Multilevel stochastic collocations with dimensionality reduction - … · 2017. 1. 27. · TUM,...

Multilevel stochastic collocations withdimensionality reduction

Ionut Farcas

TUM, Chair of Scientific Computing in Computer Science (I5)

27.01.2017

Outline

1 Motivation

2 Theoretical backgroundUncertainty modelingSparse gridsGeneralized polynomial chaos and sparse gridsMultilevel collocation methodsStochastic dimensionality reduction

3 Test scenario

4 Discussion

Motivation

problem: quantification of uncertainty in complex phenomenamultiphysics (e.g. fluid-structure interaction)plasma physics...

main challenge: “curse of dimensionality”→ “curse of resources”solution 1.1: delay the “curse of dimensionality”→ sparse gridssolution 1.2: try reducing the dimensionality→ sensitivity analysis

Motivation


main challenge: “curse of dimensionality”→ “curse of resources”

solution 1.1: delay the “curse of dimensionality”→ sparse gridssolution 1.2: try reducing the dimensionality→ sensitivity analysis

Motivation


main challenge: “curse of dimensionality”→ “curse of resources”solution 1.1: delay the “curse of dimensionality”→ sparse grids

solution 1.2: try reducing the dimensionality→ sensitivity analysis

Motivation


main challenge: “curse of dimensionality”→ “curse of resources”solution 1.1: delay the “curse of dimensionality”→ sparse gridssolution 1.2: try reducing the dimensionality→ sensitivity analysis

Uncertainty modeling

probabilistic modelingprobability space (Ω,F ,P)

θ = (θ1, θ2, . . . , θd ) vector of continuous i.i.d. random variablessupp(θi) = Γi , supp(θ) = Γ1 × Γ2 × . . .× Γd = Γ

Generalized polynomial chaos approximation

idea: represent an arbitrary random variable (of interest) as afunction of another random variable with given distribution

how: use a series expansion of orthogonal polynomialslet p = (p1, . . . ,pd ) ∈ Nd :

∑di=1 pi < P

consider d-variate orthogonal polynomials

Φp(θ) := Φp1(θ1) . . .Φpd (θd )

for simplicity, drop the multi-index subscript p and use instead ascalar index n = 1, . . . ,N, N =

(d+Pd

)orthogonality means

E[Φn(θ)Φm(θ)] =

∫ΓΦn(θ)Φm(θ)ρ(θ)dθ = γnδnm, γn ∈ R


idea: represent an arbitrary random variable (of interest) as afunction of another random variable with given distributionhow: use a series expansion of orthogonal polynomials

let p = (p1, . . . ,pd ) ∈ Nd :∑d

i=1 pi < Pconsider d-variate orthogonal polynomials

Φp(θ) := Φp1(θ1) . . .Φpd (θd )


(d+Pd


E[Φn(θ)Φm(θ)] =



idea: represent an arbitrary random variable (of interest) as afunction of another random variable with given distributionhow: use a series expansion of orthogonal polynomialslet p = (p1, . . . ,pd ) ∈ Nd :

∑di=1 pi < P

consider d-variate orthogonal polynomials

Φp(θ) := Φp1(θ1) . . .Φpd (θd )


(d+Pd


E[Φn(θ)Φm(θ)] =


Generalized polynomial chaos

let x - deterministic inputs, θ - stochastic inputs, f - modelthe gPC approximation of order N reads

f (x ,θ) ≈ fN(x ,θ) =N−1∑n=0

cn(x)Φn(θ)

gPC coefficients via projection

cn(x) =

∫Γ

f (x ,θ)Φn(θ)ρ(θ)dθ = E[f (x ,θ)Φn(θ)]

Post-processing

expectationE[f (x ,θ)] = c0(x),

variance

Var [f (x ,θ)] =N−1∑n=1

c2n(x).

total Sobol’ indices

STi (x) =

Varp[f (x ,θ)]

Var [f (x ,θ)]=

∑k∈Ap

c2k (x)

Var [f (x ,θ)],

Ap = p ∈ Nd : pi ∈ p,pi 6= 0d∑

i=1

STi (x) = 1

Sparse grid idea

problem: discretize efficiently a tensor product space

standard approach: full grid→ O(Nd ) dof, if N dof in one direction→ “curse of dimensionality”idea: delay the curse of dimensionalityuse sparse grids: weaken the assumed coupling between theinput dimensionsO(Nd )→ O(N(logN)d−1) dof

Sparse grid idea

problem: discretize efficiently a tensor product spacestandard approach: full grid→ O(Nd ) dof, if N dof in one direction→ “curse of dimensionality”

idea: delay the curse of dimensionalityuse sparse grids: weaken the assumed coupling between theinput dimensionsO(Nd )→ O(N(logN)d−1) dof

Sparse grid idea

problem: discretize efficiently a tensor product spacestandard approach: full grid→ O(Nd ) dof, if N dof in one direction→ “curse of dimensionality”idea: delay the curse of dimensionalityuse sparse grids: weaken the assumed coupling between theinput dimensionsO(Nd )→ O(N(logN)d−1) dof

Sparse grid idea

⇒

Hierarchical sparse grids ingredients

grid level l = (l1, . . . , ld ) ∈ Nd

spatial position i = (i1, . . . , id ) ∈ Nd

generic grid point ul,i = (ul1,i1 , . . . ,uld ,id )

equidistant grid with mesh size hli = 2−li , i = 1, . . . ,dbasis functions ϕl,i with support [ul,i − hl ,ul,i + hl ]

ϕl,i(u) = ϕ(u − ihl

hl

)in d-dimensions,

ϕl,i (u) =d∏

j=1

ϕlj ,ij (uj )

Hierarchical sparse grids preliminaries

Hl = spanϕl,i : 1 ≤ i ≤2l − 1 - nodal setWl = spanϕl,i : i ∈ Il -hierarchical increment set

Il = i ∈ Nd : 1 ≤ ik ≤ 2lk − 1, ik odd , k = 1 . . . d

Hl =⊗

k≤l Wk


given the hierarchical increment spaces Wl and given a level L,we can create further spaces VL

VL =⊗

k∈J Wk , for some multiindex set J

if J = l ∈ Nd : |l |∞ ≤ L - full grid spaceif J = l ∈ Nd : |l |1 ≤ L + d − 1 - standard sparse grid space



VL =⊗

k∈J Wk , for some multiindex set Jif J = l ∈ Nd : |l |∞ ≤ L - full grid space

if J = l ∈ Nd : |l |1 ≤ L + d − 1 - standard sparse grid space



VL =⊗

k∈J Wk , for some multiindex set Jif J = l ∈ Nd : |l |∞ ≤ L - full grid spaceif J = l ∈ Nd : |l |1 ≤ L + d − 1 - standard sparse grid space

Hierarchical sparse grids example

L = 5

Interpolation on hierarchical sparse grids

consider g : [0,1]d → Rthe sparse grid interpolant gI(u) of g(u) is

gI(u) =∑

l∈J ,i∈Il

αl,iϕl,i(u) (1)

αl,i are the so-called hierarchical surplusesassumeg ∈ Hmix

2 ([0,1]d ) = f : [0,1]d → R : Dl f ∈ L2([0,1]d ), |l |∞ ≤ 2,Dl f = ∂|l|1 f/∂x l1

1 . . . ∂x ldd

if full grid||g(u)− gI(u)||L2 ∈ O

(h2

L)

if sparse grid||g(u)− gI(u)||L2 ∈ O

(h2

LLd−1)

Piecewise linear basis functions

Piecewise polynomial basis functions

Spatial refinement

due to the hierarchical construction→ local refinement possibleαl,i - good measure of the interpolation error

the absolute value of αl,i - good refinement metricselect the grid points with the largest surpluses valuesadd their hierarchical descendants to Jif not all hierarchical parents exist add them

multiple grid points can be refined in one step

Spatial refinement

due to the hierarchical construction→ local refinement possibleαl,i - good measure of the interpolation errorthe absolute value of αl,i - good refinement metric

select the grid points with the largest surpluses valuesadd their hierarchical descendants to Jif not all hierarchical parents exist add them

multiple grid points can be refined in one step

Spatial refinement: Franke’s function

f (x1, x2) = 0.75 exp(− (9x1 − 2)2

4− (9x2 − 2)2

4

)+

0.75 exp(− (9x1 + 1)2

49− 9x2 + 2

10

)+

0.5 exp(− (9x1 − 7)2

4− (9x2 − 3)2

4

)−

0.2 exp(− (9x1 − 4)2 − (9x2 − 7)

)

Franke’s function

Franke’s function refinement part 1L = 5refine 20% of the grid points

Franke’s function refinement part 2L = 6refine 20% of the grid points

gPC coefficients computation

remembercn(x) =

∫Γ

f (x ,θ)Φn(θ)ρ(θ)dθ

how can we use sparse grids?let T : [0,1]d → Γ

then,

cn(x) =

∫[0,1]d

f (x ,T (u))Φn(T (u))|detJT (u)|ρ(T (u))du

intuitionΦn(T (u)) - tensor structureif f (x ,T (u)) would also have a tensor structure ...


remembercn(x) =

∫Γ


how can we use sparse grids?

let T : [0,1]d → Γ

then,

cn(x) =

∫[0,1]d




remembercn(x) =

∫Γ


how can we use sparse grids?let T : [0,1]d → Γ

then,

cn(x) =

∫[0,1]d




iff (x ,T (u)) ≈ f (x ,T (u)) =

∑l∈J ,i∈Il

αl,iϕl,i (u)

T (u) := (F−11 (u1), . . . ,F−1

d (ud )), Fi cdf of θi

then

cn(x) =

∫[0,1]d

f (x ,T (u))Φn(T (u))du

=

∫[0,1]d

( ∑l∈J ,i∈Il

αl,i(x)ϕl,i(u))Φn(T (u))du

=∑

l∈J ,i∈Il

αl,i(x)d∏

j=1

∫[0,1]

Φj(F−1j (uj))ϕlj ,ij (uj)duj

Multilevel approaches

“monolevelapproach”can we furtherreduce thecomputational cost?use multilevelapproaches

Multilevel stochastic collocation: no refinement

Multilevel stochastic collocation: with refinement

Multilevel gPC coefficients

let Mh denote the level of the deterministic domain discretizationlet Ll denote the sparse grid level

let cMh,Lln (x) denote the gPC coefficient computed using adeterministic grid of level Mhsparse grid of level Ll

then, for K + 1 levels

cMK ,LKn (x) =cM0,LK

n (x)

+ (cM1,LK−1n (x)− cM0,LK−1

n (x))+

...

+ (cMK ,L0n (x)− cMK−1,L0

n (x))

if nested sparse grids, cMK−l ,Ll−1n (x) ⊂ cMK−l ,Ll

n (x)

Stochastic dimensionality reduction

each uncertain input has a different contribution to the outputuncertainty

some inputs contribute very little→ they can be “ignored” (takenas deterministic)use sensitivity information to determine each input’s contributionin the multilevel scheme, given Kc < K and τ ∈ [0,1] (e.g. τ = 5%)

if STi (x) ≤ τ , “ignore” input i

determine the new stochastic dimensionality“project” computed result on the new (sparse) grid


each uncertain input has a different contribution to the outputuncertaintysome inputs contribute very little→ they can be “ignored” (takenas deterministic)

use sensitivity information to determine each input’s contributionin the multilevel scheme, given Kc < K and τ ∈ [0,1] (e.g. τ = 5%)




each uncertain input has a different contribution to the outputuncertaintysome inputs contribute very little→ they can be “ignored” (takenas deterministic)use sensitivity information to determine each input’s contribution

in the multilevel scheme, given Kc < K and τ ∈ [0,1] (e.g. τ = 5%)




each uncertain input has a different contribution to the outputuncertaintysome inputs contribute very little→ they can be “ignored” (takenas deterministic)use sensitivity information to determine each input’s contributionin the multilevel scheme, given Kc < K and τ ∈ [0,1] (e.g. τ = 5%)



Sparse grid projection

if input k ,1 ≤ i ≤ d is “ignored” and uk is the correspondingdeterministic value

f (x ,T (u)) =∑

l∈J ,i∈Il

αl,iϕl,i(u)

=∑

l∈J ,i∈Il

αl,i

d∏j=1

ϕlj ,ij (uj)

=∑

l∈J ,i∈Il

αl,iϕlk ,ik (uk )d−1∏j=1

ϕlj ,ij (uj)

=∑

l∈J ′,i∈I′l

α′l,iϕ′l,i(u)

Test scenario

d2ydt2 (t) + c dy

dt (t) + ky(t) = f cos(wt)y(0) = y0dydt (0) = y1

t ∈ [0,20], w = 1.05

five uncertain inputsdamping coefficient c ∼ U(0.08,0.12)spring constant k ∼ U(0.03,0.04)forcing amplitude f ∼ U(0.08,0.12)initial position y0 ∼ U(0.45,0.55)initial velocity y1 ∼ U(−0.05,0.05)

underdamped regime

Test scenario

d2ydt2 (t) + c dy

dt (t) + ky(t) = f cos(wt)y(0) = y0dydt (0) = y1

t ∈ [0,20], w = 1.05five uncertain inputs

damping coefficient c ∼ U(0.08,0.12)spring constant k ∼ U(0.03,0.04)forcing amplitude f ∼ U(0.08,0.12)initial position y0 ∼ U(0.45,0.55)initial velocity y1 ∼ U(−0.05,0.05)

underdamped regime

Tests setup

sparse grid functionality: SG++1

finite difference discretizationuniform inputs→ Legendre polynomialsmodified polynomial basis functions of deg 2tinterest = 10reference results with 32768 Gauss-Legendre nodesmultilevel approach with K = 2

cM2,L2n (x) = cM0,L2

n (x) + (cM1,L1n (x)− cM0,L1

n (x))

+ (cM2,L0n (x)− cM1,L0

n (x))

when using refinement, Li , i = 1,2 means L0 with i refinementsteps

1http://sgpp.sparsegrids.org/

Tests setup


finite difference discretization

uniform inputs→ Legendre polynomialsmodified polynomial basis functions of deg 2tinterest = 10reference results with 32768 Gauss-Legendre nodesmultilevel approach with K = 2


n (x) + (cM1,L1n (x)− cM0,L1

n (x))

+ (cM2,L0n (x)− cM1,L0

n (x))



Tests setup


finite difference discretizationuniform inputs→ Legendre polynomialsmodified polynomial basis functions of deg 2

tinterest = 10reference results with 32768 Gauss-Legendre nodesmultilevel approach with K = 2


n (x) + (cM1,L1n (x)− cM0,L1

n (x))

+ (cM2,L0n (x)− cM1,L0

n (x))



Tests setup


finite difference discretizationuniform inputs→ Legendre polynomialsmodified polynomial basis functions of deg 2tinterest = 10

reference results with 32768 Gauss-Legendre nodesmultilevel approach with K = 2


n (x) + (cM1,L1n (x)− cM0,L1

n (x))

+ (cM2,L0n (x)− cM1,L0

n (x))



Tests setup


finite difference discretizationuniform inputs→ Legendre polynomialsmodified polynomial basis functions of deg 2tinterest = 10reference results with 32768 Gauss-Legendre nodes

multilevel approach with K = 2


n (x) + (cM1,L1n (x)− cM0,L1

n (x))

+ (cM2,L0n (x)− cM1,L0

n (x))



Tests setup


finite difference discretizationuniform inputs→ Legendre polynomialsmodified polynomial basis functions of deg 2tinterest = 10reference results with 32768 Gauss-Legendre nodesmultilevel approach with K = 2


n (x) + (cM1,L1n (x)− cM0,L1

n (x))

+ (cM2,L0n (x)− cM1,L0

n (x))



Tests setup

M0 = 500,M1 = 2000,M2 = 8000reference results

Eref [y(10)] = −0.155165Varref [y(10)] = 0.0002267

error measurement

err =∣∣∣qoiref − qoi

qoiref

∣∣∣

Test case 1: no dimensionality reduction

L0 L1 L2 ref % err exp err var11 71 351 - 4.2012e-06 1.3147e-0471 351 1471 - 7.7759e-07 1.5950e-0511 31 67 20% 4.0983e-04 7.8192e-0471 191 423 20% 1.4499e-06 2.8220e-0571 230 655 30% 7.8551e-07 1.4199e-05

Test case 2: dimensionality reduction

compute Sobol’ indices for cM1,L1n (x), i.e. start with K = 1

τ = 5%

err expectation ∈ O(10−3)

L0 L1 L2 ref % ST1 ST

2 ST3 ST

4 ST5

5 71 49 - 4.1% 1.2% 56.5% 4.8% 34.0%

17 351 129 - 4.1% 1.2% 56.5% 4.8% 34.0%

5 31 13 20% 4.1% 0.65% 56.7% 4.8% 33.9%

17 191 45 20% 4.1% 1.2% 56.5% 4.8% 34.0%

17 230 67 30% 4.0% 1.2% 56.6% 4.8% 34.0%

Test case 3: dimensionality reduction

compute Sobol’ indices for cM0,L2n (x) + (cM1,L1

n (x)− cM0,L1n (x))

τ = 5%

err expectation ∈ O(10−4)

L0 L1 L2 ref % ST1 ST

2 ST3 ST

4 ST5

5 71 351 - 4.1% 1.2% 56.5% 4.8% 34.0%

17 351 1471 - 4.1% 1.2% 56.5% 4.8% 34.0%

5 31 67 20% 4.1% 1.2% 56.5% 4.8% 34.0%

17 191 423 20% 4.0% 1.2% 56.6% 4.8% 34.0%

17 230 655 30% 4.0% 1.2% 56.6% 4.8% 34.0%

Discussion

from polynomial chaos coefficients, we can compute mean,variance, Sobol’ (sensitivity) indices etc.spatially adaptive sparse grids suitable to delay the curse ofdimensionalitymultilevel ideas can further reduce the computational costuncertain inputs that contribute little to output uncertainty can beignoredall of these should be used with care

Thank you for your attention!

Multilevel stochastic collocations with dimensionality reduction - … · 2017. 1. 27. · TUM,...

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