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) − κ2(Γ(y))u = 0 in Din(y) ∪ Dout,Rout(y), uΓ(y) = 0, α(Γ(y))∇u · nΓ(y) = 0, ∂ ∂nout (u − ui) = DtN(u) − DtN(ui)

• n ∂DRout,

for every y ∈ PJ, J ∈ N, Quantity of interest: u = {u(xi)}N−1

i=0 ,

xi close to stochastic interface. Goal: Computing E[u].

r(y; ϕ) Rout Din(y) Dout,Rout (y) Γ(y) ∂DRout ui u(x0)

Model transmission problem

               − ∇ · (α(Γ(y))∇u) − κ2(Γ(y))u = 0 in Din(y) ∪ Dout,Rout(y), uΓ(y) = 0, α(Γ(y))∇u · nΓ(y) = 0, ∂ ∂nout (u − ui) = DtN(u) − DtN(ui)

• n ∂DRout,

for every y ∈ PJ, J ∈ N, Quantity of interest: u = {u(xi)}N−1

i=0 ,

xi close to stochastic interface. Goal: Computing E[u].

r(y; ϕ) Rout Din(y) Dout,Rout (y) Γ(y) ∂DRout ui u(x0)

Non-smooth parameter dependence

• 1. u(x0) not smooth across

PΓ

J (x0) := {y ∈ PJ : x0 ∈ Γ(y)} .

• 2. Discontinuities hard to locate.

⇒ multilevel Monte Carlo. To select the sequence (Ml)L−1

l=0 :

• J-independent space regularity for point evaluation
• J-independent finite element convergence of point evaluation.