A Greedy Non-Intrusive Reduced Order Model for
Fluid Dynamics
Wang Chen
1
Northwestern Polytechnical University, Xi’an 710072, P.R. China
Jan S Hesthaven
2
Swiss federal Institute of Technology in Lausanne(EPFL), CH-1015 Lausanne, Switzerland
Bai Junqiang
3
, Yasong Qiu
4
, and Yang Tihao
5
Northwestern Polytechnical University, Xi’an 710072, P.R. China
Zhang Yang
6
Xi’an Jiaotong University, Xi’an 710018, P.R. China
A greedy non-intrusive reduced order method (ROM) is proposed for parameter-
ized time-dependent problems with an emphasis on problems in fluid dynamics. The
non-intrusive ROM (NIROM) bases on a two-level proper orthogonal decomposition
(POD) to extract temporal and spatial reduced basis from a set of candidates, and
adopts the radial basis function (RBF) to approximate undetermined coefficients of
extracted reduced basis. Instead of adopting uniform or random sampling strategies,
the candidates are determined by an adaptive greedy approach to minimize the over-
all offline computational cost. Numerical studies are presented for a two-dimensional
diffusion problem as well as a lid-driven cavity problem governed by incompressible
Navier-Stokes equations. The results demonstrate that the greedy non-intrusive ROM
(GNIROM) predicts the flow field accurately and efficiently.
1
PH.D candidate, School of Aeronautics, datouroony@163.com.
2
Professor, Chair of Computational Mathematics and Simulation Science(MCSS); jan.hesthaven@epfl.ch .
3
Professor, School of Aeronautics, junqiang@nwpu.edu.cn (Corresponding Author).
4
Assistant research fellow, School of Aerospace, qiuyasong@nwpu.edu.cn
5
PH.D candidate, School of Aeronautics, xiaoyaoyangtihao@163.com .
6
Lecturer, School of Aerospace, youngz@xjtu.edu.cn
1
Nomenclature
α
θ
(k, m) = undetermined coefficients in ROM approximation, k = 1, 2, · · · K, m = 1, 2, · · · M
g
θ
(x, t) = boundary conditions
M = correlation matrix in POD
N
∗
t
= coarse temporal mesh size
N
t
= number of time steps
N
∗
x
= coarse spatial mesh size
N
x
= spatial mesh size
t = time
θ = vector of parameters in researched problem
u
θ
= exact solution of the parameterized nonlinear PDE problem
ˆu
θ
= approximated solutions with ROM
u
θ
0
(x) = initial conditions
ˆv
θ
(x, t) = auxiliary term in ROM approximation
x = spatial coordinates
∆ = error estimator of Greedy approach
Ω = physical space
∂Ω = boundary of physical space
ϕ
k
(x) = spatial basis of the parameterized nonlinear problem, k = 1, 2, · · · , K
ξ
m
(t) = temporal basis of the parameterized nonlinear problem, m = 1, 2, · · · , M
λ
k
= the k-th eigen value for Correlation matrix
µ = penalized parameter in least square approach
φ(r) = RBF kernel
γ
km
i
= RBF coefficients
σ = scaling factor of RBF
ψ(r) = RBF-QR kernel
2
I. Introduction
Although the ability to numerically model complex phenomena is sustainably improved, nu-
merical simulation in optimization, design and control with lots of degree of freedoms (DOFs) still
remains challenging due to the high computational cost. As an engineering compromise to ap-
proximate the full problem, the development of reduced-order method (ROMs) [1, 2] that enables
approximating solutions with an acceptable loss of accuracy has gained a substantial interest. ROMs
aim to approximate the full problem by projecting it onto a specific subspace with much smaller
DOFs. Once properly chosen, the reduced linear space can adequately represent the dynamics of
the full system at a substantially reduced cost. Examples of these developments include proper
orthogonal decomposition (POD) [3], Harmonic Balance approach (HB) [4], and Volterra theory [5].
Usually, ROMs extract a set of reduced basis functions from snapshots of the full solutions and
then adopt Galerkin projection [6] to approximate the full system with the reduced basis. As the
projection approaches depend on governing equations, the source codes have to be modified, leading
to the fact that the process is inconvenient for complex nonlinear problems and even impossible
when the source code is unavailable. To address this concern, there is a recent interest in non-
intrusive ROMs (NIROMs) which seeks to develop ROMs based solely on access to snapshots and
doesn’t utilize governing equations. In the context of fluid dynamics, [7] presents a second-order
Taylor series method as well as an approach based on a Smolyak sparse grid collocation method,
and applies the two NIROMs to simulate the flow past a cylinder with wind driven gyre. In [8], the
authors explore the use of empirical interpolation method [9] to recover a reduced model based on
a non-intrusive approach. Other researchers [10–13] seek to develop NIROMs by basing on POD
and radial basis function (RBF). In these approaches, RBF is used to interpolate undetermined
coefficients of reduced basis. Although there exist other nonlinear interpolation approaches such
as BP network or Kriging, RBF is more popular in non-intrusive ROMs’ development. The reason
is that RBF is flexible, convenient and accurate when approximating a function through scattered
data, demonstrated in [14]. It has also been validated in numerical cases that the non-intrusive
approaches using RBF can present a good performance in fluid dynamics, even for strong nonlinear
flow fields. For example, [11] adopts POD and RBF onto unsteady flow over an oscillating ONERA
3
M6 wing, and [12] demonstrates that the NIROM also performs well in a lock exchange problem as
well as a flow past a cylinder.
Development of ROMs for parameterized time-dependent problems is another issue. For such
problems, ROMs for time-dependent parameterized partial differential equations (PDEs) have to
approximate solutions as a function of time, spatial coordinates and a parameter vector, which turns
out to be more challenging especially when the boundary conditions vary with time. Some work on
this topic is dealing with simple parameterized time-dependent PDEs such as convection-diffusion
problems [15] and boussinesq equations [16]. To solve unsteady parameterized Navier-Stokes equa-
tions, [17] and [18] both utilize POD-Galerkin and achieve a good approximation in adaptation of
Mach and Reynolds number. [19] also adopts intrusive ROMs in patient-specific haemodynamics
with shape parameters. Discrete empirical interpolation method (DEIM) proposed in [20] is an
intrusive ROM for nonlinear problems. DEIM reduces the complexity as well as the dimension of
nonlinear terms and has proven to be efficient and accurate to approximate nonlinear problems.
Although intrusive ROMs have made great achievements in model adaption, this paper prefer to
adopt NIROMs since those don’t rely on governing equations as presented above. [21] presents
a general NIROM for parameterized time-dependent nonlinear PDEs. This approach constructs
snapshots and local basis functions for any parameter point by interpolating those from samples
and then generates hyper-surfaces to represent the dynamical system over the reduced space. [13]
proposes a NIROM consisting of two-level POD as well as RBF, which can deal with general prob-
lems with time-dependent boundary conditions. Considering that the NIROM in [13] is simpler on
implementation and requires less memory than that in [21], the former is adopted.
Besides, it is equally important to carefully selecting appropriate snapshots in model adaption.
The process to generate snapshots by a numerical solver or experiments contributes a lot to the
computational cost for ROMs. Thus, under the premise that requirements of ROMs’ approximation
accuracy are reached, fewer candidates are chosen, more efficient the ROMs perform. However,
due to the variation of possible solutions, it is impossible to determine the optimal candidates for
snapshot generation. Usually a uniform sampling or a random strategy such as Latin Hypercube
sampling [22], Centroidal voronoi tessellations (CVT) [23], Monte Carlo methods [24], etc., is used
4
to generate snapshots for ROMs. As these approaches don’t utilize the useful information from
the researched problem, they are not optimal. To guarantee the accuracy of the reduced model,
they often require an excessive number of snapshots, which leads to an unacceptable computational
cost. To deal with this difficulty, recently, some researches have been devoted to propose adaptive
sampling approaches for guiding the selection of snapshots. In [25], an algorithm that iteratively
applies surrogate-model optimization in the generation of snapshots is developed. Other related
approaches are based on a greedy algorithm. In such approaches, the snapshots are adaptively
determined by finding the locations in the predetermined parameter space where the error estimator
reaches the maximum, quantifying the quality of the ROM. In [26, 27], a POD-Greedy approach is
introduced to recover the ROMs, albeit in an intrusive manner, with a discussion of convergence
offered in [28, 29]. Patera et al. [30–33] has researched a lot on Greedy sampling for POD-Galerkin
method when approximating parameterized problems. Tan Bui-Thanh and Karen Willcox [34] also
propose a model-constrained adaptive Greedy sampling method for large-scale systems with high-
dimensional parametric input spaces [35]. However, to the authors’ knowledge, little work has been
done to develop greedy sampling for non-intrusive methods.
The paper aims to develop a greedy sampling approach for the NIROM put forward in [13],
which is applicable for time-dependent parameterized nonlinear problems. The non-intrusive ROM
employs a two-level approach to construct the spatial and temporal basis functions. And RBF is
adopted to estimate the undetermined coefficients of reduced basis because of its features stated
above. While the discussion in this paper is appropriate for problems in a general degree, the paper
mainly focus on problems and examples in the context of fluid dynamics. Different from previous
work in [13], the reduced model in this paper adaptively generate snapshots basing on a greedy
approach to minimize the overall computational cost, thus forming a greedy NIROM (GNIROM).
The structure of this paper is organized as follows. In Section II, a NIROM is developed,
including a two-level basis development and RBF approximation. The use of the RBF-QR method
[36] is also briefly discussed to improve RBF’s robustness. Afterwards, a Greedy sampling approach
is proposed in Section III to reduce the computational cost when generating the reduced basis. This
sets the stage for Section IV where the efficiency and accuracy of present ROM is confirmed through
5
