arXiv subjects

Jens Saak

Publications and source records attributed to Jens Saak.

At least 19 recordsLinked to original sources

arXiv2026

On the Solution of Large-scale Non-autonomous Differential Riccati Equations: a Numerical Study

We explore the numerical solution of large-scale non-autonomous Differential Riccati Equations (DREs). While we assume to discretize the differential operator using a Backward Differentiation Formula (BDF) of order s, we solve the generalized Algebraic Riccati Equation (gARE) resulting at each time step by different state-of-the-art methods. In particular, we compare the performance of the inexact Newton- Kleinman method with line search and the low-rank RADI iteration, considering for both methods two different initialization strategies: zero initialization and warm-start. A comprehensive panel of numerical results illustrate the potential and limitations of these methods when employed within a numerical pipeline for the solution of DREs, rather than for the isolated solution of a single gARE, as commonly considered in the existing literature.

math.NA↗

arXiv2026

Reduced rank extrapolation for multi-term Sylvester equations

We investigate the acceleration of stationary iterations for multi-term Sylvester equation by means of reduced rank extrapolation (RRE). Theoretical convergence results and implementations are provided for both small and large-scale problems. For the large-scale problems, an inexact non-stationary iteration is discussed, which makes use of low-rank matrix approximations. Numerical experiments illustrate the potential of the RRE acceleration which often leads to a substantial gain in convergence speed and therefore reducing the consumption of storage and computing time.

math.NA↗

arXiv2025

Towards a mixed-precision ADI method for Lyapunov equations

We apply mixed-precision to the low-rank Lyapunov ADI (LR-ADI) by performing certain aspects of the algorithm in a lower working precision. Namely, we accumulate the overall solution, solve the linear systems comprising the ADI iteration, and store the inner low-rank factors of the residuals in various combinations of IEEE 754 single and double precision. We empirically test our implementation on Lyapunov equations arising from first- and second-order descriptor systems. For the first-order examples, accumulating the solution in single-precision yields an almost-as-small residual as for the double-precision solution. For certain applications, like computing the H2 norm of a descriptor system, low- or mixed-precision variants of the ADI can be quite competitive

math.NA↗

arXiv2025

Towards an Efficient Shifted Cholesky QR for Applications in Model Order Reduction using pyMOR

Many model order reduction (MOR) methods rely on the computation of an orthonormal basis of a subspace onto which the large full order model is projected. Numerically, this entails the orthogonalization of a set of vectors. The nature of the MOR process imposes several requirements for the orthogonalization process. Firstly, MOR is oftentimes performed in an adaptive or iterative manner, where the quality of the reduced order model, i.e., the dimension of the reduced subspace, is decided on the fly. Therefore, it is important that the orthogonalization routine can be executed iteratively. Secondly, one possibly has to deal with high-dimensional arrays of abstract vectors that do not allow explicit access to entries, making it difficult to employ so-called `orthogonal triangularization algorithms' such as Householder QR. For these reasons, (modified) Gram-Schmidt-type algorithms are commonly used in MOR applications. These methods belong to the category of `triangular orthogonalization' algorithms that do not rely on elementwise access to the vectors and can be easily updated. Recently, algorithms like shifted Cholesky QR have gained attention. These also belong to the aforementioned category and have proven their aptitude for MOR algorithms in previous studies. A key benefit of these methods is that they are communication-avoiding, leading to vastly superior performance on memory-bandwidth-limited problems and parallel or distributed architectures. This work formulates an efficient updating scheme for Cholesky QR algorithms and proposes an improved shifting strategy for highly ill-conditioned matrices. The proposed algorithmic extensions are validated with numerical experiments on a laptop and computation server.

math.NA↗

arXiv2025

Application of operator inference to reduced-order modeling of constrained mechanical systems

Constrained mechanical systems occur in many applications, such as modeling of robots and other multibody systems. In this case, the motion is governed by a system of differential-algebraic equations (DAE), often with large and sparse system matrices. The problem dimension strongly influences the effectiveness of simulations for system analysis, optimization, and control, given limited computational resources. Therefore, we aim to obtain a simplified surrogate model with a few degrees of freedom that is able to accurately represent the motion and other important properties of the original high-dimensional DAE model. Classical model reduction methods intrusively exploit the system matrices to construct the projection of the high-fidelity model onto a low-dimensional subspace. In practice, the dynamical equations are frequently an inaccessible part of proprietary software products. In this work, we show an application of the non-intrusive operator inference (OpInf) method to DAE systems of index 2 and 3. Considering the fact that for proper DAEs there exists an ODE realization on the so-called hidden manifold, the OpInf optimization problem directly provides the underlying ODE representation of the given DAE system in the reduced subspace. A significant advantage is that only the DAE solution snapshots in a compressed form are required for identification of the reduced system matrices. Stability and interpretability of the reduced-order model is guaranteed by enforcing the symmetric positive definite structure of the system operators using semidefinite programming. The numerical results demonstrate the implementation of the proposed methodology for different examples of constrained mechanical systems, tested for various loading conditions.

math.DS↗

arXiv2025

Generalizing Reduced Rank Extrapolation to Low-Rank Matrix Sequences

Reduced rank extrapolation (RRE) is an acceleration method typically used to accelerate the iterative solution of nonlinear systems of equations using a fixed-point process. In this context, the iterates are vectors generated from a fixed-point mapping function. However, when considering the iterative solution of large-scale matrix equations, the iterates are low-rank matrices generated from a fixed-point process for which, generally, the mapping function changes in each iteration. To enable acceleration of the iterative solution for these problems, we propose two novel generalizations of RRE. First, we show how to effectively compute RRE for sequences of low-rank matrices. Second, we derive a formulation of RRE that is suitable for fixed-point processes for which the mapping function changes each iteration. We demonstrate the potential of the methods on several numerical examples involving the iterative solution of large-scale Lyapunov and Riccati matrix equations.

math.NA↗

arXiv2025

A unifying framework for ADI-like methods for linear matrix equations and beneficial consequences

We derive the alternating-directions implicit (ADI) method based on a commuting operator split and apply the results in detail to the continuous time algebraic Lyapunov equation with low-rank constant term and approximate solution, giving pointers for the Sylvester case. Previously, it has been mandatory to start the low-rank ADI for Lyapunov equations (CF-ADI, LR-ADI, G-LR-ADI) or Sylvester equations (fADI, G-fADI) with an all-zero initial value. Our approach extends the known efficient iteration schemes of low-rank increments and residuals to arbitrary low-rank initial values for all these methods. We further generalize two properties of the low-rank Lyapunov ADI to the generic ADI applied to arbitrary linear equations using a commuting operator split, namely the invariance of iterates under permutations of the shift parameters, and the efficient handling of complex shift parameters. We investigate the performance of arbitrary initial values using two outer iterations in which the low-rank Lyapunov ADI is typically called. First, we solve an algebraic Riccati equation with the Newton method. Second, we solve a differential Riccati equation with a first-order Rosenbrock method. Numerical experiments confirm that the proposed new initial value of the ADI can lead to a significant reduction in the total number of ADI steps, while also showing a 17% and 8x speed-up over the zero initial value for the two equation types, respectively.

math.NA↗

arXiv2024

Using $LDL^{T}$ factorizations in Newton's method for solving general large-scale algebraic Riccati equations

Continuous-time algebraic Riccati equations can be found in many disciplines in different forms. In the case of small-scale dense coefficient matrices, stabilizing solutions can be computed to all possible formulations of the Riccati equation. This is not the case when it comes to large-scale sparse coefficient matrices. In this paper, we provide a reformulation of the Newton-Kleinman iteration scheme for continuous-time algebraic Riccati equations using indefinite symmetric low-rank factorizations. This allows the application of the method to the case of general large-scale sparse coefficient matrices. We provide convergence results for several prominent realizations of the equation and show in numerical examples the effectiveness of the approach.

math.NA↗

arXiv2024

Efficient solution of sequences of parametrized Lyapunov equations with applications

Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of $\text{trace}(EX)$ where $X$ denotes the solution of a Lyapunov equation and $E$ is a given matrix. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a \emph{seed}, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov techniques, and it is well-suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems. Our results show that the algorithms we propose are superior to state-of-the-art techniques as they are able to remarkably speed up the computation of accurate solutions.

math.NA↗

arXiv2023

Diagonally-Addressed Matrix Nicknack: How to improve SpMV performance

We suggest a technique to reduce the storage size of sparse matrices at no loss of information. We call this technique Diagonally-Adressed (DA) storage. It exploits the typically low matrix bandwidth of matrices arising in applications. For memory-bound algorithms, this traffic reduction has direct benefits for both uni-precision and multi-precision algorithms. In particular, we demonstrate how to apply DA storage to the Compressed Sparse Rows (CSR) format and compare the performance in computing the Sparse Matrix Vector (SpMV) product, which is a basic building block of many iterative algorithms. We investigate 1367 matrices from the SuiteSparse Matrix Collection fitting into the CSR format using signed 32 bit indices. More than 95% of these matrices fit into the DA-CSR format using 16 bit column indices, potentially after Reverse Cuthill-McKee (RCM) reordering. Using IEEE 754 double precision scalars, we observe a performance uplift of 11% (single-threaded) or 17.5% (multithreaded) on average when the traffic exceeds the size of the last-level CPU cache. The predicted uplift in this scenario is 20%. For traffic within the CPU's combined level 2 and level 3 caches, the multithreaded performance uplift is over 40% for a few test matrices.

math.NA↗

arXiv2023

Towards a Benchmark Framework for Model Order Reduction in the Mathematical Research Data Initiative (MaRDI)

The race for the most efficient, accurate, and universal algorithm in scientific computing drives innovation. At the same time, this healthy competition is only beneficial if the research output is actually comparable to prior results. Fairly comparing algorithms can be a complex endeavor, as the implementation, configuration, compute environment, and test problems need to be well-defined. Due to the increase in computer-based experiments, new infrastructure for facilitating the exchange and comparison of new algorithms is also needed. To this end, we propose a benchmark framework, as a set of generic specifications for comparing implementations of algorithms using test cases native to a community. Its value lies in its ability to fairly compare and validate existing methods for new applications, as well as compare newly developed methods with existing ones. As a prototype for a more general framework, we have begun building a benchmark tool for the model order reduction (MOR) community. The data basis of the tool is the collection of the Model Order Reduction Wiki (MORWiki). The wiki features three main categories: benchmarks, methods, and software. An editorial board curates submissions and patrols edited entries. Data sets for linear and parametric-linear models are already well represented in the existing collection. Data sets for non-linear or procedural models, for which only evaluation data, or codes / algorithmic descriptions, rather than equations, are available, are being added and extended. Properties and interesting characteristics used for benchmark selection and later assessments are recorded in the model metadata. Our tool, the Model Order Reduction Benchmarker (MORB) is under active development for linear time-invariant systems and solvers.

cs.MS↗

arXiv2022

Riccati-feedback Control of a Two-dimensional Two-phase Stefan Problem

We discuss the feedback control problem for a two-dimensional two-phase Stefan problem. In our approach, we use a sharp interface representation in combination with mesh-movement to track the interface position. To attain a feedback control, we apply the linear-quadratic regulator approach to a suitable linearization of the problem. We address details regarding the discretization and the interface representation therein. Further, we document the matrix assembly to generate a non-autonomous generalized differential Riccati equation. To numerically solve the Riccati equation, we use low-rank factored and matrix-valued versions of the non-autonomous backward differentiation formulas, which incorporate implicit index reduction techniques. For the numerical simulation of the feedback controlled Stefan problem, we use a time-adaptive fractional-step-theta scheme. We provide the implementations for the developed methods and test these in several numerical experiments. With these experiments we show that our feedback control approach is applicable to the Stefan control problem and makes this large-scale problem computable. Also, we discuss the influence of several controller design parameters, such as the choice of inputs and outputs.

math.NA↗

arXiv2022

Research-Data Management Planning in the German Mathematical Community

In this paper we discuss the notion of research data for the field of mathematics and report on the status quo of research-data management and planning. A number of decentralized approaches are presented and compared to needs and challenges faced in three use cases from different mathematical subdisciplines. We highlight the importance of tailoring research-data management plans to mathematicians' research processes and discuss their usage all along the data life cycle.

math.HO↗

arXiv2022

On an integrated Krylov-ADI solver for large-scale Lyapunov equations

One of the most computationally expensive steps of the low-rank ADI method for large-scale Lyapunov equations is the solution of a shifted linear system at each iteration. We propose the use of the extended Krylov subspace method for this task. In particular, we illustrate how a single approximation space can be constructed to solve all the shifted linear systems needed to achieve a prescribed accuracy in terms of Lyapunov residual norm. Moreover, we show how to fully merge the two iterative procedures in order to obtain a novel, efficient implementation of the low-rank ADI method, for an important class of equations. Many state-of-the-art algorithms for the shift computation can be easily incorporated into our new scheme, as well. Several numerical results illustrate the potential of our novel procedure when compared to an implementation of the low-rank ADI method based on sparse direct solvers for the shifted linear systems.

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

Sustainable Research Software Hand-Over

Scientific software projects evolve rapidly in their initial development phase, yet at the end of a funding period, the completion of a research project, thesis, or publication, further engagement in the project may slow down or cease completely. To retain the invested effort for the sciences, this software needs to be preserved or handed over to a succeeding developer or team, such as the next generation of (PhD) students. Comparable guides provide top-down recommendations for project leads. This paper intends to be a bottom-up approach for sustainable hand-over processes from a developer's perspective. An important characteristic in this regard is the project's size, by which this guideline is structured. Furthermore, checklists are provided, which can serve as a practical guide for implementing the proposed measures.

cs.GL↗

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

Matrix Equations, Sparse Solvers: M-M.E.S.S.-2.0.1 -- Philosophy, Features and Application for (Parametric) Model

Matrix equations are omnipresent in (numerical) linear algebra and systems theory. Especially in model order reduction (MOR) they play a key role in many balancing based reduction methods for linear dynamical systems. When these systems arise from spatial discretizations of evolutionary partial differential equations, their coefficient matrices are typically large and sparse. Moreover, the numbers of inputs and outputs of these systems are typically far smaller than the number of spatial degrees of freedom. Then, in many situations the solutions of the corresponding large-scale matrix equations are observed to have low (numerical) rank. This feature is exploited by M-M.E.S.S. to find successively larger low-rank factorizations approximating the solutions. This contribution describes the basic philosophy behind the implementation and the features of the package, as well as its application in the model order reduction of large-scale linear time-invariant (LTI) systems and parametric LTI systems.

cs.MS↗