Article
Mathematics, Applied
A. H. Bentbib, M. El Ghomari, K. Jbilou
Summary: This research proposes a method using the global extended-rational Arnoldi method for computing approximations of matrix functions. The method projects the problem onto a low-dimensional subspace to solve large and sparse matrix computations and provides an adaptive process for obtaining shifts. Numerical examples demonstrate the good performance of the method in solving time-dependent PDEs and network analysis.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Divya Aggarwal, Samrith Ram
Summary: This paper discusses the problem of T-splitting subspaces for a linear operator T, which involves finding subspaces of a vector space V that can be expressed as the direct sum of a subspace W and the powers of T. The paper also explores the connection between this problem and Krylov spaces, as well as polynomial matrices.
FINITE FIELDS AND THEIR APPLICATIONS
(2022)
Article
Mathematics, Applied
M. A. Botchev
Summary: This paper proposes a coarse grid correction (CGC) approach to enhance the efficiency of matrix exponential and phi matrix function evaluations. The approach splits the computation into smooth and remaining parts, and handles them on different grids with relaxed stopping criterion tolerance. The CGC algorithm is validated through numerical experiments when used in combination with Krylov subspace and Chebyshev polynomial expansion methods.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Mike A. Botchev, Leonid Knizhnerman, Eugene E. Tyrtyshnikov
Summary: An efficient Krylov subspace algorithm is proposed for computing actions of the \varphi matrix function for large matrices. The algorithm is based on a reliable residual-based stopping criterion and a new efficient restarting procedure. Numerical tests demonstrate the efficiency of the approach for solving large scale evolution problems resulting from discretized time-dependent PDEs, particularly diffusion and convection-diffusion problems.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Christian Schroeder, Matthias Voigt
Summary: In standard balanced truncation model order reduction, the initial condition is typically ignored, but the proposed balancing procedure based on state shift transformation can yield a better reduced-order model with a priori error bound. Additionally, the paper discusses the construction of reduced-order models and the efficient optimization of error bounds.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Bernhard Beckermann, Alice Cortinovis, Daniel Kressner, Marcel Schweitzer
Summary: This work develops novel rational Krylov methods for updating a large-scale matrix function f(A) subject to low-rank modifications. The analysis shows the usefulness of the derived error bounds for guiding the choice of poles in the rational Krylov method. Additionally, a connection between low-rank updates of the matrix sign function and existing rational Krylov subspace methods is pointed out.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Ines Dorschky, Timo Reis, Matthias Voigt
Summary: We introduce a model reduction approach for linear time-invariant second order systems based on positive real balanced truncation. Our method guarantees to preserve asymptotic stability and passivity of the reduced order model as well as the positive definiteness of the mass and stiffness matrices. Moreover, we receive an a priori gap metric error bound. Finally we show that our method based on positive real balanced truncation preserves the structure of overdamped second order systems.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2021)
Article
Engineering, Multidisciplinary
Dionysios Panagiotopoulos, Wim Desmet, Elke Deckers
Summary: In this work, a method is proposed to accelerate the solution of multiresolution analyses of affine parametric systems by decoupling the system solution from the construction of the recycled subspace. The method follows the logic of projection-based Model Order Reduction and proves particularly beneficial for affine parametrizations involving multiple affine coefficients.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
F. S. Naranjo-Noda, J. C. Jimenez
Summary: This paper discusses the numerical computation of high dimensional multiple integrals involving matrix exponentials that can be rewritten as the product of a matrix exponential times vector. In addition to the conventional iterative methods for computing the action of the matrix exponential on a vector, iterative methods for the action of the phi-function over a vector are also considered. A Krylov-Pade approximation to the product of phi-function times vector is used to illustrate the potential for computing various high dimensional multiple integrals in applied mathematics, model identification, control engineering, and numerical methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Automation & Control Systems
Liu Dai, Zhi-Hua Xiao, Ren-Zheng Zhang, Yao-Lin Jiang
Summary: This article presents a new structure-preserving model order reduction technique based on Laguerre-Gramian for second-order systems. The proposed approach utilizes Laguerre polynomial expansion to obtain approximate low-rank decomposition of the Gramians, generates balanced systems, and constructs reduced second-order models by truncating states with small Hankel singular values. The method may lead to unstable systems, which is addressed by a modified reduction procedure and dominant subspace projection method.
TRANSACTIONS OF THE INSTITUTE OF MEASUREMENT AND CONTROL
(2022)
Article
Mathematics, Applied
M. A. Hamadi, K. Jbilou, A. Ratnani
Summary: This paper presents a data-driven approach using the Loewner framework to tackle the challenges of non-linear dynamical systems and the large amount of data. It proposes a method to compute the singular value decomposition accurately and efficiently by exploiting certain matrix equations. The accuracy and efficiency of the method are evaluated in the final section.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Luca Gemignani
Summary: This paper discusses efficient numerical methods for solving a linear system φ(A)x = b, where φ(z) is a φ-function and A is a RNxN matrix. Specifically, the computation of φ(A)-1b is of interest, with φ(z) = φ1(z) = ez-1 z ez-1 - z and φ(z) = φ2(z) = z2 being considered. Fast algorithms based on Newton's iteration and Krylov-type methods are designed for computing both φ⠃(A)-1 and φ⠃(A)-1b, with adaptations for structured matrices, such as banded and quasiseparable matrices, also studied. Numerical results are presented to demonstrate the effectiveness of the proposed algorithms.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
Peter Benner, Steffen W. R. Werner
Summary: This paper discusses the extension of frequency- and time-limited balanced truncation methods to second-order dynamical systems for practical applications. Numerical methods and modifications for large-scale sparse matrix equations are presented, along with three numerical examples for illustration.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Optics
Xiaodong Yang, Xinfang Nie, Yunlan Ji, Tao Xin, Dawei Lu, Jun Li
Summary: This paper presents a method for optimizing quantum control design using Trotter decomposition. By substituting time evolution segments with their Trotter decompositions, the computational speed can be significantly improved while maintaining an acceptable level of propagator error. Experimental results demonstrate that this strategy leads to performance improvements in gradient ascent pulse engineering and variational quantum algorithms, and it is applicable to many other quantum optimization and simulation tasks.
Article
Mathematics, Applied
Philip Saltenberger, Michel-Niklas Senn
Summary: This paper investigates the relationship between the Krylov space generated by skew-Hamiltonian matrices and Lagrangian subspaces, analyzing the set of skew-Hamiltonian matrices that can generate a Lagrangian subspace. Existence and uniqueness results are proven, and skew-Hamiltonian matrices with specific properties are identified.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
Peter Benner, Patrick Kuerschner, Zoran Tomljanovic, Ninoslav Truhar
ZAMM-ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK
(2016)
Article
Automation & Control Systems
Peter Benner, Patrick Kuerschner, Jens Saak
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2016)
Article
Mathematics, Applied
Peter Benner, Zvonimir Bujanovic, Patrick Kuerschner, Jens Saak
NUMERISCHE MATHEMATIK
(2018)
Article
Mathematics, Applied
Melina A. Freitag, Patrick Kuerschner, Jennifer Pestana
COMPUTATIONAL METHODS IN APPLIED MATHEMATICS
(2018)
Article
Automation & Control Systems
Martin Redmann, Patrick Kueschner
SYSTEMS & CONTROL LETTERS
(2018)
Article
Mathematics, Applied
Daniel Kressner, Patrick Kurschner, Stefano Massei
NUMERICAL ALGORITHMS
(2020)
Article
Mathematical & Computational Biology
Patrick Kuerschner, Sergey Dolgov, Kameron Decker Harris, Peter Benner
JOURNAL OF MATHEMATICAL NEUROSCIENCE
(2019)
Article
Mathematics, Applied
Christian Kuehn, Patrick Kurschner
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2020)
Article
Computer Science, Software Engineering
Patrick Kurschner, Melina A. Freitag
BIT NUMERICAL MATHEMATICS
(2020)
Article
Mathematics, Applied
Igor Pontes Duff, Patrick Kurschner
Summary: This paper studies model order reduction for large-scale linear systems within finite time intervals, focusing on the development of error bounds for approximated output vectors and proposing strategies for efficient balanced truncation. Numerical experiments demonstrate the effectiveness of the proposed techniques.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
Davide Palitta, Patrick Kuerschner
Summary: This paper introduces low-rank Krylov methods as a way to solve large-scale linear matrix equations. By improving the truncation steps, the convergence of the Krylov method is maintained, and this theoretical finding is validated through numerical experiments.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Jeroen Vanderstukken, Patrick Kuerschner, Ignat Domanov, Lieven de Lathauwer
Summary: In this paper, a multilinear algebra framework is proposed to solve polynomial equations systems, including those with multiple roots. The block term decomposition of the Macaulay matrix reveals the dual space of roots in each term. This method offers flexibility in numerical optimization algorithms.
SIAM JOURNAL ON MATRIX ANALYSIS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Peter Benner, Zvonimir Bujanovic, Patrick Kurschner, Jens Saak
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2020)
Article
Mathematics, Applied
Patrick Kurschner
ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS
(2019)