Adaptive ReducedBasis Generation for Time-DependentProblems · Adaptive ReducedBasis Generation for Time-DependentProblems RB Workshop, 23.6.2011 University of Paris Bernard Haasdonk

Adaptive Reduced Basis Generation forTime-Dependent Problems

RB Workshop, 23.6.2011University of Paris

Bernard HaasdonkUniversity of Stuttgart, Germany

Institute of Applied Analysis and Numerical Simulation

[email protected]

Acknowledgements:

Mario Ohlberger, Martin Drohmann (University of Münster, Germany)

Markus Dihlmann (University of Stuttgart, Germany)

23.6.2011 B. Haasdonk, RB-Workshop, Paris 2

Institute of Applied Analysis

and Numerical Simulation

Overview

RB-Schemes for time-dependent PDEs

POD-Greedy Algorithm

Motivation

Convergence

Adaptive Basis Generation

Training set adaptivity (Offline)

Adaptive P partition

Adaptive T partition

Adaptive N adaptation (Online)

Conclusion and Perspectives

1. time-evolution

2. large Param.-domain

3. less smoothness wrt. µ, t




Model Reduction with RB-Methods

Scenario:

Parametrized PDEs: geometry, material or control parameters

Multiple simulation-requests: design, optimization, control, ...

Goals of RB-Methods:

Reduced basis space by snapshots

Reduced simulation scheme, Offline/Online decomposition

Rigorous a-posteriori error estimation

References: [NP80], [PL87], [PR07], … http://augustine.mit.edu

Solution manifold and RB-space: Example reduced basis (POD):

ϕ1

ϕ6

Xh

XN

uh(·, t,µ)

uN(·, t,µ)

XN ⊂ span(uh(·; tkn ,µn))

ΦN = (ϕ1, . . . , ϕN )




RB-Method for Linear Evolution Schemes

Parametrized linear evolution equation [HO08,DHO09]

For find

s. th.

Discrete implicit/explicit scheme

For find s. th.µ ∈ P ⊂ Rp ukhKk=0 ⊂ Xh ⊂ L

2(Ω)

u0h := Ph[u0(µ)]

µ ∈ P ⊂ Rp u : [0, T ]→ X ⊂ L2(Ω)

u(0) = u0(µ)

∂tu(t) + L(µ)[u(t)] = 0

(Id + ∆tLIh

)[uk+1h ] =

(Id−∆tLEh

)[ukh] + ∆tbkh





Reduced Basis Space

Reduced Operators

Orthogonal projection

Implicit/explicit space discretization operators

RB-Evolution Scheme in Operator Form

For find s. th.µ ∈ P ⊂ Rp ukNKk=0 ⊂ XN ⊂ Xh

PN : Xh →XN

u0N := PN [u0h(µ)]

XN ⊂ span(uh(·; t,µ)) ⊂ Xh

bkN := PN [bkh]

N := dimXN dimXh

LEN := PN LEh LIN := PN LIh

(Id + ∆tLIN

)[uk+1N ] =

(Id−∆tLEN

)[ukN ] + ∆tbkN

Related: [GP05]





A-posteriori Energy Error Bound

Stability wrt. Data, independent of

Reproduction of Solutions

Relation of RB-error to Best-Approximation

ukh(µ) ∈ XN ∀k ⇒ uN(µ) = uh(µ)

∆KN,γ(µ) :=

(1

αC(γ)∑K−1k=0 ∆tk

∥∥Rk+1h

∥∥2)1/2

.|||uN(µ)− uh(µ)|||γ ≤

∆t

‖uN(µ)‖ ≤ ‖u0(µ)‖+CT

ek+1 := uk+1h (µ)− uk+1N (µ)

∥∥ek+1∥∥ ≤ C

∥∥ek∥∥+C′ inf

v∈XN

(∥∥uk+1N − v∥∥+Φ(

∥∥uk+1N − v∥∥ , ek)

)




Basis Generation

Manifold of parametric and time-dependent solutions

Trajectory-oriented view in RB generation: Time not a „normal“ parameter: causality, accessibility of snapshots only via computing trajectories

Computational effort for trajectory computation is high

Maximum information from trajectory must be used

u(µ, t0)

u(µ, tK)

u(µ′, t0)

u(µ′, tK)




POD-Greedy: Procedure

Basis generation: POD-Greedy [HO08]

Based on „Greedy“ for stationary RB problems [VPRP03]

Define initial reduced basis

Choose finite training parameter set

Iterative Extension of basis:

While

1. Find

2. Compute detailed trajectory

3. Orthogonalize trajectory

4. Add principal components as basis vectors

Mtrain ⊂ P

µ∗ := argmaxµ∈Mtrain∆N (µ)uh(µ

∗)

ΦN+k = ΦN ∪ POD(eh, k)

ΦN0⊂ Xh

ε := maxµ∈Mtrain∆N (µ) > εtol

eh := uh(µ∗)− PXN (uh(µ

∗))




POD-Greedy: Convergence Rates [Ha11]

Existing: Results for Greedy [BMPPT09,BCDDPW10]

Main Steps of Extension:

Abstract „weak“ POD-Greedy Algorithm on spaces of function sequences

Coefficient matrix representation:

Block structure

Possible repetitions of block-rows

Possible nonzero block-rows

In particular non-(block) triagonal

Selected at n-th POD-Greedy step

fn ∈ X Generated basis function at n-th POD-Greedy step

aij ∈ RK+1, (aij)k :=

⟨uki , fj

⟩X

un := (ukn)Kk=0 ∈ FT




POD-Greedy: Convergence Rates

Main Steps of Extension (cont‘d):

quantification of matrix coefficient properties

…

Extended „Flatness-Lemma“/“Delayed Comparison“ of [BCDDPW10]

Thm: „Almost optimal“ error Decay of POD-Greedy:

Algebraic (or exponential) convergence of the Kolmogorovn-widths of the solution manifold transfer to the POD-Greedy error sequence

dn(F) ≤Mn−α ⇒ σT,n(FT ) ≤ CMn−α.

‖ann‖2

W = λ0(un − PT,n−1un) ≥ γ2σ2T,n/(K + 1)

∑j∈N ‖aij‖

2

W =∑Kk=0 w

k∥∥ukn

∥∥




Adaptive Training Set Extension [HO07]

Problems of (POD-)Greedy:

Tends to overfit for small training sets

Infeasible for overly large training sets

Infeasible in absence of error estimators

Remedy

automatic training set extension




Adaptive Training Set Extension

Qualitative Results in 2D Parameter Domain Parameter domain Basis size , random validation set Resulting error (estimator) : plot of

Overfitting in uniform-fixed grid (standard greedy search) Improved uniform error distribution by adaptive approach

uniform-fixed uniform-refined adaptive-refined,

µ = (β, k) ∈ [0, 1]× [0, 5 · 10−8]N = 130

Θ = 0.05

|Mval| = 10, ρtol = 1.0

log∆(µ,ΦN)




Adaptive Training Set Extension

Quantitative Results in 3D Parameter Domain

Full 3D parameter domain

Random test set

Maximum test error

Flattening of test error curve in uniform-fixed approach Improved convergence for adaptive approach

max. test error decrease

µ = (cinit, β, k)

P := [0, 1]× [0, 1]× [0, 5 · 10−8]

|Mtest| = 1000

maxµ∈Mtest

∆(µ,ΦN )




Adaptive P-Partition [HDO10]

Adaptive P-Partition Goal: bases with desired accuracy and online runtime:

(adaptive POD)-Greedy Basis per parameter-subdomain.

If not then refine subdomain

Increased offline cost: Runtime & Storage

Considerably improved online runtime vs. accuracy

(ε ≤ εtol) ∧ (N ≤ Nmax)

εtol, Nmax

Related: [EPR09]




Adaptive P-Partition

Verification of Online Efficiency:

Considerably reduced online computation time with equal accuracy

Further orders of magnitude improvement by combination with adaptive training set extension

online time / test-error




Adaptive T-Partitioning [DDH11]

Specialized RB spaces on time-subintervals:

t

…

…

…

…

τ1 τi τΥ T0

XNi




Adaptive T-Partitioning

Online Computation of Projection Error at Basis Change:

Extended a-posteriori Error Estimators:





Idea: Aim at uniform error estimator growth over time

bisectionbisection

bisectionbisection

0 0

0

τ1

τ1

TT τ2

τ2 τ3

εtol,global εtol,global

εtol,global

εtol,1

εtol,1

εtol,2

N = NmaxN = Nmax

N < Nmax





Experiments:

Advection problem (1-parameter)

FV-Discretization: 4096 DOFs

Euler time integration: 512 time steps

Adaptive reduced basis settings:

Testing the reduced model by performing reducedsimulations for 20 randomly chosen parameters

εtol,global = 0.01,Nmax = 45





Result Diagrams:

Test error estimator over time Online runtime vs. accuracy




Online RB Dimension Adaptation

N-adaptivity in Evolution Problems [HO09] Goal: Adapt model dimension N over time

Target at equidistribution of residual-norms over time

Look-ahead prediction of error estimator determiningin-/decrease of N

ExampleError estimator evolution model order N

Attachment to error target, detection of model complexity

N-adaptive computationally faster than N-fixed




Conclusion and Perspectives

POD-Greedy

is a practical and theoretical well founded

procedure for time-dependent RB-methods

-> Kolmogorov n-widths for PDE-classes

-> Application in other

„sequence“-based problems

Adaptive Basis Generation

Adaptive basis generation approaches enable simultaneousaccuracy and online-runtime control

P and T partitioning works well -> Combined P/T adaptive basisgeneration schemes

(adaptive) grids in parameter space merely applicable to limitedparameter dimensions: -> sparse grids, meshfree

Algorithms mainly empirical, no/few analysis

-> theoretical investigations

Age: 2+

Thank you!




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Adaptive ReducedBasis Generation for Time-DependentProblems · Adaptive ReducedBasis Generation for Time-DependentProblems RB Workshop, 23.6.2011 University of Paris Bernard Haasdonk