University of Oxford Efficient white noise sampling and coupling for multilevel Monte Carlo M. Croci (Oxford), M. B. Giles (Oxford), P. E. Farrell (Oxford), M. E. Rognes (Simula) MCQMC2018 - July 4, 2018 Mathematical Institute EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Overview University of Oxford Introduction White noise sampling Mathematical Institute Numerical results Conclusions and further work
Overview University of Oxford Introduction White noise sampling Mathematical Institute Numerical results Conclusions and further work
Motivation University of Oxford The motivation of our research is the sampling of lognormal Gaussian fields. A Mat´ ern Gaussian field (approximately) satisfies a linear elliptic SPDE of the form Lu = ˙ x ∈ D , ω ∈ Ω W , + BCs , ˙ where u = u ( x , ω ) and W is spatial white noise . Other approaches can be used (with pros and cons), but we will not discuss them here. Mathematical Institute EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Motivation University of Oxford The motivation of our research is the sampling of lognormal Gaussian fields. A Mat´ ern Gaussian field (approximately) satisfies a linear elliptic SPDE of the form Lu = ˙ x ∈ D , ω ∈ Ω W , + BCs , ˙ where u = u ( x , ω ) and W is spatial white noise . Other approaches can be used (with pros and cons), but we will not discuss them here. The same techniques can be used to solve a more general class of SPDEs, e.g. N ( u ) + Lu = ˙ W , x ∈ D , ω ∈ Ω + BCs . Mathematical Institute In this case solving means to compute E [ P ( u )] for some functional P of the solution. EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Motivation University of Oxford The motivation of our research is the sampling of lognormal Gaussian fields. A Mat´ ern Gaussian field (approximately) satisfies a linear elliptic SPDE of the form Lu = ˙ x ∈ D , ω ∈ Ω W , + BCs , ˙ where u = u ( x , ω ) and W is spatial white noise . Other approaches can be used (with pros and cons), but we will not discuss them here. The same techniques can be used to solve a more general class of SPDEs, e.g. N ( u ) + Lu = ˙ W , x ∈ D , ω ∈ Ω + BCs . Mathematical Institute In this case solving means to compute E [ P ( u )] for some functional P of the solution. Common applications: finance, geology, meteorology, biology . . . EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Motivation University of Oxford The motivation of our research is the sampling of lognormal Gaussian fields. A Mat´ ern Gaussian field (approximately) satisfies a linear elliptic SPDE of the form Lu = ˙ x ∈ D , ω ∈ Ω W , + BCs , ˙ where u = u ( x , ω ) and W is spatial white noise . Other approaches can be used (with pros and cons), but we will not discuss them here. The same techniques can be used to solve a more general class of SPDEs, e.g. N ( u ) + Lu = ˙ W , x ∈ D , ω ∈ Ω + BCs . Mathematical Institute In this case solving means to compute E [ P ( u )] for some functional P of the solution. Common applications: finance, geology, meteorology, biology . . . ˙ Main issue: sampling W is hard! EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Motivation University of Oxford The motivation of our research is the sampling of lognormal Gaussian fields. A Mat´ ern Gaussian field (approximately) satisfies a linear elliptic SPDE of the form Lu = ˙ x ∈ D , ω ∈ Ω W , + BCs , ˙ where u = u ( x , ω ) and W is spatial white noise . Other approaches can be used (with pros and cons), but we will not discuss them here. The same techniques can be used to solve a more general class of SPDEs, e.g. N ( u ) + Lu = ˙ W , x ∈ D , ω ∈ Ω + BCs . Mathematical Institute In this case solving means to compute E [ P ( u )] for some functional P of the solution. Common applications: finance, geology, meteorology, biology . . . ˙ Main issue: sampling W is hard! ˙ The efficient sampling of W is the focus of this talk. EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
White noise (1D) University of Oxford 150 100 50 0 -50 Mathematical Institute -100 -150 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 WARNING! Point evaluation not defined! EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
White noise (2D) University of Oxford Mathematical Institute WARNING! Point evaluation not defined! EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
White noise (2D) University of Oxford Mathematical Institute WARNING! Point evaluation not defined! ˙ IDEA! Avoid point evaluation by integrating W . EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
White Noise (practical definition) University of Oxford Mathematical Institute ˙ Definition (Spatial White Noise W ) For any φ ∈ L 2 ( D ), define � ˙ D ˙ W φ d x . For any φ i , φ j ∈ L 2 ( D ), b i = � ˙ � W , φ � := W , φ i � , b j = � ˙ W , φ j � are zero-mean Gaussian random variables, with, � E [ b i b j ] = φ i φ j dx =: M ij , b ∼ N (0 , M ) . (1) D EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
White Noise (practical definition) University of Oxford Mathematical Institute IMPORTANT NOTE: Generalised random fields of type I and of type II. EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Finite element (FEM) framework University of Oxford When solving SPDEs (see 1 st slide) with FEM, we get (for linear problems ) Discrete weak form : find u h ∈ V h s.t. for all v h ∈ V h , a ( u h , v h ) = � ˙ W , v h � , (2) Where V h = span( { φ i } n i =0 ), (e.g. with Lagrange elements). Mathematical Institute EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Finite element (FEM) framework University of Oxford When solving SPDEs (see 1 st slide) with FEM, we get (for linear problems ) Discrete weak form : find u h ∈ V h s.t. for all v h ∈ V h , a ( u h , v h ) = � ˙ W , v h � , (2) Where V h = span( { φ i } n i =0 ), (e.g. with Lagrange elements). i u i φ i , u = [ u 0 , . . . , u n ] T , FEM linear system : u h = � Mathematical Institute A u = b ( ω ) , (3) where the entries of b are given by, � ˙ W , φ i � ( ω ) = b i ( ω ) , (4) with b ∼ N (0 , M ) as before. M is the mass matrix of V h . EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Multilevel Monte Carlo [Giles 2008] + FEM framework University of Oxford For MLMC, we have two approximation levels ℓ and ℓ − 1. For any particular ω ∈ Ω, we h , u ℓ − 1 ∈ V ℓ − 1 h , v ℓ − 1 ∈ V ℓ − 1 need to solve: find u ℓ h ∈ V ℓ s.t. for all v ℓ h ∈ V ℓ , h h h h h ) = � ˙ a ( u ℓ h , v ℓ W , v ℓ h � ( ω ) , (5) a ( u ℓ − 1 , v ℓ − 1 ) = � ˙ W , v ℓ − 1 � ( ω ) . (6) h h h Mathematical Institute EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Multilevel Monte Carlo [Giles 2008] + FEM framework University of Oxford For MLMC, we have two approximation levels ℓ and ℓ − 1. For any particular ω ∈ Ω, we h , u ℓ − 1 ∈ V ℓ − 1 h , v ℓ − 1 ∈ V ℓ − 1 need to solve: find u ℓ h ∈ V ℓ s.t. for all v ℓ h ∈ V ℓ , h h h h h ) = � ˙ a ( u ℓ h , v ℓ W , v ℓ h � ( ω ) , (5) a ( u ℓ − 1 , v ℓ − 1 ) = � ˙ W , v ℓ − 1 � ( ω ) . (6) h h h This yields the linear system � � � � � � A ℓ u ℓ b ℓ 0 = = b , A ℓ − 1 u ℓ − 1 b ℓ − 1 0 Mathematical Institute EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
Multilevel Monte Carlo [Giles 2008] + FEM framework University of Oxford For MLMC, we have two approximation levels ℓ and ℓ − 1. For any particular ω ∈ Ω, we h , u ℓ − 1 ∈ V ℓ − 1 h , v ℓ − 1 ∈ V ℓ − 1 need to solve: find u ℓ h ∈ V ℓ s.t. for all v ℓ h ∈ V ℓ , h h h h h ) = � ˙ a ( u ℓ h , v ℓ W , v ℓ h � ( ω ) , (5) a ( u ℓ − 1 , v ℓ − 1 ) = � ˙ W , v ℓ − 1 � ( ω ) . (6) h h h This yields the linear system � � � � � � A ℓ u ℓ b ℓ 0 = = b , A ℓ − 1 u ℓ − 1 b ℓ − 1 0 Mathematical Institute i } n ℓ } n ℓ − 1 i =0 ) and V ℓ − 1 = span( { φ ℓ − 1 where b ∼ N (0 , M ). Let V ℓ h = span( { φ ℓ i =0 ), then h i � � M ℓ M ℓ,ℓ − 1 � M ℓ,ℓ − 1 i φ ℓ − 1 φ ℓ M = , = dx . ( M ℓ,ℓ − 1 ) T M ℓ − 1 ij j EPSRC Centre for Doctoral Training in Industrially Focused Mathematical Modelling
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