arXiv subjects

Petar Mlinarić

Publications and source records attributed to Petar Mlinarić.

13 recordsLinked to original sources

arXiv2026

Data-Driven Model Order Reduction with pyMOR

pyMOR is a free and open-source software library of model order reduction algorithms for the Python programming language. Designed with classical model-based reduction methods for large-scale parametric partial differential equation problems in mind, algorithms in pyMOR are implemented in terms of operations on abstract VectorArray, Operator and Model interfaces, allowing for a seamless integration with external solver codes implementing the full-order model. For cases where a tight integration with the full-order model code is not feasible, data-driven model order reduction algorithms, which only require simulation or measurement data of the full-order model, are an attractive alternative. In this work we discuss the data-driven methods that have been recently added to pyMOR, show practical examples of their application using pyMOR and compare their performance with classical model-based methods. We show that pyMOR serves as a unified framework for combining model-based and data-driven methods, enabling the construction of flexible and efficient hierarchical model reduction pipelines.

cs.MS↗

arXiv2026

The tangent cone to the real determinantal variety: various expressions and a proof

The set of real matrices of upper-bounded rank is a real algebraic variety called the real generic determinantal variety. An explicit description of the tangent cone to that variety is given in Theorem 3.2 of Schneider and Uschmajew [SIAM J. Optim., 25 (2015), pp. 622-646]. The present paper shows that the proof therein is incomplete and provides a proof. It also reviews equivalent descriptions of the tangent cone to that variety. Moreover, it shows that the tangent cone and the algebraic tangent cone to that variety coincide, which is not true for all real algebraic varieties.

math.OC↗

arXiv2025

Optimal Damping for the 1D Wave Equation Using a Single Damper

Vibrational structures are susceptible to catastrophic failures or structural damages when external forces induce resonances or repeated unwanted oscillations. One common mitigation strategy is to use dampers to suppress these disturbances. This leads to the problem of finding optimal damper viscosities and positions for a given vibrational structure. Although extensive research exists for the case of finite-dimensional systems, optimizing damper positions remains challenging due to its discrete nature. To overcome this, we introduce a novel model for the damped wave equation (at the PDE level) with a damper of viscosity $\mathfrak{g}$ at position $\mathfrak{p}$ and develop a system-theoretic input/output-based analysis in the frequency domain. In this system-theoretic formulation, while we consider average displacement as the output, for input (forcing), we analyze two separate cases, namely, the uniform and boundary forcing. For both cases, explicit formulas are derived for the corresponding transfer functions, parametrized by $\mathfrak{p}$ and $\mathfrak{g}$. This explicit parametrization by $\mathfrak{p}$ and $\mathfrak{g}$ facilitates analyzing the optimal damping problem (at the PDE level) using norms such as the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ norms. We also examine limiting cases, such as when the viscosity is very large or when no external damping is present. To illustrate our approach, we present numerical examples, compare different optimization criteria, and discuss the impact of damping parameters on the damped wave equation.

math.OC↗

arXiv2024

Interpolatory Necessary Optimality Conditions for Reduced-order Modeling of Parametric Linear Time-invariant Systems

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.

math.OC↗

arXiv2024

Interpolatory $\mathcal{H}_2$-optimality Conditions for Structured Linear Time-invariant Systems

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. Under certain diagonalizability assumptions, 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.

math.NA↗

arXiv2024

IRKA is a Riemannian Gradient Descent Method

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.

math.NA↗

arXiv2023

A Unifying Framework for Interpolatory $\mathcal{L}_2$-optimal Reduced-order Modeling

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 $\mathcal{H}_2$-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.

math.NA↗

arXiv2022

$\mathcal{L}_2$-optimal Reduced-order Modeling Using Parameter-separable Forms

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.

math.NA↗

arXiv2022

Optimization-based parametric model order reduction via $\mathcal{H}_2\otimes\mathcal{L}_2$ first-order necessary conditions

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 ptimality 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.

math.OC↗

arXiv2020

Parametric model order reduction using pyMOR

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 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 arXiv:2003.00846.

cs.MS↗

arXiv2020

Clustering-Based Model Order Reduction for Nonlinear Network Systems

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.

math.NA↗

arXiv2018

Synchronization and Aggregation of Nonlinear Power Systems with Consideration of Bus Network Structures

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.

eess.SY↗

arXiv2016

Model Reduction of Linear Multi-Agent Systems by Clustering and Associated $\mathcal{H}_2$- and $\mathcal{H}_\infty$-Error Bounds

In this paper, we study a model reduction technique for leader-follower networked multi-agent systems defined on weighted, undirected graphs with arbitrary linear multivariable agent dynamics. In the network graph of this network, nodes represent the agents and edges represent communication links between the agents. Only the leaders in the network receive an external input, the followers only exchange information with their neighbors. The reduced network is obtained by partitioning the set of nodes into disjoint sets, called clusters, and associating with each cluster a single, new, node in a reduced network graph. The resulting reduced network has a weighted, symmetric, directed network graph, and inherits some of the structure of the original network. We establish a priori upper bounds on the $\mathcal{H}_2$ and $\mathcal{H}_\infty$ model reduction error for the special case that the graph partition is almost equitable. These upper bounds depend on the Laplacian eigenvalues of the original and reduced network, an auxiliary system associated with the agent dynamics, and the number of nodes that belong to the same clusters as the leaders in the network. Finally, we consider the problem of obtaining a priori upper bounds if we cluster using arbitrary, possibly non almost equitable, partitions.

math.OC↗