A Short Introduction to Reduced Basis Method for ...A Short Introduction to Reduced Basis Method for...

A Short Introduction to Reduced Basis

Method for

Parametrized Partial Di�erential

Equations

Nguyen Thanh Son

Zentrum für Technomathematik - Uni. Bremen

SCiE seminar, Bremen, 20. May 2010

Zentrum fürTechnomathematik

Fachbereich 03Mathematik/Informatik

Outline

1 Introduction

2 Reduced Basis ApproachPreliminariesApproach

3 Summary and outlook

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http://www.math.uni-bremen.de/zetem

http://www.uni-bremen.de



Outline

1 Introduction



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MotivationsIn many situations, one has to solve equations with the variation ofone or some parameters.

1 The viscous Burgers equation:

u∂u

∂x− ν ∂

2u

∂2x= 0;

2 The Helmholtz equation:

52u + k2u = 0;

3 Steady convection-di�usion equations:

D 52 c − ~v 5 c = 0.

The shape of domain considered may also contribute parameter(s)to the formulation of the problem.

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Motivations

Discretize equations by FEM, usually high order basis due tothe complexity of the geometry and the required accuracy,

Want to know about the whole or a part of solutioncorresponding to many values of parameter(s),

Such evaluations are too time-consuming since one has towork with very high order FE basis for each new value ofparameter(s),

Seek a way to reduce basis with which one works, hencereduce computing time.

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Problem Statement

The following problem and the de�nitions and hypothesesmentioned later are considered in the FEM context.Let f and l be a�ne parametric linear form, a an a�ne parametricbilinear form on X . Given µ ∈ D, �nd u(µ) ∈ X s.t.

a(u(µ), v ;µ) = f (v ;µ), ∀v ∈ X (1)

and evaluate

s(µ) = l(u(µ);µ). (2)

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Outline

1 Introduction



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Hypotheses

We only work with linear/bilinear form which are a�ne in parameter

l(v ;µ) =

Ql∑q=1

Θql (µ)lq(v), (3)

f (v ;µ) =

Qf∑q=1

Θqf (µ)f q(v), (4)

a(w , v ;µ) =Qa∑q=1

Θqa(µ)aq(w , v); (5)

in which, l = f , a are continuous on X ; a is symmetric andparametrically coercive. We call this problem compliant.

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De�nitionsInner product and norm:

(((w , v)))µ = a(w , v ;µ), |||w |||µ =√a(w ,w ;µ). (6)

Norm in X : given a �xed µ ∈ D

(w , v)X = (((w , v)))µ, ||w ||X = |||w |||µ. (7)

Coercivity constant and continuity constant

α(µ) = infw∈X

a(w ,w ;µ)

||w ||2X> 0 ∀µ ∈ D; (8)

γ(µ) = supw∈X

supw∈X

a(w , v ;µ)

||w ||X ||v ||X<∞ ∀µ ∈ D. (9)

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Idea and questions

Idea

Replace high order(N ) FE basis by a much lower order basis whichconsists of selected snapshots {u(µn), n = 1, · · · ,N}, socalledReduced Basis(RB) and decompose the whole process into O�ine

and Online stages.

Questions

How to combine these snapshots to approximate solution?

How to choose parameters points µn?

Online operation count and storage independent of N ?

How to estimate an error bound?

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Projecting on Lagrange RB space

Given parameter set {µn, n = 1, · · · ,N}, denote by un = u(µn)snapshots and then de�ne Lagrange RB space

XN = span{un, 1 ≤ n ≤ N}.

The projected problem is: seek uXN(µ) ∈ XN s.t.

a(uXN(µ), v ;µ) = f (v ;µ), ∀v ∈ XN (10)

and then evaluate

sXN(µ) = f (uXN

(µ);µ). (11)

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Greedy algorithm

We actually work with a �nite surrogate Ξtrain of D. Denote by∆XN

(µ) the RB error bound(speci�ed later), Nmax the maximal sizeof RB space. Initiating with N0, the initial size of the initial sampleS∗N0

= {µ1∗, · · · , µN0∗}, and the tolerance εtol ,min.

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Greedy(N0, S∗N0

, Ξtrain, εtol ,min)

for N = N0 + 1 : Nmax

µN∗ = arg maxµ∈Ξtrain

∆XN−1(µ);

ε∗N−1 = ∆XN−1(µ);

if ε∗N−1 ≤ εtol ,min

Nmax = N − 1;

exit;

end;

S∗N = S∗N−1 ∪ µN∗;X ∗N = X ∗N−1 + span{u(µN∗)};

end; 13 / 20





O�ine-Online decomposition

O�ine stage

Compute all ingredients for sti�ness matrix and load vector foronline stage. The operation count and storage of thesecomputations depend on N and hence expensive.

Online stage

Given any µ ∈ D, we assemble the ingredients computed in o�inestage to formulate sti�ness matrix and load vector and then, solve(10) and evaluate (11). The operation count and storage of thisprocess are independent of N and hence are not expensive .

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Error bound estimationLower bound for coercivity constant and upper bound for continuityconstant.

αLB(µ) ≡ Θmin,µa (µ) = min

q∈{1,··· ,Qa}

Θqa(µ)

Θqa(µ)

, (12)

γUB(µ) ≡ Θmax,µa (µ) = max

q∈{1,··· ,Qa}

Θqa(µ)

Θqa(µ)

. (13)

θµ(µ) ≡ γUB(µ)

αLB(µ). (14)

The residual and its Riesz representation:

r(v ;µ) = f (v ;µ)− a(uXN(µ), v ;µ). (15)

De�ne:

(e(µ), v) = r(v ;µ); (16)15 / 20





Error bound estimation

∆enN (µ) ≡ ||e(µ)||X

α12LB(µ)

, ηenN (µ) ≡∆en

N (µ)

|||e(µ)|||µ; (17)

∆sN(µ) ≡

||e(µ)||2XαLB(µ)

, ηsN(µ) ≡∆s

N(µ)

s(µ)− sN(µ); (18)

∆s,relN (µ) ≡

||e(µ)||2XαLB(µ)sN(µ)

, ηs,relN (µ) ≡∆s,rel

N (µ)

(s(µ)− sN(µ))/s(µ);

(19)

∆N , ηN are error bounds and e�ectivities respectively.

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Error bound estimation

Theorem

For given N = 1, · · · ,Nmax ,

1 ≤ ηenN (µ) ≤√θµ(µ), ∀µ ∈ D, (20)

1 ≤ ηsN(µ) ≤ θµ(µ), ∀µ ∈ D. (21)

Furthermore, for ∆s,relN (µ) ≤ 1

1 ≤ ηs,relN (µ) ≤ 2θµ(µ), ∀µ ∈ D, (22)

where the left inequality of (22) is always valid.

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Comments

The error above bounds are some how local, i.e. depend on µ,

They also developed �global bounds� for solution u(µ) which isindependent of µ,

The nonlinear and noncoercive problems require more generaland thorough treatments, for more details, see the references.

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Outline

1 Introduction



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Summary and future work

Reduced Basis method deals with Parametrized PartialDi�erential Equations: constructing reduced basis,o�ine-online decomposition and error bounds.

Reduced basis method for imcompliant, nonlinear andnoncoercive problems: Navier-Stockes equations,

Using this idea for parametric model order reduction.

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S. Deparis, �Reduced basis error bound computation ofparameter-dependent Navier-Stokes equations by the naturalnorm approach�, SIAM J. Numer. Anal., Vol. 46, No. 4, pp.2039-2067, 2008.

S. Deparis, G. Rozza, �Reduced basis method formulti-parameter dependent steady Navier-Stockes equations:application to natural convection in a cavity�, EPFL-IACSReport, 2008.

A. T. Patera, G. Rozza, Reduced Basis Approximation and A

Posteriori Error Estimation for Parametrized Partial Di�erentialEquations, MIT Pappalardo Graduate Monographs in

Mechanical Engineering, V1.0, 2007.

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A. Quarteroni, G. Rozza, �Numerical solution of parametrizedNavier-Stokes equations by reduced basis methods�, Numerical

Methods for Partial Di�erential Equations, Vol.23, No. 4, pp.923 - 948, Wiley Interscience, 2007.

K. Veroy, C. Prud'homme, D. V. Rovas, A. T. Patera,�Aposteriori error bound for reduced basis approximation ofparametrized noncoercive and nonlinear elliptic partialdi�erential equations�, AIAA Paper 2003-3847, 2003.

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