Multilevel and Multi-index Monte Carlo methods for the McKean-Vlasov equation Abdul-Lateef Haji-Ali ∗ ul Tempone † Ra´ ∗ Heriot-Watt University, United Kingdom † Alexander von Humboldt Professor, RWTH Aachen, Germany, KAUST, Saudi Arabia. DCSE Fall School on ROM and UQ November, 2019
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. 1/0
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. T. Statistics and Computing, Vol. 28(4), pp. 923–935, 2018. A McKean-Vlasov process is a stochastic process described by a SDE whose coefficients 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 )) dt + b ( X ( t ) , µ ( t )) dW ( t ) , µ ( t ) = L ( X ( t )) and µ (0) given. Goal: approximate E [ g ( X ( T ))] for some given g . 2/0
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 ∞ size) � t P � a ( X p | P ( t )) + 1 X p | P ( t ) = x 0 d t p + A ( X p | P ( t ) , X q | P ( t )) P 0 q =1 + σ W p ( t ) � t � � � X p |∞ ( t ) = x 0 p + a ( X p |∞ ( t )) + A ( X p |∞ ( t ) , y ) µ ∞ ( t )( d y ) d t 0 + σ 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) . 3/0
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 ρ ∞ ( t , x )+ div ( ρ ∞ ( t , x )( a ( x ) + ρ ∞ ( t , · ) ∗ A ( x , · ))) � σ 2 2 ∂ 2 i = i ρ ∞ ( t , x ) i with a given initial condition ρ ∞ (0 , · ) and suitable b.c. Question: What about the rate of weak convergence? � � g ( X p | P ( T )) − g ( X p |∞ ( T )) � O (1 / P ) E 4/0
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 ϑ p that follow a system of Itˆ o SDEs � � P � ϑ p + 1 d X p | P ( t ) = sin( X p | P ( t ) − X q | P ( t )) d t + σ d W p ( t ) P q =1 X p | P (0) = x 0 p where we are interested in � � 2 � � 2 P 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. 5/0
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 ϑ p that follow a system of Itˆ o SDEs � � P � ϑ p + 1 d X p | P ( t ) = sin( X p | P ( t ) − X q | P ( t )) d t + σ d W p ( t ) P q =1 X p | P (0) = x 0 p P � � � φ P = 1 where we are interested in: cos X p | P ( T ) , P p =1 � � Mean-field limit: φ P → φ ∞ = E cos( X p |∞ ( T ) ) as P ↑ ∞ � � � d X p |∞ = ϑ p + sin( X p |∞ ( t ) − y ) µ ∞ ( t , d y ) d t + σ d W p ( t ) R † Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. 5/0
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 ϑ p that follow a system of Itˆ o SDEs � � � P ϑ p + 1 T X n | N p | P − X n − 1 | N sin( X n | N p | P − X n | N N + σ ∆ W n | N = q | P ) p | P p P q =1 X 0 | N p | P = x 0 p � � P � P = 1 X N | N φ N where we are interested in: cos , p | P P p =1 � � Mean-field limit: φ P → φ ∞ = E cos( X p |∞ ( T ) ) as P ↑ ∞ � � � d X p |∞ = ϑ p + sin( X p |∞ ( t ) − y ) µ ∞ ( t , d y ) d t + σ d W p ( t ) R † Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence, Springer, Berlin, 1984. 5/0
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Objective Our objective is to build an estimator A ≈ φ ∞ with minimal work where P ( |A − φ ∞ | ≤ TOL ) ≥ 1 − ǫ for a given accuracy TOL and a given confidence level determined by 0 < ǫ ≪ 1. 6/0
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Objective Our objective is to build an estimator A ≈ φ ∞ with minimal work where P ( |A − φ ∞ | ≤ TOL ) ≥ 1 − ǫ for a given accuracy TOL and a given confidence level determined by 0 < ǫ ≪ 1. We instead impose the following, more restrictive, two constraints: Bias constraint: | E [ A ] − φ ∞ | ≤ TOL / 3 , Statistical constraint: P ( |A − E [ A ] | ≥ 2 TOL / 3) 6/0
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Simple Example Objective Our objective is to build an estimator A ≈ φ ∞ with minimal work where P ( |A − φ ∞ | ≤ TOL ) ≥ 1 − ǫ for a given accuracy TOL and a given confidence level determined by 0 < ǫ ≪ 1. We instead impose the following, more restrictive, two constraints: Bias constraint: | E [ A ] − φ ∞ | ≤ TOL / 3 , Var [ A ] ≤ (2 TOL / 3 C ǫ ) 2 . Variance constraint: assuming (at least asymptotic a ) normality of the estimator, A . Here, 0 < C ǫ is such that Φ( C ǫ ) = 1 − ǫ 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). 6/0
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Monte Carlo (MC) Monte Carlo The simplest estimator is the Monte Carlo estimator M � A MC = 1 φ 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 ω m P = p =1 and for p each particle, ω 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 . 7/0
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: � � �� � ≤ TOL � � φ N Bias( A MC ) = � φ ∞ − E such that: P 3 � � � 2 TOL � 2 φ N Var [ A MC ] = Var P and: ≤ 3 C ǫ M 8/0
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 ǫ 8/0
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 ǫ 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 8/0
Monte Carlo Methods for McKean-Vlasov [Haji-Ali, Tempone] Multilevel Monte Carlo (MLMC) – Introduction Following (Heinrich, 2001) and (Giles, 2008), For a given L ∈ N , define two hierarchies { N ℓ } L ℓ =1 and { P ℓ } L ℓ =1 satisfying P ℓ − 1 ≤ P ℓ and N ℓ − 1 ≤ N ℓ for all ℓ . Recall the telescopic decomposition � � � � � � L � P ℓ − ϕ N ℓ − 1 φ N L φ N 0 φ N ℓ φ ∞ ≈ E = E + E P L P 0 P ℓ − 1 ℓ =1 9/0
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