Dynamic datadriven model reduction: adapting reduced models from incomplete data
 Benjamin Peherstorfer^{1}Email author and
 Karen Willcox^{1}
https://doi.org/10.1186/s403230160064x
© The Author(s) 2016
Received: 28 November 2015
Accepted: 8 March 2016
Published: 21 March 2016
Abstract
This work presents a datadriven online adaptive model reduction approach for systems that undergo dynamic changes. Classical model reduction constructs a reduced model of a largescale system in an offline phase and then keeps the reduced model unchanged during the evaluations in an online phase; however, if the system changes online, the reduced model may fail to predict the behavior of the changed system. Rebuilding the reduced model from scratch is often too expensive in timecritical and realtime environments. We introduce a dynamic datadriven adaptation approach that adapts the reduced model from incomplete sensor data obtained from the system during the online computations. The updates to the reduced models are derived directly from the incomplete data, without recourse to the full model. Our adaptivity approach approximates the missing values in the incomplete sensor data with gappy proper orthogonal decomposition. These approximate data are then used to derive lowrank updates to the reduced basis and the reduced operators. In our numerical examples, incomplete data with 30–40 % known values are sufficient to recover the reduced model that would be obtained via rebuilding from scratch.
Keywords
Background
Dynamic online (near realtime) capability estimation is a pivotal component of future autonomous systems to dynamically observe, orient, decide, and act in complex and changing environments. We consider the situation where the dynamics of the system are modeled by a parametrized partial differential equation (PDE) and sensor data are generated that provide information on the current state of the system. The system dynamics are approximated by a largescale parametrized computer model, the socalled full model, resulting from the discretization of the underlying PDE. We rely on (projectionbased) model reduction [7, 29, 45] to derive a lowcost reduced model of the full model to meet the realtime demands of online capability estimation. Reduced models are typically built with onetime highcomputational costs in an offline phase and then stay unchanged while they are repeatedly evaluated in an online phase. However, in changing environments, the properties and the behavior of the system might change even during the online phase. Rebuilding the reduced model from scratch to take into account the changes in the system is often too time consuming. We therefore rely on dynamic datadriven reduced models, as introduced in [42]. Dynamic datadriven reduced models adapt directly from sensor data to changes in the underlying system, without recourse to the full model; however, the dynamic datadriven approach as presented in [42] requires sensor samples that measure the full largescale state of the system.
To model changes in the system, the parameters of the system are split into observable and latent parameters, see Fig. 1. The observable parameters are inputs to the system and therefore the values of these parameters are known. Latent parameters describe external influences on the system (e.g., damage, fatigue, erosion). The values of the latent parameters are unknown, except for the nominal latent parameters that describe the nominal state of the system (e.g., nodamage condition). Since the values of the latent parameters are unknown, a reduced model can be built in the offline phase for the nominal latent parameter only. If the latent parameters change online (e.g., the system gets damaged), the reduced model fails to predict the behavior of the system. Rebuilding the reduced model from scratch requires inferring the value of the changed latent parameters from the sensor data with a model of the changed system, then assembling the full model operators corresponding to the inferred latent parameters, and deriving the reduced model. Rebuilding from scratch therefore is often too expensive in the context of online capability estimation, see, e.g., [1, 33, 37, 42] for a discussion. The dynamic datadriven approach introduced in [42] exploits the sensor data of the system to adapt the reduced model to changes in the latent parameters online, without the computationally expensive inference step and without assembling the full model operators for the inferred latent parameters, see Fig. 2.
Our extension to handle incomplete sensor samples in the dynamic datadriven reduced model adaptation builds on gappy proper orthogonal decomposition (POD), which is a method to approximate unknown or missing values in vectorvalued data [22]. Gappy POD reconstructs the unknown values by representing the data vector as a linear combination of POD basis vectors. Applications of gappy POD in model reduction include flow field reconstruction [10, 49], acceleration of efficient approximations of nonlinear terms [5, 13, 24], and forecasting for timedependent problems [14]. In our adaptation approach, we first construct a gappy POD basis from incomplete sensor samples using an incremental POD basis generation algorithm. The missing values of the incomplete sensor samples are then approximated in the space spanned by the obtained gappy POD basis. These approximate sensor samples are used in the dynamic datadriven adaptation to derive updates to the reduced model.
This paper is organized as follows. “Preliminaries and adaptation from complete data” section introduces the full model and the dynamic datadriven adaptation. “Incomplete sensor samples” section defines incomplete sensor samples and describes the problem setup in detail. “Dynamic datadriven adaptation from incomplete sensor samples” section introduces the extension to the dynamic datadriven adaptation approach that handles incomplete sensor samples. The numerical results in “Numerical results” section demonstrate that in our examples 30–40 % of the values of the sensor samples are sufficient to recover reduced models that accurately capture the changes in the latent parameters. “Summary and future work” section gives concluding remarks.
Preliminaries and adaptation from complete data
This section briefly discusses model reduction for systems with observable and latent parameters and summarizes the dynamic datadriven adaptation approach presented in [42].
Systems with latent parameters
Classical model reduction for systems with latent parameters
Let \({\varvec{Y}}_{{\varvec{\eta }}_0} \in \mathbb {R}^{N\times M}\) be the snapshot matrix that contains as columns \(M \in \mathbb {N}\) state vectors \({\varvec{y}}_{{\varvec{\eta }}_0}({\varvec{\mu }}_1), \ldots , {\varvec{y}}_{{\varvec{\eta }}_0}({\varvec{\mu }}_M) \in \mathbb {R}^{N}\) of the full model (1) corresponding to the observable parameters \({\varvec{\mu }}_1, \ldots , {\varvec{\mu }}_M \in {\mathcal {D}}\) and the nominal latent parameter \({\varvec{\eta }}_0 \in {\mathcal {E}}\). The POD basis matrix \({\varvec{V}}_{{\varvec{\eta }}_0} \in \mathbb {R}^{N\times n}\) contains as columns the first \(n\in \mathbb {N}\) leftsingular vectors of the snapshot matrix \({\varvec{Y}}_{{\varvec{\eta }}_0}\) that correspond to the largest singular values. The POD basis vectors, i.e., the columns of the POD basis matrix \({\varvec{V}}_{{\varvec{\eta }}_0}\), span the \(n\)dimensional POD space \({\mathcal {V}}_{{\varvec{\eta }}_0}\).
Dynamic datadriven adaptation for reduced models
Incomplete sensor samples
The dynamic datadriven adaptation derives updates to a reduced model from sensor samples. We consider here the situation where we receive incomplete sensor samples, i.e., partial measurements of the state. This section mathematically defines incomplete sensor samples, and the next section develops the extension to the dynamic datadriven adaptation to handle incomplete sensor samples.
Dynamic datadriven adaptation from incomplete sensor samples
We propose an extension to the dynamic datadriven adaptation approach that handles incomplete sensor samples. Consider the adaptation from the nominal latent parameter \({\varvec{\eta }}_0\) to the latent parameter \({\varvec{\eta }}_1\) in the \(M^{\prime }\) adaptivity steps \(h = 1, \ldots , M^{\prime }\). At each adaptivity step \(h = 1, \ldots , M^{\prime }\), we receive incomplete sensor samples \({\hat{{\varvec{y}}}}^{\text {incp}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + h}) \in \mathbb {R}^{N}\) and the corresponding point selection matrices \({\varvec{P}}_h\). The point selection matrix depends on h and might change at each adaptivity step, see the discussion on future sensor technologies in “Background” section. The number of known components \(k\) is independent of h and stays constant for all \(h = 1, \ldots , M^{\prime }\).
We split the adaptivity steps \(M^{\prime }= M^{\mathrm{basis}}+ M^{\mathrm{update}}\) into \(M^{\mathrm{basis}}\in \mathbb {N}\) and \(M^{\mathrm{update}}\in \mathbb {N}\) steps. In the first \(h = 1, \ldots , M^{\mathrm{basis}}\) steps, a gappy POD basis is derived from the incomplete sensor samples \({\hat{{\varvec{y}}}}^{\text {incp}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + 1}), \ldots , {\hat{{\varvec{y}}}}^{\text {incp}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + M^{\mathrm{basis}}}) \in \mathbb {R}^{N}\). At the subsequent \(M^{\mathrm{update}}\) steps \(h = M + M^{\mathrm{basis}}+ 1, \ldots , M^{\prime }\), the missing values of the incomplete sensor samples \({\hat{{\varvec{y}}}}_{{\varvec{\eta }}_1}\left( {\varvec{\mu }}_{M + M^{\mathrm{basis}}+ h}\right) \in \mathbb {R}^{N}\) are approximated using gappy POD with the obtained gappy POD basis. The approximations of the missing values and the components in the incomplete sensor sample are combined to approximate the complete sensor sample. The dynamic datadriven adaptation is then applied to these approximate sensor samples to update the reduced model. “Deriving the gappy POD basis” section discusses the construction of the gappy POD basis and “Dynamic datadriven adaptation from approximate sensor samples” section presents the adaptation of the reduced model from the approximate sensor samples. “Computational procedure” section summarizes the procedure and presents Algorithm 1.
Deriving the gappy POD basis
In the first \(h = 1, \ldots , M^{\mathrm{basis}}\) adaptivity steps, we derive a gappy POD basis from the incomplete sensor samples. Let \(r\in \mathbb {N}\) be the dimension of the gappy POD basis with gappy POD basis matrix \({\varvec{U}}_h \in \mathbb {R}^{N\times r}\). The initial gappy POD basis matrix \({\varvec{U}}_0 \in \mathbb {R}^{N\times r}\) contains as columns the \(r\)dimensional POD basis vectors corresponding to the snapshot matrix \({\varvec{Y}}_{{\varvec{\eta }}_0}\).
We then use the approximate sensor sample \({\hat{{\varvec{y}}}}^{\text {apprx}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + 1})\) to adapt the gappy POD basis from \({\varvec{U}}_0\) to \({\varvec{U}}_1\). Consider therefore the snapshot matrix \({\varvec{Y}}_0\) and note that \({\varvec{U}}_0\) is the \(k\)dimensional POD basis derived from \({\varvec{Y}}_0\). We adapt the snapshot matrix \({\varvec{Y}}_0\) to \({\varvec{Y}}_1 \in \mathbb {R}^{N\times M}\) via a rankone update that replaces column 1 of \({\varvec{Y}}_0\) with the approximate sensor sample \({\hat{{\varvec{y}}}}^{\text {apprx}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + 1}) \in \mathbb {R}^{N}\). Since \({\varvec{Y}}_1\) is the result of a rankone update to \({\varvec{Y}}_0\), the \(k\)dimensional POD basis corresponding to \({\varvec{Y}}_1\) can be approximated in a computationally efficient manner using the incremental POD algorithm [8]. Note that this is the same approach as used in the dynamic datadriven adaptation, see “Dynamic datadriven adaptation for reduced models” section. Thus, the adapted gappy POD basis matrix \({\varvec{U}}_1\) can be derived cheaply from the basis matrix \({\varvec{U}}_0\).
At step \(h = 2\), the approximate sensor sample \({\hat{{\varvec{y}}}}^{\text {apprx}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + 2})\) is constructed with the gappy POD basis matrix \({\varvec{U}}_1\), which is then used to adapt from \({\varvec{U}}_1\) to \({\varvec{U}}_2\). This process is continued until step \(h = M^{\mathrm{basis}}\), where the gappy POD basis matrix \({\varvec{U}}_{M^{\mathrm{basis}}}\) is derived. Note that the number of columns in the snapshot matrix is fixed and that columns are replaced following the firstinfirstout principle if \(h > M\).
Dynamic datadriven adaptation from approximate sensor samples
In the \(M^{\mathrm{update}}\) steps \(h = M + M^{\mathrm{basis}}+ 1, \ldots , M^{\prime }\), we adapt the reduced model from approximate sensor samples using the dynamic datadriven adaptation. Consider therefore an adaptivity step \(h > M^{\mathrm{basis}}\), at which the incomplete sensor sample \({\hat{{\varvec{y}}}}^{\text {incp}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + h}) \in \mathbb {R}^{N}\) and the corresponding point selection matrix \({\varvec{P}}_h \in \mathbb {R}^{k\times N}\) are received. We use the gappy POD basis \({\varvec{U}}_{M^{\mathrm{basis}}}\) to derive the approximate sensor sample \({\hat{{\varvec{y}}}}^{\text {apprx}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + h})\) of the complete sensor sample with the gappy POD basis \({\varvec{U}}_{M^{\mathrm{basis}}}\). The approximate sensor sample \({\hat{{\varvec{y}}}}^{\text {apprx}}_{{\varvec{\eta }}_1}({\varvec{\mu }}_{M + h})\) is then used to adapt the reduced model with the dynamic datadriven adaptation as described in “Dynamic datadriven adaptation for reduced models” section.
Computational procedure
Algorithm 1 summarizes the dynamic datadriven adaptation that can handle incomplete sensor samples. Inputs of Algorithm 1 are the POD basis matrix \({\varvec{V}}_{h  1}\), the operators \({\tilde{{\varvec{A}}}}_{h  1}^{(1)}, \ldots , {\tilde{{\varvec{A}}}}_{h  1}^{(l_A)}\), and the righthand sides \({\tilde{{\varvec{f}}}}_{h  1}^{(1)}, \ldots , {\tilde{{\varvec{f}}}}_{h  1}^{(l_f)}\) derived at the previous adaptivity step \(h  1\). If \(h \le M^{\mathrm{basis}}\), the algorithm adapts the gappy POD basis from \({\varvec{U}}_{h  1}\) to \({\varvec{U}}_h\) using the approach presented in“Deriving the gappy POD basis” section. First, the approximate sensor sample is constructed with gappy POD. Then, the adapted basis matrix \({\varvec{U}}_h\) is computed with the incremental POD algorithm [8]. Only the gappy POD basis is adapted and the reduced model is returned unchanged. If \(h > M^{\mathrm{basis}}\), the approximate sensor sample is derived with gappy POD and \({\varvec{U}}_{M^{\mathrm{basis}}}\). The approximate sensor sample is then used with the dynamic datadriven adaptation to derive the adapted POD basis \({\varvec{V}}_h\), the adapted operators \({\tilde{{\varvec{A}}}}_{h}^{(1)}, \ldots , {\tilde{{\varvec{A}}}}_{h}^{(l_A)},\) and the adapted righthand sides \({\tilde{{\varvec{f}}}}_{h}^{(1)}, \ldots , {\tilde{{\varvec{f}}}}_{h}^{(l_f)}\).
Numerical results
This section demonstrates the dynamic datadriven adaptation from incomplete sensor samples on a model of a bending plate. The latent parameter describes damage of the plate. The damage is a local decrease of the thickness of the plate. The model is based on the Mindlin plate theory [23, 47] that takes into account transverse shear deformations but neglects important nonlinear effects such as postbuckling behavior. Therefore, the model that we use in this section is a simple description of a plate in bending. We use the plate model only to provide a proof of concept of our adaptation approach. More advanced plate models are used in realworld engineering applications. We refer to “Summary and future work” section for a discussion on further applications of our adaptation approach.
Plate problem
We consider the static analysis of a plate in bending. The plate is clamped into a frame and a load is applied. Our problem is an extension of the plate problems introduced in [23, 42, 44]. The geometry of our plate problem is shown in Fig. 3a. The spatial domain \(\Omega \in [0, 1]^2 \subset \mathbb {R}^2\) is split into two subdomains \(\Omega = \Omega _1 \cup \Omega _2\). The problem has three observable parameters \({\varvec{\mu }}= [\mu _1, \mu _2, \mu _3]^T \in {\mathcal {D}}\) with \({\mathcal {D}}= [0.05, 0.1]^2 \times [1, 100]\). The observable parameters \(\mu _1\) and \(\mu _2\) control the nominal thickness of the plate in the subdomain \(\Omega _1\) and \(\Omega _2\), respectively. The third observable parameter \(\mu _3\) defines the load on the plate.
Setup of numerical experiments
Number of sensor samples
We receive incomplete sensor samples, and therefore we use the extension to the dynamic datadriven adaptation described in “Dynamic datadriven adaptation from incomplete sensor samples” section. This means that the adaptivity steps \(h = 1, \ldots , M^{\prime }\) required for adapting from latent parameter \({\varvec{\eta }}_{i  1}\) to \({\varvec{\eta }}_i\) are split into \(M^{\mathrm{basis}}\in \mathbb {N}\) steps to derive the gappy POD basis and \(M^{\mathrm{update}}\in \mathbb {N}\) steps to update the reduced model. We chose \(M^{\mathrm{basis}}\) and \(M^{\mathrm{update}}\) conservatively in the following, because we are primarily interested in studying the effect of the number of missing components in the incomplete sensor samples onto the adaptation, rather than the number of sensor samples; see [42] for studies on the effect of the number of samples on the dynamic datadriven adaptation in the case with complete sensor samples. We set \(M^{\mathrm{basis}}= 5000\) and therefore derive the gappy POD basis from \(M^{\mathrm{basis}}= 5000\) incomplete sensor samples. We buffer 50 incomplete sensor samples and use them in the incremental basis generation procedure described in “Deriving the gappy POD basis” section.
The theory of the dynamic datadriven adaptation with complete sensor samples gives guidance on the selection of \(M^{\mathrm{update}}\). In case of complete sensor samples, setting \(M^{\mathrm{update}}= l_A \times n\) is sufficient to recover the reduced model that would be obtained via rebuilding from scratch [42]. Note that \(l_A = 4\) is the number of \({\varvec{\mu }}\)independent operators and \(n= 8\) the dimension of the POD basis space. We set \(M^{\mathrm{update}}= 5 \times l_A \times n= 160\) since we adapt from incomplete sensor samples and therefore expect that the approximation of the missing values introduces additional error into the adaptation. In total, we receive \(M^{\prime }= M^{\mathrm{basis}}+ M^{\mathrm{update}}= 5160\) incomplete sensor samples to adapt from \({\varvec{\eta }}_{i  1}\) to \({\varvec{\eta }}_i\) for \(i = 1, \ldots , 9\).
Sensor sample generation
We synthetically generate incomplete sensor samples with the full model at each step \(h = 1, \ldots , M^{\prime }\). We therefore first draw uniformly an observable parameter \({\varvec{\mu }}_{M + h}\) in \({\mathcal {D}}\) and compute the state vector \({\varvec{y}}_{{\varvec{\eta }}}({\varvec{\mu }}_{M + h})\) with the full model for the current latent parameter \({\varvec{\mu }}\). We then draw \(k\in \mathbb {N}\) unique indices uniformly in \(\{1, \ldots , N\}\) and construct the point selection matrix \({\varvec{P}}_h \in \mathbb {R}^{N\times k}\). The incomplete sensor sample is \({\hat{{\varvec{y}}}}^{\text {incp}}_{{\varvec{\eta }}}({\varvec{\mu }}_{M + h}) = {\varvec{P}}_h{\varvec{P}}^T_h{\varvec{y}}_{{\varvec{\eta }}}({\varvec{\mu }}_{M + h})\).
Error computation

A static reduced model that is built as described in “Classical model reduction for systems with latent parameters” section. The static reduced model is not adapted to changes in the latent parameter.

A rebuilt reduced model that is derived as in “Classical model reduction for systems with latent parameters” section but from \(M^{\mathrm{update}}\) complete sensor samples corresponding to the current changed latent parameter. This requires repeating the computation of the POD basis and the operator projections, which is prohibitively expensive to conduct online.

An online adaptive reduced model that is adapted to changes in the latent parameter from incomplete sensor samples with the dynamic datadriven adaptation described in Algorithm 1.
Gappy POD basis from complete sensor samples
We first consider the situation where \(\rho ^{\text {basis}}= 100~\%\) is fixed and the sensor rate \(\rho ^{\text {update}}\) varies. This means that we have available complete sensor samples (without missing components) for deriving the gappy POD basis in the first \(M^{\mathrm{basis}}\) steps but incomplete sensor samples for updating the reduced model in the final \(M^{\mathrm{update}}\) adaptivity steps.
Figures 5 and 6 demonstrate the effect of the sensor rate \(\rho ^{\text {update}}\) on the dynamic datadriven adaptation. First consider the static reduced model. As the latent parameter changes from \({\varvec{\eta }}_0\) (no damage) to \({\varvec{\eta }}_9\) (20 % decrease of thickness), the error of the static reduced model increases by three orders of magnitude. The steps in the error curve reflect the changes in the latent parameter. The error of the rebuilt reduced model stays near \(10^{4}\). Consider now the adaptive reduced model. The dimension of the gappy POD basis is set to \(r= 30\). Figure 5 shows that a sensor rate \(\rho ^{\text {update}}= 0.6~\%\) leads to an adapted reduced model with large errors. A sensor rate of \(\rho ^{\text {update}}= 0.6~\%\) means that \(k^{\text {update}}= 29\) components of the incomplete sensor sample are known, and therefore \(k^{\text {update}}< r\). This violates the condition of gappy POD that requires a fullcolumn rank \({\varvec{P}}_h^T{\varvec{U}}_{M^{\mathrm{basis}}}\), see “Deriving the gappy POD basis” section. For a slightly larger sensor rate \(\rho ^{\text {update}}= 0.8~\%\), and \(k^{\text {update}}> r\), our dynamic datadriven adaptation from incomplete sensor samples recovers the rebuilt reduced model. Figure 6 indicates that increasing the sensor rate \(\rho ^{\text {update}}\) reduces the error of the adapted reduced model in the first few adaptivity steps after a change in the latent parameter, cf. Fig. 5.
Note that the adapted reduced model achieves a slightly lower error than the rebuilt reduced model in Figs. 5 and 6. The dynamic datadriven adaptation constructs the adapted operators with an optimization problem from the sensor samples projected onto the POD space. This projection and the optimization cause the difference in the error of the adapted and the rebuilt reduced model, if the dimension of the reduced model is low. The difference decreases if the dimension of the reduced model is increased, see [42, Theorem 1].
Gappy POD basis from incomplete sensor samples
We now consider the situation where \(\rho ^{\text {basis}}< 100~\%\) and \(\rho ^{\text {update}}< 100~\%\), i.e., the gappy POD basis is derived from incomplete sensor samples and the updates to the reduced models are obtained from incomplete sensor samples as well. Figure 8 shows the effect of the sensor rate \(\rho ^{\text {basis}}\) on the adaptation. Figures 8a and b demonstrate that a sensor rate \(\rho ^{\text {basis}}= 10~\%\) is too low to recover the rebuilt reduced model with the adapted reduced model in this example. Even setting the sensor rate for the update to \(\rho ^{\text {update}}= 90~\%\) (i.e., generating the gappy POD basis from incomplete samples with \(\rho ^{\text {basis}}= 10~\%\) and updating the reduced model from approximate sensor samples with \(\rho ^{\text {update}}= 90~\%\)) cannot compensate the inadequate sensor rate \(\rho ^{\text {basis}}= 10~\%\). Increasing the sensor rate for the gappy POD basis construction to \(\rho ^{\text {basis}}= 30~\%\) leads to an adapted reduced model that recovers the rebuilt reduced model. However, with \(\rho ^{\text {basis}}= 30~\%\) there still are outliers that lead to a reduced model with a large error. Figure 9 shows that increasing the sensor rate to \(\rho ^{\text {basis}}= 70~\%\) reduces those outliers significantly. Again, increasing the dimension of the gappy POD basis from \(r= 30\) to \(r= 40\) only slightly reduces the error of the adapted reduced model, compare Fig. 9a, c, e with Fig. 9b, d, f.
Summary and future work
We proposed an extension to the dynamic datadriven adaptation that handles incomplete sensor samples, i.e., partial measurements of the largescale state. In our approach, a gappy POD basis is derived from incomplete sensor samples. The missing values of the incomplete sensor samples are approximated with gappy POD in the space spanned by the gappy POD basis. The reduced model is then adapted using the gappy POD approximations of the complete sensor samples with the dynamic datadriven adaptation. The numerical results confirm that about 30–40 % of the total number of components of the sensor samples are sufficient to recover the reduced model that would be obtained via rebuilding from scratch.
Future sensing technologies (e.g., “sensor skins”) of nextgeneration engineering systems will provide highresolution measurements. Processing these large data sets will be computationally challenging. In big data analytics, sublinear algorithms are currently developed that look at only a subset of the given data set to meet runtime requirements [48]. Our approach follows a similar paradigm. We selectively process sensor data that are most informative for deriving the update to the reduced model and ignore large parts of the received data that are irrelevant in the current situation. Our approach is applicable even if the selection of the highresolution sensor data is dynamically changing online, e.g., due to new damage events.
We considered here realtime structural assessment and decisionmaking but sensor data are available in many other applications. For example, in control, the goal is to design a controller that stabilizes a dynamical system. However, if the dynamical system passes through multiple regimes with different system characteristics, a single controller might be insufficient to stabilize the system. If sensor data, e.g., sparse measurements of the state of the dynamical system, are available, the controller can be adapted to the sensor data to take into account the changes in the underlying dynamical system. We also mention system identification as a potential application of our adaptation approach. Instead of starting with reduced operators derived in an offline phase, one could start with initial operators that have all components set to zero, and then adapt these operators to the available data. Such a system identification approach would derive a reduced model directly from data. In general, our approach is applicable to DDDAS for which massive amounts of sensor data are available.
Note that the adaptation can be repeated to adapt from \({\varvec{\eta }}_1\) to \({\varvec{\eta }}_2 \in {\mathcal {E}}\) and so on.
Declarations
Acknowledgments
BP and KW developed the methodology, performed the numerical investigations, and wrote the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was supported in part by the AFOSR MURI on multiinformation sources of multiphysics systems under Award Number FA95501510038, program manager JeanLuc Cambier, and by the United States Department of Energy Applied Mathematics Program, Awards DEFG0208ER2585 and DESC0009297, as part of the DiaMonD Multifaceted Mathematics Integrated Capability Center. Some of the numerical examples were computed on the computer cluster of the Munich Centre of Advanced Computing.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Allaire D, Chambers J, Cowlagi R, Kordonowy D, Lecerf M, Mainini L, Ulker F, Willcox K. An offline/online DDDAS capability for selfaware aerospace vehicles. Procedia Comput Sci. 2013;18:1959–68.View ArticleGoogle Scholar
 Amsallem D, Farhat C. An online method for interpolating linear parametric reducedorder models. SIAM J Sci Comput. 2011;33(5):2169–98.MathSciNetView ArticleMATHGoogle Scholar
 Amsallem D, Zahr M, Farhat C. Nonlinear model order reduction based on local reducedorder bases. Int J Numer Methods Eng. 2012;92(10):891–916.MathSciNetView ArticleGoogle Scholar
 Amsallem D, Zahr M, Washabaugh K. Fast local reduced basis updates for the efficient reduction of nonlinear systems with hyperreduction. Special issue on model reduction of parameterized systems (MoRePaS). Adv Comput Math. 2014. (accepted).Google Scholar
 Astrid P, Weiland S, Willcox K, Backx T. Missing point estimation in models described by proper orthogonal decomposition. Autom Control IEEE Trans. 2008;53(10):2237–51.MathSciNetView ArticleGoogle Scholar
 Barrault M, Maday Y, Nguyen N, Patera A. An empirical interpolation method: application to efficient reducedbasis discretization of partial differential equations. Comptes Rendus Math. 2004;339(9):667–72.MathSciNetView ArticleMATHGoogle Scholar
 Benner P, Gugercin S, Willcox K. A survey of projectionbased model reduction methods for parametric dynamical systems. SIAM Rev. 2015;57(4):483–531.MathSciNetView ArticleGoogle Scholar
 Brand M. Fast lowrank modifications of the thin singular value decomposition. Linear Algebra Appl. 2006;415(1):20–30.MathSciNetView ArticleMATHGoogle Scholar
 Brunton SL, Tu JH, Bright I, Kutz JN. Compressive sensing and lowrank libraries for classification of bifurcation regimes in nonlinear dynamical systems. SIAM J Appl Dyn Syst. 2014;13(4):1716–32.MathSciNetView ArticleMATHGoogle Scholar
 BuiThanh T, Damodaran M, Willcox K. Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition. AIAA J. 2004;42(8):1505–16.View ArticleGoogle Scholar
 Burkardt J, Gunzburger M, Lee HC. POD and CVTbased reducedorder modeling of Navier–Stokes flows. Comput Methods Appl Mech Eng. 2006;196(1–3):337–55.MathSciNetView ArticleMATHGoogle Scholar
 Carlberg K. Adaptive hrefinement for reducedorder models. Int J Numer Methods Eng. 2015;102(5):1192–210.MathSciNetView ArticleGoogle Scholar
 Carlberg K, BouMosleh C, Farhat C. Efficient nonlinear model reduction via a leastsquares petrovgalerkin projection and compressive tensor approximations. Int J Numer Methods Eng. 2011;86(2):155–81.MathSciNetView ArticleMATHGoogle Scholar
 Carlberg K, Ray J, van Bloemen Waanders B. Decreasing the temporal complexity for nonlinear, implicit reducedorder models by forecasting. Comput Methods Appl Mech Eng. 2015;289:79–103.MathSciNetView ArticleGoogle Scholar
 Chaturantabut S, Sorensen D. Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput. 2010;32(5):2737–64.MathSciNetView ArticleMATHGoogle Scholar
 Chinesta F, Ladeveze P, Cueto E. A short review on model order reduction based on proper generalized decomposition. Arch Comput Methods Eng. 2011;18(4):395–404.View ArticleGoogle Scholar
 Cui T, Marzouk YM, Willcox KE. Datadriven model reduction for the bayesian solution of inverse problems. Int J Numer Methods Eng. 2015;102(5):966–90.MathSciNetView ArticleGoogle Scholar
 Degroote J, Vierendeels J, Willcox K. Interpolation among reducedorder matrices to obtain parameterized models for design, optimization and probabilistic analysis. Int J Numer Methods Fluids. 2010;63(2):207–30.MathSciNetMATHGoogle Scholar
 Dihlmann M, Drohmann M, Haasdonk B. Model reduction of parametrized evolution problems using the reduced basis method with adaptive timepartitioning. In: Aubry D, Díez P, Tie B, Parés N, editors, Proceedings of the international conference on adaptive modeling and simulation; 2011. p. 156–67.Google Scholar
 Eftang J, Patera A. Port reduction in parametrized component static condensation: approximation and a posteriori error estimation. Int J Numer Methods Eng. 2013;96(5):269–302.MathSciNetGoogle Scholar
 Eftang J, Stamm B. Parameter multidomain hp empirical interpolation. Int J Numer Methods Eng. 2012;90(4):412–28.MathSciNetView ArticleMATHGoogle Scholar
 Everson R, Sirovich L. Karhunen–Loève procedure for gappy data. J Opt Soc Am A. 1995;12(8):1657–64.View ArticleGoogle Scholar
 Ferreira A. MATLAB codes for finite element analysis. Berlin: Springer; 2008.Google Scholar
 Galbally D, Fidkowski K, Willcox K, Ghattas O. Nonlinear model reduction for uncertainty quantification in largescale inverse problems. Int J Numer Methods Eng. 2010;81(12):1581–608.MathSciNetMATHGoogle Scholar
 Garmatter D, Haasdonk B, Harrach B. A reduced landweber method for nonlinear inverse problems. Technical report, University of Stuttgart; 2015. Available at Arxiv.Google Scholar
 Ghnatios C, Masson F, Huerta A, Leygue A, Cueto E, Chinesta F. Proper generalized decomposition based dynamic datadriven control of thermal processes. Comput Methods Appl Mech Eng. 2012;213–216:29–41.View ArticleGoogle Scholar
 Gogu C. Improving the efficiency of large scale topology optimization through onthefly reduced order model construction. Int J Numer Methods Eng. 2015;101(4):281–304.MathSciNetView ArticleGoogle Scholar
 González D, Masson F, Poulhaon F, Leygue A, Cueto E, Chinesta F. Proper generalized decomposition based dynamic data driven inverse identification. Math Comput Simul. 2012;82(9):1677–95.MathSciNetView ArticleGoogle Scholar
 Gugercin S, Antoulas A. A survey of model reduction by balanced truncation and some new results. Int J Control. 2004;77(8):748–66.MathSciNetView ArticleMATHGoogle Scholar
 Kaulmann S, Haasdonk B. Online greedy reduced basis construction using dictionaries. In: Troch I, Breitenecker F, editors, Proceedings of 7th Vienna international conference on mathematical modelling; 2012. p. 112–7.Google Scholar
 Kennedy M, O’Hagan A. Bayesian calibration of computer models. J R Stat Soc Series B (Stat Methodol). 2001;63(3):425–64.MathSciNetView ArticleMATHGoogle Scholar
 Lass O. Reduced order modeling and parameter identification for coupled nonlinear PDE systems. PhD thesis, University of Konstanz; 2014.Google Scholar
 Mainini L, Karen E. Willcox. A surrogate modeling approach to support realtime structural assessment and decisionmaking. In 10th AIAA multidisciplinary design optimization conference, AIAA SciTech. American Institute of Aeronautics and Astronautics; 2014.Google Scholar
 Lecerf M, Allaire D, Willcox K. Methodology for dynamic datadriven online flight capability estimation. AIAA J. 2015;53(10):3073–87.View ArticleGoogle Scholar
 Maday Y, Stamm B. Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces. SIAM J Sci Comput. 2013;35(6):A2417–41.MathSciNetView ArticleMATHGoogle Scholar
 Mainini L, Willcox K. Sensitivity analysis of surrogatebased methodology for real time structural assessment. In AIAA modeling and simulation technologies conference, AIAA SciTech 2015, AIAA. AIAA; 2015. Paper 1362.Google Scholar
 Mainini L, Willcox K. Surrogate modeling approach to support realtime structural assessment and decision making. AIAA J. 2015;53(6):1612–26.View ArticleGoogle Scholar
 Ohlberger M, Schindler F. Error control for the localized reduced basis multiscale method with adaptive online enrichment. SIAM J Sci Comput. 2015. (accepted).Google Scholar
 Panzer H, Mohring J, Eid R, Lohmann B. Parametric model order reduction by matrix interpolation. at Automatisierungstechnik. 2010;58(8):475–84.View ArticleGoogle Scholar
 Peherstorfer B, Butnaru D, Willcox K, Bungartz HJ. Localized discrete empirical interpolation method. SIAM J Sci Comput. 2014;36(1):A168–92.MathSciNetView ArticleMATHGoogle Scholar
 Peherstorfer B, Willcox K. Detecting and adapting to parameter changes for reduced models of dynamic datadriven application systems. Procedia Comput Sci. 2015;51:2553–62.View ArticleGoogle Scholar
 Peherstorfer B, Willcox K. Dynamic datadriven reducedorder models. Comput Methods Appl Mech Eng. 2015;291:21–41.MathSciNetView ArticleGoogle Scholar
 Peherstorfer B, Willcox K. Online adaptive model reduction for nonlinear systems via lowrank updates. SIAM J Sci Comput. 2015;37(4):A2123–50.MathSciNetView ArticleMATHGoogle Scholar
 Peherstorfer B, Willcox K, Gunzburger M. Optimal model management for multifidelity Monte Carlo estimation. Technical report 15–2, Aerospace Computational Design Laboratory, MIT; 2015.Google Scholar
 Rozza G, Huynh D, Patera A. Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch Comput Methods Eng. 2007;15(3):1–47.MathSciNetView ArticleMATHGoogle Scholar
 Sargsyan S, Brunton SL, Kutz JN. Nonlinear model reduction for dynamical systems using sparse sensor locations from learned libraries. Phys Rev E. 2015;92:033304.View ArticleGoogle Scholar
 Ventsel E, Krauthammer T. Thin plates and shells. Boca Raton: CRC Press; 2001.View ArticleGoogle Scholar
 Wang D, Han Z. Sublinear algorithms for big data applications. Berlin: Springer; 2015.View ArticleGoogle Scholar
 Willcox K. Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition. Comput Fluids. 2006;35(2):208–26.View ArticleMATHGoogle Scholar
 Zahr M, Farhat C. Progressive construction of a parametric reducedorder model for pdeconstrained optimization. Int J Numer Methods Eng. 2015;102(5):1111–35.MathSciNetView ArticleGoogle Scholar
 Zimmermann R. A locally parametrized reducedorder model for the linear frequency domain approach to timeaccurate computational fluid dynamics. SIAM J Sci Comput. 2014;36(3):B508–37.MathSciNetView ArticleMATHGoogle Scholar