balanced pod model reduction for pde systems · john singler1 1department of mathematics and...

Balanced POD Model Reduction for PDE Systems

John Singler1

1Department of Mathematics and StatisticsMissouri University of Science and Technology

John Singler (Missouri S&T) 1 / 28

Acknowledgements

Collaborators:

Belinda BattenOregon State University

Weiwei HuIMA (currently)Oklahoma State University (fall 2016)

Thanks to NSF for funding


Introduction

Balanced truncation is a popular model reduction algorithm forlinear systems of ordinary differential equations.

Recently, many researchers have focused on algorithms forapproximating the balanced truncated reduced order model forlarge scale systems and partial differential equations (PDEs).

One recent model reduction approach is balanced properorthogonal decomposition, developed by Rowley (2005).

Balanced POD is similar to standard POD: it uses simulationdata to produce modes which are used in a Petrov-Galerkinprojection to reduce the governing equations.


Introduction

Although balanced POD is a model reduction method for linearsystems, it can also be used on nonlinear systems.

One popular approach:

1 Linearize around a steady state solution of interest.

2 Compute the balanced POD modes of the linearized system.

3 Use the modes to reduce the nonlinear system.

This algorithm has been widely used in the engineering flowcontrol community for model reduction of a wide variety of fluidflow systems.

The resulting nonlinear reduced order models have been used forfeedback control.


Introduction

Belinda Batten and I extended Rowley’s algorithm to directlytreat partial differential equations (PDEs) with bounded (2009& 2012) and unbounded (2013) input and output operators.

Weiwei Hu and I (2014) modified this algorithm to treatparabolic PDEs with highly unbounded input.

Motivation: develop algorithms that can be used when matrixapproximations of the operators in the PDE are not available.

Example: It may be difficult to obtain matrix approximationsfrom complex existing simulation codes.

Bonus: Convergence theory (JRS, 2012; WH & JRS, in progress)


Goal of this talk

Balanced POD for linear PDE systems

Model problem and general frameworkBounded input and outputUnbounded input and output

Do we have any hope of understanding balanced POD fornonlinear model reduction?


Model Problem

A two dimensional convection diffusion PDE

wt = µ∇2w −~a1 · ∇w − a0 w , in Ω,

∂w

∂n= ~b · ~n u(t), on ∂Ω,

y(t) =

∫∂Ω

~c · ~n w .

Multiply by a test function in V = H1(Ω) and integrate by parts:

w(t) = Aw(t) + Bu(t), y(t) = Cw(t),

where w(t) = w(t, ·) ∈ V , and A : V → V ′, B : R→ V ′, andC : V → R are linear operators. (Here, V ′ is the dual space of V .)


Model ProblemA two dimensional convection diffusion PDE

wt = µ∇2w −~a1 · ∇w − a0 w , in Ω,

∂w

∂n= ~b · ~n u(t), on ∂Ω,

y(t) =

∫∂Ω

~c · ~n w .

The quantities Aw and Bu act on test functions v ∈ H1(Ω):

〈Aw , v〉 = −a(w , v) = −∫

Ω

µ∇w · ∇v + (~a1 · ∇w) v + a0 w v ,

〈Bu, v〉 =

∫∂Ω

~b · ~n v u.


General Framework

Unbounded input/output: We consider the general framework

d

dtw(t) = Aw(t) + Bu(t), y(t) = Cw(t).

V and H are two real Hilbert spaces with V → H

A : V → V ′ is generated by a bounded and coercive bilinearform a : V × V → R

Unbounded Input: B : Rm → V ′ and so〈Bu(t), v〉 =

∑mj=1〈bj , v〉 uj(t) for bj ∈ V ′

Unbounded Output: C : V → Rp and so y(t) = Cw(t) is givenby yi(t) = 〈ci ,w〉 for i = 1, . . . , p and ci ∈ V ′


Balanced POD: Bounded CaseBalanced truncation model reduction hinges on the Hankeloperator

[Hu](t) =

∫ ∞0

h(t + s) u(s) ds, h(τ) = CeAτB ,

where h is the impulse response.

The Hankel operator is compact and has a singular valuedecomposition

Hfk = σk gk , H∗gk = σk fk .

Each set of singular vectors

fk ⊂ L2(0,∞;Rm), gk ⊂ L2(0,∞;Rp)

is an orthonormal basis.John Singler (Missouri S&T) 10 / 28

Balanced POD: Bounded Case

The Hankel operator can be factored: H = CB, where

[Cx ](t) = CeAtx , [Bu](t) =

∫ ∞0

eAsB u(s) ds.

The balancing modes are

ϕj = σ−1/2j Bfj = σ

−1/2j

∫ ∞0

eAsB fj(s) ds,

ψj = σ−1/2j C∗gj = σ

−1/2j

∫ ∞0

eA∗sC ∗ gj(s) ds,

where σj , fj , gj are the singular values/vectors of H.



Theorem (JRS, 2012): The balancing modes are the eigenvectorsof the product of the Gramians and the modes also give thebalancing transformation: a balanced realization of (A,B ,C ) is

Abij = (Aϕj , ψi) = (ϕj ,A

∗ψi) = −a(ϕj , ψi),

Bbij = (bj , ψi), C b

ij = (ϕj , ci),

where (·, ·) is the inner product on H and

Bu =m∑j=1

bjuj , Cx = [ (x , c1), . . . , (x , cp) ]T .

The balanced reduced order model is obtained by truncation.



How do you compute this?

Rowley’s balanced POD idea: Rewrite the Lyapunov solutionmatrices in terms of simulation data and use a POD-likeprocedure to approximate the eigenvectors of the product of theGramians (the balancing modes).

An alternate approach (JRS, 2012): Rewrite the Hankel operatorin terms of the same simulation data and use a POD-likeprocedure to approximate the Hankel singular values and singularvectors.

The balancing modes can be approximated directly using theformulas above.



Here is a bit of detail:

We showed the Hankel operator can be rewritten as

[Hu](t) =

∫ ∞0

k(t, s) u(s) ds, kij(t, s) =(zi(t),wj(s)

),

where zi(t) = eA∗tci and wj(s) = eAsbj solve

zi(t) = A∗zi(t), zi(0) = ci ,

wj(t) = Awj(t), wj(0) = bj .

Approximating the above integral operator by quadrature (orsome other method) reduces the computation of the singularvalues and singular vectors of H to a matrix SVD.


Balanced POD: Unbounded Case

We extended the balanced POD algorithm to parabolic PDEsystems of the above form with B : Rm → V ′ and C : V → Rp

bounded.

Theorem (WH & JRS): If S(t) = eAt and S∗(t) = eA∗t satisfy

S : V ′ → L2(0,∞;H), S∗ : V ′ → L2(0,∞;H),

then a balanced realization of (A,B ,C ) is

Abij = −a(ϕj , ψi), Bb

ij = 〈bj , ψi〉, C bij = 〈ϕj , ci〉.

The proof relies on new balanced truncation theory developed byGuiver and Opmeer (2014).



One situation where S(t) = eAt and S∗(t) = eA∗t satisfy

S : V ′ → L2(0,∞;H), S∗ : V ′ → L2(0,∞;H)

is if the bilinear form is a sum of a symmetric form and a lowerorder form (as in the model problems).

Specifically, let a0 and a1 be the symmetric and skew-symmetricparts of the bilinear form, i.e.,

a0(u, v) =1

2

(a(u, v)+a(v , u)

), a1(u, v) =

1

2

(a(u, v)−a(v , u)

).

Then S and S∗ satisfy the above condition if there is a constantC1 such that a1(u, v) ≤ C1‖u‖V‖v‖H for all u, v ∈ V .



In the unbounded case, the balanced POD algorithm is verysimilar to the bounded case.

However, we must approximate the solution of PDEs with initialdata bj , ci ∈ V ′:

wj(t) = Awj(t), wj(0) = bj ,

zi(t) = A∗zi(t), zi(0) = ci .

The solutions of these PDEs have singularities at time t = 0.

Need to be careful with time-stepping: standard methods (e.g.,the trapezoid rule) can fail.

I have had success with adaptive solvers; Rannacher smoothingis another option (thanks to Ekkehard Sachs)


Numerical Results: Model Problem

10−4

10−2

100

102

104

−10

−5

0

5

10

15

20

ω

real(G)imag(G)|G|

10−4

10−2

100

102

104

−10

−5

0

5

10

15

20

ω

real(G)imag(G)|G|

Approximate transfer function for the 2D convection diffusion system(r = 5) versus the transfer function computed using a finite elementapproximation.

Balanced POD works very well!


A Question

In working out the convergence theory, Weiwei Hu and I cameacross this question:

For a linear parabolic PDE with initial data w0 ∈ V ′:

w(t) = Aw(t), w(0) = w0,

when does the solution satisfy w ∈ L1(0,∞;V )?

True if A is self-adjoint (normal) or if w0 is slightly smootherthan V ′.

Does anyone know any other conditions?


Nonlinear Model Reduction

Although balanced POD is a model reduction method for linearsystems, it can also be used on nonlinear systems:

One popular approach:

1 Linearize around a steady state solution of interest.

2 Compute the balanced POD modes of the linearized system.

3 Use the modes to reduce the nonlinear system.

The resulting nonlinear reduced order models have been used forfeedback control.

Can we quantify the effectiveness of these reduced ordernonlinear controllers?



Look at an easier (?) problem: The open loop (i.e., no control)nonlinear model reduction problem.


The most in-depth computational study I know of:Ilak et al. (2010) on the 1D complex Gizburg-Landau PDE

The authors found that the balanced POD reduced order modelscan be very accurate, even in the presence of complex nonlineardynamics



Question: Why can balanced POD modes built from a linear systemsuccessfully reduce a nonlinear PDE?

One reasonable explanation: For some nonlinear PDEs, the linearterms are highly important; therefore, reducing the linear terms“correctly” may give success for the whole nonlinear PDE.

Also: Standard POD has an optimal data reconstruction property,which is largely believed to be the primary reason POD modelreduction can be so successful.

Question: Does balanced POD have any data approximationproperties?



The balanced POD algorithm can be applied to two sets ofsquare integrable time-varying functions zjpj=1 and wjmj=1.

The data sets do not need to come from a linear system.

The balanced POD modes are biorthogonal:

(ϕj , ψi) = δij =

1, i = j

0, i 6= j.

The balanced POD modes give two projections:

Prx =r∑

k=1

(x , ψk)ϕk , Qrx =r∑

k=1

(x , ϕk)ψk .



Theorem (JRS, 2010):

The balanced POD modes solve the following optimal dataapproximation problem: Find biorthogonal sequences ϕk ⊂ Xand ψk ⊂ X minimizing the balanced data approximation error

Er =

∫Iz

∫Iw

∣∣(z(t)− Qrz(t),w(s)− Prw(s))∣∣2 dt ds.

Also, the data approximation error is given exactly as

Er =∑k>r

σ2k ,

and this error tends to zero as r →∞.


Nonlinear Model ReductionTheorem (JRS, 2015): If all of the BPOD eigenvalues or singularvalues are nonzero:

The individual data approximation errors are given by∫Iw

‖w(t)− Prw(t)‖2 dt =∑k>r

σk ‖ϕk‖2,∫Iz

‖z(t)− Qrz(t)‖2 dt =∑k>r

σk ‖ψk‖2,

and these errors tend to zero as r →∞.

For x in certain dense subsets of the Hilbert space:

x =∑k≥1

(x , ψk)ϕk , x =∑k≥1

(x , ϕk)ψk .


Conclusions

Balanced POD model reduction for linear PDE systems iscomputationally tractable, data-based, and has convergenceguarantees.

There are other approximate balanced truncation approaches.

Can any of these be extended to the PDE level? (True for ADI.)How do the methods behave as the mesh is refined?How do the methods perform/compare for very hard problems?(convection dominated PDEs, etc.)


There is much more work to be done!


References I

C. W. Rowley, Internat. J. Bifur. Chaos Appl. Sci. Engrg., vol.15, no. 3, 997-1013, 2005.

J. R. Singler and B. A. Batten, Int. J. Comput. Math., vol. 86,no. 2, 355–371, 2009.

J. R. Singler, Numer. Math., vol. 121, no. 1, 127–164, 2012.

J. R. Singler, J. Merritt, C. W. Ray, and B. A. Batten, Proc.ACC, 1272–1277, 2013.

W. Hu and J. R. Singler, Proc. ACC, 1680–1685, 2014.

C. Guiver and M. Opmeer, SIAM J. Control Optim., vol. 52, no.2, 1366-1401, 2014.


References II

M. Ilak, S. Bagheri, L. Brandt, C. Rowley, and D. Henningson,SIAM J. Appl. Dyn. Syst., vol. 9, no. 4, 1284-1302, 2010.

J. R. Singler, Numer. Funct. Anal. Optim., vol. 31, no. 7–9,852-869, 2010.

J. R. Singler, J. Math. Anal. Appl., vol. 421, no. 2, 1006-1020,2015.


balanced pod model reduction for pde systems · john singler1 1department of mathematics and...

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