MLMC for transmission problems with geometric uncertainties Laura - PowerPoint PPT Presentation

MLMC for transmission problems with geometric uncertainties Laura Scarabosio QUIET 2017 SISSA, Trieste, July 18-21.

Model transmission problem  − ∇ · ( α (Γ( y )) ∇ u ) − κ 2 (Γ( y )) u = 0 in D in ( y ) ∪ D out,R out ( y ) ,     � u � Γ( y ) = 0 , � α (Γ( y )) ∇ u · n � Γ( y ) = 0 ,    ∂ ∂ n out ( u − u i ) = DtN( u ) − DtN( u i ) on ∂D R out ,       for every y ∈ P J , J ∈ N ,  ∂D Rout Quantity of interest : D out,Rout ( y ) u ( x 0 ) u = { u ( x i ) } N − 1 i =0 , r ( y ; ϕ ) x i close to stochastic interface. u i R out 0 D in ( y ) Goal : Γ( y ) Computing E [ u ] .

Model transmission problem  − ∇ · ( α (Γ( y )) ∇ u ) − κ 2 (Γ( y )) u = 0 in D in ( y ) ∪ D out,R out ( y ) ,     � u � Γ( y ) = 0 , � α (Γ( y )) ∇ u · n � Γ( y ) = 0 ,    ∂ ∂ n out ( u − u i ) = DtN( u ) − DtN( u i ) on ∂D R out ,       for every y ∈ P J , J ∈ N ,  ∂D Rout Quantity of interest : D out,Rout ( y ) u ( x 0 ) u = { u ( x i ) } N − 1 i =0 , r ( y ; ϕ ) x i close to stochastic interface. u i R out 0 D in ( y ) Goal : Γ( y ) Computing E [ u ] .

Non-smooth parameter dependence 1. u ( x 0 ) not smooth across P Γ J ( x 0 ) := { y ∈ P J : x 0 ∈ Γ( y ) } . 2. Discontinuities hard to locate. ⇒ multilevel Monte Carlo. To select the sequence ( M l ) L − 1 l =0 : • J -independent space regularity for point evaluation • J -independent finite element convergence of point evaluation.