by how they perorm these two tasks. All o them are based on a (more or less drect applcaton o the Law o Large umbers, whch can be over-smpled as: b ( ( d lm a 1 ( ˆ (5- where the samples ˆ are chosen usng the PDF (. 5-1

2 ote partcularly the act that there are TWO unctons n the ntegrand (actually one uncton and one dstrbuton and the two have derent ROLES n the subsequent Monte Carlo appromaton: 1. The ( s used as the bass or choosng the ˆ value; and. The ( s used (evaluated at the chosen value or the score. The derent methods we wll study nvolve takng what s nsde the ntegral and breakng t up derent ways to satsy these two roles. In partcular, the ntegral that I presented at the begnnng o the lesson does not HAVE a PDF n t; so, you have to supply one. I we denote a normalzed probablty dstrbuton between a and b as, we can wrte the ntegral as: I b ( ( d ( (5- a From ths ramework, we can lesh out our two step approach nto: Sample usng ( between a and b (whch I desgnate usng the notaton b I a ( d ; and then Score a guess or the ntegral solutons as I strategy that has ths epected value. ( ˆ ( ˆ (or as some probablty mng Four partcular methods or perormng the ntegraton We wll now go over our partcular varatons on ths theme, each o them characterzed mathematcally by partcular choces o ( and the method o scorng: Rejecton method Averagng method Control varates method Importance samplng method Rejecton method Ths s a smlar approach to the use o rejecton methods n pckng rom a dstrbuton. It s a dart board method n whch we estmate the area under a unctonal curve by contanng the curve n a rectangular bo, pckng a pont randomly n the bo, and scorng t msses (.e., s above the curve or the ull rectangular area t hts (.e., s below the curve. As beore, we have to specy an upper bound o the uncton, ma, and then proceed by: 5-

3 1. Choosng a value o ˆ unormly between a and b.. Choosng a value o ˆ unormly between and ma. ( and scorng I. Scorng I ( ma b a ˆ ˆ otherwse. In terms o our mathematcal ramework, ths s equvalent to usng: 1 b a (or a unorm dstrbuton between a and b and scorng wth a probablty mng strategy o scorng ma wth probablty ( ˆ ( ˆ ma or scorng wth probablty ( ˆ. 1 Ths med scorng strategy obvously has the desred epected value o ma I ( ˆ ( ˆ. Eample 1: Fnd d I usng a rejecton method. (The answer s 8/. Answer: The mamum value o ths uncton n the doman s 4, so our procedure s to: 1. Choose a value o ˆ unormly between and.. Choose a value o ˆ unormly between and 4.. Score 8 ˆ s less than ( ˆ ; otherwse score. A FORTRA computer program was wrtten to solve ths problem. The results are: umber o hstores 1, , ,,

4 Averagng method Ths s a much more straght-orward approach to the problem because t uses the uncton drectly. The procedure or ths method s to: 1. Choose a value o ˆ unormly between a and b.. Score I ( ˆ ( b a. In terms o our mathematcal ramework, ths s equvalent to agan usng: 1 b a (or a unorm dstrbuton between a and b and scorng wth a drect use o I ( ˆ ( ˆ Eample : Agan nd I d usng an averagng method. Answer: The procedure s to: 1. Choose a value o ˆ unormly between and.. Score ˆ. A FORTRA computer program was wrtten to solve ths problem. The results are: umber o hstores 1, , ,, The estmated standard devatons are less than or the prevous rejecton eample. Control varates method Ths method s the rst o two methods that utlze a user-suppled second uncton, h (, whch s chosen to be a "well behaved" appromaton to (. What makes these methods so 5-4

5 powerul s that they allow the user to take use o a pror knowledge about the uncton (. In the control varates method, the ntegral soluton "begns" as the ntegral o the known uncton: I h b h( d (5-4 a and uses the Monte Carlo approach to nd an addtve correcton to ths user-suppled guess. The procedure or ths method s to: 1. Choose a value o ˆ unormly between a and b.. Score I ( ( ˆ h( ˆ ( b a I. h otce that there s O varance ntroduced through the I h part o the score, but only n the derence between h. Obvously, then a good guess wll result n a small and derence and, thereore a small varance; n the lmt o a perect guess, h, there would be no correcton and no thereore no varance. (ote that there would also be no varance ntroduced h constant snce, agan, each guess would be dentcal and correct. In terms o our mathematcal ramework, ths s equvalent to agan usng: (or a unorm dstrbuton between a and b and scorng wth 1 b a (5-5 I ( ˆ ( ˆ h E( h( ( ˆ ( ˆ h( ˆ a b ( ˆ h( d Ih ( ˆ ( ˆ h b a ( ˆ ( ˆ ( ˆ h I ( ˆ b h a d. (

6 Eample : Agan nd d I, ths tme usng a control varates method wth h (. (ote that h ( matches ( at the endponts and s smlarly unormly ncreasng. Answer: otng the ntegral o h ( over the doman (, s. Wth ths value known, the procedure s to: 1. Choose a value o ˆ unormly between and.. Score I ˆ ( ˆ. A FORTRA computer program was wrtten to solve ths problem. The results are: umber o hstores 1, , ,, The estmated standard devatons are lower than or ether o the two prevous eamples. Importance samplng method The nal method s the mportance samplng method. Ths technque s smlar to the control varates method, n that t takes advantage o a pror knowledge about the uncton (, but ders rom t n that ts correcton s multplcatve rather than addtve. In terms o our mathematcal ramework, the mportance samplng method uses the appromate uncton h as the (unnormalzed probablty dstrbuton rom whch the varables ˆ are drawn: h( h( (, (5-7 b Ih h( d a Wth the resultng score o: I ( ˆ ( ˆ h ( ˆ I ( ˆ h (

7 As wth control varates, a perect guess o h( ( would result n a zero varance soluton, ths tme because, agan, every guess would be the correct one, I h. (ote that any guess that satses h( ( constant would do just as well, snce t would normalze to the same (. Eample 4: One last tme, nd I d, ths tme usng an mportance samplng method wth the same guess that worked so well beore, h (. Answer: Snce the ntegral o h( over the doman (, s, the resultng probablty dstrbuton rom whch to pck the ˆ wll be: ( 4 Followng the drect procedure or choosng rom ths dstrbuton, we rst determne the CDF, whch s: 4 ( d 4 16 We then set ths CDF to the unorm devate: and nvert to get the ormula: 4 16 ˆ 4 As beore, a FORTRA computer program was wrtten to solve ths problem. The results are: umber o hstores 1, , ,, The estmated standard devatons are greater than those or the control varates problem (Eample, but less than those or the other two methods. 5-7

8 5. Functon substtuton usng Drac Deltas Although we have concentrated on event-based applcatons, the Monte Carlo method has a rch hstory o applcaton to varous knds o problems n mathematcs, scence and engneerng. Although Monte Carlo methods are sometmes categorzed as a numercal methods, they do not generally produce estmates o contnuous unctons themselves, lke most other numercal methods, but rather estmates o weghted ntegrals o these unctons. The man reason or ths emphass on ntegrals s that the theoretcal bass or Monte Carlo methods s the Law o Large umbers. Snce n ths law unctons appear nsde ntegrals, the typcal applcatons have smlarly been lmted to the appromaton o ntegrals o unctons (oten reerred to as talles rather than appromatons o unctons themselves, whch s the most common goal o tradtonal numercal methods. But we can move beyond ths lmtaton through the applcaton o the Law o Large umbers to each pont n the contnuous doman o a uncton to be appromated. Ths approach leads to Monte Carlo algorthms that delver estmates o complete unctons as output, rather than scalar estmates o ntegrals o these unctons. Somewhat surprsngly snce ths s smply a new way o lookng at an old subject ths approach uncovers new possbltes or the method that were always there, but were covered up by the need to reorganze problems nto ntegral orm. Etenson o the Law o Large umbers to unctons The tradtonal bass or Monte Carlo processes s the weak orm o the Law o Large umbers. When ths law s appled to a uncton o one varable, the epected value or mean value o the uncton, dened by: where ˆ s a probablty dstrbuton uncton obeyng: E ( ˆ ( ˆ ( ˆ dˆ (5-9 ( ˆ or all ˆ ( ˆ or all ˆ or whch ( ˆ can be represented n probablty as: ( ˆ dˆ 1 1 E ( ˆ lm ( ˆn (5-1 n1 5-8

9 the ˆ n are samples taken usng ( ˆ and the varance s nte,.e., ˆ ( ˆ E (. (5-11 As prevously, a caret over a varable wll denote a stochastc varable.e., a varable that s sampled rom a probablty densty uncton and a subscrpt on a stochastc varable wll denote a partcular sampled value o that varable. A Monte Carlo algorthm conssts o appromatng ths mean as closely as desred (.e., through ncreasng usng a nte stream o samples o the uncton: where a vector o samples o ˆ selected rom ( ˆ. ˆ 1 ( ˆ ( ˆ (5-1 n 1 1 T n ˆ ˆ ˆ ˆ ˆ, (5-1 Although ths s only an appromaton o the desred value, one o the strengths o the Monte Carlo method s that t also delvers an estmate o the varance o : 1 ( ˆ ˆ 1 ( ˆ n n1 (5-14 Key pont: I we apply the Law o Large umbers to every pont n the doman o a uncton (, the natural etenson o Equatons 1 and s: 1 ( E (, ˆ ( ˆ (, ˆ dˆ lm (, ˆ n (5-15 wth the result beng an estmate o the entre uncton: 1 ( (, ˆ (, ˆ (5-16 n 1 wth a correspondng estmate o the varance at each pont n the doman: n n1 5-9

10 (, ˆ 1 n n1 ˆ (, ˆ ( 1. (5-17 I wll reer to ths combnaton o appromaton and PDF as the par ˆ, ˆ a gven, ˆ wll only work wth a partcular ˆ. In Equatons 6-8, ˆ, --because, s a stochastc uncton whose non-stochastc varable s nherted rom the uncton dependency o and whose stochastc varable ˆ s ntroduced or our purposes and s sampled usng the probablty dstrbuton a uncton wth two varables (e.g.,, varable or each one o the orgnal varables (e.g.,,, ˆ, ˆ dmensonal PDF (e.g., y ˆ, ˆ. ˆ. I we are tryng to appromate y then we would most lkely need to have a stochastc y y, wth a correspondng two We wll say that we are lookng or a par, ˆ, ˆ that has the epected value o. Any such par can serve as the bass o a Monte Carlo algorthm, as long as the varance s nte at all values o at whch we need estmates o the uncton. otce, rst o all, that the complety o the tem delvered has ncreased: Equaton 4 delvers a scalar estmate o or each sample ˆ n, whereas Equaton 7 delvers a uncton o,, ˆ n, or each sample. otce also that, although there s only a smple notaton change between Equatons 4 and 7, we have moved where the known uncton s located: Equaton 4 has t nsde the ntegral, on the rght hand sde; Equaton 7 has t on the let hand sde. Ths strongly aects the work that we have to do to use t: Equaton 4 clearly shows us the par ˆ, ˆ to be used n the Monte Carlo algorthm, but Equaton 7 provdes no gudance about how an approprate par, ˆ, ˆ s to be ound, leavng us wth the task o ndng one. It smply assures s that, a par can be ound that satses Equaton 6, then that same par can be used as the bass o a Monte Carlo estmate o by averagng multple unctonal samples or all values o or whch the varance s nte. The trck to desgnng a Monte Carlo uncton appromaton usng Equaton 7 s to nd an approprate par to base t on. Med stochastc unctons usng the Drac delta Our rst stochastc appromaton s based on the Drac delta. In terms o our med stochastc, ˆ, we note that: uncton, 5-1 ˆ ˆ, ˆ ˆ (5-18

11 can easly be shown to satsy the rst equalty o Equaton 6,.e.,: E ˆ ˆ ˆ ˆ, ˆ dˆ ˆ ˆ dˆ as long as the probablty dstrbuton ˆ avods sngulartes by havng ˆ at all ponts or whch ˆ. Thereore, accordng to the rules we establshed earler, we can use t as the bass o a Monte Carlo algorthm. But, beore we do ths, let s take a look at what ths appromaton LOOKS lke. Eample. Fgure 4 shows the resultng appromaton o the uncton 1 e e, 1 ( sample uncton or 1 random values o ˆ chosen unormly rom to 1. As can be seen n the gure, the heghts o the Drac deltas ollow the shape o the curve because o the unorm dstrbuton n (although they are much smaller. 1.E+ 1.E-1 EXP(--EXP(-1 Zeroth Order Appromaton 1.E- 1.E- 1.E Fgure 4. Drac delta appromaton o sample uncton ((=1 So the answer to the queston What does t look lke? s Terrble. We are used to appromatons o unctons (e.g., least squares ts to look SOMETHIG lke the unctons they are appromatng! 5-11

12 Ths appromaton that cannot even satsy the smplest requrement o a uncton that t delvers a specc uncton value or any value o the argument wthn the doman. Because t contans the Drac delta, the stochastc uncton gven by Equaton 9 can only be practcally used to appromate nsde ntegrals. (As an asde, ths s the appromaton o that s mplct n most estng Monte Carlo algorthms, whch also serves to eplan why the unctons have to occur nsde ntegrals. Then why do we do t? Because n ts lmted applcaton (ntegraton, evaluatng the ntegral o the appromaton can be thousands or mllons or bllons o tmes aster than evaluatng the ntegral o the uncton tsel. (I ths s not true then the thousands or mllons or bllons o samples that t takes to get a good appromaton to the ntegral would not be worth t. In practce, the tradeo s OT worth t or easy ntegrals, but becomes practcal only or ether multdmensonal ntegratons or or ntegratons where the ntegral s especally dcult e.g., resonance cross secton reacton rates. Bottom Lne o Ths Secton The purpose o ths secton has been to add a mddle step to the Law o Large umbers that corresponds to the usual begnnng pont o other substtutons: Representng a unctonal appromaton that s then substtuted or the uncton tsel n ntegrals, equatons, etc. That s, gven the rst part o the Law o Large umbers: We can appromate d wth our (so-called med stochastc appromaton: n1 1 ˆ n lm ˆ ˆ and substtute the appromaton to get the second part o the Law: 1 ˆ n 1 lm d 1 ˆ n n n1 n n ˆ lm ˆ n n (5-19 (5- (5-1 But why do t ths way? Because t allows us two advantages: 1. It gves us a uncton substtuton methodology that we can use to develop Monte Carlo algorthms rom gven ntegrals, equatons, etc. 5-1

13 . By generalzng wth Equaton 6 (.e., beyond the tradtonal Drac delta appromaton we wll be able to develop hgher order Monte Carlo algorthms or appromatons. (Track length estmaton s one o these. OTE: From here on n the course, I wll generally stop wrtng the summatons over the samples I wll assume you now know what to do wth a sample and just wrte: ˆ n ˆ ˆ n n (5-5. Solvng ntegral equatons ow we apply what we have learned to an ntegral equaton. There s really nothng new here, but we have to learn to pay attenton to detal and subscrpts, etc., to develop an algorthm that can be taken by a programmer (probably you and mplemented n a computer code. I wll work through several eamples beore turnng you loose on a problem o your own. In our prevous sectons, we were gven a dened uncton, we could appromate the uncton usng: ˆ w, over a gven doman (a,b, (5- where w ˆ ˆ (5-4 Ths lessons concerns samplng rom a uncton when we do OT know the uncton eplctly (.e., or a gven, we DO OT know the value o, but we have an ntegral equaton that t satses, e.g., u du 4 (5-5 otce that the varable s n the lmts o the ntegral. Another way that the ntegral equaton mght present tsel s that the varable s n the ntegrand nstead o the lmts o the ntegral, e.g., u du (

14 The derence between ths stuaton and the one where we know the uncton s qute smple: We use the very same technque, but we substtute the RIGHT hand sde o the equaton nto the numerator nstead o the uncton tsel. For eample, usng the rst o the two equatons above, we could jump to an appromaton o the soluton o the equaton by ormally ollowng the steps rom beore, but wth substtuton: w ˆ w ˆ u du ˆ ˆ ˆ 4 (5-7 (5-8 (ote, o course, that snce the argument o s ˆ nstead o, then ˆ has to be used n the ntegral as well. O course, we are not through yet, because o the ntegral that has to be solved n the numerator. We could, o course, just ntegrate t to get: ˆ ˆ u du 4 4 (5-9 and substtute to get: w ˆ 4 ˆ (5- But that s not the purpose o ths eercse. (We use smple eamples, but make the assumpton that we have orgotten how to ntegrate! Instead, we wll peal the onon and proceed to use a Monte Carlo estmate o the numerator ntegral to create a lnked two-step algorthm. It wll use two varable samples, two weghts, two Drac deltas. So, n ths ven, we appromate the ntegrand usng: u wu u u w u uˆ uˆ ˆ (

15 ote that I have added another subscrpt to the w and varables to keep the two samples straght. Also, remember that to keep my subscrpts and superscrpts rom spreadng lke kudzu,, dstnqushng between them by ther I am gong to just denote PDFs usng the notaton arguments and trustng you to remember that each o them are derent PDFs. Substtutng ths nto the equaton or the orgnal weghts (and addng a subscrpt to the weght varable, takng advantage o how ncely Drac deltas dspose o ntegrals, we get: w ˆ ˆ u du ˆ ˆ uˆ ˆ 4 w ˆ u u u du 4 uˆ uˆ ˆ u uˆ du 4 ˆ uˆ 4, uˆ ˆ 4 ˆ, otherwse (5- Snce ths s so derent rom beore, let me make a ew comments: 1. There are TWO varable samples that have to be perormed. The nal ormula burcated nto two peces based on the way that the two samples relate to each other: uˆ ˆ or not.. It does not really matter whch order you make the two samples. You can pck ether o the varables rst. Despte the way that we pealed the onon when developng the algorthm, you can choose uˆ rst and then ˆ. I you do ths, then the burcaton s based on whether the newly chosen ˆ s greater than the prevously chosen u ˆ.. Although not apparent rom the way we dened the problem, we usually are nterested n a certan range o values. For eample, we are nterested n the value o the uncton that s non-zero only between and 4, then t would be more ecent to use a between those two values. 4. You can avod the burcaton by proper choce o the domans o the PDFs. In the eample rom above, once the ˆ s chosen, the uˆ choce can be restrcted to be between and the chosen ˆ. (Ths ollows the prevous allowance that you need not choose a varable rom a part o the doman that does not contrbute at all to the eect beng 5-15 ˆ

16 estmated; snce values o uˆ greater than ˆ do not contrbute to the ntegral, they do not have to be ncluded n the doman o u The nal algorthm s:. 1. Choose a value o ˆ randomly over the problem doman.. Choose a value u ˆ between and ˆ.. Determne uˆ and then ˆ w by evaluatng the weght ormulas 4. Do somethng useul (or nterestng wth the resultng appromaton o. The last step llustrates that the problem statement was ncomplete; t ddn t tell us what to DO wth the Monte Carlo appromaton to. Ths s mportant (or Drac appromatons because Drac deltas are useless unless put nto ntegrals so there must be one or more useul ntegrals you want to use t or, or what s the pont o appromatng t? [OTE: You only sample ( you want to ntegrate t. I you are just nterested n knowng ( at a KOW value o, then ths value takes the place o ˆ and there s no reason to do step 1.] Also, o course, the algorthm s ncomplete because t s a general algorthm. Beore actually codng t, the developer would have to decde what PDFs wll be used n the rst two steps. Developng ntegral equatons rom derental equatons Beore proceedng to solve the two orms o ntegral equatons that we have lad out, let s deal wth the act that we really don t encounter ntegral equatons as oten as derental equatons. But, we can oten convert a derental equaton nto an ntegral equaton. Ths s best shown wth a couple o eamples. Eample: Sample the derental equaton and boundary condton: d d e, ( 4 Answer: The basc technque s to ntegrate the derental equaton FROM the pont where the boundary condton s known TO the desred values o. So, we ntegrate the equaton rom to some value, we get: 5-16

17 d u du du u e du u e du u e du u 4 e du I know you are smart enough to perorm the ntegral and solve or, but that s not the pont o ths lesson. We are gong to use smple eamples and pretend that we cannot solve or the unctons eplctly, but must deal wth the ntegral equaton tsel. Let s now work wth a lttle more complcated derental equaton that results n a recurrng ntegral equaton. Eample: We want to sample a uncton that obeys the 1D slab Boltzmann equaton:,, or t, S, a, or Answer: For the postve, we can rearrange and ntegrate over rom to to get:, 1 t d S d, d 1 t, S d, d Lkewse, or negatve drectons, we can ntegrate rom to a to get:, 1 t d S d, d a a a 5-17

18 a 1 t, S d, d So, because o the way that the boundary condtons were presented, we ended up wth TWO ntegral equatons one or orward-pontng drectons and one or backward-pont drectons. a Procedure or samplng rom non-recurrng ntegral equatons Once we have a non-recurrng ntegral equaton, the development o the sample seres s a smple etenson o the method we used to perorm ntegraton: 1. Choose a value o ˆ randomly over the problem doman.. Choose values o all domans o ntegrals on the rght-hand sde o the equaton.. Determne w by evaluatng the rght-hand sde o the equaton usng the chosen ˆ and ntegral doman values. Ths s best shown by eample. Eample : Develop a samplng procedure or the equaton derved n Eample 1: u 4 e du, Answer: The steps to ollow are: 1. Choose a value o ˆ unormly over the doman (,. u. Choose a value o ˆ unormly over the range to ˆ.. Determne the weght, w, rom the equaton: w e ˆ ˆe uˆ 4 4 I you are nterested, I wrote a FORTRA code to check ths procedure. uˆ 5-18

19 5.4 Solvng lnked sets o ntegral equatons There s nothng partcularly new n ths module ecept or practce and growng condence n your ablty to work longer problems, where one Monte Carlo appromaton eeds nto another one. That s, you have a strng o lnked ntegral equatons (or can get them by ntegratng derental equatons: a( u u v t dt b( v g u a u du (5- you attack t n peces that cascade nto each other: tˆ ( t t tˆ w ˆ t t tˆ ˆ uˆ uˆ uˆ uˆ uˆ t dt a u a( u u uˆ u uˆ w ˆ t t dt uˆ uuˆ w,, ˆ ˆ t u a u uˆ w u uˆ (5-4 ˆ vˆ vˆ vˆ g( u a u du b v b( v v vˆ v vˆ ˆ g u w u u du w g u a vˆ ˆ, uˆ, vˆ vˆ 5-19 vˆ vvˆ v vˆ w v vˆ a b

20 where I have ntroduced the notaton-savng weght notaton and used the convenence uncton (nvented here or our use!: 1, abc a, b, c 1, c b a, Otherwse (5-5 A common stuaton that creates these cascades s that o hgher-order derental equatons, where the lnked equaton set comes rom reducng the nth order equaton to n rst-order equatons (wth the n boundary condtons dspersed among the resultng lnked equatons. Eample: Create a Monte Carlo algorthm to solve: Answer: We dene: y ; y 1, y ; y w u w Use o these unctons changes the orgnal equaton nto: u ; u v u ; v y v ; y 1 Integratng each o these rst-order derental equatons gves us: 5-

21 1 u d v y u d v d Usng our prevous methodology allows us to convert ths nto a Monte Carlo general algorthm: ˆ w ˆ ; w ˆ w, ˆ, ˆ u w ˆ ; w u u u ˆ u ˆ ˆ v ˆ ˆ v ˆ w,, ˆ v w ˆ ; w v v v 1wv,, y w ˆ ; w u u u v I could ( I wanted to substtute all these weghts to get the (nal w weght n terms o and all the PDFs, but I won t snce I usually code all the ntermedate weghts just lke the above equatons (to mnmze the lkely algebrac mstakes I would make combnng them. O course, lke we sad beore, ths algorthm leaves you wth y n terms o Drac deltas whch are only useul put nto ntegrals, so you must ultmately be nterested n some ntegral o y to make ths a useul eercse. But, I wll leave that part out o ths Eample. 5.5 eumann decomposton Samplng rom recurrng equatons ntroduces a complety. We cannot use the above procedure because, we try to ollow t we wll nd ourselves unable to sample the occurrences o the uncton on the rght-hand sde o the equaton. That s, the procedure requres that we sample rom (.e., on the let-hand sde. (.e., on the rght-hand sde n order to sample rom 5-1

22 However, all s not lost. For lnear occurrences o bootstrap a soluton by representng on the rght-hand sde, we can as an nnte eumann seres: 1, (5-6 substtutng ths seres or nto a nnte set o coupled equatons or the, and then samplng -- n turn -- the. TOGETHER the samples or the ndvdual. on BOTH sdes o the equaton, dvdng the resultng equaton would combne to orm a sngle sample or O course, ths procedure has an nnte number o steps or each sample o, so t wll have to be adjusted, but beore worryng about that let us rst look at an eample o how the procedure so ar would shape up. Eample: Develop an nnte samplng procedure or the recurrng equaton: d d, 1, Answer: Integratng the derental equaton over rom to (and applyng the boundary condton gves us the recurrng ntegral equaton: 1 u du, I we nsert the nnte eumann seres or the uncton on both sdes, we get: u du u du Ths can be decomposed nto the ollowng coupled equatons: 1 u du 1 5-

23 Snce the uncton sample rom 1. Sample rom by: n n 1 u du s the nnte sum o these n s:..., the procedure to. Choosng an ˆ between and. Ths nvolves usng the normalzed probablty dstrbuton 1/.. Our sample o s ound rom: ˆ w Wth w ˆ 1 ˆ 1/ 1 4. Sample rom 5. Choosng an ˆ 1 uˆ ˆ 6. Settng 7. Our sample o by: between and. Agan, the normalzed dstrbuton s 1/. usng the sample o 1 s ound rom: ˆ w ˆ 1 u du ˆ w w du ˆ ˆ ˆ 1 1/ ˆ ˆ =w u du w ˆ ˆ = Otherwse 1 8. Sample rom n n the analogous manner usng the sample o n 1 ˆ w w n n n n w ˆ ˆ Otherwse, n1, n1 n 5- :

24 ow that we have developed the "nnte procedure", let us make some observatons. 1. The IF TEST n step.c s necessary because the Drac delta sample or w ˆ les outsde the range o the ntegral, the ntegral -- and thereore our sample o 1 goes to zero because o the denton o Drac delta ntegraton.. The procedure s nnte n theory, but not nnte n practce because as soon as we pck a value o ˆn that s SMALLER than the one beore t, then wn wll go to zero. Once ths happens, o course, we can gnore the rest o the eumann steps because ther weghts wll be zero as well., etc., that consttutes. We must remember that t s not a sngle sample o our sample o the uncton sample o s, ormally: or, but ALL OF THEM together. Thereore, the 'th ˆ w n 4. Thereore, we mprove our appromaton o by takng samples, the combned best result would be: 1 wn n ˆ 1 n 5. As a practcal matter, pont means that our codng must collect data n "sample bns" --.e, whch collect data rom ndvdual eumann terms wthn a sngle sample -- and, at the end o the sample, contrbute rom the "sample bns" to the overall "soluton bns". n n Applcaton to vectors The same deas used or uncton appromaton can be used or vectors n a lnear algebra contet. The dea s that the Law o Large umbers s appled to each o the elements o a vector to gve us: 1 ˆ n 1 1 ˆ 1 M ˆ M n n lm lm n1 n1 ˆ ˆ n (

25 That s, a value o s selected rom some PDF ˆn î (probably constraned to pck ntegers rom 1 to M but ths s not requred; rom the chosen value o ˆn we deduce an ETIRE ˆ or each element. vector (the one on the rght hand sde usng separate unctons k n The Law o Large umber guarantees that ths can be the bass o a Monte Carlo process that gves us as long as EACH o the element unctons has the epected value o the element, that ˆn s, M E ˆ ˆ or each k k k m k m m1 The easest way to apply ths (and the only one I have ever seen n practce s to use: ˆ k k km m What ths means n practce s that each o the elements o the vector s gven a probablty o beng selected, and the resultng vector appromaton has n all postons EXCEPT the one k chosen (say element k, whch s gven the value. For notatonal smplcaton, I am gong to ntroduce a nde vector, k dened to be: k (5-8 (5-9 k 1 kth element (5-4 whch gves us: ˆ ˆ m m m m (5-41 Ths can easly be seen to satsy the LL equaton (two back, snce the element s chosen wth k probablty k and has the value whenever chosen, whch clearly has the epected value o k k. 5-5

26 What s not so clear s why ths would be o any value to use, snce vector operatons are generally not epensve enough to meet the general Monte Carlo crteron that you would rather deal wth the appromaton MILLIOS o tmes rather than deal wth the orgnal OCE. Lnear algebra generally sn t that tough. I our eperence s any ndcaton, the vector Monte Carlo s only used because a vector representaton (multgroup s present nsde a code that uses Monte Carlo or other reasons (spatal and drectonal complety, n our case, so we need t or compatblty. Anyway. How s t used? For the most part t s used to solve equatons o the usual lnear algebra sort: A b (5-4 where (or real problems at least, the elements o the matr and the vectors have some sort o dependence (space, drecton, etc.. We wll gnore such complcatons and just deal wth constant-value matrces and vectors. One way to teratvely solve or the vector n the above relatonshp (whch s stable under certan condtons that must be met but agan, we wll gnore ths or now s to decompose the matr A nto a dagonal and non-dagonal peces: A D (5-4 gvng us: D b (5-44 Ths can be made nto the teratve algorthm: ( ( D b D (5-45 whch s solved by creatng an ntal guess (hopeully convergence s reached. ( and then gong around and around untl Ths, o course, has nothng to do wth Monte Carlo untl we ntroduce vector appromatons. I we appromate on the rght-hand sde: ( m m w m m (

27 and plug ths nto the relaton, we get, or the net teraton: ( D b D w 1 1 D b wd m m (5-47 Although ths may look a lttle complcated wrtten ths way, t s actually smple to use. The 1 1 D b term s a vector that can be easly precalculated. The D can smlarly be precalculated, though t s a matr, not a vector. But post-multplyng a matr by m smply returns the mth column o the matr, whch we multply by a constant weght, w. O course, although ths s smple, t stll results n a vector or the new estmate, so we proceed by appromatng THIS by choosng one o ts elements usng some dscrete probablty dstrbuton, resultng n: w 1 ( 1 m 1 m m m (5-48 A derence between our use o the teratve procedure and the usual teratve use s that the Monte Carlo soluton s not lookng or a convergence to a vector VALUE, but s nstead nterested n a convergent PROCESS. That s, we just go around and around, each round gvng us an estmate o the soluton vector. And (as we have seen beore, snce we start the process wth a GUESS, t s prudent to wat awhle beore startng to keep score. A very nterestng twst o the vector process comes rom the problem o duson wthout absorpton (the LaPlace equaton, whch results n the so-called Drunken Salor problem. In ths process you: 1. Set up a grd o ponts (D n our eample. Begn a hstory at the pont whose value you want to know.. Let the partcle wander rom pont to pont wth the decson about whch way to walk beng ¼ n each o the 4 choces (ICLUDIG the drecton you just arrved rom. 4. When you reach a boundary pont, the SCORE o the hstory s the value o the boundary condton at that pont. 5. Return to step or the net hstory. 5-7

28 Chapter 5 Eercses 5-1. Desgn and code a Monte Carlo estmate o the ntegral: I e d usng the rejecton method. 5-. Desgn and code a Monte Carlo estmate o the ntegral: I e d usng the averagng method. 5-. Desgn and code a Monte Carlo estmate o the ntegral: I e d usng the control varates method. Use a guess uncton o h Desgn and code a Monte Carlo estmate o the ntegral:. I e d usng the mportance samplng method. Use a guess uncton o h Desgn, code, and run a Monte Carlo estmate o the ntegral (usng averagng method or: I ( y z d dy dz a. pseudorandom numbers b. Halton sequence usng bases,, and 5 or the three dmensons. 5-8

29 Present a plot o error (not standard devaton o the mean vs Wrte and run a code to sample the derental equaton and boundary condton: d, ( 1 Use your code to nd: a. The value o the uncton at =4; and b. The ntegral o the uncton rom to Solve or y d, gven the equaton: 1 d y ; y, y 5-8. Assume that at t=, you have a sngle atom o speces A. A decays to B wth a hal-le o 1 second and B decays to C wth a hal-le o.5 seconds. (C s stable. Estmate the epected number o atoms o C at t= seconds? Hnt: In case you are rusty wth the equatons o decay, the ntegral equatons you should start wth are: A( t e t t A B tt B( t A t e dt t C( t B t dt A B 5-9. Use the drunken salor algorthm to solve a duson problem. On a 5 cm 5 cm grd wth boundary condtons 1 (let edge, (bottom, (rght, and 4 (top, use Monte Carlo (on a.5 cm grd to estmate the value o the uncton at the pont (.5,

30 Answers to selected eercses Chapter a a b (Standard devaton depends on PDF used