Monte Carlo Methods. Monte Carlo methods

Transcription

1 ρ θ σ µ Monte Carlo Methos What is a Monte Carlo Metho? Rano walks The Metropolis rule iportance sapling Near neighbor sapling Sapling prior an posterior probability Exaple: gravity inversion The ovie philosophy This lecture follows ainly: Moesgaar an Tarantola J. Geophys. Res. 100 B an Moesgaar an Tarantola Probabilitstic approach to inverse probles in International Hanbook Earthquake an Engineering Seisology 001 Acaeic Press.

2 ρ θ σ µ Monte Carlo Methos are nae after city in Monaco principality because roulettes in Casinos rano nuber generator

3 ρ θ σ µ Monte Carlo Methos Early applications to eterination Earth s structure Press 1986.

4 ρ θ σ µ Monte Carlo Methos The goal: How can we efficiently saple a posteriori pf knowing that coputation forwar proble is expensive an coputational powers is finite!

5 ρ θ σ µ Monte Carlo Methos More technically: Given a set points in space with a probability p i attache to every point i how can we efine rano rules to select points such that probability selecting points is p i?

6 ρ θ σ µ Monte Carlo Methos Let us use peaks function Matlab to illustrate Monte Carlo techniques. The following two-iensional function is suppose to be an a posteriori probability ensity function. Each point this graph woul require at least calculation a possibly very expensive forwar proble an a coparison with ata isfit.

7 ρ θ σ µ Monte Carlo Methos Note that for ulti-iensional probles a point in oel space ay represent a coplex Earth oel. By sapling a posteriori pf we collect oels hopefully goo quality sall isfit between syntics an real ata. Eventually our eye ay ecie which oels are realistic. To avoi sapling areas low probability we introuce concept iportance sapling. What is a rano walk? We successively visit points in oel space where next point X i+1 to be visite epens on point x i. How can we choose points so that we saple pf?

8 ρ θ σ µ Rano walks The ost coon Monte Carlo sapling ethos is Metropolis sapler: At a given point rano walker is at at point x i an now we have to efine rules how to get to anor point x j. If we accept any rano ove walker woul saple no oubt at soe point whole space. Instea always accepting transition we reject ove soeties: Let fx be probability ensity function: if fx j fx i -> accept ove if fx j < fx i -> n ecie ranoly to ove to x j with probability accepting ove: P = f f x x j i P is transition probability

9 k µ θ ρ σ = Rano walks - Metropolis Rano walks - Metropolis We obtain following results for our test probability function:

10 ρ θ σ µ Rano walks - Metropolis

11 ρ θ σ µ Rano walks - Metropolis The ten thousan points visite see to represent well areas where

12 ρ θ σ µ Rano walks - Metropolis an function is now well represente.. but still at a high cost only slightly less expensive than crue Monte Carlo approach These results were obtaine with Matlab progra c_etro.

13 ρ θ σ µ Rano walks - Metropolis First oifications: Liit algorith to look in neighborhoo present point. This is calle near neighbor sapling. Here we allowe walker to ove only within 10% total size oel Space. The progra use was c_neigh. an relevant Paraeter neigh=0.1.

14 ρ θ σ µ Rano walks - Metropolis Near neighbor sapling. The progra use was c_neigh. an relevant Paraeter neigh=0..

15 ρ θ σ µ Rano walks - Convergence There There is is a crucial crucial question: question: How How any any points points o o we we have have to to visit visit until until we we have have a goo goo iea iea about about solution solution to to our our proble? proble? This This is is an an extreely extreely ifficult ifficult question question to to answer answer because: because: -- behavior behavior a rano rano walk walk algorith algorith ay ay see see straight straight forwar forwar in in D D or or 3D 3D but but ay ay behave behave very very ifferently ifferently in in systes systes with with any any iensions iensions -- reason reason why why you you use use Monte Monte Carlo Carlo Methos Methos is is because because you you on t on t know know what what function function looks looks like. like. So So how how can can you you be be sure sure you you are are sapling sapling fine fine enough enough to to get get goo goo areas areas oel oel space? space?.. soe soe se se probles probles lea lea us us to to iprove iprove techniques techniques such such as as siulate siulate annealing annealing

16 ρ θ σ µ Monte Carlo etho: gravity Inversion Inversion gravity gravity ata: ata: a a classical classical test test for for all all ories ories inversion! inversion! The The proble: proble: Fin Fin epth-epenent epth-epenent ensity ensity structure structure to to right right vertical vertical fault. fault. The The observations observations are are horizontal horizontal graients graients gravity gravity to to right right fault. fault.

17 k µ θ ρ σ = Monte Carlo etho: gravity Monte Carlo etho: gravity The forwar proble: The gravity graient at surface is given by: The forwar proble: The gravity graient at surface is given by: + = = 0 x z z z z G x x g x ρ

18 ρ θ σ µ Monte Carlo sapling prior inforation Let Let us us now now walk walk through through this this inverse inverse proble proble an an ake ake use use Monte Carlo Carlo ieas: ieas: For For any any particular particular forwar forwar proble proble first first step stepwoul be be Sapling Sapling a priori priori probability probability -- we we nee nee a pseuorano pseuorano process process our our Monte Monte Carlo Carlo approach approach to to fin fin saples saples prior prior inforation inforation -- We We know know a priori priori pf pf analytically analytically Exaple: Fro observations in a well or in any any wells we have foun that in stratifie eia istribution layer thickness is approxiately an exponential istribution an ensities have log noral istributions. We saple this prior inforation by: - Select a layer uniforly at rano - Choose a new value for layer thickness accoring to exponential istribution - Choose a value for ass ensity accoring to log-noral istribution

19 k µ θ ρ σ = Monte Carlo sapling prior inforation Monte Carlo sapling prior inforation Let us look at outcoe this process: prior probabilites Let us look at outcoe this process: prior probabilites = 0 0 exp 1 l l l l f Layer tickness k. l 0 =4k 0 log 1 exp 1 ρ ρ σ ρ ρ g Density g/c 3. ρ 0 =3.98g/c 3 σ=.58

20 ρ θ σ µ Monte Carlo sapling prior inforation a a rano rano walk walkthrough prior prior probabilities probabilities prouce prouce oels oels that that look look like like this: this:

21 ρ θ σ µ Monte Carlo sapling prior inforation we we o o not not expect expect that that fine fine layering layering is is well well resolve resolve which which is is why why it it akes akes sense sense to to look look a soo soo oels oels

22 ρ θ σ µ Gravity: experiental uncertainties The The easue easue ata ata are are assue assue to to be be containate containate by by rano rano uncorrelate uncorrelate noise. noise. To To ake ake it it a little little ore ore coplicate coplicate errors errors are are assue assue to to coe coe fro fro two two processes processes with with ifference ifference variances variances σ i an i an relative relative probabilities probabilities expresse expresse through through a: a: f ε a exp πσ ε + 1 a exp ε = σ 1 1 πσ σ

23 k µ θ ρ σ = Gravity: posterior rano walk Gravity: posterior rano walk To calculate forwar proble we nee to su over all layers: To calculate forwar proble we nee to su over all layers: + + = nl i j i j i i j x x D x log ρ this is horizontal gravity graient as a function layers with D being botto epth layer i ensity cotnrast across fault an x horizontal istanc e gravieter j. The likelihoo function L can now be calculate accoring to: + = i obs i i obs i i g a g a k L 1 1 exp 1 exp σ πσ σ πσ With out ouble errors solution woul have been = i i obs i i g k L σ exp

24 ρ θ σ µ Gravity: posterior rano walk The The rano rano walk walk through through a posteriori posteriori probability probabilityleas leas to to oels: oels:

25 ρ θ σ µ Gravity: posterior probability Note: Note: These These oels oels are are saples saples a a posteriori posteriori probability probability ensity ensity function. function. They They represent represent state state inforation inforation we we have have on on our our Earth Earth oel. oel. With With se se saples saples we we can can now now ask ask questions questions like: like: What What is is value value for for ass ass ensity ensity at at epth epth z=k z=k or or z=0k z=0k an an how how well well is is it it constraine? constraine? We We only only have have to to calculate calculate arginal arginal probabilities probabilities to to answer answer this this questions. questions. Note: Note: At At epth epth k k we we see see to to have have clearly clearly gaine gaine inforation. inforation.

26 ρ θ σ µ Gravity: posterior probability How How has has isfit isfit our our oels oels iprove iprove copare copare to to a a priori priori oels? oels? The The isfit isfit is is alost alost perfect perfect for for all all our our a a posteriori posteriori oels oels but but again again we we hit hit on on particular particular gravity gravity proble proble that that any any very very ifferent ifferent oels oels explain explain ata! ata!

27 ρ θ σ µ Gravity: posterior probability What What are are ean ean values values an an stanar stanar eviations eviations ensity ensity as as a a function function epth? epth? Here Here we we clearly clearly see see that that we we gave gave gaine gaine inforation inforation in in top top 0 0 k k!!

28 ρ θ σ µ Suary can be applie to saple a possibly high-iensional oel space efining prior an posterior probability ensity functions a physical inverse proble. The sapling a posteriori probability sees to be optial way escribing state inforation in a particular physical syste. The key to a successful Monte Carlo algorith is to efficiently walk through oel space an calculate least possible nuber oels while proviing a representative saple a posteriori probability function.