model reduction method for solving parameter-dependent...

Model reduction method for solvingparameter-dependent dynamical systems

Marie Billaud-Friess?,†

Joint work with : A. Nouy?

Work supported by GDR MOMAS

† [email protected]?GeM, Ecole Centrale de Nantes

October 6th, 2015

[email protected]

Outline

1 Motivations

2 Low-rank approximation methods

3 Proposed method

4 Numerical experiments

M. Billaud-Friess — Workshop MOMAS 2015 2/28

General context

Parameter and time dependent problem

Find u(t, ξ) : ξ × [0, T ]→ Rd such that

F(u(t, ξ), t, ξ) = 0

where ξ ∈ Ξ ⊂ Rp are parameters or random variables in measure space (Ξ, µ).

Example : Groundwater flows : benchmark couplex with uncertainties 1

Ω

D

ΩD

ΩA

ΩC

ΩMω

(∂C

∂t+ λ(ξ)C

)−∇·(de(ξ)∇C)+u(ξ)·∇C = 0

with u(ξ) = −Ki(ξ)∇H s.t.

−∇ · (K(ξ)∇H) = 0, K(ξ) =∑i

1ΩiKi(ξ)

Finite discretization Algebraic system with d very large

1. https://www.ljll.math.upmc.fr/cances/gdrmomas/ex_qualifications.html

M. Billaud-Friess — Workshop MOMAS 2015 Motivations 3/28

https://www.ljll.math.upmc.fr/cances/gdrmomas/ex_qualifications.html

Need of model reduction

Issue : Multiple parameter simulations can be expensive and intractable

• Variable of interest : s(ξ) = `(u(ξ))

• Quantity of interest : statistical moments, failure probability, sensitivity indices

Derive a surrogate problem cheaper to solve

Strategy : Model order reduction methods

• Proper generalized decomposition,

• Reduced basis with POD-Greedy strategy,

• Dynamical low-rank approach,

• Dynamical Orthogonal approximation ...

Low-rank approximation methods

u ≈r∑i=1

vi ⊗ αi, with r << d small.

M. Billaud-Friess — Workshop MOMAS 2015 Motivations 4/28

Problem setting

Parameter-dependent dynamical system

Find u : ξ × [0, T ]→ X such thatu′(t, ξ) = f(u(t, ξ), t, ξ) t ∈ (0, T ),u(0, ξ) = u0(ξ).

(1)

• ξ ∈ Ξ ⊂ Rp a (random) parameter vector ,

• X = Rd equiped with the L2 scalar product 〈·, ·〉X ,

• the state u is differentiable,

• u0 : ξ → X is the initial datum,

• f : X × [0, T ]×Ξ→ X the flux that is Lipschitz continuous with respect to u.

M. Billaud-Friess — Workshop MOMAS 2015 Low-rank approximation methods 5/28

Low-rank approximation methods : a sub-space point of view

À Ideal rank-r approximation for parametric dynamical system (1)

minv(t)∈Mr

‖v(t)− u(t, ·)‖Ξ,q, t > 0

with Mr =v =

∑ri=1 vi ⊗ αi,vi ∈ X,αi ∈ Lpµ(Ξ)

and r ∈ N such that r d.

Á Subspace point of view

minXr ⊂ X

dimXr = r

minv(t,·)∈L

pµ⊗Xr

‖v(t, ·)− u(t, ·)‖Ξ,p, t > 0

u unknown a priori Practical algorithms for computing sub-optimal approx. ur

L2-norm : Dynamical low-rank approximation [Koch07]

L∞-norm : Reduced basis approach

1 Greedy construction of Xr ⊂ X using “snapshots” of the solution

2 Low-rank approximation ur ∈ Xr by (Petrov-)Galerkin projection of (1) in Xr


Low-rank approximation with RB methods I

Low-rank approximation with time dependent reduced space Xr(t)

u(t, ξ) ≈ ur(t, ξ) =

r∑i=1

vi(t)αi(t, ξ)

Method 1 : Weak-greedy algorithm

For each t > 0, while maxξ∈Ξtrain

∆r(t, ξ) > ε or r < rmax do.

1 Find ξr+1(t) = maxξ∈Ξtrain

∆r(t, ξ).

2 Compute u(t, ξr+1(t)).

3 Set Xr+1(t) = Xr(t) + spanu(t, ξr+1(t)).

3 ∆r surrogate of the exact projection error s.t.

cr∆r(t, ·) ≤ ‖u(t, ·)− P r(t)u(t, ·)‖X ≤ Cr∆r(t, ·), cr, CR > 0

3 At each instant t, the weak-greedy algorithm converges [Bin11, Buff12]

7 Costly strategy since ξr(t) is time dependent Many calls to (1).


Low-rank approximation with RB methods II

Low-rank approximation with time independent reduced space Xr

u(t, ξ) ≈ ur(t, ξ) =

r∑i=1

viαi(t, ξ)

Strategies : Snapshot method, POD based method, POD-greedy method

Method 2 : POD-greedy algorithm [Haas11]

While maxξ∈Ξtrain

‖∆r(·, ξ)‖(0,T ),2 > ε or r < rmax do.

1 Find ξr+1 = maxξ∈Ξtrain

‖∆r(·, ξ)‖(0,T ),2.

2 Compute t 7→ u(t, ξr+1) and S(·, ξr) : t 7→ u(t, ξr+1)− P ru(t, ξr+1).

3 Set X(r+1)` = Xr` + spanPOD(S(·, ξr+1), `

).

3 Global error estimate ‖∆r(·, ξ)‖(0,T ),2

Time independent ξr with at most r calls to (1) Parameter ξr selected for given objective (e.g. GO norms)

3 Quasi-optimal convergence rates for Algorithm 2 [Haas13]7 The approximation ur is not an interpolation of u at points ξ1, . . . , ξr7 Large r when u has a rich spectral content (e.g. u solution of the advection equation)


Proposed approach

Goal : Design a time-dependent low-rank approximation method for solvingparametric dynamical systems

u(t, ξ) ≈ ur(t, ξ) =r∑i=1

vi(t)αi(t, ξ)

Specifications : A method combining the best features of method 1 & 2

1 Time dependent reduced space Xr(t) ⊂ X by snapshots of the solutionselected with a greedy algorithm

2 Low-rank approximation ur(t) by solving the reduced dynamical systemobtained by (Petrov-)Galerkin projection of (1) on Xr(t)

3 Rigorous a-posteriori error estimation ‖ur(t, ·)− u(t, ·)‖X ≤ ∆r(t, ·)

M. Billaud-Friess — Workshop MOMAS 2015 Proposed method 9/28

Constructing the reduced space

Method 3 : T-greedy algorithm

While maxξ∈Ξtrain

‖∆r(·, ξ)‖(0,T ),2 > ε or r < rmax do.

1 Find ξr+1 = arg maxξ∈Ξtrain

‖∆r(t, ξ)‖(0,T ),2.

2 Compute t 7→ u(t, ξr+1) and S(·, ξr) : t 7→ u(t, ξr+1)− P ru(t, ξr+1).

3 For each time t > 0 proceed as follows.

• Set vr+1(t) =Sr+1(t)

‖Sr+1(t)‖X.

• Construct Xr+1(t) = Xr(t) + spanvr+1(t).

Proposition

Assuming that dimXr(t) = r for t > 0, then

1 the applications vi : t 7→ vi(t), i = 1, . . . , r exists and are differentiable,

2 the resulting family of functions vi(t)ri=1 is orthonormal for each t > 0.


Projection strategy

Notations : Here ξ ∈ Ξ is fixed

• V r(t) ∈ Rd×r = [v1(t), . . . ,vr(t)] s.t. V Tr (t)V r(t) = Ir, P r(t) = V r(t)V

Tr (t),

• αr ∈ Rr,ur ∈ Rd s.t. ur(t, ξ) = V r(t)αr(t, ξ)⇔ αr(t, ξ) = V r(t)Tur(t, ξ).

¬ The Galerkin projection

V r(t)Tu′r(t, ξ) = V r(t)

Tf(ur(t), ξ, t)

The low-rank approximation ur

u′r(t, ξ) = P r(t)f(ur(t, ξ), ξ, t) + (Id − P r(t))V′r(t)αr(t, ξ) ur(0) = P r(0)u0

Remark : Dependence in time of Xr(t) taken into account by means of V ′r(t)

® The reduced dynamical system

α′r(t, ξ) = fr(αr(t, ξ), ξ, t) α0r = V T

r (0)u0 a

with fr(αr(t, ξ), ξ, t) = V r(t)Tf(ur(t, ξ), ξ, t)− V r(t)

TV ′r(t)αr(t, ξ)

a. For parameter independent initial condition α0r = (1, 0, . . . , 0).


Error analysisLinear case

Summary : For f(u(t), t) = Au(t), the approximation error e(t) = ur(t)− u(t) fort > 0 is

e′(t) = Ae(t)︸︷︷︸flux error

+Qr(t)Aur(t)︸︷︷︸flux projection

−Qr(t)V′r(t)αr(t)︸︷︷︸

basis dynamics

e(0) = Qr(0)u0.

where Qr(t) = P r(t)− Id ∈ Rd×d

Proposition

The approximation error e satisfies ‖e(t)‖X ≤ ∆r(t),∀t ∈ [0, T ], where

∆r(t) :=

[sups∈[0,t]

‖r(s)‖X1− e−‖A‖X t

‖A‖X+ ‖e(0)‖X

]et‖A‖X

with ‖A‖X = λmax(AAT ) and r(s) = Qr(t)(Aur(t)− V ′r(t)αr(t))

Remark :

• Error estimate valid for time independent reduced space [Haas11]

• Coarse error estimate valid only for linear cases


Error analysisGeneral case

Summary : The approximation error e(t) = ur(t)− u(t) for t > 0 ise′(t) = (f(ur(t), t)− f(u(t), t))︸︷︷︸

flux error

+Qr(t)f(ur(t), t)︸︷︷︸flux projection

−Qr(t)V′r(t)αr(t)︸︷︷︸

basis dynamics

e(0) = Qr(0)u0.

Proposition (general case)

The approximation error e satisfies ‖e(t)‖X ≤ ∆r(t), ∀t ∈ [0, T ], where

∆r(t) :=∫ t

0γ(s)e

∫ ts β(τ)dτds + ‖e(0)‖Xe

∫ t0 β(τ)dτds,

γ(t) = ‖Qr(t)(f(ur(t), t)− V ′r(t)αr(t))‖X ,

β(t) = LX [f ](ur(t)).

In practice : A first order linearization of f is used [Wir14]

LX [f ](yr(t)) ≤ LX [J(ur(t))] + o(‖e‖X)

with J(ur) ∈ Rd×d the Jacobian matrix of f at ur.


Test case 1

Deterministic dvection equation with periodic boundary conditions

∂

∂tu(x, t) + a

∂

∂xu(x, t) = 0, u0(x) =

1√2πe− (x+0.06)2

0.052

• Spatial domain Ω = (−1, 1) and time interval I = (0, 1).

• The advection velocity is a = 1.325.

Discretization :

1 Finite difference upwind scheme on a regular mesh Ω = xid+1i=0 where

xi = −1 + iδx and δx = 2d+1

with d = 200 nodes.

d

dtu(t) = Au(t), u(0) = (u0)di=1.

2 Time integration with explicit Euler scheme on uniform discretizationI = tkKk=1, with tk = kδt and δt = δx/2

uk+1 = (Id + δtA)uk.

M. Billaud-Friess — Workshop MOMAS 2015 Numerical experiments 14/28

Deterministic problem

Comparison : Confrontation of two projection methods with

1 fixed reduced space Xr = spanPOD`(u0, . . . ,uK)2 time dependent reduced space Xk

r = spanuk

Method n1

dim(Xr) = ` E2 E∞

1 0.99862 15 0.99592 110 0.95843 0.4123220 0.070717 0.01070550 1.6532e-10 2.9288e-11100 4.9305e-13 4.2383e-14

Tab.1 : Relative error

Ep = ‖ur − u‖(0,T ),p/‖u‖(0,T ),p

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

−1

0

1

2

3

4

5

x

uur for ` = 1ur for ` = 5ur for ` = 10ur for ` = 20ur for ` = 50ur for ` = 100

Fig.1 : Approximation compared tothe exact solution at T = 1

1 For large r, fine precision limits due to POD

2 With r = 1, E2 = 1.9235e-15, E∞ =3.9074 e-15 interpolant method


Test case 2

Advection-diffusion eq. with homogeneous Dirichlet boundary conditions

∂

∂tu(x, t, ξ) + a(ξ)

∂

∂xu(x, t, ξ)− µ(ξ)

∂

∂x2u(x, t, ξ) = 0, u0(x) = x(2− x)2e2x.

• Spatial domain Ω = (0, 2) and time interval I = (0, 1),

• µ(ξ) = 0.5(2 + cos(πξ)2) and a(ξ) = 0.1 sin(πξ) with ξ ∼ N (0, 1).

Discretization :

1 Finite difference scheme on a regular mesh with d = 200 nodes

d

dtu(t) = A(ξ)u(ξ, t), u(0) = u0

2 Time integration with implicit Euler scheme with K = 200

(IX − δtA(ξ))uk+1(ξ) = uk(ξ)


First results

Results for T-greedy algorithm r = 10

ξ = 0 ξ = 0.2 ξ = −0.5

0 2 · 10−24 · 10−26 · 10−28 · 10−2 0.10

0.5

1

1.5

2

t

x

0 2 · 10−24 · 10−26 · 10−28 · 10−2 0.10

0.5

1

1.5

2

tx

0 2 · 10−24 · 10−26 · 10−28 · 10−2 0.10

0.5

1

1.5

2

t

x

ξ = 0 ξ = 0.2 ξ = −0.5

0 0.5 1 1.5 20

2

4

6

8

x

uur

0 0.5 1 1.5 20

2

4

6

8

x

uur

0 0.5 1 1.5 20

2

4

6

8

x

uur

Fig.3 : Surface contours of ur (top) and comparison to the exact solution at T = 1 (bottom)


T-greedy vs. POD-greedyGreedy algorithm stopped for r = rmax

Data : Use of ∆NL, with #Ξtrain = 30 and ` = 1 for POD-greedy

0 5 10 15 2010−14

10−11

10−8

10−5

10−2

rank r

sup E2

sup E∞mean E2

mean E∞

0 5 10 15 20

10−7

10−5

10−3

10−1

rank r

sup E2

sup E∞mean E2

mean E∞

Fig. 4 : Relative error with respect to r : for T-greedy (left) and POD-greedy (right) methods

• Methods provide approximation with good keeping r small.

• POD-greedy converges more slowly than T-greedy.

• Coarser approximation for POD-greedy that requires richer reduced space.


Results for r = 10 and ξ = 0.2

0 50 100 150 20010−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

time index k

‖e‖X‖∆NL‖X‖∆L‖X

0 50 100 150 200

2

4

6

8

10

time index k

κNLκL

0 50 100 150 20010−4

10−3

10−2

10−1

100

time index k

‖e‖X‖∆NL‖X‖∆L‖X

0 50 100 150 200100

101

102

103

time index k

κNLκL

Fig. 5 : T-greedy (top) and POD-greedy (above)algorithm

Denoting the effectivity index as

κL,NL = ‖∆L,NL‖X/‖e‖X

¬ For T-greedy algorithm : efficienterror estimates for with ∆NL thesharpest.

maxt‖κL(k)‖X max

k‖κNL(k)‖X

9.9393 3.72878

For the POD-greedy algorithm :coarse error estimates with ∆NL

the sharpest.

maxk‖κL(k)‖X max

k‖κNL(k)‖X

572.101 47.0514


T-greedy vs. POD-greedyGreedy alg. with stopping criterion in precision ε

Data : Use of ∆NL, with #Ξtrain = 30 and ` = 1 for POD-greedy

Strong greedy algorithmε T-G POD-G

1e-1 5 71e-2 6 111e-3 8 151e-4 9 191e-5 11 231e-6 12 27

weak-greedy algorithmε T-G POD-G

1e-1 6 121e-2 7 171e-3 8 211e-4 11 251e-5 12 -1e-6 14 -

Tab. 3 : Reduced space dimension with respect to ε with exact error criterion (left), witherror estimate criterion (right)

• Small reduced spaces with T-greedy.

• Influence of the error estimator especially for the POD-greedy algorithm


Test case 3

Burgers’s equation with homogeneous Dirichlet boundary conditions

∂

∂tu(x, t, ξ)− µ(ξ)

∂2

∂x2u(x, t, ξ) + u(t, x, ξ)

∂

∂xu(x, t, ξ) = b(x)Ts(t, ξ), u(x, 0) = 0,

• Domain Ω = (0, 1) and time interval I = (0, 1)

• µ(ξ) = ξ with ξ ∼ U(0.01, 0.06).

• Source terme s(t) = (s1(t), s2(t)), b(x) = (b1(x), b2(x)) with

s1(t) = sin(4πt) s2(t) = 1[0.2,0.4](t),

b1(x) = 4e−( x−0.20.03

)21[0.1,0.3](x), b2(x) = 4 · 1[0.6,0.7](x)

Discretization : Finite difference scheme on a regular mesh with d = 200 nodes andtime integration with semi-implicit Euler scheme on uniform discretization of I withK = 300

(IX − dtA(ξ))uk+1(ξ) = uk(ξ) + dt[f(uk) +Bs(tk)]


First results


ξ = 0.01 ξ = 0.035 ξ = 0.06

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

t

x

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

tx

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

t

x

ξ = 0.01 ξ = 0.035 ξ = 0.06

0 0.2 0.4 0.6 0.8 1

0

0.1

0.2

0.3

uur

0 0.2 0.4 0.6 0.8 1

0

5 · 10−2

0.1

0.15

uur

0 0.2 0.4 0.6 0.8 1

0

5 · 10−2

0.1

uur

Fig.7 : Surface contours of ur (top) and comparison to the exact solution at T = 1 (bottom)


Study of the T-greedy algorithmGreedy algorithm stopped for r = rmax

Data : #Ξtrain = 30 and use of ∆NL


0 2 4 6 8 10 12 1410−12

10−9

10−6

10−3

100

rank r

sup E2sup E∞mean E2mean E∞

0 100 200 30010−14

10−11

10−8

10−5

10−2

time index k

ξ = 0.01ξ = 0.035ξ = 0.06

0 100 200 3000

50

100

time index k

κN

L

ξ = 0.01ξ = 0.035ξ = 0.06

Fig. 8 Relative error with respect to r Fig.9 : Error est. (left) and effectivity index (right)

• Good convergence with rank

• Error estimate more pessimistic forthis problem

• Influence of ξ on the error estimate

ξ maxt‖κNL(t)‖X

0.01 110.0640.035 26.85780.06 25.7706


Study of the T-greedy algorithmGreedy alg. with stopping criterion in precision ε

Strong greedy alg. weak-greedy alg.ε ξ = 0.01 ξ = 0.035 ξ = 0.06 ξ = 0.01 ξ = 0.035 ξ = 0.06

1e-1 4 4 4 7 7 61e-2 5 5 5 8 7 61e-3 7 8 7 9 9 81e-4 8 9 8 11 10 91e-5 10 11 9 12 12 101e-6 10 12 10 13 11 11

Tab. 6 : Reduced space dimension with respect to ε

• Richer reduced spaces needed (e.g. for ξ = 0.01)


Test case 4

Multi-parameter convection-reaction-diffusion with dirichlet boundary conditions

∂

∂tu(x, t, ξ)−µ(ξ)4u(x, t, ξ)+c(x, ξ)·∇u(x, t, ξ)+κ(ξ)u(x, t, ξ) = b(x), u(x, 0) = 0

• Domain Ω = (0, 1)2 and time interval I = (0, 0.03)

• µ(ξ) = 1 + 0.2ξ1, c(x, ξ) = 250(1 + 0.2ξ2)(x2− 12, 1

2−x1)T , κ(ξ) = 10(1 + 0.2ξ3)

ξ = (ξ1, . . . , ξ3) where ξi are i.i.d. ξi ∼ U(−1, 1).

• Source term b(x) = 1(0.7,0.8)2(x).

Discretization :

• P1 finite element approximation with d = 2212 nodes

M d

dtu(t, ξ) = A(ξ)u(t, ξ) + b⇒ d

dtu(t, ξ) = A(ξ)u(t, ξ) + b,

with M = LT L and Lu = u.

• Semi-implicit Euler scheme with K = 100

(Id − dtA(ξ))uk+1(ξ) = uk(ξ) + dt b


First results, r = 10 and ξ = (−0.2244, 0.6744, 0.5326)

Fig.10 : ur for t = T2

(left) and t = T (right)

Fig.11 : Absolute error u− ur


Conclusions & perspectives

T-greedy method : A low-rank approximation method, with sub-space point of view,for solving parameter dependent dynamical systems that can be interpreted as a RBmethod involving time-dependent reduced space.

3 Practical method with efficient error estimates, validated for both linear andnon-linear cases

3 An alternative to POD-greedy method

Improvement and further explorations

• Efficient treatment of non-linearity : combining EIM and T-greedy strategies

• Analysis of the T-greedy algorithm

Thank you for your attention

Any questions ?

M. Billaud-Friess — Workshop MOMAS 2015 Conclusions 27/28

Bibliography

[Bin11] P. Binev, A. Cohen, W. Dahmen, R. De Vore, G. Petrova, P. WojtaszczykConvergence Rates for Greedy Algorithms in Reduced Basis Algorithms. SIAMJournal on Mathematical Analysis, 2011.

[Buff12] A. Buffa, Y. Maday, A. Patera, C. Prud’homme, G. Turicini A priori convergence ofthe greedy algorithm for the parametrized reduced basis. ESAIM-Math. Model.Numer. Anal., 2012

[Haas11] B. Haasdonk, M. Ohlberger, M. Efficient reduced models and a posteriori errorestimation for parametrized dynamical systems by offline/online decomposition.Mathematical and Computer Modelling of Dynamical Systems, 2011.

[Haas13] B. Haasdonk Convergence Rates of the POD-Greedy method. M2AN Math.Model. Numer. Anal., 2013.

[Hom07] C. Homescu, L.R. Petzold, R. Serban Error Estimation for Reduced-Order Modelsof Dynamical Systems. j-SIAM-REVIEW 2007.

[Koch07] O. Koch, C. Lubich Dynamical Low-Rank Approximation. SIAM Journal onMatrix Analysis and Applications, 2007.

[Sap09] T.P. Sapsis, P.F.J. Lermusiaux Dynamically orthogonal field equations forcontinuous stochastic dynamical systems. Physica D : Nonlinear Phenomena, 2009.

[Wir14] D. Wirtz, D.C. Sorensen, B. Haasdonk A-posteriori error estimation for DEIMreduced nonlinear dynamical systems. SIAM J. Sci. Comput., 2014.

M. Billaud-Friess — Workshop MOMAS 2015 Conclusions 28/28

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