reduced basis methods for nonlinear parametrized …reduced basis methods for nonlinear parametrized...

Reduced Basis Methods for

Nonlinear Parametrized Partial Differential Equations

M. Grepl

Advisory Board MeetingRWTH Aachen University

July 12, 2011

Aachen Institute for Advanced Studyin Computational Engineering Science

AICES Progress since 2009

• Overview of the AICES project

• New Principal Investigators

• New Young Researchers

• New facilities

• Recent highlights

• Goals and schedule of Advisory Board meeting

Outline

2

• Excellence Initiative Graduate School

• First period Nov 2006–Oct 2012

• Annually 1 M! + overhead

• Proposed for 2013–2017

• Funded personnel growth:

Overview of the AICES Project

3

0

4

8

12

16

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28

32

36

40

1 3 3 3 3 3 3

1 1 3 3 4 41

1 1 1 11

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912

18 201

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02/200712/2007

12/200812/2010

Service Team

Junior Research Group Leaders

Adjunct Professors

Doctoral Fellows

Master Students

Postdoctoral Research Associates

• Computational engineering science is maturing

• “Old” challenges are still here:• complexity increasing intricacy of analyzed systems• multiscale interacting scales considered at once• multiphysics interacting physical phenomena

• AICES concentrates on areas of synthesis:• model identification and discovery supported by

model-based experimental analysis (MEXA)• understanding scale interaction and scale integration• optimal design and operation of engineered systems

• Inspiration: Collaborative Research Center 540• established in 1999, continued until 2009• Marquardt: coordinator, 6 AICES PIs: project leads

Overview of AICES Academic Aims

SFB 540

4

Workshop on Model Order Reduction of Transport-dominated Phenomena

Brandenburg-Berlin Academy of Sciences and Humanities

May 19-20, 2015

Acknowledgments

Collaborators:

I Eduard Bader

I Mark Karcher

I Dirk Klindworth

I Robert O’Connor

I Mohammad Rasty

I Karen Veroy

Sponsors:

I Excellence Initiative of the German federal and state governments

I German Research Foundation through Grant GSC 111

M. Grepl (RWTH Aachen) 1

Outline

PART I: A Reduced Basis Primer

I Introduction

I Key Ingredients

I Extensions & References

PART II: Nonlinear (and Nonaffine) Problems

I Empirical Interpolation Method

I Nonlinear Elliptic Problems

I Extensions & References

I Numerical Results

Nonlinear Reaction Diffusion Systems


Part I

A Reduced Basis Primer


Introduction

The Reduced Basis Method

is a model order reduction technique that provides

rapid and reliable approximation of solutions

to parametrized partial differential equations (µPDEs)

for design and optimization, control, parameter estimation, . . .

— i.e., for parameter space exploration.



FE SPACE

a(u(µ), v;µ) = f(v;µ), for all v ∈ X



FE SPACEHIGH-DIMENSIONAL




FE SPACE

SNAPSHOTS

u(µi)

HIGH-DIMENSIONAL

XN = spanu(µi) , i = 1, . . . , N



FE SPACE

SNAPSHOTS

u(µi)

EXACT SOLUTION

u(µ)

HIGH-DIMENSIONAL




FE SPACE

u(µi)SNAPSHOTS

(1) APPROXIMATIONuN(µ)

EXACT SOLUTION

u(µ)

HIGH-DIMENSIONAL

a(uN(µ), v;µ) = f(v;µ), for all v ∈ XN .



FE SPACE

u(µi)SNAPSHOTS

(2) ERROR BOUND

∆N(µ)

(1) APPROXIMATIONuN(µ)

EXACT SOLUTION

u(µ)

HIGH-DIMENSIONAL

‖u(µ)− uN(µ)‖X ≤ ∆uN

(µ) :=‖rN(·;µ)‖X ′

αLB(µ)



FE SPACE

u(µi)SNAPSHOTS

(2) ERROR BOUND

∆N(µ)

uN(µ)(1) APPROXIMATION

EXACT SOLUTION

u(µ)

HIGH-DIMENSIONAL

(3) OFFLINE-ONLINE COMPUTATIONAL DECOMPOSITION

a(w, v;µ) =Q∑q=1

θq(µ) aq(w, v), ∀w, v ∈ X︸︷︷︸µ-DEPENDENTCOEFFICIENTS

︸︷︷︸µ-INDEPENDENTBILINEAR FORMS



FE SPACE

u(µi)SNAPSHOTS

(2) ERROR BOUND

∆N(µ)

uN(µ)(1) APPROXIMATION

EXACT SOLUTION

u(µ)

HIGH-DIMENSIONAL

(3) OFFLINE-ONLINE COMPUTATIONAL DECOMPOSITION(4) GREEDY ALGORITHM

µn+1 = arg maxµ∈Ds

∆un(µ)

‖uN(µ)‖X


Problem Statement

We wish to compute, for any parameter µ ∈ D,

s(µ) = `(u(µ);µ) OUTPUT

where u(µ) ∈ X satisfies

a(u, v;µ) = f(v;µ), for all v ∈ X , FEMN (µ)

and

f(v;µ), `(v;µ) are bounded linear functionals

a(·, ·;µ) : X × X → R is continuous and coercive

for all µ ∈ D.


(1) Approximation

We letXN = span

u(µi)︸︷︷︸ , i = 1, . . . , N

SNAPSHOTS

and compute our approximation as

sN(µ) = `(uN(µ);µ)

where uN(µ) ∈ XN satisfies

a(uN , v;µ) = f(v;µ), for all v ∈ XN . ROMN(µ)


(1) Approximation (Algebraic Formulation)

We let

XN =[ζ1︸︷︷︸ ζ2︸︷︷︸ . . . ζN︸︷︷︸ ] s.t. uN(µ) = XNuN(µ)

X -ORTHONORMAL SNAPSHOTS ζi “=” u(µi)


sN(µ) = L(µ)TXN︸︷︷︸ uN(µ)

= LTN

(µ) uN(µ)

where uN(µ) ∈ RN satisfies

XTNA(µ)XN︸︷︷︸ uN(µ) = XT

NF (µ)︸︷︷︸

AN(µ) uN(µ) = FN(µ)

How do we quantify the error?


(2) Error Estimation

Let α(µ) be the coercivity constant of a

α(µ) := infv∈X

a(v, v;µ)

‖v‖2Xand define residual rN(v;µ) := f(v;µ)− a(uN , v;µ), ∀v ∈ X .

The error, eN(µ) := u(µ)− uN(µ), then satisfies

a(eN , v;µ) = rN(v;µ), ∀ v ∈ X ,

and, for any µ ∈ D and αLB(µ) ≤ α(µ), we have

‖eN(µ)‖X ≤‖rN(·;µ)‖X ′αLB(µ)︸︷︷︸

=: ∆uN(µ)

and |s− sN | ≤ ‖`‖X ′∆uN︸︷︷︸

=: ∆`N(µ)

How do we compute uN , sN ,∆uN,∆`

Nefficiently?


(3) Offline/Online Decomposition

We assume a(v, w;µ) =Q∑q=1

θq(µ) aq(v, w)︸︷︷︸µ-DEPENDENTCOEFFICIENTS


so that AN(µ)︸︷︷︸RB-MATRIX

= XTNA(µ)︸︷︷︸

FE-MATRIX

XN =Q∑q=1

θq(µ) XTNAqXN︸︷︷︸

µ-INDEPENDENT


(3) Offline/Online Decomposition – Example

Affine Parameter Dependence: Example 1

Parameters:µ = ( k0, k1, k2, k3, k4︸︷︷︸

THERMALCONDUCTIVITIES

)

Governing Equation:

−ki∇2yi = 0

in Ωi, i = 0 to 4

The matrix A(µ) then represents the bilinear form

a(v, w;µ) =4∑i=0

ki︸︷︷︸θq(µ)

∫Ωi

∇v · ∇w︸︷︷︸“=”Aq




x1

x2

1

PARAMETRIZED DOMAIN

Ω(µ) µ

Bilinear Form on Parameter-Dependent Domain

a(v, w;µ) =

∫Ω(µ)

(∂v

∂x1

∂w

∂x1

+∂v

∂x2

∂w

∂x2

)︸︷︷︸

∇v·∇w

dΩ(µ), v, w ∈ H1(Ω(µ))

After (Affine) Mapping to a Reference Domain

a(v, w;µ) =

∫Ω

(∂v

∂x1

∂w

∂x1

+1

µ2

∂v

∂x2

∂w

∂x2

)µdΩ, v, w ∈ H1(Ω)




x2Ω(µ)

x1 x1

x2µ

1

PARAMETRIZED DOMAIN

1

REFERENCE DOMAIN

Ω 1

Bilinear Form on Parameter-Dependent Domain

a(v, w;µ) =

∫Ω(µ)

(∂v

∂x1

∂w

∂x1

+∂v

∂x2

∂w

∂x2

)︸︷︷︸

∇v·∇w

dΩ(µ), v, w ∈ H1(Ω(µ))

After (Affine) Mapping to a Reference Domain

a(v, w;µ) =

∫Ω

(∂v

∂x1

∂w

∂x1

+1

µ2

∂v

∂x2

∂w

∂x2

)µdΩ, v, w ∈ H1(Ω)


(3) Offline/Online Decomposition

We assume a(v, w;µ) =Q∑q=1

θq(µ) aq(v, w)︸︷︷︸µ-DEPENDENTCOEFFICIENTS


so that AN(µ)︸︷︷︸RB-MATRIX

= XTNA(µ)︸︷︷︸

FE-MATRIX

XN =Q∑q=1

θq(µ) XTNAqXN︸︷︷︸

µ-INDEPENDENT

Since all required quantities can be decomposed in this manner, we can

OFFLINE: Form and store µ-independent quantities

at cost O(N ∗)

ONLINE: For any µ ∈ D, compute approx and error bounds

at cost O(QN2 +N3) and O(Q2N2)

How do we choose the snapshots?


(4) (Weak) Greedy Algorithm

Given X2 = spanu(µ1), u(µ2), how do we choose µ3?

µ3 = arg maxµ∈Ds

∆2(µ)

‖u2(µ)‖X

where Ds ⊂ D is a finite train sample

X3 = spanu(µ1) u(µ2) u(µ3)

Key points:

I ∆n(µ) is sharp and inexpensive to compute (online)

I Error bounds enable choice of good approximation spaces




µ1

∆n

‖un‖X

µ2


∆2(µ)

‖u2(µ)‖X



Key points:






∆n

‖un‖X

µ2µ1


∆2(µ)

‖u2(µ)‖X



Key points:






µ3

∆n

‖un‖X

µ2µ1


∆2(µ)

‖u2(µ)‖X



Key points:




Summary

The reduced basis method provides

accurate uN ≈ u APPROX

reliable ∆N ≥ ‖u− uN‖X ERR EST

efficient surrogates cost O(N∗) DECOMP

N small GREEDY

to solutions of parametrized PDEs

for the many-query, real-time,

and slim-computing contexts.


Extensions & References

Review Articles: [RHP08], [QRM11], . . .

Book: [HRS15]

Reduced Basis Methods for:

I elliptic coercive [PRV+02], [PRVP02], . . .

I elliptic noncoercive [VPRP03], [MPR02], . . .

I saddle point (Stokes) [Rov03], [RV07], [GV12], . . .

I parabolic [GP05], [RMM06], [HO08], [UP14], . . .

I hyperbolic (transport-dominated) [PR07], [HKP11], [DPW14], . . .

Disclaimer:

I List contains – to the best of my knowledge – only the first paperson these topics and is thus not meant to be exhaustive

I References to nonaffine and nonlinear problems to follow . . .M. Grepl (RWTH Aachen) 16


References by topic for:

I error bounds (adjoint techniques) [PRV+02], [GP05], . . .

I stability factor lower bounds [VRP02], [HRSP07], [HKC+10], [CHMR09],

I a priori convergence [MPT02], . . .

I greedy algorithm

original papger [VPRP03]

convergence analysis [BMP+12], [BCD+11], [DPW12], [Haa13], . . . variations (hpRB, ...) [EPR10], [BTWGvBW07], [BTWG08], . . .

I geometry parametrization [RHP08], [LR10], . . .

I multiscale problems [Ngu08]

I stochastic problems [BBM+09], [BBL+10]

I . . .

Disclaimer: see previous slide . . .M. Grepl (RWTH Aachen) 17

Part II

Nonlinear (and Nonaffine) Problems


Nonlinear Problems

Distinguish between problems involving

I quadratic nonlinearities

– (Petrov-)Galerkin projection

– A posteriori error bounds: Brezzi-Rappaz-Raviart [BRR80]

– Offline/online: dominant online cost O(N4)

– Ref.: [VPP03], [VP05], [Dep08], [DL12], [RG12], [MPU14], [Yan14], . . .

I higher-order or nonpolynomial nonlinearities

– Additional approximation of nonlinearity (EIM)

– A posteriori error estimation (in general no bounds)

– Offline-online decomposition possible

– References: to follow . . .


Problem Statement

We wish to compute, for any parameter µ ∈ D,

s(µ) = `(u(µ);µ)

where u(µ) ∈ X satisfies

a(u, v;µ) +

∫Ωgnl(u;x;µ) v = f(v), ∀v ∈ X ,

andf(v;µ), `(v;µ) are bounded linear functionals

a(·, ·;µ) : X × X → R is continuous and coercive

and the nonlinearity satisfies

– gnl : R× Ω×D → R continuous;

– gnl(u1;x;µ) ≤ gnl(u2;x;µ), ∀u1 ≤ u2;

– u gnl(u;x;µ) ≥ 0, ∀u ∈ R, for any x ∈ Ω, µ ∈ D.


RB-Galerkin Approximation

We letXN = span

u(µi)︸︷︷︸ , i = 1, . . . , n

SNAPSHOTS


sN(µ) = `(uN(µ);µ)

where uN(µ) ∈ XN satisfies

a(uN , v;µ) +

∫Ωgnl(uN ;x;µ) v = f(v), ∀ v ∈ XN .

Can we still compute uN efficiently?


RB-Galerkin Approximation

Sample Computation:

We expand uN(µ) =

N∑j=1

uN j(µ) ζj ,

and obtain (v = ζi, 1 ≤ i ≤ N )∫Ωgnl(uN(µ);x;µ) ζi =

∫Ωgnl

(N∑j=1

uN j(µ) ζj;x;µ

)ζi

Issues

I Online assembly requires N -dependent cost

I Similar to nonaffine problems: assume gna is nonaffine, then

b(ζi;µ) =

∫Ωgna(x;µ) ζi e.g. e−(x−µ

σ)2

requires N -dependent cost.


Empirical Interpolation Method

Main Idea:

gna(x;µ) ≈ gnaM

(x;µ) =M∑m=1

ϕMm(µ)︸︷︷︸EIM

qm(x)︸︷︷︸Collateral RB

Then: b(ζi;µ) =

∫Ωgna(x;µ) ζi ≈

∫ΩgnaM

(x;µ) ζi

=M∑m=1

ϕMm(µ)

∫Ωqm(x) ζi ,

Issues

I How do we choose XgM = spanqm(x), m = 1, . . . ,M?

I Can we compute the coefficients ϕMm(µ) efficiently?

I How do we quantify the (interpolation) error introduced?



Greedy Procedure [MNPP07]

Initialize: Choose µg1 ∈ D and set

xT1 = arg maxx∈Ω|g(x;µg1)|, q1 =

g(x;µg1)

g(xT1 ;µg1).

For 1 ≤M ≤Mmax − 1:

I Sample Set

µgM+1 ≡ arg maxµ∈Ξ

gtrain

‖g(·;µ)− gM(·;µ)‖L∞(Ω)

I Interpolation Points

xTM+1

= arg maxx∈Ω|rM+1(x)|,

where rM+1(x) = g(x;µgM+1)− gM(x;µgM+1)

I EIM Space

qM+1(x) =rM+1(x)

rM+1(xTM+1)



Interpolation Points, Samples, and Spaces:

T gM = xT1 ∈ Ω, . . . , xTM∈ Ω,

SgM ≡ µg1 ∈ D, . . . , µgM ∈ D, and associated

XgM = spanq1, . . . , qM, 1 ≤M ≤Mmax,

chosen by greedy procedure.

Approximation: for given µ ∈ D

g(x;µ) ≈ gM(x;µ) =M∑m=1

ϕMm(µ) qm(x),

whereM∑m=1

qm(xTn) ϕMm(µ) = g(xTn ;µ), 1 ≤ n ≤M.

Note: BMnm = qm(xTn) lower triangular and invertible.

How do we quantify the (interpolation) error?



“Next Point” Estimator

Given an approximation gM(x;µ) for M ≤Mmax − 1, we define

εM(µ) ≡ |g(xTM+1

;µ)− gM(xTM+1

;µ)|

and, if g( · ;µ) ∈ XgM+1, we have

‖g( · ;µ)− gM( · ;µ)‖L∞(Ω) ≤ εM(µ).

Note

I The estimator εM(µ) is very cheap to evaluate

I In general g( · ;µ) 6∈ XgM+1, and hence our estimator εM(µ) is

indeed a lower bound

I An a posteriori error bound exists [EGP10], but does not extend tothe nonlinear case


EIM-RB-Galerkin Approximation

Given µ ∈ D, we evaluate

sN,M(µ) = `(uN,M(µ);µ)

where uN,M(µ) ∈ XN satisfies

a(uN,M , v;µ) +

∫Ωg

nl,uN,MM (x;µ) v = f(v), ∀ v ∈ XN .

and

gnl,uN,MM (x;µ) ≡

M∑m=1

ϕMm(µ)qm(x),

withM∑j=1

BMij ϕM j(µ) = gnl(uN,M(xTi ;µ);xTi ;µ), 1 ≤ i ≤M.

Admits offline/online treatment:online cost (per Newton iteration) O(M2 +MN2 +N3)

How do we quantify the (RB+EIM) error?


Error Estimation

Define the dual norm

ϑqM = supv∈X

∫Ω qM+1 v

‖v‖Xand the residual, for all v ∈ X ,

rN,M(v;µ) := f(v)− a(uN,M , v;µ)−∫

Ωg

nl,uN,MM (x;µ) v.

If gnl(uN,M ;x;µ) ∈ XgM+1, the error, eN,M := u− uN,M , is

bounded by

‖eN,M(µ)‖X ≤1

αLB(µ)

(‖rN,M(·;µ)‖X ′ + εM(µ)ϑq

M

).

Note

I Admits offline/online treatment: online cost O(M2N2)

I In general gnl(uN,M ;x;µ) 6∈ XgM+1, and hence the bound is

more like an estimator


Error Estimation

Define the dual norm

ϑqM = supv∈X

∫Ω qM+1 v

‖v‖Xand the residual, for all v ∈ X ,

rN,M(v;µ) := f(v)− a(uN,M , v;µ)−∫

Ωg

nl,uN,MM (x;µ) v.

If gnl(uN,M ;x;µ) ∈ XgM+1, the error, eN,M := u− uN,M , is

bounded by

‖eN,M(µ)‖X ≤1

αLB(µ)

(‖rN,M(·;µ)‖X ′ + εM(µ)ϑq

M

).

Note

I Admits offline/online treatment: online cost O(M2N2)

I In general gnl(uN,M ;x;µ) 6∈ XgM+1, and hence the bound is

more like an estimatorM. Grepl (RWTH Aachen) 28

Nonlinear Parabolic Problems

We wish to compute, for any µ ∈ D,

s(t;µ) = `(u(t;µ))

where u(t;µ) ∈ Y , satisfies u(0;µ) = 0

m(u(t;µ), v;µ) + a(u(t;µ), v;µ)

+

∫Ωgnl(u(t;µ);x;µ) v = b(v)u(t), ∀v ∈ Y.

Key Ingredients:

I FE[x]-FD[t] truth approximation

I POD/Greedy algorithm to construct XN and XgM

I Error bound/estimator for space-time energy norm

I Online cost: O(M2 +MN2 +N3)† plus O(KM2N2).

† Cost per Newton iteration per timestep.



Empirical Interpolation Method:

I methodology [BMNP04], [GMNP07], [MNPP07], . . .

I error bounds [EGP10], [EGPR13], . . .

I generalized EIM [MM13], [MMPY15], . . .

I variations [CS10], . . .

Related Methods:

I best points interpolation [NPP08], . . .

I adaptive cross approximation [Beb00], [Beb11], . . .

I gappy POD [ES95], [BTDW04], [Wil06], [GFWG10], . . .

I missing point estimation [AWWB08], . . .

For a comparison of these methods: [BMS14]



RB for nonaffine problems:

I elliptic [BMNP04], [GMNP07], [Ngu07], [Roz09], . . .

I parabolic [GMNP07], [Gre12a], [KGV12], . . .

RB for nonlinear problems:

I elliptic [GMNP07], [NP08], [CTU09], . . .

I parabolic [GMNP07], [Gre12a], [DHO12], [Gre12b], . . .

Issues

I Convergence and stability of EIM-RB approximation

I Greedy EIM algorithm requires (truth) solution at all µ ∈ Ds

I Simultaneous EIM + RB [DP15]


Model Problem

Given µ = (µ1, µ2) ∈ D ≡ [0.01, 10]2, evaluate Ω =]0, 1[2

sk(µ) =

∫Ωuk(µ)

where uk(µ) ∈ X , 1 ≤ k ≤ K, satisfies u0(µ) = 0

1∆tm(uk(µ)− uk−1(µ), v) + a(uk(µ), v)

+

∫Ωgnl(uk(µ);x;µ) v = b(v) sin(2πtk), ∀ v ∈ X ,

with gnl(uk(µ);x;µ) = µ1eµ2 uk(µ) − 1

µ2.

Truth Approximation:

I Space: X ⊂ X e ≡ H10(Ω) with dimension N = 2601;

I Time: I = (0, 2], ∆t = 0.01, and thus K = 200.M. Grepl (RWTH Aachen) 32

Sample Results

Truth solution u(tk;µ) at time tk = 25∆t and

µ = (0.01, 0.01) µ = (10, 10)

0

0.5

1

0

0.5

1−1.5

−1

−0.5

0

0.5

1

1.5

x(1)

Solution for µ = (0.01,0.01), tk = 25 ∆ t

x(2)

yk (µ)

0

0.5

1

0

0.5

1−1.5

−1

−0.5

0

0.5

1

1.5

x(1)

Solution for µ = (10,10), tk = 25 ∆ t

x(2)

yk (µ)

b(v) = 100∫Ω v sin(2πx1) sin(2πx2)


Convergence: Energy Norm

0 5 10 15 20 25 30 35 40 45 50 5510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

N

ε N,M

,max

,rel

y

M = 10M = 20M = 40M = 60M = 80M = 100

0 5 10 15 20 25 30 35 40 45 50 5510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

N

∆ N,M

,max

,rel

y

M = 20M = 40M = 70M = 100M = 130M = 160

Results for sample Ξtest ∈ D of size 225.

I “Plateau” in curves for M fixed.

I “Knees” reflect balanced contribution of both error terms.

I Sharp bounds require conservative choice of M .


Convergence: Energy Norm & Output

N M εyN,M,max,rel ∆yN,M,max,rel ηyN,M

1 40 3.83E – 01 1.15E + 00 2.445 60 1.32E – 02 4.59E – 02 2.4310 80 9.90E – 04 3.41E – 03 2.1020 100 9.40E – 05 4.16E – 04 2.7740 140 3.36E – 06 8.75E – 06 1.64

N M εsN,M,max,rel ∆sN,M,max,rel ηsN,M

1 40 9.99E – 01 2.49E + 01 14.15 60 5.35E – 03 1.00E + 00 13010 80 2.57E – 04 7.42E – 02 14620 100 1.43E – 05 9.06E – 03 43640 140 2.85E – 06 1.90E – 04 205

Results for sample Ξtest ∈ D of size 225.


Online Computational Times

N M sN,M(µ, tk) ∆sN,M(µ, tk) s(µ, tk)

1 40 5.42E – 05 9.29E – 05 15 60 9.67E – 05 8.58E – 05 110 80 1.19E – 04 9.37E – 05 120 100 1.71E – 04 1.05E – 04 140 140 3.15E – 04 1.35E – 04 1

Average CPU times for sample Ξtest ∈ D of size 225.

I Computational savings O(103) for ∆sN,M,max,rel < 1%.

I But offline stage much more expensive than for linear case.


Nonlinear Reaction-Diffusion Systems

General formulation: [Gre12b]

∂y(x, t;µ)

∂t= ∇(D(µ)∇y(x, t;µ)) + f(y(x, t;µ);µ)

Self-ignition of a coal stockpile∂T (x,t)∂t

= ∇2T (x, t) + βΦ2 (c(x, t) + 1) e−γ/(T (x,t)+1),

∂c(x,t)∂t

= Le∇2c(x, t)− Φ2 (c(x, t) + 1) e−γ/(T (x,t)+1),

where

γ : Arrhenius number,

β : Prater temperature,

Le : Lewis number,

Φ : Thiele modulus.

s1

s2

T (x, t)

c(x, t)

0.50.20 1 x

1

I Nonlinearity not monotonic: a posterior error bounds not valid.



I Very complex dynamic behavior (depending on parameters).

I Here: µ ≡ γ ∈ [12, 12.6], all other parameters fixed, N = 501

Temperature and Concentration: γ = 12.0

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4Outputs, µ = 12

Time t

Out

put

s1(t;µ): Temperature

s2(t;µ): Concentration

1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

Temperature s1(t;µ)

Con

cent

ratio

n s 2(t

;µ)

Phase Plot: µ = 12.0





Temperature and Concentration: γ = 12.5

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4Outputs, µ = 12.5

Time t

Out

put



1.2 1.4 1.6 1.8 2 2.2 2.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Con

cent

ratio

n s 2(t

;µ)






Temperature and Concenctration: γ = 12.58

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

3.5

4Outputs, µ = 12.58

Time t

Out

put



1.2 1.4 1.6 1.8 2 2.2 2.40

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4


Con

cent

ratio

n s 2(t

;µ)





I Here: µ ≡ γ ∈ [12, 12.6], all other parameters fixed, N = 501NT = Nc = 16, M = 36: tRB/tFEM = 5.12E – 03

FEM solution and RB approximation: γ = 12.6

0 1 2 3 4 5 60

1

2

3

4

Time t

Out

put s

1(t;µ

)

Temperature, µ = 12.6

s1(t;µ)

s1,N

(t;µ)

0 1 2 3 4 5 610

−8

10−6

10−4

10−2

Time t

Rel

. Err

or in

s1(t

;µ)

0 1 2 3 4 5 60

0.20.40.60.8

1

Time t

Out

put s

2(t;µ

)

Concentration, µ = 12.6

s2(t;µ)

s2,N

(t;µ)

0 1 2 3 4 5 610

−8

10−6

10−4

10−2

Time t

Rel

. Err

or in

s2(t

;µ)





FEM, RB, and output interpolation: γ = 12.5

0 1 2 3 4 5 60

1

2

3

4

Time t

Ou

tpu

t s

1(t

;µ)


0 1 2 3 4 5 610

−8

10−6

10−4

10−2

100

Time t

Re

l. E

rro

r in

s1(t

;µ)

s1(t;µ)

s1,N

(t;µ)

s1,int

(t;µ)

RB

Int.

0 1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Time t

Ou

tpu

t s

2(t

;µ)


0 1 2 3 4 5 610

−8

10−6

10−4

10−2

100

Time t

Re

l. E

rro

r in

s2(t

;µ)

s2(t;µ)

s2,N

(t;µ)

s2,int

(t;µ)

RB

Int.





FEM, RB, and output interpolation: γ = 12.5

0.5 0.55 0.6 0.65 0.7 0.75 0.80

1

2

3

4

Time t

Ou

tpu

t s

1(t

;µ)


0.5 0.55 0.6 0.65 0.7 0.75 0.810

−8

10−6

10−4

10−2

100

Time t

Re

l. E

rro

r in

s1(t

;µ)

s1(t;µ)

s1,N

(t;µ)

s1,int

(t;µ)

RB

Int.

0.5 0.55 0.6 0.65 0.7 0.75 0.80

0.2

0.4

0.6

0.8

1

Time t

Ou

tpu

t s

2(t

;µ)


0.5 0.55 0.6 0.65 0.7 0.75 0.810

−8

10−6

10−4

10−2

100

Time t

Re

l. E

rro

r in

s2(t

;µ)

s2(t;µ)

s2,N

(t;µ)

s2,int

(t;µ)

RB

Int.


References I

[AWWB08] P. Astrid, S. Weiland, K. Willcox, and T. Backx. Missing point estimation in models described by properorthogonal decomposition. IEEE Transactions on Automatic Control, 53(10):2237–2251, 2008.

[BBL+10] S. Boyaval, C. Le Bris, T. Lelievre, Y. Maday, N.C. Nguyen, and A.T. Patera. Reduced basis techniquesfor stochastic problems. Arch. Comput. Method. E., 17:435–454, 2010.

[BBM+09] S. Boyaval, C. Le Bris, Y. Maday, N.C. Nguyen, and A.T. Patera. A reduced basis approach forvariational problems with stochastic parameters: Application to heat conduction with variable robincoefficient. Comput. Methods Appl. Mech. Engrg., 198(41-44):3187 – 3206, 2009.

[BCD+11] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk. Convergence rates forgreedy algorithms in reduced basis methods. SIAM J. Math. Anal., 43(3):1457–1472, 2011.

[Beb00] M. Bebendorf. Approximation of boundary element matrices. Numer. Math., 86(4):565–589, 2000.

[Beb11] M. Bebendorf. Adaptive cross approximation of multivariate functions. Constr. Approx., 34(2):149–179,2011.

[BMNP04] M. Barrault, Y. Maday, N.C. Nguyen, and A.T. Patera. An ’empirical interpolation’ method: applicationto efficient reduced-basis discretization of partial differential equations. C. R. Math., 339(9):667 – 672,2004.

[BMP+12] A. Buffa, Y. Maday, A.T. Patera, C. Prud’homme, and G. Turinici. A priori convergence of the greedyalgorithm for the parametrized reduced basis method. ESAIM: Math. Model. Num., 46(03):595–603, 42012.

[BMS14] M. Bebendorf, Y. Maday, and B. Stamm. Comparison of some reduced representation approximations. InA. Quarteroni and G. Rozza, editors, Reduced Order Methods for Modeling and ComputationalReduction, volume 9 of MS&A - Modeling, Simulation and Applications, pages 67–100. SpringerInternational Publishing, 2014.

[BRR80] F. Brezzi, J. Rappaz, and P. A. Raviart. Finite dimensional approximation of nonlinear problems. Numer.Math., 38(1):1–30, 1980. 10.1007/BF01395805.


References II

[BTDW04] T. Bui-Thanh, M. Damodaran, and K. Willcox. Aerodynamic data reconstruction and inverse designusing proper orthogonal decomposition. AIAA Journal, 42(8):1505–1516, 2004.

[BTWG08] T. Bui-Thanh, K. Willcox, and O. Ghattas. Model reduction for large-scale systems withhigh-dimensional parametric input space. SIAM J. Sci. Comput., 30:3270–3288, October 2008.

[BTWGvBW07] T. Bui-Thanh, K. Willcox, O. Ghattas, and B. van Bloemen Waanders. Goal-oriented, model-constrainedoptimization for reduction of large-scale systems. J. Comput. Phys., 224(2):880 – 896, 2007.

[CHMR09] Y. Chen, J.S. Hesthaven, Y. Maday, and J. Rodrıguez. Improved successive constraint method based aposteriori error estimate for reduced basis approximation of 2d maxwell’s problem. ESAIM: Math. Model.Num., null:1099–1116, 10 2009.

[CS10] S. Chaturantabut and D.C. Sorensen. Nonlinear model reduction via discrete empirical interpolation.SIAM J. Sci. Comput., 32(5):2737–2764, 2010.

[CTU09] C. Canuto, T. Tonn, and K. Urban. A-posteriori error analysis of the reduced basis method for non-affineparameterized nonlinear pde’s. SIAM J. Numer. Anal., 47:2001–2022, 2009.

[Dep08] S. Deparis. Reduced basis error bound computation of parameter-dependent Navier-Stokes equations bythe natural norm approach. SIAM Journal of Numerical Analysis, 46(4):2039–2067, 2008.

[DHO12] M. Drohmann, B. Haasdonk, and M. Ohlberger. Reduced basis approximation for nonlinear parametrizedevolution equations based on empirical operator interpolation. SIAM J. Sci. Comput., 34(2):A937–A969,2012.

[DL12] S. Deparis and A.E. Løvgren. Stabilized reduced basis approximation of incompressible three-dimensionalNavier-Stokes equations in parametrized deformed domains. J. Sci. Comput., 50:198–212, 2012.10.1007/s10915-011-9478-2.

[DP15] C. Daversin and C. Prud’homme. Simultaneous empirical interpolation and reduced basis method fornon-linear problems. Technical report, arXiv:1504.06131 [math.AP], 2015.


References III

[DPW12] R. DeVore, G. Petrova, and P. Wojtaszczyk. Greedy algorithms for reduced bases in Banach spaces.Constr. Approx., 2012.

[DPW14] W. Dahmen, C. Plesken, and G. Welper. Double greedy algorithms: Reduced basis methods for transportdominated problems. ESAIM: Mathematical Modelling and Numerical Analysis, 48:623–663, 5 2014.

[EGP10] J.L. Eftang, M.A. Grepl, and A.T. Patera. A posteriori error bounds for the empirical interpolationmethod. C. R. Math., 348:575–579, 2010.

[EGPR13] J.L. Eftang, M.A. Grepl, A.T. Patera, and E.M. Rønquist. Approximation of parametric derivatives bythe empirical interpolation method. Foundations of Computational Mathematics, 13(5):763–787, 2013.

[EPR10] J.L. Eftang, A.T. Patera, and E.M. Rønquist. An ”hp” certified reduced basis method for parametrizedelliptic partial differential equations. SIAM J. Sci. Comput., 32:3170–3200, 2010.

[ES95] R. Everson and L. Sirovich. Karhunen-Loeve procedure for gappy data. JOSA A, 12(8):1657–1664, 1995.

[GFWG10] D. Galbally, K. Fidkowski, K. Willcox, and O. Ghattas. Non-linear model reduction for uncertaintyquantification in large-scale inverse problems. Int. J. Numer. Meth. Engng, 81(12):1581–1608, 2010.

[GMNP07] M.A. Grepl, Y. Maday, N.C. Nguyen, and A.T. Patera. Efficient reduced-basis treatment of nonaffine andnonlinear partial differential equations. ESAIM: Math. Model. Num., 41(3):575–605, 2007.

[GP05] M.A. Grepl and A.T. Patera. A posteriori error bounds for reduced-basis approximations of parametrizedparabolic partial differential equations. ESAIM: Math. Model. Num., 39(1):157–181, 2005.

[Gre12a] M.A. Grepl. Certified reduced basis methods for nonaffine linear time-varying and nonlinear parabolicpartial differential equations. M3AS: Mathematical Models and Methods in Applied Sciences, 22(3):40,2012.

[Gre12b] M.A. Grepl. Model order reduction of parametrized nonlinear reaction-diffusion systems. Computers andChemical Engineering, 43(0):33 – 44, 2012.


References IV

[GV12] A.-L. Gerner and K. Veroy. Certified reduced basis methods for parametrized saddle point problems.SIAM J. Sci. Comput., 34(5):A2812–A2836, 2012.

[Haa13] B. Haasdonk. Convergence rates of the pod-greedy method. ESAIM: Math. Model. Num., 47:859–873, 42013.

[HKC+10] D. B. P. Huynh, D. J. Knezevic, Y. Chen, J.S. Hesthaven, and A.T. Patera. A natural-norm successiveconstraint method for inf-sup lower bounds. Comput. Methods Appl. Mech. Engrg.,199(29-32):1963–1975, 2010.

[HKP11] D. P. B. Huynh, D. J. Knezevic, and A. T. Patera. A laplace transform certified reduced basis method;application to the heat equation and wave equation. C. R. Math., 349:401–405, 2011.

[HO08] B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrizedlinear evolution equations. ESAIM: Math. Model. Num., 42(02):277–302, 2008.

[HRS15] J.S. Hesthaven, G. Rozza, and B. Stamm. Certified Reduced Basis Methods for Parametrized Problems.Springer Briefs in Mathematics. Springer, 2015.

[HRSP07] D. B. P. Huynh, G. Rozza, S. Sen, and A. T. Patera. A successive constraint linear optimization methodfor lower bounds of parametric coercivity and inf-sup stability constants. C. R. Math., 345(8):473–478,2007.

[KGV12] D. Klindworth, M.A. Grepl, and G. Vossen. Certified reduced basis methods for parametrized parabolicpartial differential equations with non-affine source terms. Comput. Methods Appl. Mech. Engrg.,209-212(0):144–155, 2012.

[LR10] T. Lassila and G. Rozza. Parametric free-form shape design with pde models and reduced basis method.Comput. Methods Appl. Mech. Engrg., 199(23-24):1583–1592, 2010.


References V

[MM13] Y. Maday and O. Mula. A generalized empirical interpolation method: Application of reduced basistechniques to data assimilation. In Analysis and Numerics of Partial Differential Equations, volume 4 ofSpringer INdAM Series, pages 221–235. Springer Milan, 2013.

[MMPY15] Y. Maday, O. Mula, A.T. Patera, and M. Yano. The generalized empirical interpolation method:Stability theory on hilbert spaces with an application to the stokes equation. Computer Methods inApplied Mechanics and Engineering, 287(0):310 – 334, 2015.

[MNPP07] Y. Maday, N.C. Nguyen, A.T. Patera, and G.S.H. Pau. A general, multipurpose interpolation procedure:the magic points. Communications on Pure and Applied Analysis (CPAA), 8:383 – 404, 2007.

[MPR02] Y. Maday, A. T. Patera, and D. V. Rovas. A blackbox reduced-basis output bound method fornoncoercive linear problems. Studies in Mathematics and its Applications, D. Cioranescu and J. L. Lions,eds., Elsevier Science B.V., 31:533–569, 2002.

[MPT02] Y. Maday, A.T. Patera, and G. Turinici. Global a priori convergence theory for reduced-basisapproximations of single-parameter symmetric coercive elliptic partial differential equations. C. R. Math.,335(3):289 – 294, 2002.

[MPU14] M.Yano, A.T. Patera, and K. Urban. A space-time hp-interpolation-based certified reduced basis methodfor burgers’ equation. Mathematical Models and Methods in Applied Sciences, 0(0):1–33, 2014.

[Ngu07] N. C. Nguyen. A posteriori error estimation and basis adaptivity for reduced-basis approximation ofnonaffine-parametrized linear elliptic partial differential equations. J. Comput. Phys., 227:983–1006,December 2007.

[Ngu08] N.C. Nguyen. A multiscale reduced-basis method for parametrized elliptic partial differential equationswith multiple scales. J. Comput. Phys., 227(23):9807 – 9822, 2008.

[NP08] N. C. Nguyen and J. Peraire. An efficient reduced-order modeling approach for non-linear parametrizedpartial differential equations. Int. J. Numer. Methods Eng., 76:27–55, 2008.


References VI

[NPP08] N. C. Nguyen, A. T. Patera, and J. Peraire. A ”best points” interpolation method for efficientapproximation of parametrized functions. Int. J. Numer. Methods Eng., 73(4):521–543, 2008.

[PR07] Anthony T. Patera and Einar M. Ronquist. Reduced basis approximation and a posteriori error estimationfor a boltzmann model. Comput. Methods Appl. Mech. Engrg., 196(29-30):2925 – 2942, 2007.

[PRV+02] C. Prud’homme, D. V. Rovas, K. Veroy, L. Machiels, Y. Maday, A. T. Patera, and G. Turinici. Reliablereal-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J.Fluid. Eng., 124(1):70–80, 2002.

[PRVP02] C. Prud’homme, D.V. Rovas, K. Veroy, and A.T. Patera. A mathematical and computational frameworkfor reliable real-time solution of parametrized partial differential equations. ESAIM: Math. Model. Num.,36:747–771, 2002.

[QRM11] A. Quarteroni, G. Rozza, and A. Manzoni. Certified reduced basis approximation for parametrized partialdifferential equations and applications. Journal of Mathematics in Industry, 1(3):1–49, 2011.

[RG12] M. Rasty and M.A. Grepl. Efficient reduced basis solution of quadratically nonlinear diffusion equations.In Proceedings of 7th Vienna Conference on Mathematical Modelling – MATHMOD 2012, 2012.accepted.

[RHP08] G. Rozza, D.B.P. Huynh, and A.T. Patera. Reduced basis approximation and a posteriori error estimationfor affinely parametrized elliptic coercive partial differential equations. Archives of ComputationalMethods in Engineering, 15(3):229–275, 2008.

[RMM06] D. V. Rovas, L. Machiels, and Y. Maday. Reduced-basis output bound methods for parabolic problems.IMA J. Numer. Anal., 26(3):423–445, 2006.

[Rov03] D.V. Rovas. Reduced-Basis Output Bound Methods for Parametrized Partial Differential Equations. PhDthesis, Massachusetts Institute of Technology, February 2003.


References VII

[Roz09] G. Rozza. Reduced basis methods for stokes equations in domains with non-affine parameter dependence.Computing and Visualization in Science, 12:23–35, 2009. 10.1007/s00791-006-0044-7.

[RV07] G. Rozza and K. Veroy. On the stability of the reduced basis method for Stokes equations inparametrized domains. Comput. Methods Appl. Mech. Engrg., 196(7):1244 – 1260, 2007.

[UP14] K. Urban and A.T. Patera. An improved error bound for reduced basis approximation of linear parabolicproblems. Math. Comp., 83:1599–1615, 2014.

[VP05] K. Veroy and A. T. Patera. Certified real-time solution of the parametrized steady incompressibleNavier-Stokes equations: rigorous reduced-basis a posteriori error bounds. Internat. J. Numer. MethodsFluids, 47:773–788, 2005.

[VPP03] K. Veroy, C. Prud’homme, and A.T. Patera. Reduced-basis approximation of the viscous burgersequation: rigorous a posteriori error bounds. C. R. Math., 337(9):619–624, 2003.

[VPRP03] K. Veroy, C. Prud’homme, D. V. Rovas, and A. T. Patera. A posteriori error bounds for reduced-basisapproximation of parametrized noncoercive and nonlinear elliptic partial differential equations. InProceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003. AIAA Paper 2003-3847.

[VRP02] K. Veroy, D.V. Rovas, and A.T. Patera. A posteriori error estimation for reduced-basis approximation ofparametrized elliptic coercive partial differential equations: ”convex inverse” bound conditioners. ESAIM:Contr. Optim. Ca., 8:1007–1028, 2002. Special Volume: A tribute to J.L. Lions.

[Wil06] K. Willcox. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition.Computers and Fluids, 35(2):208 – 226, 2006.

[Yan14] M. Yano. A space-time petrov–galerkin certified reduced basis method: Application to the boussinesqequations. SIAM J. Sci. Comput., 36(1):A232–A266, 2014.


Thank you for your attention!

For questions or comments:

[email protected]


Part III

Extra Slides


Offline/Online Decomposition: Error Bounds

Crucial ingredient: Dual norm of residual ‖rN(·;µ)‖X ′

We expand uN(µ) =N∑j=1

uN j(µ) ζj

and obtain from the definition of the residual and affine dependence

rN(v;µ) = f(v)− a(uN(µ), v;µ)

= f(v)− a(

N∑n=1

uN n(µ)ζn, v;µ

)= f(v)−

N∑n=1

uN n(µ) a(ζn, v;µ

)= f(v)−

N∑n=1

uN n(µ)Qa∑q=1

θqa(µ) aq(ζn, v

)For simplicity, we assume here that f(v) does not depend on µ.



Riesz representation:

(e(µ), v)X = rN(v;µ)

= f(v)−Qa∑q=1

N∑n=1

θqa(µ)uN n(µ)aq(ζn, v

),

Linear Superposition:

⇒ e(µ) = C +Qa∑q=1

N∑n=1

θqa(µ)uN n(µ)Aqn

where

(C, v)X = f(v), ∀v ∈ X ;

(Aqn, v)X = −aq(ζn, v), ∀v ∈ X ,1 ≤ n ≤ N, 1 ≤ q ≤ Qa.



Thus

‖e(µ)‖2X =

(C +

Qa∑q=1

N∑n=1

θqa(µ)uN n(µ)Aqn , ·)X

=

(C, C)X +Qa∑q=1

N∑n=1

θqa(µ)uN n(µ)

2(C,Aqn)X +Qa∑q′=1

N∑n′=1

θq′a (µ)uN n′(µ)(Aqn,A

q′

n′)X



Thus

‖e(µ)‖2X =

(C +

Qa∑q=1

N∑n=1

θqa(µ)uN n(µ)Aqn , ·)X

= (C, C)X +Qa∑q=1

N∑n=1

θqa(µ)uN n(µ)


N∑n′=1


q′

n′)X



Offline: once, parameter independent

I Compute C,Aqn, 1 ≤ n ≤ Nmax, 1 ≤ q ≤ Qa, from

(C, v)X = f(v), ∀v ∈ X ;

(Aqn, v)X = −aq(ζn, v), ∀v ∈ X ,1 ≤ n ≤ N, 1 ≤ q ≤ Qa.

I Form/Store (C, C)X , (C,Aqn)X , (Aqn,Aq′

n′)X ,

1 ≤ n, n′ ≤ Nmax, 1 ≤ q, q′ ≤ Qa.

Complexity depends on N , Qa, and N .



Online: many times, for each new µ(and associated solution uN(µ))

I Evaluate

‖e(µ)‖2X = (C, C)X +Qa∑q=1

N∑n=1

θqa(µ)uN n(µ)


N∑n′=1


q′

n′)X

– O(Q2

aN2)

Complexity depends on N , Qa, but not N .



Summary of computational cost:

OFFLINE —

O(QaNmaxN •) + O(Q2aN

2maxN )

solve Poisson problems form µ-independent inner products;

ONLINE —

O(Q2aN

2)

evaluate ‖e(µ)‖X -sum;

Online cost is independent of N .


reduced basis methods for nonlinear parametrized …reduced basis methods for nonlinear parametrized...

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