my presentation at university of nottingham "fast low-rank methods for solving stochastic...

Low-rank tensors for PDEs withuncertain coefficients

Alexander Litvinenko

Center for UncertaintyQuantification


Center for Uncertainty Quantification Logo Lock-up

http://sri-uq.kaust.edu.sa/

Extreme Computing Research Center, KAUST

Alexander Litvinenko Low-rank tensors for PDEs with uncertain coefficients

http://sri-uq.kaust.edu.sa/

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The structure of the talk

Part I (Stochastic Forward Problem):1. Motivation2. Elliptic PDE with uncertain coefficients3. Discretization and low-rank tensor approximations

Part II (Stochastic Inverse Problem via Bayesian Update):1. Bayesian update surrogate2. Examples

Part III (Quantification of uncertainties in aerodynamics)

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My interests and collaborations




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Motivation to do Uncertainty Quantification (UQ)

Motivation: there is an urgent need to quantify and reduce theuncertainty in multiscale-multiphysics applications.

UQ and its relevance: Nowadays computational predictions areused in critical engineering decisions. But, how reliable arethese predictions?

Example: Saudi Aramco currently has a simulator,GigaPOWERS, which runs with 9 billion cells. How sensitiveare these simulations w.r.t. unknown reservoir properties?

My goal is systematic, mathematically founded, develop-ment of UQ methods and low-rank algorithms relevant forapplications.




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PDE with uncertain coefficient

Consider− div(κ(x , ω)∇u(x , ω)) = f (x , ω) in G × Ω, G ⊂ Rd ,u = 0 on ∂G,

where κ(x , ω) - uncertain diffusion coefficient.

1. Efficient Analysis of High Dimensional Data in TensorFormats, Espig, Hackbusch, A.L., Matthies and Zander,2012.2. Efficient low-rank approximation of the stochasticGalerkin matrix in tensor formats, Wahnert, Espig, Hack-busch, A.L., Matthies, 2013.3. Polynomial Chaos Expansion of random coefficientsand the solution of stochastic partial differential equationsin the Tensor Train format, Dolgov, Litvinenko, Khoromskij,Matthies, 2016.

0 0.5 1-20

0

20

40

60

50 realizations of the solution u,

the mean and quantiles




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Canonical and Tucker tensor formats

[Pictures are taken from B. Khoromskij and A. Auer lecture course]

Storage: O(nd )→ O(dRn) and O(Rd + dRn).




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Karhunen Loeve and Polynomial Chaos Expansions

Apply bothTruncated Karhunen Loeve Expansion (KLE):κ(x , ω) ≈ κ0(x) +

∑Lj=1 κjgj(x)ξj(θ(ω)), where

θ = θ(ω) = (θ1(ω), θ2(ω), ..., ),ξj(θ) = 1

κj

∫G (κ(x , ω)− κ0(x)) gj(x)dx .

Truncated Polynomial Chaos Expansion (PCE)κ(x , ω) ≈

∑α∈JM,p

κ(α)(x)Hα(θ),

ξj(θ) ≈∑

α∈JM,pξ(α)j Hα(θ).




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Discretization of elliptic PDE

Ku = f, where

K:=∑L

`=1K` ⊗⊗M

µ=1∆`µ, K` ∈ RN×N , ∆`µ ∈ RRµ×Rµ ,u:=

∑rj=1 uj ⊗

⊗Mµ=1 ujµ, uj ∈ RN , ujµ ∈ RRµ ,

f:=∑R

k=1 f k ⊗⊗M

µ=1 gkµ, f k ∈ RN and gkµ ∈ RRµ .

(Wahnert, Espig, Hackbusch, Litvinenko, Matthies, 2011)




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Numerical Experiments

2D L-shape domain, N = 557 dofs.Total stochastic dimension is Mu = Mk + Mf = 20, there are|JM,p| = 231 PCE coefficients

u =231∑j=1

uj,0 ⊗20⊗µ=1

ujµ ∈ R557 ⊗20⊗µ=1

R3.

Tensor u has 320 · 557 ≈ 2 · 1012 entries ≈ 16 TB of memory.




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Level sets

Now we compute ui : ui > b ·maxi u,i := (i1, ..., iM+1)

for b ∈ 0.2, 0.4, 0.6, 0.8.

I The computing time for each b was 10 minutes.I Intermediate ranks of sign(b‖u‖∞1− u) and of rank(uk )

were less than 24.




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Part II

Part II: Bayesian update

We will speak about Gauss-Markov-Kalman filter for theBayesian updating of parameters in a computational model.

Multiple publications with Bojana V. Rosic, Elmar Zander, Oliver Pajonk and H.G. Matthies from TU Braunschweig,

Germany.

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Numerical computation of NLBU

Look for ϕ such that q(ξ) = ϕ(z(ξ)), z(ξ) = y(ξ) + ε(ω):

ϕ ≈ ϕ =∑α∈Jp

ϕαΦα(z(ξ))

and minimize ‖q(ξ)− ϕ(z(ξ))‖2L2, where Φα are polynomials

(e.g. Hermite, Laguerre, Chebyshev or something else).Taking derivatives with respect to ϕα:

∂

∂ϕα〈q(ξ)− ϕ(z(ξ)),q(ξ)− ϕ(z(ξ))〉 = 0 ∀α ∈ Jp

Inserting representation for ϕ, solve linear system for ϕα.




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Numerical computation of NLBU

Finally, the assimilated parameter qa will be

qa = qf + ϕ(y)− ϕ(z), (1)

z(ξ) = y(ξ) + ε(ω),ϕ =

∑β∈Jp

ϕβΦβ(z(ξ))




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Example: 1D elliptic PDE with uncertain coeffs

−∇ · (κ(x , ξ)∇u(x , ξ)) = f (x , ξ), x ∈ [0,1]

+ Dirichlet random b.c. g(0, ξ) and g(1, ξ).3 measurements: u(0.3) = 22, s.d. 0.2, x(0.5) = 28, s.d. 0.3,x(0.8) = 18, s.d. 0.3.

I κ(x, ξ): N = 100 dofs, M = 5, number of KLE terms 35, beta distribution for κ, Gaussian covκ, cov.length 0.1, multi-variate Hermite polynomial of order pκ = 2;

I RHS f (x, ξ): Mf = 5, number of KLE terms 40, beta distribution for κ, exponential covf , cov. length 0.03,multi-variate Hermite polynomial of order pf = 2;

I b.c. g(x, ξ): Mg = 2, number of KLE terms 2, normal distribution for g, Gaussian covg , cov. length 10,multi-variate Hermite polynomial of order pg = 1;

I pφ = 3 and pu = 3




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Example: Updating of the parameter

0 0.5 10

0.5

1

1.5

0 0.5 10

0.5

1

1.5

Figure: Original and updated parameter κ.

Collaboration with Y. Marzouk, MIT, and TU Braunschweig. Wetry to build an equivalent of KLD for PCE expansion.Collaborate with H. Najm, Sandia Lab. We try to compare ourtechnique with his advanced MCMC technique for chemicalcombustion eqn.




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Example: updating of the solution u

0 0.5 1-20

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Figure: Original and updated solutions, mean value plus/minus 1,2,3standard deviations. Number of available measurements 0,1,2,3,5

[graphics are built in the stochastic Galerkin library sglib, written by E. Zander in TU Braunschweig]




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Part III: My contribution to MUNA

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Example: uncertainties in free stream turbulence

α

v

v

u

u’

α’

v1

2

Random vectors v1(θ) and v2(θ) model free stream turbulence




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Example: 3sigma intervals

Figure: 3σ interval, σ standard deviation, in each point of RAE2822airfoil for the pressure (cp) and friction (cf) coefficients.




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Mean and variance of density, tke, xv, zv, pressure




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Domain decomposition

Application of domain decomposition and Hierarchical matricesfor solving multi-scale problems.

(a)macroscopic scale (b)microscopic scale (c)molecular scale

Ω

v

T

repeated cells

v

. . . .

.

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mean value

hH

TH Th




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Conclusion

IntroducedI Low-rank tensor methods to solve elliptic PDEs with

uncertain coefficients,I Post-processing in low-rank tensor format, computing level

setsI Bayesian update surrogate ϕ (as a linear, quadratic,...

approximation)I Quantification of uncertainties in Numerical AerodynamicsI Domain decomposition and Hierarchical matrices for

multiscale problems




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Thank you

Thank you!




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My experience since 2002




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Literature

1. PCE of random coefficients and the solution of stochastic partialdifferential equations in the Tensor Train format , S. Dolgov, B. N.Khoromskij, A. Litvinenko, H. G. Matthies, 2015/3/11,arXiv:1503.032102. Efficient analysis of high dimensional data in tensor formats, M.Espig, W. Hackbusch, A. Litvinenko, H.G. Matthies, E. Zander SparseGrids and Applications, 31-56, 40, 20133. Application of hierarchical matrices for computing theKarhunen-Loeve expansion, B.N. Khoromskij, A. Litvinenko, H.G.Matthies, Computing 84 (1-2), 49-67, 31, 20094. Efficient low-rank approximation of the stochastic Galerkin matrixin tensor formats, M. Espig, W. Hackbusch, A. Litvinenko, H.G.Matthies, P. Waehnert, Comp. & Math. with Appl. 67 (4), 818-829,20125. Numerical Methods for Uncertainty Quantification and BayesianUpdate in Aerodynamics, A. Litvinenko, H. G. Matthies, Book”Management and Minimisation of Uncertainties and Errors inNumerical Aerodynamics”, pp 265-282, 2013




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Literature

1. A. Litvinenko and H. G. Matthies, Inverse problems anduncertainty quantificationhttp://arxiv.org/abs/1312.5048, 2013

2. L. Giraldi, A. Litvinenko, D. Liu, H. G. Matthies, A. Nouy, Tobe or not to be intrusive? The solution of parametric andstochastic equations - the ”plain vanilla” Galerkin case,http://arxiv.org/abs/1309.1617, 2013

3. O. Pajonk, B. V. Rosic, A. Litvinenko, and H. G. Matthies, ADeterministic Filter for Non-Gaussian Bayesian Estimation,Physica D: Nonlinear Phenomena, Vol. 241(7), pp.775-788, 2012.

4. B. V. Rosic, A. Litvinenko, O. Pajonk and H. G. Matthies,Sampling Free Linear Bayesian Update of PolynomialChaos Representations, J. of Comput. Physics, Vol.231(17), 2012 , pp 5761-5787




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http://arxiv.org/abs/1312.5048

http://arxiv.org/abs/1309.1617

my presentation at university of nottingham "fast low-rank methods for solving stochastic...

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