IGA-based Multi-Index Stochastic Collocation for Uncertainty Quantification J. Beck 1 , L. Tamellini 2 , R. Tempone 1 , 3 1 King Abdullah University of Science and Technology, Saudi Arabia 2 CNR-IMATI, Italy 3 Alexander von Humboldt Professor in Mathematics of Uncertainty Quantification, RWTH Aachen University, Germany DCSE Fall School on ROM and UQ, November 4–8, 2019 1 / 28
Outline Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions 2 / 28
Outline Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions 3 / 28
The uncertainty quantification problem → PDE solution u ( x ) PDE solution u ( x , y ) ◮ The parameter y = [ y 1 , y 2 , y 3 . . . ] is a set of model parameters, e.g., in the equation, external forces, geometry, initial and boundary conditions, ◮ The values of y are uncertain (experimental measures, limited knowledge on system properties). ◮ y can be modeled as a random vector and with N components ⇒ u is a random function , u ( y ). Goal (forward UQ): Compute statistical quantities for u , e.g., to assess how the uncertainty on y reflects on u : E [ u ], V ar [ u ], Pr ( u > u 0 ). 4 / 28
Working example: elliptic PDE with a random coefficient Mathematical model: � − div[ a ( x ) ∇ u ( x )] = f ( x ) x ∈ B , u ( x ) = 0 x ∈ ∂ B , Quantity of Interest: u at some ˆ x 5 / 28
Working example: elliptic PDE with a random coefficient Parameters y: N independent uniform rand. var on [ − 1 , 1] describe randomness in diffusion coefficient (more later) Mathematical model: � − div[ a ( x , y ) ∇ u ( x , y )] = f ( x ) x ∈ B , u ( x , y ) = 0 x ∈ ∂ B , Quantity of Interest: E y [ u (ˆ x , y )], i.e., the expected value of u at ˆ x ∈ B 5 / 28
Working example: elliptic PDE with a random coefficient Parameters y: N independent uniform rand. var on [ − 1 , 1] describe randomness in diffusion coefficient (more later) Mathematical model: � − div[ a ( x , y ) ∇ u ( x , y )] = f ( x ) x ∈ B , u ( x , y ) = 0 x ∈ ∂ B , Quantity of Interest: E y [ u (ˆ x , y )], i.e., the expected value of u at ˆ x ∈ B Challenge: N can be VERY large 5 / 28
Solve PDE using IGA Say we want to compute the heat diffusion in a body shaped as shown Need to define the mesh : Solve PDE using IGA on a grid with discretization level α = ( α 1 , α 2 , . . . , α d ), i.e., having 2 α 1 × 2 α 2 × · · · 2 α d elements. Here d = 3. 6 / 28
Random input data Let us assume the body is not homogeneous due to imperfections , which are not known precisely, and so we describe thermal conductivity as random fluctuations around some nominal value. Below are three possible realizations of such random conductivity: Here, y enters in the description of the thermal conductivity as: a ([ ρ, θ, z ] , y ) = 5+ y 1 sin (2 θ ) sin ( π ( ρ − 1)) sin ( π z ) + 0 . 8 y 2 sin (4 θ ) sin ( π ( ρ − 1)) sin ( π z ) + 0 . 7 y 3 sin (6 θ ) sin ( π ( ρ − 1)) sin ( π z ) + . . . 0 . 2 y 8 sin (16 θ ) sin ( π ( ρ − 1)) sin ( π z ) + . . . The more parameters included, the more complex fluctuations can be modeled. 7 / 28
Naive method to compute E [ u (ˆ x , · )] - Full tensorization E [ · ] is an integral, and if u varies smoothly wrt to y ( u ( y ) is an analytic func.) → use Gauss quadrature with M 1 × M 2 × . . . × M N points ( “stochastic collocation” ) M 1 × M 2 × ... � E [ u (ˆ x , · )] ≈ ω j u α (ˆ x , y j ) j =1 ◮ Exploits an interpolant of u ( y ) on the collocation points { y j } 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 collocation points y j mesh for the PDE Expensive for large N : Cost ∼ O ( � N j =1 M j ) 8 / 28
Outline Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions 9 / 28
Idea: use successive corrections PDE solver accuracy F ◮ C , coarse discretization P ◮ Q , refine stochastic collocation only ◮ P , refine PDE mesh only ◮ F , fine discretization in C Q both ← our goal Stochastic Collocation 10 / 28
Idea: use successive corrections Write a telescopic equality: F = F PDE solver accuracy + C − C F P + C − C + P − P + Q − Q C Q Stochastic Collocation 11 / 28
Idea: use successive corrections Rearrange it: F = C PDE solver accuracy + P − C F P + Q − C + F − P − Q + C C Q Stochastic Collocation 11 / 28
Idea: use successive corrections Interpret as a sum of corrections F = C PDE solver accuracy + ∆ PDE F P + Q − C + F − P − Q + C C Q Stochastic Collocation 11 / 28
Idea: use successive corrections Interpret as a sum of corrections F = C PDE solver accuracy + ∆ PDE F P + ∆ stoc + F − P − Q + C C Q Stochastic Collocation 11 / 28
Idea: use successive corrections Interpret as a sum of corrections F = C PDE solver accuracy + ∆ PDE F P + ∆ stoc + ∆ mixed C Q Stochastic Collocation 11 / 28
Idea: use successive corrections i.e., a hierarchical decomp. of F : F = C PDE solver accuracy + ∆ PDE F P + ∆ stoc + ∆ mixed C Q Stochastic Collocation 11 / 28
Idea: use successive corrections Strategy: drop ∆ mixed F ≈ C PDE solver accuracy + ∆ PDE + ∆ stoc + ✘✘✘ ∆ mixed Sparsification components Rationale: mixed correction are expensive to compute (they in- volve F ) but small if u regular! Stochastic Collocation Formula: F ≈ P + Q − C Big computational saving, small loss in accuracy 11 / 28
One step further: Multi-Index Stochastic Collocation For PDE in R d , d > 1 with N > 1 parameters: specify discretization for each spatial direction and parameter independently ! → Hence, the name “Multi-Index” Instead of the fine discretization... 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 ... we combine linearly many “small-ish” discretizations over space and parameters. 12 / 28
One step further: Multi-Index Stochastic Collocation Fix e.g., d = 3 (in space), N = 2 (in probability) Full-tensor approximation of E [ u ]: G β 1 ,β 2 ,β 3 , M 1 , M 2 G β,β,β, M , M = Fine approx. sol. with a mesh with 2 β × 2 β × 2 β elements and M × M points in probability space G α,α,α, m , m = Coarse approx. sol. with a mesh with 2 α × 2 α × 2 α elements and m × m points in probability space 13 / 28
One step further: Multi-Index Stochastic Collocation G β,β,β, M , M = G α,α,α, m , m + correction along x 1 + correction along x 2 + correction along x 3 + correction along y 1 + correction along y 2 + all mixed correction . . . 14 / 28
One step further: Multi-Index Stochastic Collocation G β,β,β, M , M = G α,α,α, m , m + G β,α,α, m , m − G α,α,α, m , m + correction along x 2 + correction along x 3 + correction along y 1 + correction along y 2 + all mixed correction . . . 14 / 28
One step further: Multi-Index Stochastic Collocation G β,β,β, M , M = G α,α,α, m , m + G β,α,α, m , m − G α,α,α, m , m + G α,β,α, m , m − G α,α,α, m , m + correction along x 3 + correction along y 1 + correction along y 2 + all mixed correction . . . 14 / 28
One step further: Multi-Index Stochastic Collocation G β,β,β, M , M = G α,α,α, m , m + G β,α,α, m , m − G α,α,α, m , m + G α,β,α, m , m − G α,α,α, m , m + G α,α,β, m , m − G α,α,α, m , m + correction along y 1 + correction along y 2 + all mixed correction . . . 14 / 28
One step further: Multi-Index Stochastic Collocation G β,β,β, M , M = G α,α,α, m , m + G β,α,α, m , m − G α,α,α,, m , m + G α,β,α, m , m − G α,α,α, m , m + G α,α,β, m , m − G α,α,α, m , m + G α,α,α, M , m − G α,α,α, m , m + correction along y 2 + all mixed correction . . . 14 / 28
One step further: Multi-Index Stochastic Collocation G β,β,β, M , M = G α,α,α, m , m + G β,α,α, m , m − G α,α,α, m , m + G α,β,α, m , m − G α,α,α, m , m + G α,α,β, m , m − G α,α,α, m , m + G α,α,α, M , m − G α,α,α, m , m + G α,α,α, m , M − G α,α,α, m , m + all mixed correction . . . 14 / 28
One step further: Multi-Index Stochastic Collocation G β,β,β, M , M = G α,α,α, m , m + ∆ [1 0 0 0 0] G + ∆ [0 1 0 0 0] G + ∆ [0 0 1 0 0] G + ∆ [0 0 0 1 0] G + ∆ [0 0 0 0 1] G + all mixed correction (here drop those of “low profit”) . . . Mixed corrections, e.g., ∆ [1 1 0 0 0] G or ∆ [0 1 1 0 1] G 14 / 28
One step further: Multi-Index Stochastic Collocation ◮ Evaluate either form with corrections, or final linear combination (“combination technique”), e.g., G β,β,β, M , M ≈ G β,α,α, m , m + G α,β,α, m , m + G β,α,α, M , m + · · · i.e., a linear combination of several “small” approximations ◮ Anisotropic meshes in space are also used ◮ Multi-Index Monte Carlo, Multi-Index Stochastic Collocation etc ◮ When only isotropic meshes are considered (i.e. α 1 = α 2 = . . . ), the methods are called “Multilevel” (Multilevel Monte Carlo,, Multilevel Stochastic Collocation ...) 14 / 28
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