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SIAM J. SCI. COMPUT.
Vol. 38, No. 5, pp. B837–B865
c 2016 Society for Industrial and Applied Mathematics
STRUCTURE-PRESERVING MODEL REDUCTION FOR
NONLINEAR PORT-HAMILTONIAN SYSTEMS∗
S. CHATURANTABUT† , C. BEATTIE‡ , AND S. GUGERCIN‡
Abstract. This paper presents a structure-preserving model reduction approach applicable to
large-scale, nonlinear port-Hamiltonian systems. Structure preservation in the reduction step ensures
the retention of port-Hamiltonian structure which, in turn, ensures the stability and passivity of the
reduced model. Our analysis provides a priori error bounds for both state variables and outputs.
Three techniques are considered for constructing bases needed for the reduction: one that utilizes
proper orthogonal decompositions, one that utilizes H2 /H∞ -derived optimized bases, and one that
is a mixture of the two. The complexity of evaluating the reduced nonlinear term is managed
efficiently using a modification of the discrete empirical interpolation method that also preserves
port-Hamiltonian structure. The efficiency and accuracy of this model reduction framework are
illustrated with two examples: a nonlinear ladder network and a tethered Toda lattice.
Key words. nonlinear model reduction, proper orthogonal decomposition, port-Hamiltonian,
H2 approximation, structure preservation
AMS subject classifications. 37M05, 65P10, 93A15
DOI. 10.1137/15M1055085
1. Introduction and background. The modeling of complex physical systems
often involves systems of coupled partial differential equations, which, upon spatial
discretization, lead to dynamical models and systems of ordinary differential equations
with very large state-space dimension. This motivates model reduction methods that
produce low-dimensional surrogate models capable of mimicking the input/output
behavior of the original system model. Such reduced-order models could then be
used as proxies, replacing the original system model in various computationally intensive contexts that are sensitive to system order, for example, as a component in a
larger simulation. Dynamical systems frequently have structural features that reflect
underlying physics and conservation laws characteristic of the phenomena modeled.
Reduced models that do not share such key structural features with the original system
may produce response artifacts that are “unphysical,” and as a result, such reduced
models may be unsuitable for use as dependable surrogates for the original system,
even if they otherwise yield high response fidelity. The key system feature that we
wish to retain in our reduced models will be port-Hamiltonian structure. In a certain
sense, this will be an expression of system passivity.
1.1. Port-Hamiltonian systems. Models of dynamic phenomena may be constructed within a system-theoretic network modeling paradigm that formalizes the
interconnection of naturally specified subsystems. If the core dynamics of subsystem
∗ Submitted to the journal’s Computational Methods in Science and Engineering section January
12, 2016; accepted for publication (in revised form) July 12, 2016; published electronically October
26, 2016. A portion of this work, contained in sections 2, 2.1, and 2.2, appeared previously in [3].
http://www.siam.org/journals/sisc/38-5/M105508.html
Funding: This work was supported in part by NSF through grant DMS-1217156. The work
of the second author was supported in part by the Einstein Stiftung Berlin. The work of the third
author was supported in part by the Alexander von Humboldt Stiftung.
† Department of Mathematics and Statistics, Thammasat University, Pathumthani, 12120 Thailand ([email protected]).
‡ Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123 ([email protected],
[email protected]).
B837
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B838
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
components are described by variational principles (e.g., least-action or virtual work),
the aggregate system model typically has structural features that characterize it as a
port-Hamiltonian system. While greater generality is both possible and useful (see,
in particular, the review article [30] and the monographs [8] and [16]), it will suffice to consider realizations of finite-dimensional nonlinear port-Hamiltonian (NLPH)
systems that appear as
(1.1)
ẋ = (J − R) ∇x H(x) + Bu(t),
y = BT ∇x H(x),
where x ∈ Rn is the n-dimensional state vector; H : Rn → [0, ∞) is a continuously
differentiable scalar-valued vector function—the Hamiltonian, describing the internal
energy of the system as a function of state; J = −JT ∈ Rn×n is the structure matrix
describing the interconnection of energy storage elements in the system; R = RT ≥ 0
is the n × n dissipation matrix describing energy loss in the system; and B ∈ Rn×m
is the port matrix describing how energy enters and exits the system. We will always
assume that the Hamiltonian is bounded below, and so, without loss of generality,
strictly positive, i.e., H(x) > 0 for all x.
The family of systems characterized by (1.1) generalizes the classical notion of
Hamiltonian systems which would be expressed in our notation as ẋ = J∇x H(x).
The analogue of conservation of energy for Hamiltonian systems becomes for (1.1)
Z t1
y(t)T u(t) dt,
(1.2)
H(x(t1 )) − H(x(t0 )) ≤
t0
which is to say, the change in the internal energy of the system, as measured by H,
is bounded by the total work done on the system. The presumed positivity of H
conforms with its use as a “supply function” in the sense of Willems [31], and so
NLPH systems are always stable and passive. Furthermore, the class of NLPH systems defined as in (1.1) is closed under power-conserving interconnection—connecting
port-Hamiltonian systems together produces an aggregate system that must also be
port-Hamiltonian, and hence a fortiori, must be both stable and passive. This last
fact provides compelling motivation to preserve port-Hamiltonian structure when producing low-order surrogate models intended to be used as proxies for systems of the
sort defined by (1.1).
We assume in all that follows that the matrices J, R, and B are constant; however,
this assumption is adopted here largely for convenience. J, R, and B may each
depend on the state vector, x, and input vector, u, and may also carry an explicit
time dependence. This allows for a rich variety of nonlinearities for which the key
dissipation inequality (1.2) still holds, and for which the underlying dynamical system
remains both passive and stable. The model reduction strategies that we describe
below can be adapted with negligible modification in this elaborated setting. One
should note the fundamental focus rests on retaining the dissipation inequality (1.2),
thus assuring passivity and stability of the reduced model. “Classical” Hamiltonian
systems have additional symmetries that are associated with an underlying Poisson
structure capable of distinguishing canonical position and momentum coordinates.
This requirement is relaxed for port-Hamiltonian systems. Preserving these additional
symmetries in a reduced model when present in the original model may also be of
interest; this more restrictive framework will not be considered here.
1.2. Petrov–Galerkin reduced models. Most model reduction approaches
involve some variation of a Petrov–Galerkin projective approximation to the equations
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
B839
describing the system dynamics. This proceeds by choosing two subspaces of Rn : an rdimensional trial subspace, Vr ⊂ Rn , and an r-dimensional test subspace, Wr ⊂ Rn .
It is convenient and nonrestrictive in practice to assume additionally that Vr and
Wr have a “generic orientation” with respect to one another so that neither subspace
contains any nontrivial vectors that are orthogonal to all vectors in the other subspace.
The evolution of an associated reduced-order model may be described in the following
(initially indirect) way:
(1.3)
Find a trajectory, vr (t), contained in Vr such that
v̇r (t) − (J − R) ∇x H(vr ) − Bu(t) ⊥ Wr ;
the associated output is yr (t) = BT ∇x H(vr ).
The dynamics described by (1.3) can be represented directly as a dynamical system evolving in a state space of reduced dimension r once bases are chosen for the
two subspaces Vr and Wr . Let Ran(M) denote the range of a matrix M. Define
matrices Vr , Wr ∈ Rn×r so that Vr = Ran(Vr ) and Wr = Ran(Wr ). We can represent reduced system trajectories as vr (t) = Vr xr (t) with xr (t) ∈ Rr for each t; the
Petrov–Galerkin approximation (1.3) can be rewritten as
WrT [Vr ẋr (t) − (J − R) ∇x H(Vr xr ) − Bu(t)] = 0
and yr (t) = BT ∇x H(Vr xr ).
Since Vr and Wr are assumed to have a generic orientation with respect to one another,
WrT Vr is invertible, and we may choose bases for Vr and Wr such that WrT Vr = I.
This leads to a state-space representation of a reduced-order nonlinear dynamical
system approximating (1.1):
(1.4)
ẋr = WrT (J − R) ∇x H(Vr xr ) + WrT Bu(t),
yr = BT ∇x H(Vr xr ).
Typically r n, and (1.4) describes a reduced-order model for the original system
(1.1). There are two shortcomings that may be anticipated. First, (1.4) will not have
the form of (1.1) unless special subspaces are chosen, and so (1.4) will not typically
be an NLPH system and passivity may be lost. Second, if the Hamiltonian function,
H, is nonquadratic, each evaluation of ∇x H(Vr xr ) in (1.4) occurring in the course
of a simulation will likely require a lifting of xr to Rn (implicit in the formation of
Vr xr ), and so direct simulation of (1.4) is still likely to have complexity proportional
to n r; little or no savings may be realized from reducing the system order.
We consider each of these issues in subsequent sections. In section 2, a structurepreserving model reduction approach for large-scale NLPH systems will be introduced.
This approach is built upon Petrov–Galerkin projections that are modified to ensure
that the resulting reduced system retains port-Hamiltonian structure, and thus stability and passivity. Three types of reduced-order bases used to define these projections
will be considered: (i) one based on the proper orthogonal decomposition (POD),
(ii) one derived from H2 -optimal approaches for a related linear problem (which we
refer to as “H2ε bases”), and (iii) hybrid H2ε -POD bases that combine both types.
The bases (i) and (ii) were originally considered in [3]. Numerical experiments in section 2.4 illustrate that the hybrid H2ε -POD bases significantly outperform the other
two. In section 2.5, we develop corresponding error analyses and bounds for reduced
states and outputs. In order to resolve the “lifting bottleneck” described above, we
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B840
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
develop, in section 3, a variant of the discrete empirical interpolation method (DEIM)
[6] that incorporates the structure-preserving model reduction approach of section 2.
In section 3.4, the efficiency and accuracy of our approach are illustrated with two
examples: a nonlinear ladder network and a tethered Toda lattice. Corresponding a
priori error bounds for states and outputs are derived in section 3.5.
2. Preserving port-Hamiltonian structure in reduced models. The process of obtaining a reduced model from an original full-order model can be viewed as
one of identifying and preserving high-value portions of the state space, i.e., portions
of the state space that contribute substantively to the system response. We proceed
with the following heuristics: Suppose we have identified two r-dimensional subspaces,
Vr and Wr , that are “high-value” in the sense that for “most” input signal profiles,
u(t), in (1.1), we have

• The associated trajectory, x(t), stays “close” to Vr , so that



x(t) ≈ vr (t) for some vr (t) ∈ Vr , and

High-value
(2.1)
• the internal force, ∇x H(x), stays “close” to Wr , so that
PH-spaces: 



∇x H(x(t)) ≈ wr (t) for some wr (t) ∈ Wr .
Evidently, if Vr = Ran(Vr ) and Wr = Ran(Wr ), then vr (t) = Vr xr (t) for some
trajectory xr ∈ Rr , and wr (t) = Wr fr (t) for some choice of fr ∈ Rr . Exactly what
comprises “most” input signal profiles and how one measures “closeness” in (2.1) will
vary depending on context; we consider different possibilities later in this section. If
either the trajectory, x(t), or internal force, ∇x H(x), fails to stay “close” to any rdimensional subspace, then it may not make sense to seek reduced models of order r.
This simply reflects limitations on the accuracy achievable using reduced models of
dimension no greater than r.
For the time being, note that if x(t) ≈ Vr xr (t), then plausibly,
∇x H(Vr xr (t)) ≈ ∇x H(x(t)) ≈ Wr fr (t).
Significantly, these statements amount to assertions about the subspaces, Vr and Wr ,
and do not constrain the choice of bases for the subspaces. As long as Vr and Wr
have a generic orientation with respect to one another, bases may be chosen so that
WrT Vr = I. With this in mind, note further that
(2.2)
fr (t) = VrT Wr fr (t) ≈ VrT ∇x H(Vr xr (t)) = ∇xr Hr (xr (t)),
where we have introduced a reduced Hamiltonian, Hr (xr ) = H(Vr xr ). Thus,
(2.3)
∇x H(Vr xr (t)) ≈ Wr ∇xr Hr (xr (t)).
Substituting Vr xr (t) for x(t) and Wr ∇xr Hr (xr (t)) for ∇x H(x(t)) in (1.1) and
then multiplying by WrT leads to a state-space representation of a reduced portHamiltonian approximation:
(2.4)
ẋr = (Jr − Rr )∇xr Hr (xr ) + Br u(t),
yr (t) = BTr ∇xr Hr (xr ),
with Hr (xr ) = H(Vr xr ), Jr = WrT JWr , Rr = WrT RWr , and Br = WrT B. Note
that Hr : Rr → [0, ∞) is continuously differentiable, Jr = −JTr , and Rr = RTr ≥ 0,
so (2.4) retains the structure of (1.1) and is a port-Hamiltonian system.
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
B841
The earlier works [10] and [26] also offer structure-preserving model reduction
methods for NLPH systems. These papers exploit the Kalman decomposition and
balanced truncation in deriving reduced models of NLPH systems. Obtaining the
Kalman decomposition of the full-order original system or balancing it is computationally demanding for nonlinear systems of even modest order; see, e.g., [11, 26] and
references therein. Such approaches are infeasible for the problem class we consider,
which may have thousands of state variables. In what follows, we develop approaches
that remain feasible for this problem class; they depend on the construction of lowdimensional projecting subspaces motivated by the heuristics in (2.1) .
2.1. POD subspaces. The proper orthogonal decomposition (POD) is a natural approach to producing high-value modeling spaces as described in (2.1). POD
is a popular approach to (unstructured) model reduction [18, 27] which we adapt to
our setting as follows: Fix a square integrable input signal, u(t), for the system (1.1).
The corresponding trajectory, x(t), will then also be square integrable. Denoting
orthogonal projections, P and Q, we consider the minimization problem
Z ∞
argmin
P? = rank(P)=r
k (I − P) x(t)k2 dt
0
and
Q? =
argmin
rank(Q)=r
Z
∞
k (I − Q) ∇x H(x(t))k2 dt.
0
We would like to take Vr = Ran(P? ) and Wr = Ran(Q? ), but this is not a computationally tractable approach as it stands. If the integrals are truncated and then
approximated with a trapezoid rule, one arrives at a characterization of POD subspaces which is incorporated into our first method, summarized as Algorithm 1. As
our numerical results show (and consistent with common experience), the use of POD
in Algorithm 1 provides subspaces Vr and Wr that can be very effective in capturing
dynamic features that are present in the original sampled system response. However,
as an empirical method, it is incapable of providing information about dynamic response features that are absent in the sampled system response, but that could have
been present had a different choice of input profile been made. One can mitigate
this difficulty somewhat by extending POD by including a representative sampling of
input profiles, but this leaves open the question of what constitutes a representative
sampling of input profiles. A different approach leads to our next class of subspaces.
2.2. Hε2 -optimal subspaces. We next consider a choice of subspaces, Vr and
Wr , that are optimal (in a sense we will describe) for all possible input profiles that are
sufficiently small, though it will often be the case that this choice is effective for larger
input profiles as well. In contrast to the previous POD approach, no simulations need
to be performed in order to derive these approximating subspaces, and in particular,
no choice of actual inputs is necessary. The subspaces that are generated provide
a near-optimal reduction for the corresponding linearized port-Hamiltonian model.
Linearization is used here only as a tool to obtain useful information that is encoded
into Vr and Wr , which are in turn used for the reduction of the nonlinear system
(1.1).
Any (virtual) input profile, u(t), may be scaled to have sufficiently small magnitude, and so, assuming either null initial conditions or sufficiently large t, the resulting
trajectory, x(t), will be small as well. Linear terms in the internal forcing then dom-
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S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
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Algorithm 1. Structure-preserving POD reduction of NLPH systems (POD-PH).
1:
Generate a trajectory x(t), and collect snapshots:
X = [x(t0 ), x(t1 ), x(t2 ), . . . , x(tN )].
2:
Simultaneously collect associated force snapshots:
F = [∇x H(x(t0 )), ∇x H(x(t1 )), . . . , ∇x H(x(tN ))].
3:
e r , for a
Truncate an SVD of the snapshot matrix, X, to get a POD basis, V
e r x̃r (t)).
“high-value” subspace of the state space (x(t) ≈ V
4:
f r , spanning a second “highTruncate an SVD of F to get a second POD basis, W
f r f̃r (t).
value” subspace approximating the range of ∇x H(x(t)) ≈ W
5:
fr →
er →
Change bases W
7 Wr and V
7 Vr such that WrT Vr = I.
6:
With Vr and Wr determined in this way, the POD-PH reduced port-Hamiltonian
system is then specified by (2.4).
inate so that ∇x H(x) ≈ Qx for some symmetric positive-definite matrix Q ∈ Rn×n .
Indeed, Q = ∇2 H(0), the Hessian matrix for H(x) evaluated at the equilibrium state,
x = 0. This leads to an ancillary linear port-Hamiltonian system:
ẋ = (J − R) Q x + Bu(t),
(2.5)
y = BT Q x.
We proceed to construct projecting subspaces, Vr and Wr , that would produce,
via the reduction described in (2.4), an effective reduced port-Hamiltonian model
approximating (2.5):
(2.6)
ẋr = (Jr − Rr )Qr xr + Br u(t),
yr (t) = BTr Qr xr ,
where Jr = WrT JWr , Rr = WrT RWr , Qr = VrT QVr , and Br = WrT B. Observe
that Jr = −JTr , Rr = RTr ≥ 0, and Qr = QTr > 0. The quality of the subspaces is
interpreted now as how well (2.6) approximates (2.5). This in turn may be interpreted
as a rational approximation problem: We associate the linear dynamical systems (2.5)
and (2.6) with their transfer functions,
(2.7)
G(s) = BT (sQ−1 − (J − R))−1 B and
−1
Gr (s) = BTr (sQ−1
Br ,
r − (Jr − Rr ))
respectively. If Gr (s) approximates G(s) well with respect to some (appropriately
chosen) norm, then the reduced system outputs will approximate the full-order system
outputs uniformly well over all inputs with bounded energy (square integrable); our
model reduction problem has been reduced to a rational approximation problem.
We wish to find a degree-r rational function, Gr (s), having the structure given in
(2.7) that also approximates G(s) well. Tangential rational interpolation provides a
useful tool for this: Given r interpolation points σ1 , . . . , σr in the complex plane with
corresponding tangent directions {b1 , . . . , br } ∈ Cm , construct an rth-order system,
Gr , so that Gr is port-Hamiltonian and
(2.8)
Gr (σi )bi = G(σi )bi
for i = 1, . . . , r.
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
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A solution to this problem was given in [13] and [14].
Theorem 2.1. Given the system (2.5), interpolation points σ1 , . . . , σr , and tangent directions b1 , . . . , br , construct
e r = [(σ1 I − (J − R)Q)−1 Bb1 , . . . , (σr I − (J − R)Q)−1 Bbr ].
V
e T QV
e r = RT R, assign Vr = V
e r R−1 , and then
Define the Cholesky factorization of V
r
construct Wr = QVr . Set
(2.9)
Jr = WrT JWr , Qr = VrT QVr = Ir , Rr = WrT RWr , and Br = WrT B.
Then the reduced model,
(2.10)
Gr :
ẋr = (Jr − Rr )Qr xr + Br u,
yr = BTr Qr xr ,
is port-Hamiltonian (hence stable and passive) and also satisfies the interpolation
conditions (2.8).
For information on transfer function interpolation in the special case of singleinput/single-output port-Hamiltonian systems, see [23, 13, 24]. For an overview of
model reduction methods for linear port-Hamiltonian systems, see [22].
2.2.1. Hε2 port-Hamiltonian approximation. Port-Hamiltonian approximations of reduced order may be constructed using Theorem 2.1 once shifts, {σi }, and
tangent directions, {bi }, are chosen, but this does not give any information on how
best to choose {σi } and {bi }. We discuss issues related to finding effective approximations with respect to the H2 norm: The H2 norm of G is defined as
kGkH2 =
1
2π
Z
∞
−∞
2
kG(ıω)kF dω
1/2
.
Let Gr (s) minimize the H2 error kG − Gr kH2 over all possible degree-r rational funcPr
c bT
tions, and suppose that Gr (s) has a partial fraction expansion Gr (s) = k=1 k bk .
s−λk
Then, as shown in [12], the interpolation conditions
(2.11)
bk )bk = Gr (−λ
bk )bk
G(−λ
for k = 1, . . . , r
are necessary conditions for H2 optimality (there are additional conditions that must
also be satisfied in general). In other words, the optimal H2 approximant Gr is a
tangential interpolant to G at the mirror images of the reduced-order poles. For the
full set of necessary conditions required for optimality, see [12].
A method was introduced in [14] that produces an interpolatory reduced-order
port-Hamiltonian system satisfying the conditions given in (2.11). Since the interpobk and the tangential directions bk depend on the reduced model to
lation points −λ
be computed, an iterative process is used to correct the interpolation points and tangential directions until the desired conditions in (2.11) are obtained. These constitute
only a subset of the necessary conditions required for H2 -optimality. The remaining
degrees of freedom are used in maintaining port-Hamiltonian structure. An algorithm
that accomplishes this was introduced in [14]. Algorithm 2 uses this methodology to
construct the model reduction bases for the second structure-preserving model reduction of NLPH systems.
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Algorithm 2. Structure-preserving H2ε reduction of NLPH systems (H2ε -PH).
1:
Linearize the NLPH system (1.1) to obtain a linear port-Hamiltonian system of
the form in (2.5).
2:
Make an initial selection of shifts {σi }r1 , and tangent directions {bi }r1 .
3:
while (not converged) do
4:
b r = [(σ1 I − (J − R)Q)−1 Bb1 , . . . , (σr I − (J − R)Q)−1 Bbr ].
V
5:
b r L−1 with V
b T QV
b r = LT L (so Qr = VT QVr = Ir ).
Set Vr = V
r
r
6:
Set Wr = QVr (so VrT Wr = Ir ).
7:
Set Jr = WrT JWr , Rr = WrT RWr , and Br = WrT B.
8:
Calculate left eigenvectors: zTi (Jr − Rr ) = λi zTi .
9:
Set σi ←− −λi and bi ←− BTr zi for i = 1, . . . , r.
10:
end while
{ Calculate final H2ε -PH bases: }
11:
b r = [(σ1 I − (J − R)Q)−1 Bb1 , . . . , (σr I − (J − R)Q)−1 Bbr ].
Find V
12:
b r = LT L.
b r L−1 with V
b T QV
Set Vr = V
r
13:
Set Wr = QVr .
14:
With Vr and Wr determined in this way, construct the reduced NLPH system
using (2.4).
Note that we use the linearized port-Hamiltonian model only to obtain the H2ε -PH
model reduction subspaces. Once Vr and Wr are obtained using the H2ε approach
outlined in Algorithm 2, we use these subspaces to reduce the original nonlinear system
as shown in (2.4). See [21, 25] for other approaches that derive useful information
from linearized systems in order to reduce nonlinear systems.
2.3. A hybrid POD -Hε2 approach. The model reduction subspaces that POD
provides are effective in capturing the dynamics that are represented in the original
snapshot data, but naturally they will miss features that are absent in this data.
To resolve this issue at least partially, we have proposed the use of asymptotically
optimal subspaces that are uniformly accurate for input profiles that are sufficiently
small. These subspaces are generated from a near-optimal reduction of a linearized
port-Hamiltonian model using an approach proposed in [14]. As the trajectory moves
away from the linearization point, the efficiency of these subspaces in capturing the
true dynamics will likely degrade. Therefore, we propose combining the subspaces
resulting from POD (i.e., directly obtained from a simulation of the NLPH system)
with the asymptotically optimal subspaces resulting from the linearized model. The
structure-preserving reduction is applied as in (2.4) with the aggregate subspace.
b rb = [b
b2 , . . . , v
brb] be the POD-PH basis
For a given reduced dimension r, let V
v1 , v
of dimension rb obtained from Algorithm 1, and let V̄r̄ = [v̄1 , v̄2 , . . . , v̄r̄ ] be the H2ε PH basis of dimension r̄ obtained from Algorithm 2, where rb and r̄ are positive
integers with rb + r̄ = r. Then, the hybrid basis Vr of dimension r can be obtained
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
B845
b rb V̄r̄ ] ∈ Rn×r . The other
from the orthonormal basis of concatenated matrix [V
projection basis Wr can be constructed similarly by combining POD-PH and H2ε -PH
basis vectors. This leads to a third algorithm for structure-preserving model reduction
of NLPH systems, Algorithm 3, using POD-H2ε subspaces.
Algorithm 3. Structure-preserving POD-H2ε reduction of NLPH (POD-H2ε -PH).
1:
Pick rb and r̄ = r so that rb + r̄ = r.
2:
b rb and W
c rb of dimension rb using Algorithm 1.
Obtain the POD-PH bases V
3:
Obtain the H2ε -PH bases V̄r̄ and W̄r̄ of dimension r̄ using Algorithm 2.
4:
e r of dimension n × r as the orthonormal basis of the concatenated
Construct V
b rb V̄r̄ ] ∈ Rn×r .
matrix [V
5:
f r of dimension n × r as the orthonormal basis of the concatenated
Construct W
c rb W̄r̄ ] ∈ Rn×r .
matrix [W
6:
fr →
er →
Change bases W
7 Wr and V
7 Vr such that WrT Vr = I.
7:
With Vr and Wr determined in this way, the POD-H2ε reduced port-Hamiltonian
system is then specified by (2.4).
2.4. An illustrative example. To illustrate the structure-preserving model
reduction techniques described in section 2, we consider an N -stage nonlinear ladder
network (see Figure 1) producing reduced models for three cases of bases: (i) POD
bases (Algorithm POD-PH), (ii) H2ε bases (Algorithm H2ε -PH), and (iii) hybrid PODH2ε bases (Algorithm POD-H2ε -PH). The system has two inputs and two outputs. The
two inputs are a voltage signal applied to the left-hand terminal pair and a current
injection across the right-hand terminal pair. The symmetrically paired outputs are
the induced current across the left-hand terminal pair and the induced voltage signal
across the right-hand terminal pair. For simplicity, we assume that each stage of
the ladder network is built from identical components and that the current injection
from the right is zero. Resistors and inductors are assumed to behave linearly. The
capacitors have a nonlinear capacitance-voltage characteristic of the form Ck (V ) =
C0 V0
V0 +V .
Fig. 1. Ladder network circuit topology (capacitors are nonlinear).
Inductors and capacitors are evidently the energy storage elements of the circuit,
so we take as state variables the magnetic fluxes in the inductors, {φk (t)}N
k=1 , and the
charges on the capacitors, {Qk }N
k=1 , with labels referring to the stages k = 1, . . . , N ,
respectively, where they occur. The energy stored in the stage k (linear) inductor
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B846
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
1
φ2k . To determine the energy
may be expressed in terms of its magnetic flux as 2 L
0
stored in the nonlinear capacitors, note first that the charge on a capacitor may be
expressed as a function of the voltage, V , held across the capacitor:
Z V
V
Qk (V ) =
C(v) dv = C0 V0 log 1 +
,
V0
0
which may be inverted to obtain
Vk (Qk ) = V0 exp
Qk
C0 V0
−1 .
The energy stored in the capacitor at stage k of the circuit is then given by
Z Qk
Qk
Vk (q) dq = C0 V02 exp
− 1 − Qk V0 ,
C0 V0
0
and the total energy stored in stage k is then
Qk
1 2
H [k] (φk , Qk ) = C0 V02 exp
− 1 − Qk V0 +
φ .
C0 V0
2 L0 k
The Hamiltonian for this system is
H(Q1 , . . . , QN , φ1 , . . . , φN ) =
N
X
H [k] (φk , Qk ).
k=1
We order the state variables so that
0
S J = −S
, where S is an upper
T
0
−1 on the superdiagonal; R = G00 I
x = [Q1 , . . . , QN , φ1 , . . . , φN ]T and therefore
bidiagonal matrix with 1 on the diagonal and
0 ; and B = eN +1 , eN , where ek denotes
R I
0
the kth column of the identity. Consider the particular case of a 50-stage (N = 50)
circuit with parameters L0 = 2µH, V0 = 1V, R0 = 1Ω, G0 = 10µf.
We applied a voltage pulse to the left port of the network (Gaussian pulse windowed to 3µsec with a magnitude of 3V, σ of 0.5) and observed the output voltage
at the right port. The output is displayed as a solid green trace in Figure 2 (color
available online). The induced response of the linearized full-order network is also
displayed (green dashed line) for comparison. Notice that nonlinearity sharpens the
peak of the response and significantly reduces dispersion. The POD basis sets are
generated from uniformly sampled snapshots x(t) and ∇x H(x(t)) of this full-order
system with Gaussian impulse training input. These basis sets are then used to construct the POD-PH reduced system. In the cases of H2ε -PH bases, the procedure
described in Algorithm 2 is used. We also use Algorithm 3 to generate the PODH2ε -PH bases. All three reduced models are then simulated for the same Gaussian
impulse training input, which was used for generating the POD snapshot as well as
a different one, a sinusoidal input. First we investigate the accuracy of POD-PH and
H2ε -PH for r = 6. Figure 2 illustrates that the two structure-preserving nonlinear
reduced models capture the output of the original nonlinear system very accurately
for both types of excitations.
Next, we consider the effect on the accuracy of the state-space solutions of using
different proportions of POD-PH and H2ε -PH basis vectors in the hybrid POD-H2ε PH basis. Results for a wide range of r values are illustrated in Figure 3, depicting
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B847
MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
True
response
6
Volts
8
6
Linearized
full order
4
4
2
2
POD−PH
r=6
0
0
H2−PH
−2
−2
r=6
−4
0
2
4
6
Time (µ−seconds)
8
10
0
2
4
6
Time (µ−seconds)
−4
10
8
Fig. 2. Ladder network: Time responses of the reduced-order systems from Gaussian pulse
(left) and from sinusoidal input (right).
Average Relative Error of outputs
0
Average Relative Error of State solutions
10
r=4
r=6
r=10
r=12
r=14
r=16
0
r=4
r=6
r=10
r=12
r=14
r=16
10
Avg Rel Error (inf−norm)
Avg Rel Error (inf−norm)
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response
to sinusoid
Output response to sinusoid inputOutput
(POD from
Gaussian
pulse) input (POD from Gaussian pulse)
Volts
Output response to Gaussian pulse input (POD from Gaussian pulse)
8
−1
10
−1
10
−2
10
−2
0
2
4
6
8
number of POD bases: rpod
10
12
14
16
10
0
2
4
6
8
number of POD bases: rpod
10
12
14
16
Fig. 3. Ladder network: Average relative errors of outputs and state variables of structurepreserving reduced systems (2.4) using hybrid bases with different numbers of POD and H2ε vectors.
the relative error in the output. In these figures, the colors correspond to a fixed order
r (color available online). The x-axis is the number of POD-PH basis vectors for that
given order, so, e.g., the blue line corresponds to relative error for r = 16. For that
line, the value corresponding to x = 12 means that for the r = 16 model, r̂ = 12
POD-PH basis vectors are combined with r̄ = r − r̂ = 4 H2ε -PH basis vectors. The
most accurate approximation resulted from reduced systems with bases that combined
roughly equal numbers of POD-PH and H2ε -PH basis vectors. Figure 4 shows that
reduced systems constructed from these hybrid bases (using equal numbers of PODPH and H2ε -PH vectors) can give much more accurate approximations than those
constructed with POD-PH alone or H2ε -PH alone. For r = 12, say, a reduced system
with a hybrid basis using r̂ = r̄ = 6 (red dashed line) produced roughly 10 times
smaller error for both state variables and outputs than those constructed using only
POD-PH (green solid lines) or only H2ε -PH bases (blue solid lines).
2.5. An a priori error bound for NLPH-reduced models. Fix a reduction
order, r, and let Vr = Ran(Vr ) and Wr = Ran(Wr ) denote r-dimensional reduction
subspaces used in creating reduced NLPH systems as in (2.4). We provide an error
analysis here that bounds the deviation between the true state trajectory of an NLPH
system and that provided by a reduced NLPH system of the form given in (2.4).
This leads in turn to a bound on the error between the true system output and the
reduced NLPH system output. Typically, the reduction subspaces Vr and Wr will be
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B848
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
Average Relative Error
Average Relative Error of output (different input than the training input)
0
10
−1
−1
10
−2
10
−3
10
4
10
−2
H2−PH
Hε2−PH
POD − PH
POD − PH
ε
ε
POD & Hε2−PH
6
10
POD & H2−PH
8
10
12
14
4
6
−3
8
10
12
10
14
r
r
Average Relative Error of State variable (same input as the training input)
Average Relative Error of State variable(different input than training input)
0
0
10
Average Relative Error
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Average Relative Error of output (same input as the training input)
0
10
10
−1
−1
10
10
ε
Hε2−PH
H2−PH
POD−PH
POD−PH
ε
POD & Hε2−PH
POD & H2−PH
−2
10
4
−2
6
8
10
12
14
4
6
8
r
10
12
10
14
r
Fig. 4. Ladder network: Average relative errors of outputs and state variables of reduced
systems using bases from (i) H2ε , (ii) POD, (iii) combination of POD and H2ε (with same number of
basis vectors for both POD and H2ε ). The POD-PH reduced systems are constructed using Gaussian
impulse training input. The left and the right plots, respectively, use the impulse input (same as the
training input for POD) and the sinusoidal input (different from the training input for POD).
chosen consistently with the heuristics of (2.1), but the bounds we derive apply more
generally.
Let Q ∈ Rn×n be a symmetric positive-definite matrix, and define apweighted
inner product on Rn as hx, ziQ = xT Qz, with a related norm, kxkQ = hx, xiQ .
We leave the choice of Q open for the time being; however, the choices Q = I and
Q = ∇2 H(x0 ) at a locally stable equilibrium point x0 will have particular merit.
For a mapping F : Rn → Rn , we define the associated Lipschitz constant and
logarithmic Lipschitz constant of F relative to Q (see [28]) as
kF(u) − F(v)kQ
ku − vkQ
u6=v
(2.12) LQ [F] = sup
hu − v, F(u) − F(v)iQ
,
ku − vk2Q
u6=v
and LQ [F] = sup
respectively. Note that LQ [F] could be negative and −LQ [F] ≤ LQ [F] ≤ LQ [F].
e r, W
f r ∈ Rn×r such that
Suppose Q-orthogonal bases for Vr and Wr are chosen: V
e r ) and Wr = Ran(W
f r ), with V
e T QV
e r = I and W
f T QW
f r = I. Consider
Vr = Ran(V
r
r
n
n
e rV
eTQ
Q-orthogonal projectors ΠV : R → Vr , ΠW : R → Wr defined by ΠV = V
r
T
frW
f Q. Define an ancillary state-space projection as Pr = Vr WT .
and ΠW = W
r
r
For a given (true) system trajectory, x(t), the best approximation (relative to the
Q norm) that is available by a path in Vr is given by ΠV x(t), and so the state-space
error x(t) − Vr xr (t) is bounded pointwise below as
k(I − ΠV )x(t)kQ ≤ kx(t) − Vr xr (t)kQ
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
B849
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for all t ≥ 0. The error associated with the reduced internal force, ∇xr Hr , is bounded
similarly:
k(I − ΠW )∇x H(x(t))kQ ≤ k∇x H(x(t)) − Wr ∇xr Hr (xr (t))kQ .
Let the optimal state-space and internal force residual vectors be defined as
εx (t) = (I − ΠV )x(t)
and εF (t) = (I − ΠW )∇x H(x(t)).
The squared residual state-space and internal force errors integrated over [0, T ] will
be denoted as
Z T
Z T
Ex =
kεx (t)k2Q dt =
k(I − ΠV )x(t)k2Q dt
0
0
and
Z
EF =
T
kεF (t)k2Q
Z
T
dt =
0
k(I − ΠW )∇x H(x(t))k2Q dt.
0
We seek to bound the state-space error, x − Vr xr , and output error, y − yr , in
terms of Ex , EF , and the deviation between the projected and reduced initial condition.
Note that when the reduction spaces, Vr and Wr , are generated via a POD approach
(e.g., Algorithm 1), then the aggregate errors Ex , EF are approximately minimized
and can be expressed directly as the sum of neglected singular values of the snapshot
matrices associated with x(t) and ∇x H(x(t)), respectively. As the step size decreases,
Vr and Wr converge to subspaces that minimize Ex and EF , respectively (see, e.g.,
[17]). Nonetheless, as illustrated in section 2.4, neither the state-space error induced
by the resulting reduced system nor the corresponding output error will generally be
minimized with this choice.
Theorem 2.2. Suppose that Q ∈ Rn×n is symmetric positive-definite, and that
a reduced port-Hamiltonian system as in (2.4) is constructed to approximate the fullorder NLPH system (1.1) using reduction bases Vr , Wr ∈ Rn×r defined so that
VrT QVr = Ir and VrT Wr = Ir . (Note that it may or may not be the case that
Wr = QVr .) Suppose further that F(x) = ∇x H(x) in (1.1) is Lipschitz continuous.
Denote A = J − R and Pr = Vr WrT . Then
Z
(2.13)
0
T
2
kx(t) − Vr xr (t)kQ dt ≤ Cx Ex + CF EF + C0 kWrT x(0) − xr (0)k2 ,
Z
(2.14)
T
2
bx Ex + C
bF EF + C
b0 kWrT x(0) − xr (0)k2 ,
ky(t) − yr (t)k dt ≤ C
0
where
α = LQ [Pr APTr F],
β = kPr AkQ kPTr kQ ,
δ = 2kBT Q−1 Bk kPTr k2Q ,
Z
cα (t) =
γ = LQ [F] kPr kQ ,
t
e2ατ dτ,
Z
Cα (t) =
0
2
Cx = (2βγ)2 Cα (T ) + 2 kPr kQ ,
CF = (2β)2 Cα (T ),
t
cα (τ ) dτ,
0
C0 = 2 cα (T ), and
bx = δ · LQ [F]2 Cx , C
bF = δ · (1 + LQ [F]2 CF ), C
b0 = δ · LQ [F]2 C0 .
C
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B850
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
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Proof. First note that
(I − Pr )x(t) = (I − Pr )(I − ΠV )x(t) = (I − Pr )εx (t)
(2.15)
(2.16)
and
(I − PTr )F(t) = (I − PTr )(I − ΠW )F(t) = (I − PTr )εF (t).
The state-space error can be separated into the sum of a component in Ker(Pr )
(i.e., orthogonal to Wr ) and a component contained in Ran(Pr ) = Vr :
x(t) − Vr xr (t) = ρ(t) + Vr θ(t),
where ρ(t) = x(t) − Vr WrT x(t) = (I − Pr )εx (t) ∈ Wr⊥ and θ(t) = WrT x(t) − xr (t).
Then
Z T
Z T
Z T
(2.17)
kx(t) − Vr xr (t)k2Q dt ≤ 2
kρ(t)k2Q dt + 2
kθ(t)k2 dt.
0
0
0
For the first term, we may estimate immediately
Z
T
kρ(t)k2Q dt ≤
0
Z
T
k(I − Pr )εx (t)k2Q dt ≤ kPr k2Q Ex ,
0
where we have made use of the identity kI − Pr kQ = kPr kQ (see [29]).
To bound the second term of (2.17), note that
θ̇(t) = WrT ẋ(t) − ẋr (t) = Ar VrT F(Vr WrT x(t)) − F(Vr xr (t)) + η(t),
where Ar = Jr − Rr (see (2.9)) and η(t) = WrT (J − R)[F(x(t)) − Wr VrT F(Vr WrT x(t))].
d
d
kθ(t)k = 12 dt
kθ(t)k2 = hθ(t), θ̇(t)i, so we have
Note that kθ(t)k dt
d
kθ(t)k =
dt
θ(t)
, θ̇(t)
kθ(t)k
=
h
i
θ(t)
, Ar VrT F(Vr WrT x(t)) − F(Vr xr (t)) + η(t) .
kθ(t)k
Observe that
D
h
iE | hθ(t), η(t)i | = Vr θ(t), Vr WrT (J − R) F(x(t)) − Wr VrT F(Vr WrT x(t))
Q
h
i
≤ kVr θ(t)kQ Pr A F(x(t)) − PTr F(Pr x(t)) Q
≤ kθ(t)k kPr AkQ k(I − PTr )F(x(t)) + PTr (F(x(t)) − F(Pr x(t)))kQ
≤ kθ(t)k kPr AkQ · k(I − PTr )εF (t)kQ + kPTr kQ kF(x(t)) − F(Pr x(t))kQ
≤ kθ(t)k kPr AkQ · kI − PTr kQ kεF (t)kQ + kPTr kQ LQ [F]k(I − Pr )x(t)kQ
≤ kθ(t)k kPr AkQ kPTr kQ · (kεF (t)kQ + LQ [F] kPr kQ kεx (t)kQ )
≤ kθ(t)k · β (kεF (t)kQ + γkεx (t)kQ )
and
D
h
iE
θ(t), Ar VrT F(Vr WrT x(t)) − F(Vr xr (t))
D
h
iE
= Vr θ(t), Pr APTr F(Vr WrT x(t)) − F(Vr xr (t))
≤
LQ [Pr APTr F]
·
kVr θ(t)k2Q
Q
2
= α kθ(t)k ,
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B851
MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
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where we make use of the fact that kI − Pr kQ = kPr kQ and kI − PTr kQ = kPTr kQ .
For (2.14), we find
ky − yr k = kBT F(x) − BT Wr VrT F(Vr xr )k
= BT Q−1/2 Q1/2 F(x) − PTr F(Vr xr ) ≤ kBT Q−1/2 k k(I − PTr )εF + PTr (F(x) − F(Vr xr )) kQ
≤ kBT Q−1 Bk1/2 k(I − PTr )εF kQ + kPTr (F(x) − F(Vr xr )) kQ
≤ kBT Q−1 Bk1/2 kPTr kQ (kεF kQ + k (F(x) − F(Vr xr )) kQ )
≤ kBT Q−1 Bk1/2 kPTr kQ (kεF kQ + LQ [F]kx − Vr xr kQ ) .
Thus,
T
Z
ky(t) − yr (t)k2 dt ≤ 2kBT Q−1 Bk kPTr k2Q
0
Z
T
kεF (t)k2Q + LQ [F]2 kx − Vr xr k2Q dt
0
≤ 2kBT Q−1 Bk kPTr k2Q
· EF + LQ [F]2 Cx Ex + CF EF + C0 kWrT x(0) − xr (0)k2
,
and (2.14) follows.
The bounds given in Theorem 2.2 may not be explicitly computable in general.
However, they do provide insight into how each component contributing to the reduced
model influences overall accuracy. For example, from (2.13) and (2.14), one may note
that using WrT x0 as the initial condition for the reduced system eliminates an error
source arising from the initial condition. If POD bases are used to determine the
reduction spaces, Vr and Wr , these bounds are approximately minimized as well,
though they may not ultimately minimize the actual reduced model error.
3. Structure-preserving model reduction with DEIM. The performance
of projection-based reduced models of nonlinear systems can be degraded by the
need to lift the reduced state to the full state dimension in order to evaluate the
nonlinear term. For example, consider the evaluation of VrT ∇x H(Vr xr ) in (2.2). If
∇x H(x) is nonlinear in x, it is likely that VrT ∇x H(Vr xr ) cannot be precomputed
explicitly as a map from Rr to Rr without an intermediate lifting to Rn . The order
of complexity required to evolve the reduced system will then remain at least O(n),
and there may be little benefit seen in the use of a reduced model. Several approaches
have been proposed to address this difficulty; see, e.g., [1, 2, 4, 6, 9]. We resolve
the complexity issue by developing a variant of the discrete empirical interpolation
method (DEIM) [6], which is itself a discrete variant of the empirical interpolation
method introduced in [2]. Since DEIM (as presented in [6]) does not typically preserve
port-Hamiltonian structure, we develop here a structure-preserving variant of DEIM
together with associated error estimates. There have been other structure-preserving
methods developed recently that employ methods resolving the lifting bottleneck; e.g.,
see [5] for an approach that preserves Lagrangian structure in structural dynamics
using gappy POD.
3.1. The discrete empirical interpolation method. The “lifting bottleneck” in the reduction of large-scale nonlinear models as described above is resolved
by DEIM through the approximation of the nonlinear system function via interpolation. This approximation is done in such a way as to not require a prolongation of the
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B852
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
reduced state variables (“lifting”) back to the original high-dimensional state-space.
Only a few selected entries of the original nonlinear term need be evaluated at each
time step.
In particular, let f : D 7→ Rn be a nonlinear vector-valued function defined on a
domain D ⊆ Rn . Let Um = [u1 , . . . , um ] ∈ Rn×m have rank m, and define Em =
[e℘1 , . . . , e℘m ] ∈ Rn×m with the index set {℘1 , . . . , ℘m } output from Algorithm 4
n
using the input basis {ui }m
i=1 . e℘j ∈ R denotes the ℘j th column of the n × n identity
matrix.
The DEIM approximation of order m ≤ n for f in span{Um } is given by
(3.1)
b
f (τ ) := P f (τ ),
where
P = Um (ETm Um )−1 ETm .
Note that in the original work [6], Um is assumed to have orthonormal columns, yet
linear independence suffices for P to be well defined, and that is all that we require
here.
Algorithm 4. DEIM [6].
n
INPUT: {u` }m
`=1 ⊂ R linearly independent
OUTPUT: ℘
~ = [℘1 , . . . , ℘m ]T ∈ Rm
1:
[|ρ|, ℘1 ] = max{|u1 |}.
2:
U = [u1 ], E = [e℘1 ], ℘
~ = [℘1 ].
3:
for ` = 2 to m do
4:
Solve (ET U)c = ET u` for c.
5:
r = u` − Uc.
|ρ| = |v℘ | = maxi=1,...,n {|vi |}, with the
6:
[|ρ|, ℘` ] = max{|r|}.
smallest index taken for ℘ in case of ties.
℘
~
7:
U ← [U u` ], E ← [E e℘` ], ℘
~←
.
℘`
8: end for
We adapt an error bound for the DEIM function approximation derived in [6] for
the case that Um has Q-orthonormal columns.
Lemma 3.1. If UTm QUm = Im , then kf (τ ) − b
f (τ )kQ ≤ kPkQ EQ (f (τ ), Um ),
where EQ (f (τ ), Um ) is the best Q norm approximation error for f (τ ) from Ran(Um ).
The invertibility of ET U at the end of each cycle of the DEIM procedure is
verified in [6], where it is shown that each DEIM interpolation index is selected to
limit the stepwise growth of the factor k(ETm Um )−1 k = kPkQ in the error bound. This
will be used in the next section to assess the accuracy of the state variables in the
DEIM reduced system. DEIM shown in Algorithm 4 uses LU with partial pivoting
for interpolation indices. A new selection operator for DEIM based on the pivoted
QR has been recently introduced by Drmač and Gugercin [7]. Our numerical results
in this paper are based the original implementation in Algorithm 4. We also refer
the reader to [19] for a recently introduced localized version of DEIM and to [20]
for an online adaptivity approach to DEIM that adjusts the DEIM subspace and the
interpolation indices with online low-rank updates.
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
B853
Consider the NLPH system in (1.1), where J, R, and B are constant and F(x) =
∇x H(x) is nonlinear. In the reduced system (2.4), we use the approximation (2.3):
∇x H(x) ≈ Wr VrT ∇x H(x) for x ∈ Ran(Vr ). DEIM can be applied directly to F
(using Wr instead of U) to obtain an approximation in the form of
F(Vr xr ) ≈ Wr (ET Wr )−1 ET F(Vr xr ),
allowing us to evaluate the nonlinear term with low complexity. However, this approach will not preserve the underlying ph structure; it generally will not produce a
passive system; and indeed, the reduced system might no longer be stable. We modify
deim to overcome these shortcomings.
3.2. The DEIM Hamiltonian. We continue to assume that the source of nonlinearity in the system (1.1) lies in the Hamiltonian gradient, ∇x H(x), and we focus
on approximating this nonlinear term in a way that is consistent with the PH structure, so that the complexity does not depend on the original full-order dimension.
This restriction comes largely without loss of generality since additional state-space
dependence of J, R, and B can be accommodated with usual DEIM-based approaches
with no threat to the underlying port-Hamiltonian structure.
We first identify a linear component of ∇x H(x), or equivalently, a quadratic
component of H(x):
(3.2)
H(x) = 21 xT Qx + h(x),
where Q is an n × n positive-definite, constant matrix. A typical choice may be Q =
∇2x H(x0 ) at an equilibrium point, ∇x H(x0 ) = 0. Similar strategies are considered
in [15] to maintain high accuracy. Once Q is selected, (3.2) determines h(x), and
∇x h(x) then captures the remaining nonlinear portion of ∇x H(x).
We select a new modeling basis, Um , orthogonalized with respect to Q, so
that ∇x h(x(t)) ≈ Um gm (t) and UTm QUm = I. There will be a variety of choices
for Um ; we consider a couple of them below. Using Um , we calculate DEIM indices, ℘1 , . . . , ℘m , with Algorithm 4, define the associated DEIM projection, P =
Um (ETm Um )−1 ETm , and finally, introduce a “DEIM Hamiltonian”:
(3.3)
1
b
H(x)
= xT Q x + h(PT x).
2
b
b
Observe that ∇x H(x)
= Qx + P∇x h(PT x), so the error induced by ∇x H(x)
is
b
∇x H(x) − ∇x H(x)
= ∇x h(x) − P∇x h(PT x).
b
If Q is chosen well, then ∇x H(x)
can exactly recover the “linear part” of ∇x H(x). Observe that the evaluation of the remaining nonlinear term, P∇x h(PT x), only involves
the evaluation of m elements of ∇x h on only m nonzero arguments—the remaining
arguments having only m nonzero values.
b
Observe that trivially ∇x H(x) = ∇x H(x)
when m = n. However, the approxb
imation ∇x H(x) ≈ ∇x H(x)
can be effective even for significantly smaller m n.
Nonetheless, the enforced symmetry in this approximation appears to require some
additional considerations.
Let Ω ⊂ Rn be a compact convex set containing each of the trajectories x(t) and
T
P x(t) for 0 ≤ t ≤ T on its interior. Define
Z
◦
K =
∇x h(x)∇x h(x)T Q dν(x),
Ω
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B854
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
with ν denoting Lebesgue measure in Rn . K◦ is Q-self-adjoint and positive-definite
with associated eigenpairs K◦ u◦k = σk◦ u◦k for σ1◦ ≥ σ2◦ ≥ · · · ≥ σn◦ ≥ 0 and Q◦
orthonormal eigenvectors, u◦T
i Quj = δij . For any choice of 0 < m ≤ n, designate the
◦
◦
dominant modes as Um = [u1 , . . . , u◦m ] and the complementary subdominant modes
e ◦ = [u◦ , . . . , u◦ ]. Observe that if Ω shrank to x(t), U◦ could be considered
as U
m
n
m
m+1
as the limiting case of a usual POD basis of dimension m.
We have in that case
◦
K◦ U◦m = U◦m diag(σ1◦ , σ2◦ , . . . , σm
)
e◦ = U
e ◦ diag(σ ◦ , . . . , σ ◦ ).
and K◦ U
m
m
m+1
n
If EQ (∇x h(x), U◦m ) denotes the (local) best Q norm approximation error to ∇x h(x)
out of U◦m (as defined in Lemma 3.1), then
◦(m)
εQ
2
Z
EQ (∇x h(x), U◦m )2 dν(x) =
=
2
k(I − U◦m U◦T
m Q)∇x h(x)kQ dν(x)
Ω
Ω
Z
Z
T
◦
◦T
trace (I − U◦m U◦T
m Q)∇x h(x)∇x h(x) (I − QUm Um )Q dν(x)
=
ZΩ
=
trace ∇x h(x)∇x h(x)T Q(I − U◦m U◦T
m Q) dν(x)
Ω
n
X
= trace K◦ (I − U◦m U◦T
Q)
=
σk◦ .
m
k=m+1
Lemma 3.2. Suppose that h(x) is continuously differentiable in Ω. Construct the
DEIM projection using U◦m with P◦ = U◦m (ETm U◦m )−1 ETm . Then for each 0 ≤ t ≤ T ,
◦(m)
|h(x(t)) − h(PT◦ x(t))| ≤ k(I − PT◦ )x(t)kQ−1 · εQ
(3.4)
.
e m to be complementary to those of Em so that
Proof. Define the columns of E
e m ] is an n × n permutation matrix. Observe that Ran(PT ) = Ran(Em ) and
[Em , E
◦
e◦
Ran(I − PT◦ ) = Ker(PT◦ ) = Ker(U◦T
m ) = Ran(QUm ). Hence,
e ◦ ζ2
x = PT◦ x + (I − PT◦ )x = Em ζ 1 + QU
m
for continuous functions given as
−1 ◦T
ζ 1 (x) = (U◦T
Um x
m Em )
1
and
e T QU
e ◦ )−1 E
e T x.
ζ 2 (x) = (E
m
m
m
1
e ◦ ζ 2 = Q− 2 (I − PT )x and kζ 2 k = k(I − PT )xkQ−1 .
Note that Q 2 U
m
◦
◦
For each fixed x ∈ Ω, define the path connecting x and PT◦ x:
e ◦ ζ 2 | θ ∈ [0, 1] }.
C(x) = {θx + (1 − θ)PT◦ x | θ ∈ [0, 1] } = {Em ζ 1 + θ QU
m
e ◦ ζ 2 ), observe that
If we define f (ζ 1 , ζ 2 ) = h(x) = h(Em ζ 1 + U
m
e ◦T Q∇x h(Em ζ + QU
e ◦ .ζ ).
∇ζ 2 f (ζ 1 , ζ 2 ) = U
1
m
m 2
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
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Integrating ∇ζ 2 f along C(t), we find
|h(x) −
h(PT◦ x)|
Z
◦T
e
= |f (ζ 1 , ζ 2 ) − f (ζ 1 , 0)| = Um Q∇x h(ξ) · dξ
C(x)
Z 1
e ◦T
e◦
= ζ T2 U
m Q∇x h(Em ζ 1 + θ QUm ζ 2 ) dθ 0
1
Z
= kζ 2 k
2
e ◦T
e◦
kU
m Q∇x h(Em ζ 1 + θ QUm ζ 2 )k dθ
21
0
1
2
2
e ◦T
kU
Q∇
h(ξ)k
dν(ξ)
x
m
Z
≤ kζ 2 k
Ω
≤ k(I −
PT◦ )x(t)kQ−1
◦(m)
· εQ
.
Thus, a modest extension of the usual POD basis is sufficient to produce DEIM
approximations and associated DEIM Hamiltonians that converge uniformly to the
true Hamiltonian with a rate related to the decay rate of σk◦ . Stronger hypotheses
on the approximating modes appear to be necessary in order to assure rapid convergence of the corresponding gradients. Towards that end, we assume that h is twice
continuously differentiable, and define
Z
Z
2
T
K=
∇x h(x)∇x h(x) Q dν(x) +
∇2x h(x) Q dν(x).
Ω
Ω
K is Q-self-adjoint and positive-definite with associated eigenpairs K uk = σk uk for
σ1 ≥ σ2 ≥ · · · ≥ σn ≥ 0 and Q-orthonormal eigenvectors, uTi Quj = δij . Analogously to what has gone before, designate dominant modes as Um = [u1 , . . . , um ] and
e m = [um+1 , . . . , un ], with
subdominant modes as U
K Um = Um diag(σ1 , σ2 , . . . , σm ),
em = U
e m diag(σm+1 , . . . , σn ),
KU
(m)
and an associated error, εQ , defined similarly:
n
2 Z
X
(m)
εQ
=
k(I − Um UTm Q)∇x h(ξ)k2Q + k(I − Um UTm Q)∇2x h(ξ)k2Q dν(ξ) =
σk .
Ω
k=m+1
e m , as
Lemma 3.3. Given the modeling basis, Um , and the basis completion, U
described above,
√
(m)
k∇x h − P ◦ (∇x h) ◦ PT kL2 (Ω) ≤ 2kPkQ M εQ ,
q
kET
m QEm k
−1
where M = max{1, kPkQ
n−m+1 supx∈Ω kxkQ }.
Proof. Note that
Z
k∇x h − P ◦ (∇x h) ◦ PT k2L2 (Ω) =
k∇x h(x) − P∇x h(PT x)k2Q dν(x)
Ω
Z Z
2
T
≤2
k (I − P) ∇x h(x)k2Q dν(x)
P ∇x h(x) − ∇x h(P x) dν(x) + 2
Q
Ω
Ω
Z 2
T
T
≤ 2 kPk2Q kETm QEm k
Em ∇x h(x) − ∇x h(P x) dν(x)
Ω
Z
e Tm Q∇x h(x)k2 dν(x).
+ 2kI − Pk2Q
kU
Ω
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B856
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
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Consider the function f (ζ 1 , ζ 2 ) as defined in the proof of Lemma 3.2, but now adapted
to the new basis, Um . Note that
e m ζ 2 ).
∇ζ 1 f (ζ 1 , ζ 2 ) = ETm ∇x h(Em ζ 1 + QU
Since ∇x h is continuously differentiable in an open neighborhood of C(t), we have
that
ETm (∇x h(x) − ∇x h(PT x)) = ∇ζ 1 f (ζ 1 , ζ 2 ) − ∇ζ 1 f (ζ 1 , 0)
Z 1
e m ζ 2 ) · QU
e m ζ 2 dθ
= ∇ζ 1
∇x h(Em ζ 1 + θ QU
0
Z 1
T
2
e
e
Em ∇x h(Em ζ 1 + θ QUm ζ 2 ) QUm dθ ζ 2 ,
=
0
and so
Z 2
T
T
Em ∇x h(x) − ∇x h(P x) dν(x)
Ω
Z 1Z
e m k2 dν(x) dθ
≤ sup kζ 2 (x)k2
kETm ∇2x h(θ x + (1 − θ)PT x) QU
x∈Ω
0
Ω
≤ sup k(I − PT )xk2Q−1
x∈Ω
≤
kPk2Q
sup
x∈Ω
kxk2Q−1
1
Z
Z
0
1
Z
e m k2 | det(θ I + (1 − θ)PT )| dν(ξ) dθ
kETm ∇2x h(ξ) QU
Ω
T
Z
| det(θ I + (1 − θ)P )| dθ
0
kPk2Q
sup kxk2Q−1
≤
n − m + 1 x∈Ω
e Tm Q∇2x h(ξ)Em k2 dν(ξ)
kU
Ω
Z
e Tm Q∇2x h(ξ)k2
kU
dν(ξ).
Ω
The interchange of order of integration is justified in the first inequality since the
integrand is uniformly bounded on the joint domain [0, 1]×Ω. In the second inequality,
we have introduced the (linear) change of variable ξ = [θ I + (1 − θ)PT ]x ∈ Ω for all
θ ∈ [0, 1]. The third inequality incorporates the observation kI − PT kQ−1 = kPkQ .
The last inequality uses the elementary observation that the Jacobian matrix for the
change-of-variables has exactly two eigenvalues: λ = 1 with multiplicity m, and λ = θ
with multiplicity n − m.
3.3. Preserving port-Hamiltonian Structure with DEIM. Equipped with
b
the DEIM Hamiltonian, H(x),
we apply our previously described structure-preserving
b r (xr ) = H(V
b r xr ). As before,
approach: we define a reduced DEIM Hamiltonian, H
b
b
we expect that ∇x H(Vr xr (t)) ≈ Wr ∇xr Hr (xr (t)), and the DEIM-reduced portHamiltonian approximation becomes
(3.5)
b r (xr ) + Br u(t),
ẋr = (Jr − Rr )∇xr H
b r (xr )
yr (t) = BTr ∇xr H
with Jr = WrT JWr , Rr = WrT RWr , Br = WrT B, and
b r (xr ) = VT ∇x H(V
b r xr ) = xr + VT P∇x h(PT Vr xr ).
∇xr H
r
r
In the evaluation of VrT P∇x h(PT Vr xr ), the r × m matrix, VrT Um , may be precomputed and ∇x h is evaluated only on arguments having m nonzero values. Only those
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
B857
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Algorithm 5. POD-DEIM structure-preserving DEIM reduction of NLPH systems.
1:
Select a positive-definite matrix Q (nominally approximating ∇2x H(x0 )) and define h(x) = H(x) − 12 xT Qx.
2:
Generate the trajectory x(t) (or ensemble of trajectories) and collect snapshots:
X = [x(t0 ), x(t1 ), x(t2 ), . . . , x(tN )] .
F = [∇x H(x(t0 )), ∇x H(x(t1 )), . . . , ∇x H(x(tN ))] .
G = [∇x h(x(t0 )), ∇x h(x(t1 )), . . . , ∇x h(x(tN ))] .
3:
Truncate the SVDs of the snapshot matrices, X, F, and G to obtain, respectively,
e r, W
f r , and Um used to approximate
the POD basis matrices V
x(t) ≈ Vr xr (t),
∇x H(x) ≈ Wr fr (t),
and ∇x h(x(t)) ≈ Um gm (t).
Choose bases so that VrT QVr = I, VrT Wr = I, and UTm QUm = I.
4:
From Algorithm 4, calculate DEIM indices, ℘1 , . . . , ℘m , and the DEIM projection,
P = Um (ET Um )−1 ET .
5:
The DEIM-reduced port-Hamiltonian approximation becomes
b r (x̂ r ) + Br u(t),
x̂˙ r = (Jr − Rr )∇x̂ r H
b r (x̂ r ),
ŷ r = BTr ∇x̂ r H
with Jr = WrT JWr , Rr = WrT RWr , Br = WrT B, and
b r (x̂ r ) = VrT ∇x H(V
b r x̂ r ) = xr + VrT P∇x h(PT Vr x̂ r ).
∇x̂ r H
m values associated with the DEIM indices need to be evaluated. Algorithm 5 summarizes the steps for constructing a structure-preserving POD-DEIM reduced system.
Note that we can apply this structure-preserving DEIM approach to bases Vr
and Wr associated with H2ε -PH bases obtained from Algorithm 2, instead of PODPH bases obtained from Algorithm 1. In the numerical examples of section 3.4, we
consider both strategies, referring to the variant using Algorithm 2 as H2ε -DEIM-PH.
3.4. Numerical examples. We consider two system models: the nonlinear ladder network from section 2.4, and a Toda lattice model with exponential interactions.
We will illustrate the performance of the structure-preserving DEIM-based model reduction approaches introduced above using the POD-DEIM-PH, H2ε -DEIM-PH, and
the hybrid POD-H2ε -PH bases.
3.4.1. Ladder network. The N -stage nonlinear ladder network studied in section 2.4 will be considered again. We will investigate here the effect of incorporating
the structure-preserving DEIM approximation in reducing the nonlinear term. As
in section 2.4, we use a Gaussian pulse as the training input and test the reduced
models on both the Gaussian pulse and a sinusoidal input. Figure 5 shows that PODDEIM-PH and H2ε -DEIM-PH reduced systems capture the behavior of the original
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B858
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
Output response to Gaussian pulse input (POD from Gaussian pulse)
Output response to sinusoid input (POD from Gaussian pulse)
8
8
Volts
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True response
6
Linearized
full order
6
4
POD−PH
r=6
4
2
POD−DEIM−PH
r=6, m=24
2
H2−PH ROM
0
0
r=6
−2
−2
H2−DEIM PH ROM
r = 6,m =24
−4
0
2
4
6
Time (µ−seconds)
8
10
0
2
4
6
Time (µ−seconds)
8
−4
10
Fig. 5. Ladder network: ROM responses to Gaussian pulse (left) and sinusoidal input (right).
Output: Full, Hε2, Hε2−DEIM−PH: r =6 with m =6,12,24,48
8
Output: Full, POD, POD−DEIM−PH:r =6 with m=6,12,24,48
8
Full: n=100
7
Hε2: r=6
6
POD:r=6
6
Hε2−DEIM−PH
5
r=6, m= 6
4
Hε2−DEIM−PH
3
r=6,m= 12
2
Hε2−DEIM−PH
1
0
−1
−2
0
Full(n=100)
7
2
4
6
t
8
10
4
POD−DEIM−PH
r=6,m=12
3
2
r=6,m= 24
1
Hε2−DEIM−PH
0
r=6,m = 48
POD−DEIM−PH
r=6,m=6
5
POD−DEIM−PH
r=6,m=24
POD−DEIM−PH
r=6,m=48
−1
−2
0
2
4
6
8
10
t
Fig. 6. Ladder network: Outputs from H2ε -PH reduced system (dimension r = 6) and H2ε DEIM-PH reduced systems ( H2ε dimension r = 6 with DEIM dimensions m = 6, 12, 14, 48)—
similarly for the plot of POD-PH and POD-DEIM-PH reduced systems.
output responses accurately for both inputs, repeating the success of the POD-PH
and H2ε -PH reduced models. Notice from Figure 5 that the linearized system does not
give a good approximation to the true output response of the nonlinear system, as
pointed out in section 2. Notice also that the accuracy of the POD-DEIM-PH reduced
system (r = 6, m = 24) is very close to the accuracy of the POD-PH reduced system
(r = 6); the POD-PH reduced system is slightly more accurate, as expected. However, the POD-DEIM-PH reduced system has much lower computational complexity,
as explained in the previous section; thus the structure-preserving DEIM approximation reduces the computational complexity yet does not degrade the accuracy of the
reduced model. As the DEIM dimension increases, the POD-DEIM-PH yields the
same accuracy as POD-PH, as shown in Figure 6 for a fixed POD dimension of r = 6.
Figure 7 shows the convergence of the average relative errors. Analogous to the previous numerical test in section 2.4, we also consider projection bases constructed using
the hybrid POD-H2ε -PH approach combined with DEIM approximation. As one may
see in Figure 8, this hybrid approach is more accurate than those constructed using
only POD bases (green lines) or only H2ε bases (blue lines). See color online.
3.4.2. Toda lattice. A Toda lattice model describes the motion of a chain of
particles, each one connected to its nearest neighbors with “exponential springs.” The
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B859
MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
Average Relative Error of State solution
Average Relative Error of Output response
2
0.7
Hε −PH
2
ε
2
0.5
Error
Error
POD−PH
POD−DEIM−PH
0.6
Hε −PH
2
ε
H −DEIM−PH
2
1.5
1
H −DEIM−PH
0.4
0.3
0.2
0.5
0.1
0
4
6
8
10
12
0
4
14
6
8
r
10
12
14
r
Fig. 7. Ladder network: Average relative errors of the reduced systems constructed from PODPH, POD-DEIM-PH, H2ε -PH, and H2ε -DEIM-PH bases (with dimensions r = 2, 4, . . . , 18 and DEIM
dimension m = 3r).
Average Relative Error of State variable
0
10
Average Relative Error
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POD−PH
POD−DEIM−PH
Hε2−PH
Hε2−DEIM−PH
POD−PH
POD−DEIM−PH
−1
10
POD&Hε2
POD&Hε2−DEIM−PH
4
6
8
10
12
14
r
Fig. 8. Ladder network: Average relative errors using the POD-PH, POD-DEIM-PH, H2ε -PH,
H2ε -DEIM-PH, and the hybird POD-H2ε -PH and POD-H2ε -DEIM-PH bases (with r = 2, 4, . . . , 18
and DEIM dimension m = 3r).
equations of motion for the N -particle Toda lattice with such exponential interactions
can be written in the form of a nonlinear port-Hamiltonian system as in (1.1) with
J=
0
−I
I
0
∈R
n×n
, R=
0
0
0 diag(γ1 , . . . , γN )
∈R
n×n
, B=
0
e1
∈ Rn ,
where n = 2N and
x=
q
p
∈ Rn , q = [q1 , . . . , qN ]T , p = [p1 , . . . , pN ]T ,
with qj and pj being, respectively, the displacement of the jth particle from its equilibrium position and the momentum of the jth particle for j = 1, . . . , N . In this
example, we use N = 1000 (i.e., the system dimension is n = 2000), x0 = 0, and
γj = 0.1 for j = 1, . . . , N . The corresponding nonlinear Hamiltonian is given by
H(x) = H([q; p]) =
N
X
1
k=1
2
p2k
+
N
−1
X
exp(qk − qk+1 ) + exp(qN ) − q1 − N.
k=1
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B860
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
Output: Full N=1000 (full dim: n =2000), u(t) = 0.1sin(t)
Output: Full N=1000 (full dim: n=2000), u(t) = 0.1
0.1
0.1
Full system: n=2000
POD−PH: r=100
POD−DEIM−PH:r=m=100
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0.09
0.08
0.05
0.07
0
0.06
0.05
−0.05
0.04
0.03
−0.1
0.02
Full system: n=2000
POD−PH: r=100
POD−DEIM−PH:r=m=100
0.01
0
0
20
40
60
80
100
−0.15
0
20
40
60
80
100
t
t
Fig. 9. Toda lattice: The outputs from inputs u(t) = 0.1 and u(t) = 0.1 sin(t).
Relative Error Output: u(t) = 0.1
0
POD
10
POD−DEIM−PH:
m=r
−2
10
Online CPU times relative to the full model: u(t) = 0.1
0.12
0.1
POD−DEIM−PH:
m=m1
−4
10
0.08
−6
10
POD−DEIM−PH
m=m2
−8
0.06
10
0.04
−10
10
60
70
80
90
r
100
110
120
60
70
80
90
r
100
110
120
Fig. 10. Toda lattice with input u(t) = 0.1: Relative errors of the outputs and the online
CPU times for POD-PH and POD-DEIM reduced systems with POD basis dimension r and DEIM
dimension m = r, m1 , m2 , where m1 = r + ceil(r/3), m2 = r + ceil(2r/3). The full-order system
has order n = 2000.
The decomposition H(x) = 21 xT Qx + h(x) is used with

Q=
Q0
0
0
I
,
1 −1
0
 −1
2
−1


 0 −1
2

..
..
Q0 = 

.
.



 0
···
0
0
···
..
.
..
.
−1
0
0
0
0
0
−1
0
2 −1
−1
2






,





R1
PN −1
and h(x) = k=1 (qk − qk+1 )3 ϕ(qk − qk+1 ), where ϕ(z) = 21 0 θ2 e(1−θ)z dθ. The
system was excited with two different inputs: u(t) = 0.1 and u(t) = 0.1 sin(t). Figure 9
shows that the outputs due to both inputs are accurately approximated by the outputs
of the POD-PH and POD-DEIM-PH reduced systems. The average relative errors and
the simulation times relative to the simulation time of the full model are illustrated
in Figures 10 and 11. Notice that the accuracy of the POD-DEIM-PH reduced model
captures that of the POD-PH model as the DEIM dimension m increases. Note also
that the POD-DEIM-PH reduced model cuts the simulation time by nearly 96% while
retaining accuracy.
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B861
MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
Relative Error Output: u(t) = 0.1sin(t)
−2
POD
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10
−4
POD−DEIM−PH:
m=r
−6
POD−DEIM−PH:
m=m1
−8
POD−DEIM−PH
m=m2
10
10
Online CPU times relative to the full model: u(t)=0.1sin(t)
0.07
0.06
0.05
0.04
10
0.03
0.02
−10
10
70
80
90
100
110
120
70
80
90
r
100
110
120
r
Fig. 11. Toda lattice with input u(t) = 0.1 sin(t): Relative errors of the outputs and the online
CPU times for POD-PH and POD-DEIM-PH reduced systems with POD basis dimension r and
DEIM dimension m = r, m1 , m2 , where m1 = r + ceil(r/3), m2 = r + ceil(2r/3). The full-order
system has order n = 2000.
3.5. An a priori error bound for PH-preserving DEIM reduction. We
derive error bounds for a DEIM-based reduced-order model preserving PH structure
by estimating additional errors that occur by introducing the symmetrized-DEIM
approximations into the structure-preserving reduction framework of section 2. In
particular, suppose that reduction bases, Vr and Wr , have been chosen in some
manner (e.g., as described in section 2.1, 2.2, or 2.3, say), and then are used to
produce a reduced model that preserves the PH structure of the original system:
ẋr = (Jr − Rr )∇xr Hr (xr ) + Br u(t),
(3.6)
yr = BTr ∇xr Hr (xr ).
We then symmetrically “sparsify” the nonlinear interactions in the Hamiltonian gradient evaluation and introduce a further DEIM reduction, as described in section 3.3
and Algorithm 5, in order to produce
b r (x̂ r ) + Br u(t),
x̂˙ r = (Jr − Rr )∇x̂ r H
(3.7)
b r (x̂ r ),
ŷ r = BTr ∇x̂ r H
b has been defined in (3.3).
where x(0) = x̂ (0) and the DEIM Hamiltonian, H,
Theorem 3.4. Let xr (t) and yr (t) be the state trajectory and output associated
with the reduced port-Hamiltonian system (3.6). Suppose a DEIM basis of order m
has been chosen and used to define a DEIM projection P and a DEIM Hamiltonian
b
H(x)
as introduced in (3.3). Recalling the discussion of section 3.2, suppose Ω ⊂ Vr
contains the trajectories Vr xr (t) and PT Vr xr , and that εh > 0 satisfies
sup k∇x h(ξ) − P∇x h(PT ξ)kQ ≤ εh .
ξ∈Ω
Let x̂ r (t) and ŷ r (t) be the state trajectory and output associated with the DEIMreduced port-Hamiltonian system (3.7).
Then with
α = LQ [G] − ρmin ,
β = kPr APTr kQ ,
G(ξ) = Pr APTr P∇x h(PT ξ),
γ = 1 + LQ [P ∇x h ◦ PT ],
ρmin = min λ(Rr ),
δ = kBk kPr k,
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B862
S. CHATURANTABUT, C. BEATTIE, AND S. GUGERCIN
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we have, for α 6= 0,
β
kxr (t) − x̂ r (t)k ≤ α
(eα t − 1) εh ,
kyr (t) − ŷ r (t)k ≤ δ 1 + βαγ (eα t − 1) εh ,
(3.8)
whereas for α = 0,
kxr (t) − x̂ r (t)k ≤ β t εh ,
kyr (t) − ŷ r (t)k ≤ δ (1 + β t ) εh .
Proof. Let er (t) := xr (t) − x̂ r (t) be the difference between state trajectories
generated by (3.6) and (3.7). Define f (x) = ∇x h(x) and observe that
h
i
er (t)
er (t)
b
, ėr (t) =
, Ar ∇xr Hr (xr ) − ∇x̂ r Hr (x̂ r )
ker (t)k
ker (t)k
er (t)
T
T
, Ar er + Vr ∇x h(Vr xr ) − P ∇x h(P Vr x̂ r )
=
.
ker (t)k
d
ker (t)k =
dt
We have
er , Ar er + VrT ∇x h(Vr xr ) − P ∇x h(PT Vr x̂ r )
D
E
= − her , Rr er i + Vr er , Pr APTr ∇x h(Vr xr ) − P ∇x h(PT Vr x̂ r )
Q
D
E
T
T
T
= − her , Rr er i + Vr er , Pr APr P ∇x h(P Vr xr ) − P ∇x h(P Vr x̂ r )
Q
D
E
T
T
+ Vr er , Pr APr ∇x h(Vr xr ) − P ∇x h(P Vr xr )
Q
≤ −ρmin ker k2 + LQ [G] ker k2
+ ker k kPr APTr kQ k∇x h(Vr xr ) − P ∇x h(PT Vr xr )kQ .
Thus,
d
ker (t)k ≤ (LQ [G] − ρmin ) ker (t)k
dt
+ kPr APTr kQ k∇x h(Vr xr (t)) − P ∇x h(PT Vr xr (t))kQ .
If α = LQ [G] − ρmin = 0, then trivially,
kxr (t) − x̂ r (t)k ≤ kPr APTr kQ
t
Z
k∇x h(Vr xr (τ )) − P ∇x h(PT Vr xr (τ ))kQ dτ ≤ β t εh .
0
If α 6= 0, then by Gronwell’s inequality,
kxr (t) − x̂ r (t)k ≤ kPr APTr kQ
t
Z
eα(t−τ ) k∇x h(Vr xr (τ )) − P ∇x h(PT Vr xr (τ ))kQ dτ
0
≤
β αt
e − 1 εh .
α
This expression holds with a positive bound regardless of whether α > 0 or α < 0.
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MODEL REDUCTION FOR PORT-HAMILTONIAN SYSTEMS
B863
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The resulting difference in output maps may be directly bounded:
b r (x̂ r ) kyr − ŷ r k = BTr ∇xr Hr (xr ) − ∇x̂ r H
T
T
= Br xr − x̂ r + Vr ∇x h(Vr xr ) − P ∇x h(PT Vr x̂ r ) ≤ kBTr er k + kBTr VrT ∇x h(Vr xr ) − P ∇x h(PT Vr xr ) k
+ kBTr VrT P ∇x h(PT Vr xr ) − P ∇x h(PT Vr x̂ r ) k
≤ kBk kPr k 1 + LQ [P ∇x h ◦ PT ] kxr (t) − x̂ r (t)k + kBk kPr k εh ,
which leads to the second conclusions, respectively, for α 6= 0 and α = 0.
4. Conclusions. We have introduced a structure-preserving projection-based
model reduction framework for large-scale multi-input/multi-output nonlinear portHamiltonian systems. We constructed projection subspaces using three different approaches: POD, H2 -based, and a hybrid POD-H2 –based. We showed that for the
same reduced order, the hybrid basis significantly outperforms the other two. We
introduced a modification of DEIM within this framework to approximate the nonlinear part of the Hamiltonian gradient symmetrically. In all cases, the resulting
reduced system preserves port-Hamiltonian structure, and thus retains the stability
and passivity of the original system. We have derived the corresponding a priori error
bounds of the state variables and outputs by using an application of generalized logarithmic norms for unbounded nonlinear operators. The effectiveness of the proposed
approaches was shown on a nonlinear ladder network and a Toda lattice model.
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