Branching for PDEs Xavier Warin CEMRACS July Xavier Warin - PowerPoint PPT Presentation

Branching for PDEs Xavier Warin CEMRACS July  Xavier Warin Branching for PDEs CEMRACS July  1 / 94

The StOpt Library Table of contents 1 The StOpt Library 2 Branching for KPP equations 3 Generalization of KPP equations 4 General polynomial driver in u [2] 5 General driver f ( u ) [3] 6 Recall on Malliavin weights 7 Unbiased simulation of SDE for linear PDE [7] [8] 8 Semi linear equations Xavier Warin Branching for PDEs CEMRACS July  2 / 94 Re-normalization of ghost method [9], [10] 9

The StOpt Library A C++ toolbox with python interface Regression methods for conditional expectations : Local with Linear, Constant per mesh approximation , Local adaptive to the distribution with Linear, Constant per mesh approximation , Global polynomial (Hermite, Canonical, Tchebychev), Sparse grids, Interpolation methods (linear, Monotone Legendre, sparse grids ) Xavier Warin Branching for PDEs CEMRACS July  3 / 94

The StOpt Library Provide a framework to solve complex optimization problems General HJB equations with deterministic Semi Lagrangian methods, Non linear Stochastic Optimization problems with stocks : Regressions with Monte Carlo for non controlled processes, Stochastic Monte Carlo quantization for controlled process, Some Linear problems with stocks in high dimension : Stochastic Dual Dynamic Programming Method. Parallelization : Message Passing (MPI), Multi-threaded, Using vectorized matrix/array library Eigen (INRIA): Xavier Warin Branching for PDEs CEMRACS July  4 / 94

The StOpt Library An open source library Developed during the ANR Caesars. Gitlab site : https://gitlab.com/stochastic-control/StOpt , Documentation : https://hal.archives-ouvertes.fr/hal-01361291 Python installer (Windows, Linux available at Labo FiME web site : https://www.fime-lab.org/ Try it and avoid to redevelop (most of the time less efficiently) even if branching not currently available. Xavier Warin Branching for PDEs CEMRACS July  5 / 94

Table of contents The StOpt Library 1 Branching for KPP equations 2 Generalization of KPP equations 3 General polynomial driver in u [2] 4 General driver f ( u ) [3] 5 Recall on Malliavin weights 6 Unbiased simulation of SDE for linear PDE [7] [8] 7 Semi linear equations 8 Re-normalization of ghost method [9], [10] 9 10 The full non linear case Xavier Warin Branching for PDEs CEMRACS July  6 / 94

Branching for KPP equations Table of contents 1 The StOpt Library 2 Branching for KPP equations 3 Generalization of KPP equations 4 General polynomial driver in u [2] 5 General driver f ( u ) [3] 6 Recall on Malliavin weights 7 Unbiased simulation of SDE for linear PDE [7] [8] 8 Semi linear equations Xavier Warin Branching for PDEs CEMRACS July  7 / 94 Re-normalization of ghost method [9], [10] 9

Branching for KPP equations The KPP equation McKean’s formulation [1] Equation to solve in R d : ∂ t u + L u + β f ( u ) = 0 , L u = µ · Du + 1 2 σσ ⊤ : D 2 u u ( T , . ) = g with µ ∈ R d , σ ∈ M d , the non linear term : f ( u ) = u 2 − u and notation A : B := Trace ( AB ⊤ ) . Xavier Warin Branching for PDEs CEMRACS July  8 / 94

Branching for KPP equations Using Ito.. Consider the process for W t a d dimensional Brownian motion : dX 0 , x = µ dt + σ dW t t X 0 , x = x 0 Supposing regularity of the solution : �� T � � ) e − β T � e − β s ( ∂ t u + L u − β u ) ( s , X 0 , x u ( T , X 0 , x = u ( 0 , x ) + E ) ds E s T 0 �� T � � ) e − β T � � ) 2 � g ( X 0 , x β e − β s u ( s , X 0 , x u ( 0 , x ) = E + E ds s T 0 if τ a R.V. following an exponential law with parameter β : E [ 1 τ> T ] = e − β T = 1 − cdf ( τ ) , ρ τ ( s ) = β e − β s Xavier Warin Branching for PDEs CEMRACS July  9 / 94

Branching for KPP equations Introducing a Poisson process τ ( 1 ) with intensity β Considering the integral as an expectation � τ ( 1 ) ) 2 � g ( X 0 , x T ) 1 τ ( 1 ) > T + 1 τ ( 1 ) < T u ( τ ( 1 ) , X 0 , x u ( 0 , x ) = E 0 , x � � ψ ( τ ( 1 ) , X 0 , x = E 0 , x τ ( 1 ) ) (1) where ψ ( t , x ) = g ( x ) 1 t > T + 1 t < T u ( t , x ) 2 Introduce the Poisson processes τ ( 1 , 1 ) , τ ( 1 , 2 ) (2 particles) by independence � � � � u ( t , x ) 2 = E t , x ψ ( t + τ ( 1 , 1 ) , X t , x ψ ( t + τ ( 1 , 2 ) , X t , x E t , x t + τ ( 1 , 1 ) t + τ ( 1 , 2 ) � � ψ ( t + τ ( 1 , 1 ) , X t , x t + τ ( 1 , 1 ) ) ψ ( t + τ ( 1 , 2 ) , X t , x = E t , x t + τ ( 1 , 2 ) ) (2) Xavier Warin Branching for PDEs CEMRACS July  10 / 94

Branching for KPP equations By recursion Plugging (2) in (1), introducing T ( 1 ) = T ∧ τ ( 1 ) , T ( 1 , j ) = T ∧ ( T ( 1 ) + τ ( 1 , j ) ) , j = 1 , 2 � 1 T ( 1 ) = T g ( X 0 , x u ( 0 , x ) = E 0 , x T ( 1 ) ) + 1 T ( 1 ) < T  � T ( 1 , j ) ) 2 � 2 � 1 T ( 1 , j ) = T g ( X 0 , x T ( 1 , j ) ) + 1 T ( 1 , j ) < T u ( T ( 1 , j ) , X 0 , x  j = 1 Recursion till all particles arrive at date T . Xavier Warin Branching for PDEs CEMRACS July  11 / 94

Branching for KPP equations Kpp tree At date T ( 1 ) ( 1 ) generates ( 1 , 1 ) and ( 1 , 2 ) , At date T ( 1 , 1 ) , ( 1 , 1 ) generates ( 1 , 1 , 1 ) and ( 1 , 1 , 2 ) , At date T ( 1 , 1 , 1 ) , ( 1 , 1 , 1 ) generates ( 1 , 1 , 1 , 1 ) and ( 1 , 1 , 1 , 2 ) At date T ( 1 , 2 ) , ( 1 , 2 ) generates ( 1 , 2 , 1 ) and ( 1 , 2 , 2 ) , At date T ( 1 , 2 , 2 ) , ( 1 , 2 , 2 ) generates ( 1 , 2 , 2 , 1 ) and Figure: Galton-Watson tree for KPP ( 1 , 2 , 2 , 2 ) , Xavier Warin Branching for PDEs CEMRACS July  12 / 94

Branching for KPP equations Notations k = ( k 1 , k 2 , ..., k n − 1 , k n ) , k i ∈ { 1 , 2 } particle of generation n k − = ( k 1 , k 2 , ..., k n − 1 ) its ancestor , ( 1 ) − = ∅ K n t set of all living particles of generation n at date t . K t := ∪ n ≥ 1 K n t set of all living particles at date t . n K t (resp. K t ) set of all particles (resp. of generation n ) alive before time t τ k Poisson process associated to particle k = ( k 1 , ..., k n ) , Branching times per particle k � T k − + τ k � T k := ∧ T , T ∅ = 0 Xavier Warin Branching for PDEs CEMRACS July  13 / 94

Branching for KPP equations Kpp tree ( 1 , 1 , 1 , 2 ) ancestor : ( 1 , 1 , 1 , 2 ) − = ( 1 , 1 , 1 ) , K 3 T = { ( 1 , 2 , 1 ) , ( 1 , 1 , 2 ) } K 4 T = { ( 1 , 1 , 1 , 1 ) , ( 1 , 1 , 1 , 2 ) , ( 1 , 2 , 2 , 1 ) , ( 1 , 2 , 2 , 2 ) } Figure: Galton-Watson tree for KPP K T the 10 particles. Xavier Warin Branching for PDEs CEMRACS July  14 / 94

Branching for KPP equations PDE representation d -dimensional Brownian motion ( W k t ) k ∈K T , for k ∈ K T \ K T , dynamic for T k − ≤ t < T k X k = X k − T k − + µ ( t − T k − ) + σ W k t t − T k − Sample estimator: � g ( X k ˆ u ( 0 , x ) = T ) k ∈K T The number of particles in K T is finite a.s if || g || ∞ < 1, u ∈ C 1 , 2 ([ 0 , T ] × R d ) : u ( 0 , x ) ∈ L 1 ∩ L 2 , ˆ � � ˆ u ( 0 , x ) = E u ( 0 , x ) Xavier Warin Branching for PDEs CEMRACS July  15 / 94

Generalization of KPP equations Table of contents 1 The StOpt Library 2 Branching for KPP equations 3 Generalization of KPP equations 4 General polynomial driver in u [2] 5 General driver f ( u ) [3] 6 Recall on Malliavin weights 7 Unbiased simulation of SDE for linear PDE [7] [8] 8 Semi linear equations Xavier Warin Branching for PDEs CEMRACS July  16 / 94 Re-normalization of ghost method [9], [10] 9

Generalization of KPP equations First extension of KPP Non linear PDE ∂ t u + L u + β f ( u ) = 0 , u ( T , . ) = g with N � p k u k − u f ( u ) = i = 0 N � p k = 1 , 0 ≤ p k ≤ 1 i = 0 Xavier Warin Branching for PDEs CEMRACS July  17 / 94

Generalization of KPP equations Feynman Kac : Supposing regularity of the solution : �� T � � � � T ) e − β ( T − t ) � N � g ( X x β e − β ( s − t ) p i u ( s , X x s ) i u ( t , x ) = E + E ds t i = 1 Introduce for particle k , ( I k ) k ∈K T \K T random such that P ( I k = l ) = p l . � τ ( 1 ) ) I ( k ) � g ( X ( 1 ) T ) 1 τ ( 1 ) > T + 1 τ ( 1 ) < T u ( τ ( 1 ) , X ( 1 ) u ( 0 , x ) = E 0 , x same estimator � g ( X k u ( 0 , x ) = E [ T )] k ∈K T Xavier Warin Branching for PDEs CEMRACS July  18 / 94

Generalization of KPP equations Tree generalization : f ( u ) = p 0 + p 1 u + p 2 u 2 + p 3 u 3 ( 1 ) , ( 1 , 3 ) generate 3 particles (probability p 3 ), ( 1 , 1 ) dies without children (prob p 0 ), ( 1 , 2 ) generates one son (prob p 1 ), Figure: Galton-Watson tree ( 1 , 2 , 1 ) , and ( 1 , 3 , 3 ) generates 2 sons for generalized KPP (prob p 2 ) Xavier Warin Branching for PDEs CEMRACS July  19 / 94

General polynomial driver in u [2] Table of contents 1 The StOpt Library 2 Branching for KPP equations 3 Generalization of KPP equations 4 General polynomial driver in u [2] 5 General driver f ( u ) [3] 6 Recall on Malliavin weights 7 Unbiased simulation of SDE for linear PDE [7] [8] 8 Semi linear equations Xavier Warin Branching for PDEs CEMRACS July  20 / 94 Re-normalization of ghost method [9], [10] 9