a short introduction to reduced basis method for ...a short introduction to reduced basis method for...
TRANSCRIPT
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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Zentrum fürTechnomathematik
Fachbereich 03Mathematik/Informatik
Outline
1 Introduction
2 Reduced Basis ApproachPreliminariesApproach
3 Summary and outlook
3 / 20
Zentrum fürTechnomathematik
Fachbereich 03Mathematik/Informatik
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
2 Reduced Basis ApproachPreliminariesApproach
3 Summary and outlook
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Fachbereich 03Mathematik/Informatik
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
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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
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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
2 Reduced Basis ApproachPreliminariesApproach
3 Summary and outlook
19 / 20
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Fachbereich 03Mathematik/Informatik
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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