Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation ul Tempone † Abdul-Lateef Haji-Ali ∗ Ra´ ∗ Mathematical Institute, University of Oxford, United Kingdom † King Abdullah University of Science and Technology (KAUST), Saudi Arabia. CEMRACS, Luminy, France July 21, 2017
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Motivation Particle Systems in the Mean-field Particle systems are a collection of coupled, usually identical and simple, models that can be used to model complicated phenomena. Molecular dynamics, Crowd simulation, Oscillators, . . . Certain stochastic particles systems have a mean-field limit when the number of particles increase. Such limits can be useful to understand their complicated phenomena. 1/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Motivation Main reference This presentation is based on the manuscript • ”Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation” by A. L. Haji Ali and R. Tempone. arXiv:1610.09934 , 2016. To appear in Statistics and Computing. A McKean-Vlasov process is a stochastic process described by a SDE whose coe ffi cients depend on the distribution of the solution itself. They relate to the Vlasov model for plasma evolution and were first studied by Henry McKean in 1966. For 0 < t T the process X ( t ) solves dX ( t ) = a ( X ( t ) , µ ( t )) dW ( t ) + b ( X ( t ) , µ ( t )) dt , µ ( t ) = L ( X ( t )) and µ (0) given. Goal: approximate E [ g ( X ( T ))] for some given g . 2/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Motivation Convergence to the mean-field For particles X p | P , p = 1 , . . . , P , (evolving in a system of size P ) define ”shadow” particles X p | ∞ , (evolving in a system of 1 size) 0 1 Z t P X @ a ( X p | P ( t )) + 1 X p | P ( t ) = x 0 A d t p + A ( X p | P ( t ) , X q | P ( t )) P 0 q =1 + � W p ( t ) Z t ✓ Z ◆ X p | ∞ ( t ) = x 0 p + a ( X p | ∞ ( t )) + A ( X p | ∞ ( t ) , y ) µ ∞ ( t )( d y ) d t 0 + � W p ( t ) , with µ ∞ ( t ) the marginal distribution for X p | ∞ ( t ) for any p . Consistency: The initial values x 0 p , p = 1 , . . . , P , are i.i.d. from µ ∞ (0) . 3/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Motivation For t > 0 , and all x , the pdf of the marginal distribution µ ∞ ( t ) of the infinite size system satisfies a nonlinear Fokker Planck equation @ t ⇢ ∞ ( t , x )+ div ( ⇢ ∞ ( t , x )( a ( x ) + ⇢ ∞ ( t , · ) ⇤ A ( x , · ))) X � 2 2 @ 2 i = i ⇢ ∞ ( t , x ) i with a given initial condition ⇢ ∞ (0 , · ) and suitable b.c. Question: What about the rate of weak convergence? ⇥ ⇤ E g ( X p | P ( T )) � g ( X p | ∞ ( T )) . . . . 4/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Kuramoto oscillator model † For p = 1 , 2 , . . . , P consider equally coupled oscillators with intrinsic natural frequencies # p that follow a system of Itˆ o SDEs ! P X # p + 1 d X p | P ( t ) = sin( X p | P ( t ) � X q | P ( t )) d t + � d W p | P ( t ) P q =1 X p | P (0) = x 0 p | P where we are interested in ! 2 ! 2 X P X P � � � � 1 1 Total order = cos X p | P ( T ) + sin X p | P ( T ) ; P P p =1 p =1 a real number between zero and one that measures the level of synchronization of the coupled oscillators. † Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. 5/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Kuramoto oscillator model † For p = 1 , 2 , . . . , P consider equally coupled oscillators with intrinsic natural frequencies # p that follow a system of Itˆ o SDEs ! P X # p + 1 d X p | P ( t ) = sin( X p | P ( t ) � X q | P ( t )) d t + � d W p | P ( t ) P q =1 X p | P (0) = x 0 p | P X P � � � P = 1 where we are interested in: cos X p | P ( T ) , P p =1 ⇥ ⇤ Mean-field limit: � P ! � ∞ = E cos( X p | ∞ ( T ) ) as P " 1 ✓ Z ◆ d X p | ∞ = # p + sin( X p | ∞ ( t ) � y ) µ ∞ ( t , d y ) d t + � d W p | P ( t ) R † Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. 5/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Kuramoto oscillator model † , Euler-Maruyama For p = 1 , 2 , . . . , P consider equally coupled oscillators with intrinsic natural frequencies # p that follow a system of Itˆ o SDEs ! X P # p + 1 T X n | N p | P � X n − 1 | N sin( X n | N p | P � X n | N N + � ∆ W n | N = q | P ) p | P p | P P q =1 X 0 | N p | P = x 0 p | P P ⇣ ⌘ X P = 1 X N | N � N where we are interested in: cos , p | P P p =1 ⇥ ⇤ Mean-field limit: � P ! � ∞ = E cos( X p | ∞ ( T ) ) as P " 1 ✓ ◆ Z d X p | ∞ = # p + sin( X p | ∞ ( t ) � y ) µ ∞ ( t , d y ) d t + � d W p | P ( t ) R † Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. 5/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Objective Our objective is to build an estimator A ⇡ � ∞ with minimal work where P ( |A � � ∞ | TOL ) � 1 � ✏ for a given accuracy TOL and a given confidence level determined by 0 < ✏ ⌧ 1. 6/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Objective Our objective is to build an estimator A ⇡ � ∞ with minimal work where P ( |A � � ∞ | TOL ) � 1 � ✏ for a given accuracy TOL and a given confidence level determined by 0 < ✏ ⌧ 1. We instead impose the following, more restrictive, two constraints: Bias constraint: | E [ A ] � � ∞ | TOL / 3 , Statistical constraint: P ( |A � E [ A ] | � 2 TOL / 3) 6/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Objective Our objective is to build an estimator A ⇡ � ∞ with minimal work where P ( |A � � ∞ | TOL ) � 1 � ✏ for a given accuracy TOL and a given confidence level determined by 0 < ✏ ⌧ 1. We instead impose the following, more restrictive, two constraints: Bias constraint: | E [ A ] � � ∞ | TOL / 3 , Var [ A ] (2 TOL / 3 C ✏ ) 2 . Variance constraint: assuming (at least asymptotic a ) normality of the estimator, A . Here, 0 < C ✏ is such that Φ ( C ✏ ) = 1 � ✏ 2 , where Φ is the c.d.f. of a standard normal random variable. a N. Collier, A.-L. Haji-Ali, E. von Schwerin, F. Nobile, and R. Tempone. “A continuation multilevel Monte Carlo algorithm”. BIT Numerical Mathematics, 55(2):399-432, (2015). 6/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Monte Carlo (MC) Monte Carlo The simplest (and most popular) estimator is the Monte Carlo estimator M X A MC = 1 � N P ( ω m P ) . M m =1 For a given P , N and M that we can choose to satisfy the error � � P constraints and minimize the work. Here ω m ! m P = p =1 and for p each particle, ! m p denotes the independent, identically distributed (i.i.d.) samples of the set of underlying random variables that are used in calculating X N | N p | P , 1 p P . 7/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Monte Carlo (MC) Monte Carlo work complexity In our 1D example, we can check (at least numerically) that Work( A MC ) , Minimize total work: � i� h � TOL � � � N Bias( A MC ) = such that: � � ∞ � E P 3 ⇥ ⇤ ✓ 2 TOL ◆ 2 � N Var [ A MC ] = Var P and: 3 C ✏ M 8/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Monte Carlo (MC) Monte Carlo work complexity In our 1D example, we can check (at least numerically) that � MNP 2 � Minimize total work: Work( A MC ) = O � N − 1 � � P − 1 � TOL Bias( A MC ) = O + O such that: 3 � P − 1 � ✓ 2 TOL ◆ 2 Var [ A MC ] = O and: M 3 C ✏ 8/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Monte Carlo (MC) Monte Carlo work complexity In our 1D example, we can check (at least numerically) that � MNP 2 � Minimize total work: Work( A MC ) = O � N − 1 � � P − 1 � TOL Bias( A MC ) = O + O such that: 3 � P − 1 � ✓ 2 TOL ◆ 2 Var [ A MC ] = O and: M 3 C ✏ In this case, we choose � TOL − 1 � � TOL − 1 � � TOL − 1 � P = O , N = O , M = O � TOL − 4 � and the total cost of a naive Monte Carlo is O . Observe: The cost of a “single cloud” naive method with M = 1 � TOL − 5 � is O 8/34
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Multilevel Monte Carlo (MLMC) – Introduction Following (Heinrich, 2001) and (Giles, 2008), For a given L 2 N , define two hierarchies { N ` } L ` =1 and { P ` } L ` =1 satisfying P ` − 1 P ` and N ` − 1 N ` for all ` . Recall the telescopic decomposition h i h i L h i X P ` � ' N ` − 1 � N L � N 0 � N ` = E + � ∞ ⇡ E E P L P 0 P ` − 1 ` =1 9/34
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