arXiv subjects

Björn Liljegren-Sailer

Publications and source records attributed to Björn Liljegren-Sailer.

11 recordsLinked to original sources

arXiv2026

Reducing Training Complexity in Empirical Quadrature-Based Model Reduction via Structured Compression

Model order reduction seeks to approximate large-scale dynamical systems by lower-dimensional reduced models. For linear systems, a small reduced dimension directly translates into low computational cost, ensuring online efficiency. This property does not generally hold for nonlinear systems, where an additional approximation of nonlinear terms --known as complexity reduction-- is required. To achieve online efficiency, empirical quadrature and cell-based empirical cubature are among the most effective complexity reduction techniques. However, the offline training of these methods operates on a matrix whose dimension scales with both the snapshot count and the reduced model dimension, and can become a computational bottleneck at larger scale. Existing strategies such as parallelization and randomized linear algebra reduce the cost of processing this matrix but do not reduce its dimension. In this paper, we introduce a preprocessing approach based on a specific structured compression of the training data. Crucially, our approach ensures that no operation scales concurrently with the snapshot count, the reduced model dimension, and the problem dimension. Overall, this yields roughly an order-of-magnitude reduction in offline computational cost and memory requirements, thereby enabling the application of the complexity reduction methods to larger-scale problems. Accuracy is preserved, as indicated by our error analysis and demonstrated through numerical examples.

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arXiv2023

Operator splitting for port-Hamiltonian systems

The port-Hamiltonian approach presents an energy-based modeling of dynamical systems with energy-conservative and energy-dissipative parts as well as an interconnection over the so-called ports. In this paper, we apply an operator splitting that treats the energy-conservative and energy-dissipative parts separately. This paves the way for linear equation solvers to exploit the respective special structures of the iteration matrices as well as the multirate potential in the different right-hand sides. We illustrate the approach using test examples from coupled multibody system dynamics.

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arXiv2023

Data-driven and low-rank implementations of Balanced Singular Perturbation Approximation

Balanced Singular Perturbation Approximation (SPA) is a model order reduction method for linear time-invariant systems that guarantees asymptotic stability and for which there exists an a priori error bound. In that respect, it is similar to Balanced Truncation (BT). However, the reduced models obtained by SPA generally introduce better approximation in the lower frequency range and near steady-states, whereas BT is better suited for the higher frequency range. Even so, independently of the frequency range of interest, BT and its variants are more often applied in practice, since there exist more efficient algorithmic realizations thereof. In this paper, we aim at closing this practically-relevant gap for SPA. We propose two novel and efficient algorithms that are adapted for different settings. Firstly, we derive a low-rank implementation of SPA that is applicable in the large-scale setting. Secondly, a data-driven reinterpretation of the method is proposed that only requires input-output data, and thus, is realization-free. A main tool for our derivations is the reciprocal transformation, which induces a distinct view on implementing the method. While the reciprocal transformation and the characterization of SPA is not new, its significance for the practical algorithmic realization has been overlooked in the literature. Our proposed algorithms have well-established counterparts for BT, and as such, also a comparable computational complexity. The numerical performance of the two novel implementations is tested for several numerical benchmarks, and comparisons to their counterparts for BT as well as the existing implementations of SPA are made.

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arXiv2023

A time domain a posteriori error bound for balancing-related model order reduction

The aim in model order reduction is to approximate an input-output map described by a large-scale dynamical system with a low-dimensional and cheaper-to-evaluate reduced order model. While high fidelity can be achieved by a variety of methods, only a few of them allow for rigorous error control. In this paper, we propose a rigorous error bound for the reduction of linear systems with balancing-related reduction methods. More specifically, we consider the simulation over a finite time interval and provide an a posteriori adaption of the standard a priori bound for Balanced Truncation and Balanced Singular Perturbation Approximation in that setting, which improves the error estimation while still yielding a rigorous bound. Our result is based on an error splitting induced by a Fourier series approximation of the input and a subsequent refined error analysis. We make use of system-theoretic concepts, such as the notion of signal generator driven systems, steady-states and observability. Our bound is also applicable in the presence of nonzero initial conditions. Numerical evidence for the sharpness of the bound is given.

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arXiv2022

Certified Reduced Basis Method for the Damped Wave Equations on Networks

In this paper we present a reduced basis method which yields structure-preservation and a tight a posteriori error bound for the simulation of the damped wave equations on networks. The error bound is based on the exponential decay of the energy inside the system and therefore allows for sharp bounds without the need of regularization parameters. The fast convergence of the reduced solution to the truth solution as well as the tightness of the error bound are verified numerically using an academic network as example.

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arXiv2022

On port-Hamiltonian approximation of a nonlinear flow problem on networks

This paper deals with the systematic development of structure-preserving approximations for a class of nonlinear partial differential equations on networks. The class includes, for example, gas pipe network systems described by barotropic Euler equations. Our approach is guided throughout by energy-based modeling concepts (port-Hamiltonian formalism, theory of Legendre transformation), which provide a convenient and general line of reasoning. Under mild assumptions on the approximation, local conservation of mass, an energy bound, and the inheritance of the port-Hamiltonian structure can be shown. Our approach is not limited to conventional space discretization but also covers complexity reduction of the nonlinearities by inexact integration. Thus, it can serve as a basis for structure-preserving model reduction. Combined with an energy stable time integration, we numerically demonstrate the applicability and good stability properties of the approach using the Euler equations as an example.

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arXiv2022

Effective error estimation for model reduction with inhomogeneous initial conditions

A priori error bounds have been derived for different balancing-related model reduction methods. The most classical result is a bound for balanced truncation and singular perturbation approximation that is applicable for asymptotically stable linear time-invariant systems with homogeneous initial conditions. Recently, there have been a few attempts to generalize the balancing-related reduction methods to the case with inhomogeneous initial conditions, but the existing error bounds for these generalizations are quite restrictive. Particularly, it is required to restrict the initial conditions to a low-dimensional subspace, which has to be chosen before the reduced model is constructed. In this paper, we propose an estimator that circumvents this hard constraint completely. Our estimator is applicable to a large class of reduction methods, whereas the former results were only derived for certain specific methods. Moreover, our approach yields to significantly more effective error estimation, as also will be demonstrated numerically.

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arXiv2021

On snapshot-based model reduction under compatibility conditions for a nonlinear flow problem on networks

This paper is on the construction of structure-preserving, online-efficient reduced models for the barotropic Euler equations with a friction term on networks. The nonlinear flow problem finds broad application in the context of gas distribution networks. We propose a snapshot-based reduction approach that consists of a mixed variational Galerkin approximation combined with quadrature-type complexity reduction. Its main feature is that certain compatibility conditions are assured during the training phase, which make our approach structure-preserving. The resulting reduced models are locally mass conservative and inherit an energy-bound and port-Hamiltonian structure. We also derive a well-posedness result for them. In the training phase, the compatibility conditions pose challenges, we face constrained data approximation problems as opposed to the unconstrained training problems in the conventional reduction methods. The training of our model order reduction consists of a principal component analysis under a compatibility constraint and, notably, yields reduced models that fulfill an optimality condition for the snapshot data. The training of our quadrature-type complexity reduction involves a semi-definite program with combinatorial aspects, which we approach by a greedy procedure.

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arXiv2021

Input-tailored system-theoretic model order reduction for quadratic-bilinear systems

In this paper we suggest a moment matching method for quadratic-bilinear dynamical systems. Most system-theoretic reduction methods for nonlinear systems rely on multivariate frequency representations. Our approach instead uses univariate frequency representations tailored towards user-pre-defined families of inputs. Then moment matching corresponds to a one-dimensional interpolation problem, not to multi-dimensional interpolation as for the multivariate approaches, i.e., it also involves fewer interpolation frequencies to be chosen. Comparing to former contributions towards nonlinear model reduction with univariate frequency representations, our approach shows profound differences: Our derivation is more rigorous and general and reveals additional tensor-structured approximation conditions, which should be incorporated. Moreover, the proposed implementation exploits the inherent low-rank tensor structure, which enhances its efficiency. In addition, our approach allows for the incorporation of more general input relations in the state equations - not only affine-linear ones as in existing system-theoretic methods - in an elegant way. As a byproduct of the latter, also a novel modification for the multivariate methods falls off, which is able to handle more general input-relations.

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arXiv2018

Stability preserving approximations of a semilinear hyperbolic gas transport model

We consider the discretization of a semilinear damped wave equation arising, for instance, in the modeling of gas transport in pipeline networks. For time invariant boundary data, the solutions of the problem are shown to converge exponentially fast to steady states. We further prove that this decay behavior is inherited uniformly by a class of Galerkin approximations, including finite element, spectral, and structure preserving model reduction methods. These theoretical findings are illustrated by numerical tests.

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arXiv2017

On structure-preserving model reduction for damped wave propagation in transport networks

We consider the discretization and subsequent model reduction of a system of partial differential-algebraic equations describing the propagation of pressure waves in a pipeline network. Important properties like conservation of mass, dissipation of energy, passivity, existence of steady states, and exponential stability can be preserved by an appropriate semi-discretization in space via a mixed finite element method and also during the further dimension reduction by structure preserving Galerkin projection which is the main focus of this paper. Krylov subspace methods are employed for the construciton of the reduced models and we discuss modifications needed to satisfy certain algebraic compatibility conditions; these are required to ensure the well-posedness of the reduced models and the preservation of the key properties. Our analysis is based on the underlying infinite dimensional problem and its Galerkin approximations. The proposed algorithms therefore have a direct interpretation in function spaces; in principle, they are even applicable directly to the original system of partial differential-algebraic equations while the intermediate discretization by finite elements is only required for the actual computations. The performance of the proposed methods is illustrated with numerical tests and the necessity for the compatibility conditions is demonstrated by examples.

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