Variation of the Balanced POD Algorithm
for Model Reduction of Linear Systems*
John R. Singler1
Abstract— We present a variation on an existing model
reduction algorithm for linear systems based on balanced
proper orthogonal decomposition (POD). In contrast to many
computational approaches to balanced truncation, the algorithm variation approximates the reduced order model directly
without first transforming the linear system. The algorithm
is applicable to large-scale finite dimensional systems and a
class of infinite dimensional systems. The algorithm variation
is compared to the original balanced POD algorithm on example
partial differential equation systems.
I. I NTRODUCTION
Balanced truncation is a classical method for model reduction of linear systems of ordinary differential equations
[1], [2]. The standard balanced truncation algorithms are
only applicable for systems of moderate size, and a large
amount of recent research has focused on the development
and analysis of algorithms for balanced truncation (and the
associated Lyapunov equations) of very high dimensional
systems; see, e.g., [3], [4], [5], [6] and the many references
therein.
Balanced POD was originally introduced by Rowley in
[7] as a balanced truncation algorithm for high dimensional
linear systems of ordinary differential equations. Rowley’s
balanced POD algorithm uses similar ideas to the method of
snapshots for standard POD computations [8], however there
are now two separate datasets. Balanced POD has now been
widely used for linearized fluid flow model reduction and
control problems; see, e.g., [9], [10], [11].
In [12], we extended Rowley’s balanced POD algorithm
to an infinite dimensional setting that includes a class of
partial differential equations (PDEs). In [13], we proved
convergence of two data-based variations of the algorithm
in [12]. The data-based algorithm variations were designed
to be used when matrix approximations of the infinite dimensional operators are not available. This case can occur, e.g.,
when using an existing simulation code for the computations
and access to the source code is difficult or restricted, or
when spatial discretization of the PDE system yields matrix
approximations of the operators that are not appropriate for
existing matrix-based balanced truncation algorithms.
In [13], we related the balanced POD modes (or balancing
transformation) to the Hankel operator of the system, instead
of the product of the system Gramians as in Rowley’s
*This work was supported in part by the National Science Foundation
under grant DMS-1217122.
1 J. Singler is with the Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409, USA,
[email protected]
original paper. In this work, we combine this line of reasoning with existing results from infinite dimensional balanced
truncation theory [14], [15] to produce another variation on
Rowley’s balanced POD algorithm. The algorithm variation
is transformation-free, i.e., the balanced reduced order model
is approximated directly, as opposed to first transforming the
system to balanced coordinates and then truncating as is done
in the above works.
II. T WO M ODEL P ROBLEMS
To test the algorithm variation, we consider two example
PDE systems from [13]. The first problem is a simple
one dimensional hyperbolic PDE system that has a transfer
function that can be evaluated exactly for comparison. The
second problem is a two dimensional parabolic PDE system
whose transfer function must be approximated.
A. Model Problem 1
First, consider the following simple one dimensional hyperbolic PDE
wt = −a(x)wx + b(x)u(t),
0 < x < 1,
t > 0,
with input u(t), boundary condition
w(t, 0) = 0,
and output
Z
y(t) =
1
c(x)w(t, x) dx.
0
We consider a(x) = β − αx with β > α > 0, and assume
b(x) and c(x) are square integrable.
This problem can be formulated as a linear system
ẇ(t) = Aw(t) + Bu(t),
y(t) = Cw(t)
2
(1)
as follows. Let X be the RHilbert space L (0, 1) with standard
1
inner product (f, g) = 0 f (x)g(x) dx. The operators B :
1
R → X and C : X → R1 are defined by [Bu](x) = b(x)u
and Cw = (w, c). The operator A : D(A) ⊂ X → X is
given by [Aw](x) = −a(x)wx (x), with domain D(A) =
HL1 (0, 1). The space HL1 is the set of functions w ∈ X such
that w(0) = 0, w is (weakly) differentiable, and wx ∈ X.
We also need the Hilbert adjoint operator A∗ : D(A∗ ) ⊂
X → X, which is given by [A∗ z](x) = (a(x)z(x))x , with
1
1
D(A∗ ) = HR
(0, 1). The space HR
is defined similarly to
1
HL , except the boundary condition is now z(1) = 0.
For a(x) given above, it can be checked that the exact
transfer function G(s) = C(sI − A)−1 B is given by
Z 1
Z x
c(x) (β−αx)s/α
b(v) (β−αv)−1−s/α dv dx.
G(s) =
0
0
Later, we take β = 0.5, α = 0.4, b(x) = 1−x, and c(x) = x,
and then these integrals can be computed exactly.
B. Model Problem 2
Next, consider the parabolic PDE given by
wt = µ(wxx + wyy ) + c1 (x, y)wx + c2 (x, y)wy + b(x, y)u(t),
over the spatial domain Ω = [0, 1] × [0, 1], with Dirichlet
boundary conditions on Γ0 (the bottom, right, and top walls):
A. Assumptions and Notation
We consider the following general framework. Let X be
a real separable Hilbert space with inner product (·, ·) and
corresponding norm k · k = (·, ·)1/2 . Assume the operator
A : D(A) ⊂ X → X generates an exponentially stable C0 semigroup eAt over X, and the operators B : Rm → X and
C : X → Rm are both finite rank and bounded. The latter
assumption implies that B and C take the form
Bu =
m
X
Cx = [ (x, c1 ), . . . , (x, cp ) ]T ,
bj uj ,
(2)
j=1
w(t, x, 0) = 0,
w(t, 1, y) = 0,
w(t, x, 1) = 0,
and a Neumann boundary condition on the left wall:
wx (t, 0, y) = 0.
The system output is
Z
η(t) =
c(x, y)w(t, x, y) dx dy.
Ω
where each bj and cj are in X and u = [ u1 , . . . , um ]T is
a vector in Rm (see [16, Theorem 6.1]).
Also, for a Hilbert space X, let L2 (0, ∞; X) be the Hilbert
space of all functions x such that x(t) ∈ X for all t > 0
with finite norm
Z ∞
1/2
kx(t)k2 dt
.
kxkL2 (0,∞;X) =
0
B. The Hankel Operator and the Balanced Realization
We assume µ is a positive constant, the convection coefficients c1 (x, y) and c2 (x, y) are bounded, and the functions
b(x, y) and c(x, y) are square integrable over Ω.
This problem can also be written as a linear system of
the form (1). Take the Hilbert space X to be L2 (Ω), the
space of square integrable functions
defined over Ω, with
R
standard inner product (f, g) = Ω f (x, y)g(x, y) dx dy. The
operators B : R1 → X and C : X → R1 are defined
similarly to the first model problem. For the linear operator
A, we place the problem in weak form. Let V be the Hilbert
space
1
V = {v ∈ H (Ω) : v = 0 on Γ0 },
with inner product (v, w)V = (vx , wx ) + (vy , wy ). Next,
multiply the PDE by a test function v ∈ V and integrate by
parts to yield
(wt , v)X = a(w, v) + (b, v)X u,
The Hankel operator H : L2 (0, ∞; Rm ) → L2 (0, ∞; Rp )
of the linear system (A, B, C) is defined by
Z ∞
[Hu](t) = [CBu](t) =
CeA(t+s) Bu(s) ds,
0
where the controllability operator C : X → L2 (0, ∞; Rp )
and the observability operator B : L2 (0, ∞; Rm ) → X are
defined by
Z ∞
At
[Cx](t) = Ce x, Bu =
eAs Bu(s) ds.
0
In our earlier works [12], [13] we provided alternate forms
for these operators.
Proposition 1: Under the above assumptions, the operators C and B defined above are also given by
[Cx](t) = [ (x, z1 (t)), . . . , (x, zp (t)) ]T ,
Z ∞X
m
Bu =
uj (s) wj (s) ds.
0
where the bilinear form a : V × V → R is given by
(3)
(4)
j=1
∗
a(w, v) = µ(w, v)V + (c1 wx , v)X + (c2 wy , v)X .
The operator A : D(A) ⊂ X → X is (roughly) defined by
(Aw, v) = −a(w, v),
for all w ∈ D(A) and v ∈ V .
See [13] for details.
III. R EPRESENTATIONS OF THE H ANKEL O PERATOR AND
THE BALANCED R EALIZATION
We derive the transformation-free balanced POD algorithm variation by using an alternate representation of the
Hankel operator of the system as well as different representations of the balanced realization.
where zi (t) = eA t ci and wj (t) = eAt bj are in L2 (0, ∞; X)
and are the unique solutions of the linear evolution equations
żi (t) = A∗ zi (t),
zi (0) = ci ,
(5)
ẇj (t) = Awj (t),
wj (0) = bj ,
(6)
for i = 1, . . . , p and j = 1, . . . , m.
Corollary 1: Under the above assumptions, the Hankel
operator H : L2 (0, ∞; Rm ) → L2 (0, ∞; Rp ) of the system
(A, B, C) is given by
Z ∞
[Hu](t) =
k(t, s) u(s) ds,
(7)
0
where the p × m kernel function k(t, s) has ij entries
kij (t, s) = zi (t), wj (s) ,
∗
and zi (t) = eA t ci and wj (t) = eAt bj are the unique
solutions of the linear evolution equations (5) and (6).
For the class of systems (A, B, C) considered here, the
Hankel operator is known to be trace class (or nuclear),
and therefore compact [17, Theorem 4]. Therefore, there
exist singular values σ1 ≥ σ2 ≥ · · · ≥ 0 (with repetitions
according to multiplicity) and corresponding singular vectors
{fk } ⊂ L2 (0, ∞; Rm ) and {gk } ⊂ L2 (0, ∞; Rp ) satisfying
Aϕj = lim+
∗
Hfk = σk gk ,
H gk = σk fk .
h→0
2
The singular vectors are orthonormal with respect to the L
inner product, i.e.,
Z ∞
(fj , fk )L2 (0,∞;Rm ) =
fjT (t) fk (t) dt = δjk ,
0
Z ∞
(gj , gk )L2 (0,∞;Rp ) =
gjT (t) gk (t) dt = δjk .
0
As in [13], we use the Hankel singular values and vectors
to define the balancing transformation. If σj is nonzero,
define the jth balancing modes ϕj and ψj in X by
Z ∞X
m
−1/2
−1/2
fj,k (t) wk (t) dt, (8)
ϕj = σj
Bfj = σj
0
−1/2 ∗
ψj = σj
−1/2
k=1
p
∞X
Z
C gj = σj
0
gj,k (t) zk (t) dt,
(9)
k=1
where fj,k and gj,k are the kth components of fj and gj .
As in the finite dimensional case, the balancing modes are
the eigenvectors of the products of the Gramians, and the
balancing modes provide the balancing transformation.
Theorem 1 ([13]): Suppose the above assumptions hold,
the Hankel singular values are distinct, and the Hilbert space
is infinite dimensional. Then the balancing modes satisfy
{ϕi } ⊂ D(A), {ψi } ⊂ D(A∗ ), and a balanced realization
(Ab , B b , C b ) of the system (A, B, C) is given by
Abij = (Aϕj , ψi ) = (ϕj , A∗ ψi ),
b
Bij
= (bj , ψi ),
b
Cij
= (ϕj , ci ).
(10)
(11)
The balanced POD algorithms in [13] are based on this
representation of the balanced realization. Once the balancing modes are approximated, the entries of B b and C b are
straightforward to construct. If a matrix approximation of
A or A∗ is not available, then there are two options. First,
for parabolic problems with the A operator derived from a
bilinear form a : V × V → R (as in the second model
problem), the entries of Ab can be computed as
Abij = −a(ϕj , ψi )
(12)
if the bilinear form is available for computations and the
computed balancing modes are in the Hilbert space V .
Otherwise, the quantity Aϕj can be approximated using
−1/2
Aϕj = −σj
B f˙j + Bfj (0)
(13)
m
m
Z ∞ X
X
−1/2
= −σj
wk (t) f˙j,k (t) dt +
wk (0) fj,k (0) ,
0
k=1
or A∗ ψi can be approximated using a similar formula. The
derivatives of the Hankel singular vectors can be approximated by finite differences or other similar approaches, or
using an indirect method proposed and analyzed in [13].
Remark 1: The approach of directly approximating Aϕj
for a linearized fluid flow system was also taken in [10]. In
that work, the authors used the approximation
k=1
eAδ ϕj − ϕj
eAh ϕj − ϕj
≈
h
δ
for small δ > 0. They approximated eAδ ϕj by simulating the
linear PDE for 0 ≤ t ≤ δ with initial data ϕj . This approach
will work, however it may require manual tuning of δ to
ensure accuracy. Using expression (13) does not require any
tuning, and it also requires less computational cost.
In this work, we construct the balanced truncation directly
using the Hankel singular values and singular vectors. This
can be done using the following result.
Theorem 2 ([14], [15]): Under the assumptions of Theorem 1, a balanced realization (Ab , B b , C b ) of the system
(A, B, C) is given by
σ 1/2 Z ∞
j
Abij =
giT (t) ġj (t) dt,
σi
0
σ 1/2 Z ∞
i
f˙iT (t) fj (t) dt,
(14)
=
σj
0
1/2
1/2
B b = [σ1 f1 (0), σ2 f2 (0), . . .]T ,
Cb =
1/2
1/2
[σ1 g1 (0), σ2 g2 (0), . . .],
(15)
(16)
and Ab can also be expressed as
1
(σi σj ) 2
σj fiT (0)fj (0) − σi giT (0)gj (0) , i 6= j,
2
2
σi − σj
(17)
1
1
(18)
Abii = − fiT (0)fi (0) = − giT (0)gi (0).
2
2
The representation (14)-(16) was proved by Curtain and
Glover in [14]. A similar formula to the alternate expression
for Ab given in (17)-(18) was proved by Glover, Curtain,
and Partington in [15, proof of Lemma 4.3] for the output
normal realization; their argument is easily modified for the
standard balanced realization to prove (17)-(18). This result
has also been recently extended to cover a wider class of
systems [18].
Below, we use a variation of the balanced POD algorithm
to approximate both the Hankel singular values and singular
vectors and then use Theorem 2 to approximate the balanced
truncation. As far as the author is aware, Theorem 2 has
not been used to approximate the balanced truncation of
a PDE system except for [19]. (Theorem 2 has been used
for theoretical investigations of balanced truncation; see,
e.g., [15], [20], [21].) In [19], we used a continuous time
eigensystem realization algorithm (ERA) to approximate the
Hankel singular values and singular vectors. A numerical
example showed the ERA successfully approximated the
balanced truncated reduced order model, however balanced
Abij =
POD (using the representation in Theorem 1) was more
accurate in certain cases.
The goal of this work is to compare balanced POD computations using the different representations of the balanced
realization using Theorems 1 and 2.
Remark 2: Discrete time ERA [22] uses the singular value
decomposition of a Hankel matrix and a representation of
the (discrete time) balanced realization that has a similar
form to that of Theorem 2. This method has been used to
reduce continuous time PDE systems in [23], [24]. In fact,
it is shown in [24] that discrete time ERA and discrete time
balanced POD are equivalent for finite dimensional systems
when all computations are exact. Of course, for complicated
systems there will always be errors and the approaches
will differ. Furthermore, continuous time balanced POD and
ERA will yield different results from their discrete time
counterparts—even if the continuous time nature of the
system is taken into account in the discrete time algorithms.
A thorough comparison of all of these approaches needs to
be performed to see if and when one approach is preferable
over the others; this work is one step toward this goal.
IV. BALANCED POD
We summarize our approach to balanced POD from [13]
for two collections of time varying functions {zi (t)} and
{wj (t)} in L2 (0, ∞; X). Although these functions can be ar∗
bitrary, we consider the data {zi , wj } given by zi (t) = eA t ci
At
and wj (t) = e bj , i.e., the unique solutions of the linear
evolution equations (5) and (6). In this case, the balanced
POD of {zi , wj } consists of the Hankel singular values,
Hankel singular vectors, and balancing modes for the system
(A, B, C). These quantities can be used to form the balanced
realization (Ab , B b , C b ) using either the representation in
Theorem 1 derived from the balancing transformation, or
the transformation-free representations in Theorem 2.
As with continuous time POD, the balanced POD of
two datasets can be approximated using many methods. In
[13], we extended the method of snapshots and quadrature
approaches for standard POD to balanced POD; here, we
briefly outline the quadrature approach.
The main idea is to approximate the Hankel operator with
quadrature—this reduces the balanced POD computations to
a matrix singular value decomposition. This is a different
approach than was taken in earlier works [7], [12]; this balanced POD variation provides the same approximation of the
Hankel singular values and balanced POD modes as in the
earlier works, however we now also obtain approximations
to the Hankel singular vectors at the quadrature points. For
simplicity, we only consider the case of a single function in
each dataset; the case of multiple functions is similar and
can be treated by “stacking” the data as in the earlier works.
w
w Nw
z
Let {αiz , τiz }N
i=1 and {αi , τi }i=1 be the weights and
nodes of two quadrature rules. Apply the quadrature rules
to the equations Hf = σg and H∗ g = σf and evaluate at
the quadrature nodes to obtain the approximate equations
Nw
X
z
Hf (τi ) ≈
αjw z(τiz ), w(τjw ) f (τjw ) ≈ σg(τiz ),
j=1
H∗ g (τiw ) ≈
Nz
X
αjz w(τiw ), z(τjz ) g(τjz ) ≈ σf (τiw ).
j=1
Replace the above approximate equations by equalities and
multiply the resulting equations by (αiz )1/2 and (αiw )1/2 ,
respectively. Define the scaled quantities
azi = (αiz )1/2 z(τiz ),
uj = (αjw )1/2 f (τjw ),
w 1/2
aw
w(τjw ),
j = (αj )
vi = (αiz )1/2 g(τiz ),
and let Γ be the Nz × Nw matrix with ij entries Γij =
(azi , aw
j ). This gives
Γu = σv,
ΓT v = σu.
Therefore, the singular value decomposition of Γ gives
approximations to the nonzero singular values of H and the
corresponding singular vectors evaluated at the quadrature
nodes. The balancing modes (8)-(9) are approximated using
quadrature on the integrals:
− 12
Nw
X
− 12
Nz
X
ϕk ≈ σk
− 21
αjw f (τjw ) w(τjw ) = σk
j=1
ψk ≈ σk
i=1
Nw
X
uk,j aw
j , (19)
j=1
− 21
αiz g(τiz ) z(τiz ) = σk
Nz
X
vk,i azi .
(20)
i=1
Except for simple cases, we will not have the exact data
{zi , wj }, but we will have approximate data {ziN , wjN }. In
[13], we proved that if the approximate data converges to the
exact data in L2 (0, ∞; X), then the singular values converge
and the singular vectors and balancing modes corresponding
to distinct singular values converge.
V. T HE A LGORITHM VARIATION
We now combine balanced POD and the representations of
the balanced realization in Theorems 1 and 2 to approximate
the balanced truncation (Ar , Br , Cr ), i.e., the rth order
truncation of the balanced realization (Ab , B b , C b ).
Balanced POD Algorithms for Balanced Truncation:
1) For i = 1, . . . , p, compute approximations ziN (t) and
∗
wjN (t) to the solutions zi (t) = eA t ci and wj (t) =
eAt bj of the linear differential equation (5)-(6).
2) Compute approximations {σkN , fkN , gkN } of the Hankel
singular values and singular vectors, and, if needed,
N
compute approximations {ϕN
j , ψi } to the balancing
modes, e.g., by the balanced POD quadrature approach
presented above.
3) Choose r and approximate the balanced truncated
model (Ar , Br , Cr ) by either
(MA ) using equations (10)-(11) from Theorem 1, and
computing Aϕj using (13), or computing A∗ ψi
similarly;
(Ma ) using equations (10)-(11) from Theorem 1, and
computing Ab using a bilinear form a : V ×V →
R as in (12);
(H0 ) using equations (17)-(18) and (15)-(16) from
Theorem 2;
(Hd ) using equations (14)-(16) from Theorem 2.
Notes:
A
a
• In step 3, (M ) and (M ) are the balanced POD
algorithms using the balancing modes from [13]. The
algorithm (Ma ) is only applicable for parabolic problems derived from a bilinear form (as in the second
model problem), and one must be able to evaluate the
bilinear form.
0
d
• In step 3, (H ) and (H ) are the transformation-free
variations of the balanced POD algorithm. The variation
(H0 ) uses only approximations to the Hankel singular
vectors evaluated at t = 0, while (Hd ) also uses
approximations to integrals containing time derivatives
of the Hankel singular vectors.
0
d
• The variations (H ) and (H ) require less computation:
specifically, the balancing modes are not computed, and
the inner products in the realization (10)-(11) are not
computed.
• The balancing transformation can be ill-conditioned,
even in the finite dimensional case [25]. Since (H0 )
and (Hd ) do not rely on a balancing transformation,
they may avoid possible errors due to an ill-conditioned
transformation.
b
d
• The integral representation (14) of A in (H ) with i = j
can be integrated exactly to give the representation (18)
in (H0 ). Therefore, in our numerical experiments with
the integral representation, we used the latter formula
when i = j.
VI. N UMERICAL R ESULTS
We report numerical results for the two model problems
outlined in Section II-B. We compare transfer functions of
the model problems and the balanced POD reduced order
models using the H∞ norm, which is the largest singular
value of the function on the imaginary axis. Since we
consider systems with only one input and one output, the
H∞ norm of a transfer function G(s) is the maximum value
of |G(iω)| for ω real.
For the computations, we approximated all time integrals
using the trapezoid rule. Also, we approximated the time
derivatives of the singular vectors using two different methods: second order finite differences and an indirect approach
from [13] (again with the trapezoid rule to approximate the
time integrals).
A. Model Problem 1
We begin with the first model problem, the 1D hyperbolic
PDE system with transfer function that can be evaluated
exactly.
To approximate the solution of the PDEs (5) and (6), we
used a simple first order accurate discontinuous Galerkin
method (with piecewise constant basis functions) for the
spatial discretization and forward Euler for the time discretization. We used an equal spacing ∆x between the spatial
nodes, and an equal time spacing ∆t = β∆x for forward
Euler. We integrated the discrete equations until tmax + 1/2,
where tmax = α−1 ln(β(β − α)) (see [13] for more details).
The approximate H∞ norm between the exact transfer
function G and the transfer function of the balanced POD
reduced order model Gr is shown in Table I for various
values of r and N , the number of equally spaced nodes
in the discontinuous Galerkin computation. (For the H∞
norm computation, we chose ω in the interval 10−3 ≤ ω ≤
102 .) We found nearly identical results for all approaches,
including using the two approaches to approximating the
derivatives of the singular vectors. We observed similar
behavior for other cases.
TABLE I
A PPROXIMATE TRANSFER FUNCTION ERRORS kG − Gr k∞ FOR MODEL
PROBLEM 1 WITH VARIOUS VALUES OF r AND N , THE NUMBER OF
EQUALLY SPACED NODES .
method
r = 3, N = 500
r = 10, N = 1000
1.93 × 10−3
2.00 × 10−2
7.64 × 10−3
1.93 × 10−3
10−2
10−3
1.93 × 10−3
2.00 ×
H0
Hd
2.00 ×
10−2
r = 5, N = 500
10−3
MA
7.64 ×
7.64 ×
For this problem, the transformation free algorithm variation using the Hankel singular values and singular vectors
and the representation of Theorem 2 works just as well as the
approach using the balancing modes and the representation
of Theorem 1.
B. Model Problem 2
For the second model problem, we find that the different
approaches can give quite different results.
For our experiments, we take µ = 0.1 and convection
coefficients
c1 (x, y) = x sin(2πx) sin(πy),
c2 (x, y) = y sin(πx) sin(2πy).
We also take b(x, y) = 5 if x ≥ 1/2 and b(x, y) = 0
otherwise, and c(x, y) = 5 for all x, y. We use piecewise
bilinear finite elements for the spatial discretization and the
trapezoid rule for time stepping. We stopped time stepping
when the L2 norm of the approximate solution was less than
10−3 .
Since we do not have an exact transfer function for this
problem, we compare the transfer function of the balanced
truncation of the matrix approximating system (using only
finite elements for the spatial discretization) and the transfer
function of the balanced POD reduced order model. We used
31 equally spaced finite element nodes in each coordinate
direction for all of the computations.
The approximate H∞ norm errors are shown in Table II
for r = 3 and various values of the time step ∆t. (For the
H∞ norm computation, we chose ω in the interval 10−4 ≤
ω ≤ 104 .) Most of the algorithm variations give accurate
approximations with similar errors, except the approaches
d
(MA
f ) and (Hf ) which approximate the time derivatives of
the Hankel singular vectors using finite differences. The
(MA
f ) approach fails completely, even though the similar
approach (MA
i ), where the time derivatives of the Hankel
singular vectors are approximated using the indirect method,
does give an accurate approximation. It is unclear why
the indirect methods are superior to the finite difference
approximations—also see (Hdf ) and (Hdi )—to the time derivatives for this example.
TABLE II
A PPROXIMATE TRANSFER FUNCTION ERRORS FOR MODEL PROBLEM 2
WITH r = 3 AND VARIOUS VALUES OF ∆t. T HE TIME DERIVATIVES
WERE APPROXIMATED BY EITHER SECOND ORDER FINITE DIFFERENCES
( SUBSCRIPT f ) OR THE INDIRECT METHOD ( SUBSCRIPT i) OF [13].
method
1
30
−2
10
r = 3, ∆t =
Ma
1.93 ×
MA
f
7.48 × 10+2
MA
i
H0
4.93 ×
10−2
4.93 ×
10−2
Hdf
Hdi
1
60
−3
10
r = 3, ∆t =
8.71 ×
9.07 × 100
1
100
−3
10
r = 3, ∆t =
2.88 ×
2.91 × 100
8.73 ×
10−3
2.76 × 10−3
8.68 ×
10−3
2.71 × 10−3
2.61 × 10−1
5.50 × 10−2
2.28 × 10−2
10−2
10−3
2.71 × 10−3
4.93 ×
8.68 ×
For r = 4, most of the errors behave similarly; see
Table III. One difference is that the approach (Ma ) using
the bilinear form is the most accurate for all values of the
time step ∆t. However, as the time step ∆t is refined, all
d
of the other approaches except (MA
f ) and (Hf ) are nearly as
accurate.
TABLE III
A PPROXIMATE TRANSFER FUNCTION ERRORS kG∆t
r − Gr k∞ FOR
MODEL PROBLEM 2 WITH r = 4 AND VARIOUS VALUES OF ∆t.
method
1
30
10−3
r = 4, ∆t =
1
60
10−3
r = 4, ∆t =
1
100
10−3
r = 4, ∆t =
Ma
7.86 ×
MA
f
2.16 × 10+1
2.87 × 10+2
6.41 × 100
MA
i
6.27 × 10−2
1.32 × 10−2
3.90 × 10−3
H0
6.27 ×
10−2
1.32 ×
10−2
3.85 × 10−3
8.58 ×
10−2
5.50 ×
10−1
4.98 × 10−2
1.32 × 10−2
3.85 × 10−3
Hdf
Hdi
6.27 × 10−2
4.43 ×
2.47 ×
VII. C ONCLUSIONS
We presented a variation on the balanced POD algorithm for balanced model reduction of a linear system.
The variation computes the reduced order model directly
without computing a balancing transformation or requiring
access to (approximating) system matrices. This saves some
computational work and avoids potential errors due to an
ill-conditioned balancing transformation. The algorithm variation worked well on two example problems, although we
did find that balanced POD with a transformation gave higher
accuracy with coarser time steps for one example.
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