IGA-based Multi-Index Stochastic Collocation for Uncertainty - - PowerPoint PPT Presentation

IGA-based Multi-Index Stochastic Collocation for Uncertainty Quantification

• J. Beck1, L. Tamellini2, R. Tempone1,3

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

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Outline

Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions

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Outline

Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions

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The uncertainty quantification problem

PDE solution u(x) → PDE solution u(x, y)

◮ The parameter y = [y1, y2, y3 . . .] 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

• n 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], Var[u], Pr(u > u0).

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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

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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: Ey[u(ˆ x, y)], i.e., the expected value of u at ˆ x ∈ B

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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: Ey[u(ˆ x, y)], i.e., the expected value of u at ˆ x ∈ B Challenge: N can be VERY large

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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 = (α1, α2, . . . , αd), i.e., having 2α1 × 2α2 × · · · 2αd elements. Here d = 3.

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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+ y1 sin (2θ) sin (π(ρ − 1)) sin (πz) + 0.8y2 sin (4θ) sin (π(ρ − 1)) sin (πz) + 0.7y3 sin (6θ) sin (π(ρ − 1)) sin (πz) + . . . 0.2y8 sin (16θ) sin (π(ρ − 1)) sin (πz) + . . .

The more parameters included, the more complex fluctuations can be modeled.

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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 M1 × M2 × . . . × MN points (“stochastic collocation”) E[u(ˆ x, ·)] ≈

M1×M2×...

• j=1

ωjuα(ˆ x, yj)

◮ Exploits an interpolant of u(y) on the collocation points {yj}

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

mesh for the PDE collocation points yj Expensive for large N: Cost ∼ O(N

j=1 Mj)

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Outline

Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation

F P Q C

◮ C, coarse discretization ◮ Q, refine stochastic

collocation only

◮ P, refine PDE mesh only ◮ F, fine discretization in

both ← our goal

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation

F P Q C

Write a telescopic equality: F =F + C − C + C − C + P − P + Q − Q

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation

F P Q C

Rearrange it: F =C + P − C + Q − C + F − P − Q + C

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation

F P Q C

Interpret as a sum of corrections F =C + ∆PDE + Q − C + F − P − Q + C

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation

F P Q C

Interpret as a sum of corrections F =C + ∆PDE + ∆stoc + F − P − Q + C

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation

F P Q C

Interpret as a sum of corrections F =C + ∆PDE + ∆stoc + ∆mixed

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation

F P Q C

i.e., a hierarchical decomp. of F: F =C + ∆PDE + ∆stoc + ∆mixed

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Idea: use successive corrections

PDE solver accuracy Stochastic Collocation Sparsification components

Strategy: drop ∆mixed F ≈C + ∆PDE + ∆stoc +✘✘✘ ∆mixed Rationale: mixed correction are expensive to compute (they in- volve F) but small if u regular! Formula: F ≈ P + Q − C Big computational saving, small loss in accuracy

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One step further: Multi-Index Stochastic Collocation

For PDE in Rd, 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 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

... we combine linearly many “small-ish” discretizations over space and parameters.

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One step further: Multi-Index Stochastic Collocation

Fix e.g., d = 3 (in space), N = 2 (in probability) Full-tensor approximation of E[u]: Gβ1,β2,β3,M1,M2 Gβ,β,β,M,M = Fine approx. sol. with a mesh with 2β × 2β × 2β elements and M × M points in probability space Gα,α,α,m,m = Coarse approx. sol. with a mesh with 2α × 2α × 2α elements and m × m points in probability space

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One step further: Multi-Index Stochastic Collocation

Gβ,β,β,M,M =Gα,α,α,m,m + correction along x1 + correction along x2 + correction along x3 + correction along y1 + correction along y2 + all mixed correction . . .

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One step further: Multi-Index Stochastic Collocation

Gβ,β,β,M,M =Gα,α,α,m,m + Gβ,α,α,m,m − Gα,α,α,m,m + correction along x2 + correction along x3 + correction along y1 + correction along y2 + all mixed correction . . .

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One step further: Multi-Index Stochastic Collocation

Gβ,β,β,M,M =Gα,α,α,m,m + Gβ,α,α,m,m − Gα,α,α,m,m + Gα,β,α,m,m − Gα,α,α,m,m + correction along x3 + correction along y1 + correction along y2 + all mixed correction . . .

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One step further: Multi-Index Stochastic Collocation

Gβ,β,β,M,M =Gα,α,α,m,m + Gβ,α,α,m,m − Gα,α,α,m,m + Gα,β,α,m,m − Gα,α,α,m,m + Gα,α,β,m,m − Gα,α,α,m,m + correction along y1 + correction along y2 + all mixed correction . . .

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One step further: Multi-Index Stochastic Collocation

Gβ,β,β,M,M =Gα,α,α,m,m + Gβ,α,α,m,m − Gα,α,α,,m,m + Gα,β,α,m,m − Gα,α,α,m,m + Gα,α,β,m,m − Gα,α,α,m,m + Gα,α,α,M,m − Gα,α,α,m,m + correction along y2 + all mixed correction . . .

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One step further: Multi-Index Stochastic Collocation

Gβ,β,β,M,M =Gα,α,α,m,m + Gβ,α,α,m,m − Gα,α,α,m,m + Gα,β,α,m,m − Gα,α,α,m,m + Gα,α,β,m,m − Gα,α,α,m,m + Gα,α,α,M,m − Gα,α,α,m,m + Gα,α,α,m,M − Gα,α,α,m,m + all mixed correction . . .

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One step further: Multi-Index Stochastic Collocation

Gβ,β,β,M,M =Gα,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

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One step further: Multi-Index Stochastic Collocation

◮ Evaluate either form with corrections, or final linear combination

(“combination technique”), e.g., Gβ,β,β,M,M ≈ Gβ,α,α,m,m + Gα,β,α,m,m + Gβ,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. α1 = α2 = . . .), the

methods are called “Multilevel” (Multilevel Monte Carlo,, Multilevel Stochastic Collocation ...)

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Sparse grids using the combination technique

Use a linear combination of coarser approximations (i.e., extract most of the information from these)

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

;

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Sparse grids using the combination technique

Use a linear combination of coarser approximations (i.e., extract most of the information from these)

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

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Sparse grids using the combination technique

Use a linear combination of coarser approximations (i.e., extract most of the information from these)

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

A collocation approach (over several sparse grids) → Independent solves, code reuse, parallel implementation!

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MISC estimator

3) Define MISC estimator, AMISC, as E[u] ≈ AMISC =

• (

α, β)∈I

c

α, βG α, β,

for some index set I ∈ Nd+N.

◮ The choice of multi-index set I is critical ◮ Avoid refine all directions simultaneously

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A-priori algorithm to build I (PDE-informed)

For the elliptic PDE we introduced earlier, these estimates typically hold error( α, β) ∼

N

• j=1

exp(−gj2βj )

• u is y-analytic, 2βj pts per dir.

×

d

• i=1

exp(−riαi)

• = hri

i ,ri =solver rate

work( α, β) ∼

N

• j=1

2βj

C.C. points are nested

×

d

• i=1

exp(γiαi)

• = hγi

i

,solver scales algebraically in nDoF

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A-priori algorithm to build I (PDE-informed)

For the elliptic PDE we introduced earlier, these estimates typically hold error( α, β) ∼

N

• j=1

exp(−gj2βj )

• u is y-analytic, 2βj pts per dir.

×

d

• i=1

exp(−riαi)

• = hri

i ,ri =solver rate

work( α, β) ∼

N

• j=1

2βj

C.C. points are nested

×

d

• i=1

exp(γiαi)

• = hγi

i

,solver scales algebraically in nDoF

Because of the double exponent in β the profit decay is faster wrt y than x; then the set I =

• [

α β] ∈ Nd+N

+

: error( α, β) work( α, β) > ε

• and it automatically balances errors wrt

x and y (cf. motivations!)

correction in probability correction in space

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Convergence result for MISC

Given the estimates above (Haji-Ali, Nobile, Tamellini, Tempone, 2016): If N < ∞ Err ≤ C work−s1 log(work)s2 with s1, s2 > 0 depending on the worst spatial direction and independent of N (but C does); here s1 = mini=1,...,d

ri γi .

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

19 / 28

A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

19 / 28

A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

19 / 28

A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

19 / 28

A posteriori algorithm to build index set

Algorithm (Gerstner & Griebel, 2003):

Given [ α∗ β∗] = [1 1], I = {[1 1]}, repeat

• 1. Add to the MISC estimator the neighbors
• f [

α∗ β∗] (feasible wrt to I)

• 2. Assess (a posteriori) work and error

contribution of each multi-idx just added

• 3. set as new [

α∗ β∗] the idx with the highest profit Profit = error work

Multi-index set

• β

2 4 6 8 1 2 3 4 5 6 7 8 I reduced margin

• curr. idx
• α

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Outline

Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions

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Test I - diffusion equation with random diffusion coeff.

• −div[a(x, y)∇u(x, y)] = 1

x ∈ B, u(x, y) = 0 x ∈ ∂B,

• yi ∼ uniform(−1, 1)
• Quantity of Interest (QoI):

E[

• Ω u(x, ·)dx]
• IGA with NURBS of degree p = 2 and

maximal continuity C 1 Diffusion coeff

a([ρ, θ, z], y) = exp

• y1 sin (2θ) sin (π(ρ − 1)) sin (πz) +

4 10 y2 sin (8θ) sin (π(ρ − 1)) sin (πz) + 1 10 y3 sin (32θ) sin (π(ρ − 1)) sin (πz)

• 21 / 28

Test I - diffusion equation with random diffusion coeff.

10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 2 10 3 10 4 10 5 10 6 MIMC MLMC MISC MIMC realizations MLMC realizations TOL-2 - MIMC/MLMC TOL-2.75 - MC TOL-1/4 - MISC 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 MIMC MLMC MISC Error = TOL

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Test I - diffusion equation with random diffusion coeff.

1 2 3 4 5 6 7 8 9 10 0 10 1 10 2 10 3 10 4

TOL1 TOL2 TOL3 TOL4 TOL5

Figure: Collocation allocation with respect to d

j=1(αj − 1)

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Test II - linear elasticty, random Young modulus / Poission ratio

◮

• D

2µ∇su : ∇sv+

• D

λ div(u) div(v) =

• Σ2

P·v, ∀v ∈ [H1

Σ1(D)]d ◮ ∇su = ∇u+∇T u 2

, P = [0, 0, 0.1]

◮ boundary conditions: clamped bottom, free surface otherwise ◮ λ =

E ν (1 + ν)(1 − 2ν), µ = E 2(1 + ν)

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Test II - linear elasticty, random Young modulus / Poission ratio

◮

• D

2µ∇su : ∇sv+

• D

λ div(u) div(v) =

• Σ2

P·v, ∀v ∈ [H1

Σ1(D)]d ◮ ∇su = ∇u+∇T u 2

, P = [0, 0, 0.1]

◮ boundary conditions: clamped bottom, free surface otherwise ◮ λ =

E ν (1 + ν)(1 − 2ν), µ = E 2(1 + ν)

◮ E ∼ uniform(105e9, 120e9), ν ∼ uniform(0.265, 0.34) (titanium)

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Test II - linear elasticty, random Young modulus / Poission ratio

◮

• D

2µ∇su : ∇sv+

• D

λ div(u) div(v) =

• Σ2

P·v, ∀v ∈ [H1

Σ1(D)]d ◮ ∇su = ∇u+∇T u 2

, P = [0, 0, 0.1]

◮ boundary conditions: clamped bottom, free surface otherwise ◮ λ =

E ν (1 + ν)(1 − 2ν), µ = E 2(1 + ν)

◮ E ∼ uniform(105e9, 120e9), ν ∼ uniform(0.265, 0.34) (titanium) ◮ target: displacement of point P (in red in picture)

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Test II - linear elasticty, random Young modulus / Poission ratio

10 -7 10 -6 10 -5 10 -4 10 0 10 1 10 2 10 3 10 4 MIMC MIMC MIMC realizations TOL-2 - MIMC/MLMC TOL-0.4 - MISC TOL-3 - MC 10 -7 10 -6 10 -5 10 -4 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 MIMC MISC Error = TOL

25 / 28

Outline

Introduction to Uncertainty Quantification Multi-Index Stochastic Collocation (MISC) method Numerical tests Conclusions

26 / 28

Conclusions

◮ Uncertainty Quantification is a fast-growing topic, at the verge of

Engineering, Applied Mathematics, Statistics and Scientific Computing

27 / 28

Conclusions

◮ Uncertainty Quantification is a fast-growing topic, at the verge of

Engineering, Applied Mathematics, Statistics and Scientific Computing

◮ UQ analysis: typically repeatedly solving PDEs for different

combination of random inputs. This can be very computationally expensive

27 / 28

Conclusions

◮ Uncertainty Quantification is a fast-growing topic, at the verge of

Engineering, Applied Mathematics, Statistics and Scientific Computing

◮ UQ analysis: typically repeatedly solving PDEs for different

combination of random inputs. This can be very computationally expensive

◮ Multilevel/Multi-Index methods mitigate this cost by exploiting

hierarchies of meshes, thereby extracting as much information as possible from the coarser ones

27 / 28

Conclusions

◮ Uncertainty Quantification is a fast-growing topic, at the verge of

Engineering, Applied Mathematics, Statistics and Scientific Computing

◮ UQ analysis: typically repeatedly solving PDEs for different

combination of random inputs. This can be very computationally expensive

◮ Multilevel/Multi-Index methods mitigate this cost by exploiting

hierarchies of meshes, thereby extracting as much information as possible from the coarser ones

◮ the adaptive selection of the corrections in MISC guarantees

balancing between physical and stochastic errors

27 / 28

Conclusions

◮ Uncertainty Quantification is a fast-growing topic, at the verge of

Engineering, Applied Mathematics, Statistics and Scientific Computing

◮ UQ analysis: typically repeatedly solving PDEs for different

combination of random inputs. This can be very computationally expensive

◮ Multilevel/Multi-Index methods mitigate this cost by exploiting

hierarchies of meshes, thereby extracting as much information as possible from the coarser ones

◮ the adaptive selection of the corrections in MISC guarantees

balancing between physical and stochastic errors

◮ Multi-index methods can be naturally combined with IGA solvers,

due to their tensor structure

27 / 28

Conclusions

◮ Uncertainty Quantification is a fast-growing topic, at the verge of

Engineering, Applied Mathematics, Statistics and Scientific Computing

◮ UQ analysis: typically repeatedly solving PDEs for different

combination of random inputs. This can be very computationally expensive

◮ Multilevel/Multi-Index methods mitigate this cost by exploiting

hierarchies of meshes, thereby extracting as much information as possible from the coarser ones

◮ the adaptive selection of the corrections in MISC guarantees

balancing between physical and stochastic errors

◮ Multi-index methods can be naturally combined with IGA solvers,

due to their tensor structure

◮ In the numerical tests, the IGA-based MISC method shows the best

performance and exhibits the theoretical rates, namely TOL−1/4 and TOL−0.4, respectively, in comparison with the rate of MLMC/MIMC, TOL−2 (i.e., the optimal sampling rate).

27 / 28

Thanks for your attention

• J. Beck, L. Tamellini, R. Tempone.

IGA-based multi-index stochastic collocation for random PDEs on arbitrary

• domains. Computer Methods in Applied Mechanics and Engineering, 351:

130–350, 2019.

• J. Beck, L. Tamellini, G. Sangalli.

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