Publications

Preprints

2024

Petar Mlinarić, Peter Benner, and Serkan Gugercin. Interpolatory necessary optimality conditions for reduced-order modeling of parametric linear time-invariant systems. arXiv preprint 2401.10047, 2024. doi:10.48550/arXiv.2401.10047.
[abstract▼] [BibTeX▼]

Interpolatory necessary optimality conditions for \(\mathcal H_2\)-optimal reduced-order modeling of non-parametric linear time-invariant (LTI) systems are known and well-investigated. In this work, using the general framework of \(\mathcal L_2\)-optimal reduced-order modeling of parametric stationary problems, we derive interpolatory \(\mathcal H_2 \otimes \mathcal L_2\)-optimality conditions for parametric LTI systems with a general pole-residue form. We then specialize this result to recover known conditions for systems with parameter-independent poles and develop new conditions for a certain class of systems with parameter-dependent poles.
```
@unpublished{MliBG24,
    author = "Mlinarić, Petar and Benner, Peter and Gugercin, Serkan",
    title = "Interpolatory Necessary Optimality Conditions for Reduced-order Modeling of Parametric Linear Time-invariant Systems",
    year = "2024",
    note = "arXiv preprint 2401.10047",
    doi = "10.48550/arXiv.2401.10047"
}
```

2023

Petar Mlinarić, Christopher Beattie, Zlatko Drmač, and Serkan Gugercin. IRKA is a Riemannian gradient descent method. arXiv preprint 2311.02031, 2023. doi:10.48550/arXiv.2311.02031.
[abstract▼] [BibTeX▼]

The iterative rational Krylov algorithm (IRKA) is a commonly used fixed-point iteration developed to minimize the \(\mathcal H_2\) model order reduction error. In this work, IRKA is recast as a Riemannian gradient descent method with a fixed step size over the manifold of rational functions having fixed degree. This interpretation motivates the development of a Riemannian gradient descent method utilizing as a natural extension variable step size and line search. Comparisons made between IRKA and this extension on a few examples demonstrate significant benefits.
```
@unpublished{MliBDG23,
    author = "Mlinarić, Petar and Beattie, Christopher and Drmač, Zlatko and Gugercin, Serkan",
    title = "{IRKA} is a {R}iemannian Gradient Descent Method",
    year = "2023",
    note = "arXiv preprint 2311.02031",
    doi = "10.48550/arXiv.2311.02031"
}
```
Petar Mlinarić, Peter Benner, and Serkan Gugercin. Interpolatory \(\mathcal H_2\)-optimality conditions for structured linear time-invariant systems. arXiv preprint 2310.10618, 2023. doi:10.48550/arXiv.2310.10618.
[abstract▼] [BibTeX▼]

Interpolatory necessary optimality conditions for \(\mathcal H_2\)-optimal reduced-order modeling of unstructured linear time-invariant (LTI) systems are well-known. Based on previous work on \(\mathcal L_2\)-optimal reduced-order modeling of stationary parametric problems, in this paper we develop and investigate optimality conditions for \(\mathcal H_2\)-optimal reduced-order modeling of structured LTI systems, in particular, for second-order, port-Hamiltonian, and time-delay systems. We show that across all these different structured settings, bitangential Hermite interpolation is the common form for optimality, thus proving a unifying optimality framework for structured reduced-order modeling.
```
@unpublished{MliBG23,
    author = "Mlinarić, Petar and Benner, Peter and Gugercin, Serkan",
    title = "Interpolatory {\(\mathcal H\_2\)}-optimality Conditions for Structured Linear Time-invariant Systems",
    year = "2023",
    note = "arXiv preprint 2310.10618",
    doi = "10.48550/arXiv.2310.10618"
}
```

Journal Publications

2023

Petar Mlinarić and Serkan Gugercin. \(\mathcal L_2\)-optimal reduced-order modeling using parameter-separable forms. SIAM J. Sci. Comput., 45(2):A554–A578, April 2023. doi:10.1137/22M1500678.
[abstract▼] [full text] [code] [BibTeX▼]

We provide a unifying framework for \(\mathcal L_2\)-optimal reduced-order modeling for linear time-invariant dynamical systems and stationary parametric problems. Using parameter-separable forms of the reduced-model quantities, we derive the gradients of the \(\mathcal L_2\) cost function with respect to the reduced matrices, which then allows a non-intrusive, data-driven, gradient-based descent algorithm to construct the optimal approximant using only output samples. By choosing an appropriate measure, the framework covers both continuous (Lebesgue) and discrete cost functions. We show the efficacy of the proposed algorithm via various numerical examples. Furthermore, we analyze under what conditions the data-driven approximant can be obtained via projection.
```
@article{MliG23,
    author = "Mlinarić, Petar and Gugercin, Serkan",
    title = "{\(\mathcal L\_2\)}-optimal Reduced-order Modeling Using Parameter-separable Forms",
    journal = "SIAM J. Sci. Comput.",
    year = "2023",
    month = "April",
    volume = "45",
    number = "2",
    pages = "A554--A578",
    doi = "10.1137/22M1500678",
    customlinkcode = "https://github.com/pmli/l2-opt-rom-ex"
}
```
Petar Mlinarić and Serkan Gugercin. A unifying framework for interpolatory \(\mathcal L_2\)-optimal reduced-order modeling. SIAM J. Numer. Anal., 61(5):2133–2156, September 2023. doi:10.1137/22M1516920.
[abstract▼] [full text] [code] [BibTeX▼]

We develop a unifying framework for interpolatory \(\mathcal L_2\)-optimal reduced-order modeling for a wide classes of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for \(\mathcal L_2\)-optimal model order reduction and leads to the interpolatory conditions for \(\mathcal H_2 \otimes \mathcal L_2\)-optimal model order reduction of multi-input/multi-output parametric dynamical systems. Moreover, we derive novel interpolatory optimality conditions for rational discrete least-squares minimization and for \(\mathcal L_2\)-optimal model order reduction of a class of parametric stationary models. We show that bitangential Hermite interpolation appears as the main tool for optimality across different domains. The theoretical results are illustrated on two numerical examples.
```
@article{MliG23b,
    author = "Mlinarić, Petar and Gugercin, Serkan",
    title = "A Unifying Framework for Interpolatory {\(\mathcal L\_2\)}-optimal Reduced-order Modeling",
    journal = "SIAM J. Numer. Anal.",
    year = "2023",
    month = "September",
    volume = "61",
    number = "5",
    pages = "2133--2156",
    doi = "10.1137/22M1516920",
    customlinkcode = "https://github.com/pmli/l2-opt-interp-ex"
}
```

2022

Manuela Hund, Tim Mitchell, Petar Mlinarić, and Jens Saak. Optimization-based parametric model order reduction via \(\mathcal H_2 \otimes \mathcal L_2\) first-order necessary conditions. SIAM J. Sci. Comput., 44(3):A1554–A1578, June 2022. doi:10.1137/21M140290X.
[abstract▼] [full text] [code] [BibTeX▼]

In this paper, we generalize existing frameworks for \(\mathcal H_2 \otimes \mathcal L_2\)-optimal model order reduction to a broad class of parametric linear time-invariant systems. To this end, we derive first-order necessary optimality conditions for a class of structured reduced-order models and then, building on those, propose a stability-preserving optimization-based method for computing locally \(\mathcal H_2 \otimes \mathcal L_2\)-optimal reduced-order models. We also make a theoretical comparison to existing approaches in the literature and, in numerical experiments, show how our new method, with reasonable computational effort, produces stable optimized reduced-order models with significantly lower approximation errors.
```
@article{HunMMetal22,
    author = "Hund, Manuela and Mitchell, Tim and Mlinarić, Petar and Saak, Jens",
    title = "Optimization-based Parametric Model Order Reduction via {\(\mathcal H\_2 \otimes \mathcal L\_2\)} First-order Necessary Conditions",
    journal = "SIAM J. Sci. Comput.",
    year = "2022",
    month = "June",
    volume = "44",
    number = "3",
    pages = "A1554--A1578",
    doi = "10.1137/21M140290X",
    customlinkcode = "https://zenodo.org/record/5710777"
}
```

2018

Hidde-Jan Jongsma, Petar Mlinarić, Sara Grundel, Peter Benner, and Harry L. Trentelman. Model reduction of linear multi-agent systems by clustering with \(\mathcal H_2\) and \(\mathcal H_\infty \) error bounds. Math. Control Signals Systems, 30(6):1–38, April 2018. doi:10.1007/s00498-018-0212-6.
[abstract▼] [BibTeX▼]

In the recent paper [Monshizadeh et al., Projection-Based Model Reduction of Multi-Agent Systems Using Graph Partitions, IEEE Trans Control Netw Syst 1(2):145-154, 2014], model reduction of leader-follower multi-agent networks by clustering was studied. For such multi-agent networks, a reduced order network is obtained by partitioning the set of nodes in the graph into disjoint sets, called clusters, and associating with each cluster a single, new, node in a reduced network graph. In Monshizadeh et al. (2014), this method was studied for the special case that the agents have single integrator dynamics. For a special class of graph partitions, called almost equitable partitions, an explicit formula was derived for the \(\mathcal H_2\) model reduction error. In the present paper, we will extend and generalize the results from Monshizadeh et al. (2014) in a number of directions. Firstly, we will establish an a priori upper bound for the \(\mathcal H_2\) model reduction error in case that the agent dynamics is an arbitrary multivariable input-state-output system. Secondly, for the single integrator case, we will derive an explicit formula for the \(\mathcal H_\infty\) model reduction error. Thirdly, we will prove an a priori upper bound for the \(\mathcal H_\infty\) model reduction error in case that the agent dynamics is a symmetric multivariable input-state-output system. Finally, we will consider the problem of obtaining a priori upper bounds if we cluster using arbitrary, possibly non almost equitable, partitions.
```
@article{JonMGetal18,
    author = "Jongsma, Hidde-Jan and Mlinarić, Petar and Grundel, Sara and Benner, Peter and Trentelman, Harry L.",
    title = "Model reduction of linear multi-agent systems by clustering with {\(\mathcal H\_2\)} and {\(\mathcal H\_\infty\)} error bounds",
    journal = "Math. Control Signals Systems",
    year = "2018",
    month = "April",
    volume = "30",
    number = "6",
    pages = "1--38",
    doi = "10.1007/s00498-018-0212-6"
}
```

Conference Proceedings

2019

Linus Balicki, Petar Mlinarić, Stephan Rave, and Jens Saak. System-theoretic model order reduction with pyMOR. Proc. Appl. Math. Mech., November 2019. doi:10.1002/pamm.201900459.
[abstract▼] [BibTeX▼]

This paper shows recent developments in pyMOR, in particular the addition of system-theoretic methods. All methods are implemented using pyMOR's abstract interfaces, which allows the application to partial differential equation (PDE) models implemented with third-party libraries. We demonstrate this by applying balanced truncation to a PDE model discretized in FEniCS.
```
@article{BalMRetal19,
    author = "Balicki, Linus and Mlinarić, Petar and Rave, Stephan and Saak, Jens",
    title = "System-theoretic model order reduction with {pyMOR}",
    journal = "Proc. Appl. Math. Mech.",
    year = "2019",
    month = "November",
    volume = "19",
    number = "1",
    doi = "10.1002/pamm.201900459"
}
```

2018

Manuela Hund, Petar Mlinarić, and Jens Saak. An \(\mathcal H_2 \otimes \mathcal L_2\)-optimal model order reduction approach for parametric linear time-invariant systems. Proc. Appl. Math. Mech., December 2018. doi:10.1002/pamm.201800084.
[abstract▼] [full text] [BibTeX▼]

So far, \(\mathcal H_2 \otimes \mathcal L_2\)-optimal model order reduction (MOR) of linear time-invariant systems, preserving the affine parameter dependence, was only considered for special cases by Baur et al in 2011. In this contribution, we present necessary conditions for an \(\mathcal H_2 \otimes \mathcal L_2\)-optimal parametric reduced order model, for general affine parametric systems resembling the special case investigated by Baur et al.
```
@article{HunMS18,
    author = "Hund, Manuela and Mlinarić, Petar and Saak, Jens",
    title = "An {\(\mathcal H\_2 \otimes \mathcal L\_2\)}-Optimal Model Order Reduction Approach for Parametric Linear Time-Invariant Systems",
    journal = "Proc. Appl. Math. Mech.",
    year = "2018",
    month = "December",
    volume = "18",
    number = "1",
    doi = "10.1002/pamm.201800084"
}
```
Petar Mlinarić, Takayuki Ishizaki, Aranya Chakrabortty, Sara Grundel, Peter Benner, and Jun-ichi Imura. Synchronization and aggregation of nonlinear power systems with consideration of bus network structures. In European Control Conference (ECC), 2266–2271. November 2018. doi:10.23919/ECC.2018.8550528.
[abstract▼] [full text] [BibTeX▼]

We study nonlinear power systems consisting of generators, generator buses, and non-generator buses. First, looking at a generator and its bus' variables jointly, we introduce a synchronization concept for a pair of such joint generators and buses. We show that this concept is related to graph symmetry. Next, we extend, in two ways, the synchronization from a pair to a partition of all generators in the networks and show that they are related to either graph symmetry or equitable partitions. Finally, we show how an exact reduced model can be obtained by aggregating the generators and associated buses in the network when the original system is synchronized with respect to a partition, provided that the initial condition respects the partition. Additionally, the aggregation-based reduced model is again a power system.
```
@inproceedings{MliICetal18,
    author = "Mlinarić, Petar and Ishizaki, Takayuki and Chakrabortty, Aranya and Grundel, Sara and Benner, Peter and Imura, Jun-ichi",
    title = "Synchronization and Aggregation of Nonlinear Power Systems with Consideration of Bus Network Structures",
    booktitle = "European Control Conference (ECC)",
    year = "2018",
    month = "November",
    pages = "2266--2271",
    doi = "10.23919/ECC.2018.8550528"
}
```

2016

Peter Benner, Sara Grundel, and Petar Mlinarić. Stability preserving model reduction for linearly coupled linear time-invariant systems. Proc. Appl. Math. Mech., 16(1):817–818, October 2016. doi:10.1002/pamm.201610397.
[abstract▼] [BibTeX▼]

We develop a stability preserving model reduction method for linearly coupled linear time-invariant (LTI) systems. The method extends the work of Monshizadeh et al. for multi-agent systems with identical LTI agents. They propose using Bounded Real Balanced Truncation to preserve a sufficient condition for stability of the coupled system. Here, we extend this idea to arbitrary linearly coupled LTI systems using the sufficient condition for stability introduced by Reis and Stykel. The model reduction error bounds for this method also follow from results of Reis and Stykel, which allows the adaptive choice of reduced orders. We demonstrate the method on Reis's and Stykel's coupled string-beam example.
```
@article{BenGM16,
    author = "Benner, Peter and Grundel, Sara and Mlinarić, Petar",
    title = "Stability preserving model reduction for linearly coupled linear time-invariant systems",
    journal = "Proc. Appl. Math. Mech.",
    year = "2016",
    month = "October",
    volume = "16",
    number = "1",
    pages = "817--818",
    doi = "10.1002/pamm.201610397"
}
```
Petar Mlinarić, Sara Grundel, and Peter Benner. Clustering-based model order reduction for multi-agent systems with general linear time-invariant agents. In 22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS), 230–235. Minneapolis, MN, USA, July 2016. URL: http://hdl.handle.net/11299/181518.
[abstract▼] [full text] [BibTeX▼]

In this paper, we extend our clustering-based model order reduction method for multi-agent systems with single-integrator agents to the case where the agents have identical general linear time-invariant dynamics. The method consists of the Iterative Rational Krylov Algorithm, for finding a good reduced order model, and the QR decomposition-based clustering algorithm, to achieve structure preservation by clustering agents. Compared to the case of single-integrator agents, we modified the QR decomposition with column pivoting inside the clustering algorithm to take into account the block-column structure. We illustrate the method on small and large-scale examples.
```
@inproceedings{MliGB16,
    author = "Mlinarić, Petar and Grundel, Sara and Benner, Peter",
    title = "Clustering-Based Model Order Reduction for Multi-Agent Systems with General Linear Time-Invariant Agents",
    booktitle = "22nd International Symposium on Mathematical Theory of Networks and Systems (MTNS)",
    address = "Minneapolis, MN, USA",
    year = "2016",
    month = "July",
    pages = "230--235",
    URL = "http://hdl.handle.net/11299/181518"
}
```

2015

Petar Mlinarić, Sara Grundel, and Peter Benner. Efficient model order reduction for multi-agent systems using QR decomposition-based clustering. In 54th IEEE Conference on Decision and Control (CDC), 4794–4799. Osaka, Japan, December 2015. doi:10.1109/CDC.2015.7402967.
[abstract▼] [full text] [BibTeX▼]

In this paper we present an efficient model order reduction method for multi-agent systems with Laplacian-based dynamics. The method combines an established model order reduction method and a clustering algorithm to produce a graph partition used for reduction, thus preserving structure and consensus. By the Iterative Rational Krylov Algorithm, a good reduced order model can be found which is not necessarily structure preserving. However, based on this we can efficiently find a partition using the QR decomposition with column pivoting as a clustering algorithm, so that the structure can be restored. We illustrate the effectiveness on an example from the open literature.
```
@inproceedings{MliGB15,
    author = "Mlinarić, Petar and Grundel, Sara and Benner, Peter",
    title = "Efficient model order reduction for multi-agent systems using {QR} decomposition-based clustering",
    booktitle = "54th IEEE Conference on Decision and Control (CDC)",
    address = "Osaka, Japan",
    year = "2015",
    month = "December",
    pages = "4794--4799",
    doi = "10.1109/CDC.2015.7402967"
}
```

Book Chapters

2021

Peter Benner, Sara Grundel, and Petar Mlinarić. Clustering-based model order reduction for nonlinear network systems. In Peter Benner, Tobias Breiten, Heike Faßbender, Michael Hinze, Tatjana Stykel, and Ralf Zimmermann, editors, Model Reduction of Complex Dynamical Systems, volume 171 of International Series of Numerical Mathematics, pages 75–96. Birkhäuser, Cham, August 2021. doi:10.1007/978-3-030-72983-7_4.
[abstract▼] [full text] [code] [BibTeX▼]

Clustering by projection has been proposed as a way to preserve network structure in linear multi-agent systems. Here, we extend this approach to a class of nonlinear network systems. Additionally, we generalize our clustering method which restores the network structure in an arbitrary reduced-order model obtained by projection. We demonstrate this method on a number of examples.
```
@incollection{BenGM21,
    author = "Benner, Peter and Grundel, Sara and Mlinarić, Petar",
    editor = "Benner, Peter and Breiten, Tobias and Fa{\ss}bender, Heike and Hinze, Michael and Stykel, Tatjana and Zimmermann, Ralf",
    title = "Clustering-Based Model Order Reduction for Nonlinear Network Systems",
    booktitle = "Model Reduction of Complex Dynamical Systems",
    series = "International Series of Numerical Mathematics",
    publisher = {Birkh{\"a}user, Cham},
    year = "2021",
    month = "August",
    volume = "171",
    pages = "75--96",
    doi = "10.1007/978-3-030-72983-7\_4",
    customlinkcode = "https://zenodo.org/record/3924653"
}
```
Petar Mlinarić, Stephan Rave, and Jens Saak. Parametric model order reduction using pyMOR. In Peter Benner, Tobias Breiten, Heike Faßbender, Michael Hinze, Tatjana Stykel, and Ralf Zimmermann, editors, Model Reduction of Complex Dynamical Systems, volume 171 of International Series of Numerical Mathematics, pages 357–367. Birkhäuser, Cham, August 2021. doi:10.1007/978-3-030-72983-7_17.
[abstract▼] [full text] [code] [BibTeX▼]

pyMOR is a free software library for model order reduction that includes both reduced basis and system-theoretic methods. All methods are implemented in terms of abstract vector and operator interfaces, which allows a direct integration of pyMOR's algorithms with a wide array of external PDE solvers. In this contribution, we give a brief overview of the available methods and experimentally compare them for the parametric instationary thermal-block benchmark defined in [Rave, Saak: A non-stationary thermal-block benchmark model for parametric model order reduction, 2020].
```
@incollection{MliRS21,
    author = "Mlinarić, Petar and Rave, Stephan and Saak, Jens",
    editor = "Benner, Peter and Breiten, Tobias and Fa{\ss}bender, Heike and Hinze, Michael and Stykel, Tatjana and Zimmermann, Ralf",
    title = "Parametric model order reduction using {pyMOR}",
    booktitle = "Model Reduction of Complex Dynamical Systems",
    series = "International Series of Numerical Mathematics",
    publisher = {Birkh{\"a}user, Cham},
    year = "2021",
    month = "August",
    volume = "171",
    pages = "357--367",
    doi = "10.1007/978-3-030-72983-7\_17",
    customlinkcode = "https://zenodo.org/record/3928528"
}
```

Thesis

2020

Petar Mlinarić. Structure-preserving model order reduction for network systems. Dissertation, Otto von Guericke University, Magdeburg, Germany, 2020. doi:10.25673/33570.
[abstract▼] [BibTeX▼]

This thesis considers structure-preserving system-theoretic model order reduction for certain structured input-output systems, particularly network systems. In the first part, our focus lies on the clustering-based approach to reduce network systems. Therein, we begin by considering clustering-based model order reduction for linear multi-agent systems. This approach finds a reduced model whose dynamics evolve over a smaller network. To measure the reduction error, we use the \(\mathcal H_2\)-norm and consider the \(\mathcal H_2\)-optimal clustering problem. Since clustering is generally a difficult combinatorial problem, we propose a framework based on relaxing the discrete problem to find an \(\mathcal H_2\)-suboptimal clustering. Following on this, we directly extend the framework to a class of nonlinear multi-agent systems. Next, we derive upper bounds for \(\mathcal H_2\) and \(\mathcal H_\infty\) clustering-based reduction errors for linear multi-agent systems based on almost equitable partitions. These results generalize work for multi-agent systems with single-integrator agents. Using a similar approach for power systems, which are a special class of nonlinear multi-agent systems, we find conditions for exact clustering-based reduction. Additionally, we study subsystem reduction for network systems. We propose a balancing-based approach that guarantees stability preservation under a small-gain condition. Furthermore, we consider the \(\mathcal H_2\)-optimal subsystem reduction problem. We derive Gramian-based first-order necessary \(\mathcal H_2\)-optimality conditions and use a gradient-based optimization method to fulfill them. Finally, we apply the structure-preserving \(\mathcal H_2\)-optimal model reduction approach for network systems to other structured systems. In particular, we consider \(\mathcal H_2\)-optimal model order reduction of second-order systems, port-Hamiltonian systems, time-delay systems and \(\mathcal H_2 \otimes \mathcal L_2\)-optimal model order reduction of parametric systems. Here, we also derive Gramian-based \(\mathcal H_2\)-optimality conditions and use an optimization approach to construct a reduced model. For some structured systems, we also derive interpolatory \(\mathcal H_2\)-optimality conditions under additional assumptions on the reduced model.
```
@phdthesis{Mli20,
    author = "Mlinarić, Petar",
    title = "Structure-preserving model order reduction for network systems",
    school = "Otto von Guericke University",
    address = "Magdeburg, Germany",
    year = "2020",
    doi = "10.25673/33570",
    type = "{D}issertation"
}
```