Publications | Randomized low rank algorithms and applications to parameter dependent problems

This page will contain papers and manuscripts resulting from the research done on the project.

Published papers

2023

$[DOI]$ L. Grubišić, R. Hiptmair, and D. Renner, “Detecting Near Resonances in Acoustic Scattering,” J. Sci. Comput., vol. 96, iss. 3, 2023.
[Bibtex]

@article{article,
author = {Grubi\v{s}i\'{c}, Luka and Hiptmair, Ralf and Renner, Diego},
title = {Detecting Near Resonances in Acoustic Scattering},
year = {2023},
issue_date = {Sep 2023},
publisher = {Plenum Press},
address = {USA},
volume = {96},
number = {3},
issn = {0885-7474},
url = {https://doi.org/10.1007/s10915-023-02284-5},
doi = {10.1007/s10915-023-02284-5},
abstract = {We propose and study a method for finding quasi-resonances for a linear acoustic transmission problem in frequency domain. Starting from an equivalent boundary-integral equation we perform Galerkin boundary element discretization and look for the minima of the smallest singular value of the resulting matrix as a function of the wave number k. We develop error estimates for the impact of Galerkin discretization on singular values and devise a heuristic adaptive algorithm for finding the minima in prescribed k-intervals. Our method exclusively relies on the solution of eigenvalue problems for real k, in contrast to alternative approaches that rely on extension to the complex plane.},
journal = {J. Sci. Comput.},
month = {jul},
numpages = {24},
keywords = {Acoustic scattering, Quasi-resonances, Boundary integral equations, Boundary element method, Singular values, Wielandt matrix, Zero finding, 35J05, 65N38, 65F15}
}

$[DOI]$ L. Grubišić, M. Lazar, I. Nakić, and M. Tautenhahn, “Optimal Control of Parabolic Equations – A Spectral Calculus Based Approach,” SIAM Journal on Control and Optimization, vol. 61, iss. 5, pp. 2802-2825, 2023.
[Bibtex]

@article{article,
author = {Grubi\v{s}i\'{c}, Luka and Lazar, Martin and Naki\'{c}, Ivica and Tautenhahn, Martin},
title = {Optimal Control of Parabolic Equations – A Spectral Calculus Based Approach},
journal = {SIAM Journal on Control and Optimization},
volume = {61},
number = {5},
pages = {2802-2825},
year = {2023},
doi = {10.1137/21M1449762},
URL = {https://doi.org/10.1137/21M1449762},
eprint = {https://doi.org/10.1137/21M1449762},
abstract = { Abstract. In this paper we consider a constrained parabolic optimal control problem. The cost functional is quadratic and it combines the distance of the trajectory of the system from the desired evolution profile together with the cost of a control. The constraint is given by a term measuring the distance between the final state and the desired state towards which the solution should be steered. The control enters the system through the initial condition. We present a geometric analysis of this problem and provide a closed-form expression for the solution. This approach allows us to present the sensitivity analysis of this problem based on the resolvent estimates for the generator of the system. The numerical implementation is performed by exploring efficient rational Krylov approximation techniques that allow us to approximate a complex function of an operator by a series of linear problems. Our method does not depend on the actual choice of discretization. The main approximation task is to construct an efficient rational approximation of a generalized exponential function. It is well known that this class of functions allows exponentially convergent rational approximations, which, combined with the sensitivity analysis of the closed form solution, allows us to present a robust numerical method. Several case studies are presented to illustrate our results. }
}

$[DOI]$ D. Kressner and I. Šain Glibić, “Singular quadratic eigenvalue problems: linearization and weak condition numbers,” BIT, vol. 63, iss. 1, 2023.
[Bibtex]

@article{article,
author = {Kressner, Daniel and \v{S}ain Glibi\'{c}, Ivana},
title = {Singular quadratic eigenvalue problems: linearization and weak condition numbers},
year = {2023},
issue_date = {Mar 2023},
publisher = {BIT Computer Science and Numerical Mathematics},
address = {USA},
volume = {63},
number = {1},
issn = {0006-3835},
url = {https://doi.org/10.1007/s10543-023-00960-4},
doi = {10.1007/s10543-023-00960-4},
abstract = {The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are exceptional and standard eigenvalue solvers, such as the QZ algorithm, tend to yield good accuracy despite the inevitable presence of roundoff error. Recently, Lotz and Noferini quantified this phenomenon by introducing the concept of δ-weak eigenvalue condition numbers. In this work, we consider singular quadratic eigenvalue problems and two popular linearizations. Our results show that a correctly chosen linearization increases δ-weak eigenvalue condition numbers only marginally, justifying the use of these linearizations in numerical solvers also in the singular case. We propose a very simple but often effective algorithm for computing well-conditioned eigenvalues of a singular quadratic eigenvalue problems by adding small random perturbations to the coefficients. We prove that the eigenvalue condition number is, with high probability, a reliable criterion for detecting and excluding spurious eigenvalues created from the singular part.},
journal = {BIT},
month = {feb},
numpages = {25},
keywords = {Singular eigenvalue problems, Polynomial eigenvalue problem, Linearization, Weak condition number, 65F30, 65F15, 65F35, 15A18}
}

2022

$[DOI]$ Z. Drmač and B. Peherstorfer, “Learning Low-Dimensional Dynamical-System Models from Noisy Frequency-Response Data with Loewner Rational Interpolation,” in Realization and Model Reduction of Dynamical Systems: A Festschrift in Honor of the 70th Birthday of Thanos Antoulas, C. Beattie, P. Benner, M. Embree, S. Gugercin, and S. Lefteriu, Eds., Springer International Publishing, 2022, p. 39–57.
[Bibtex]

@Inbook{article,
author = {Drma{\v{c}}, Zlatko and Peherstorfer, Benjamin},
editor = {Beattie, Christopher and Benner, Peter and Embree, Mark and Gugercin, Serkan and Lefteriu, Sanda},
title = {Learning Low-Dimensional Dynamical-System Models from Noisy Frequency-Response Data with {L}oewner Rational Interpolation},
bookTitle = {Realization and Model Reduction of Dynamical Systems: A Festschrift in Honor of the 70th Birthday of {T}hanos {A}ntoulas},
year = {2022},
publisher = {Springer International Publishing},
pages = {39--57},
isbn = {978-3-030-95157-3},
DOI = {10.1007/978-3-030-95157-3_3},
url = {https://doi.org/10.1007/978-3-030-95157-3_3}
}

$[DOI]$ Z. Drmač and I. Šain Glibić, “An Algorithm for the Complete Solution of the Quartic Eigenvalue Problem,” Acm transactions on mathematical software, vol. 48, iss. 1, pp. 1-34, 2022.
[Bibtex]

@article{article,
author = {Drma\v{c}, Zlatko and \v{S}ain Glibi\'{c}, Ivana},
title = {An Algorithm for the Complete Solution of the Quartic Eigenvalue Problem},
journal = {Acm transactions on mathematical software},
year = {2022},
pages = {1-34},
keywords = {Eigenvalues, eigenvectors, infinite eigenvalues, linearization, nonlinear eigenvalue problem, Orr–Sommerfeld equation, quadratification, quartic eigenvalue problem, QZ algorithm},
volume = {48},
number = {1},
issn = {0098-3500},
DOI = {10.1145/3494528},
}

$[DOI]$ P. Jorkowski, K. Schmidt, C. Schenker, L. Grubišić, and R. Schuhmann, “Adapted Contour Integration for Nonlinear Eigenvalue Problems in Waveguide Coupled Resonators,” IEEE transactions on antennas and propagation, vol. 70, iss. 1, pp. 499-513, 2022.
[Bibtex]

@article{article,
author = {Jorkowski, Philipp and Schmidt, Kersten and Schenker, Carla and Grubi\v{s}i\'{c}, Luka and Schuhmann, Rolf},
title = {Adapted Contour Integration for Nonlinear Eigenvalue Problems in Waveguide Coupled Resonators},
journal = {IEEE transactions on antennas and propagation},
year = {2022},
pages = {499-513},
DOI = {10.1109/TAP.2021.3111413},
keywords = {Eigenvalues and eigenfunctions, Transmission line matrix methods, Resonators, Mathematical model, Optical waveguides, Resonant frequency, Numerical models},
volume = {70},
number = {1},
issn = {0018-926X},
}

$[DOI]$ G. A. Mensah, A. Orchini, P. Buschmann, and L. Grubišić, “A subspace-accelerated method for solving nonlinear thermoacoustic eigenvalue problems,” Journal of sound and vibration, vol. 520, p. 5, 2022.
[Bibtex]

@article{article,
author = {Mensah, Georg A. and Orchini, Alessandro and Buschmann, Philip and Grubi\v{s}i\'{c}, Luka},
title = {A subspace-accelerated method for solving nonlinear thermoacoustic eigenvalue problems},
journal = {Journal of sound and vibration},
year = {2022},
pages = {5},
DOI = {10.1016/j.jsv.2021.116553},
chapter = {116553},
keywords = {Thermoacoustics, Subspace method, Reduced order model, Nonlinear eigenvalue problem},
volume = {520},
issn = {0022-460X},
}

$[DOI]$ C. Engstrom, S. Giani, and L. Grubišić, “Higher Order Composite DG approximations of Gross– Pitaevskii ground state: Benchmark results and experiments,” Journal of computational and applied mathematics, vol. 400, p. 15, 2022.
[Bibtex]

@article{article,
author = {Engstrom, Christian and Giani, Stefano and Grubi\v{s}i\'{c}, Luka},
title = {Higher Order Composite DG approximations of Gross– Pitaevskii ground state: Benchmark results and experiments},
journal = {Journal of computational and applied mathematics},
year = {2022},
pages = {15},
DOI = {10.1016/j.cam.2021.113652},
chapter = {113652},
keywords = {Gross–Pitaevskii eigenvalue problem, Discontinuous Galerkin finite element approximations, Composite finite elements},
volume = {400},
issn = {0377-0427},
}

$[DOI]$ C. Engstrom, S. Giani, and L. Grubišić, “A spectral projection based method for the numerical solution of wave equations with memory,” Applied mathematics letters, vol. 127, p. 9, 2022.
[Bibtex]

@article{article,
author = {Engstrom, Christian and Giani, Stefano and Grubi\v{s}i\'{c}, Luka},
title = {A spectral projection based method for the numerical solution of wave equations with memory},
journal = {Applied mathematics letters},
year = {2022},
pages = {9},
DOI = {10.1016/j.aml.2021.107844},
chapter = {107844},
keywords = {Inverse Laplace transform, Spectral projection, Wave equation with delay, Finite element approximation},
volume = {127},
issn = {0893-9659},
}

$[DOI]$ S. Čanić, L. Grubišić, D. Lacmanović, M. Ljulj, and J. Tambača, “Optimal design of vascular stents using a network of 1D slender curved rods,” Computer methods in applied mechanics and engineering, vol. 394, p. 32, 2022.
[Bibtex]

@article{article,
author = {\v{C}ani\'{c}, Sun\v{c}ica and Grubi\v{s}i\'{c}, Luka and Lacmanovi\'{c}, Domagoj and Ljulj, Matko and Tamba\v{c}a, Josip},
title = {Optimal design of vascular stents using a network of 1D slender curved rods},
journal = {Computer methods in applied mechanics and engineering},
year = {2022},
pages = {32},
DOI = {10.1016/j.cma.2022.114853},
chapter = {114853},
keywords = {Optimal stent design, Vascular stents, Computational algorithm, Cypher stent},
volume = {394},
issn = {0045-7825},
}

$[DOI]$ Z. Bujanović, D. Kressner, and C. Schröder, “Iterative refinement of Schur decompositions,” Numerical algorithms, 2022.
[Bibtex]

@article{article,
author = {Bujanovi\'{c}, Zvonimir and Kressner, Daniel and Schr\"{o}der, Christian},
title = {Iterative refinement of {S}chur decompositions},
journal = {Numerical algorithms},
year = {2022},
DOI = {10.1007/s11075-022-01327-6},
keywords = {Schur decomposition, iterative refinement, mixed precision, eigenvalue computation},
}

2021

$[DOI]$ Z. Drmač, “Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices,” SeMA Journal. Boletin de la Sociedad Espanñola de Matemática Aplicada, vol. 78, pp. 53-92, 2021.
[Bibtex]

@article{article,
author = {Drma\v{c}, Zlatko},
title = {Numerical methods for accurate computation of the eigenvalues of Hermitian matrices and the singular values of general matrices},
journal = {SeMA Journal. Boletin de la Sociedad Espan\~{n}ola de Matem\'{a}tica Aplicada},
year = {2021},
pages = {53-92},
DOI = {10.1007/s40324-020-00229-8},
keywords = {Backward error, Condition number, Eigenvalues, Hermitian matrices, Jacobi method, LAPACK, Perturbation theory, Rank revealing decomposition, Singular value decomposition},
volume = {78},
issn = {2254-3902},
}

$[DOI]$ N. Bosner, “Parallel Prony’s method with multivariate matrix pencil approach and its numerical aspects,” SIAM journal on matrix analysis and applications, vol. 42, iss. 2, pp. 635-658, 2021.
[Bibtex]

@article{article,
author = {Bosner, Nela},
title = {Parallel Prony's method with multivariate matrix pencil approach and its numerical aspects},
journal = {SIAM journal on matrix analysis and applications},
year = {2021},
pages = {635-658},
DOI = {10.1137/20M1343658},
keywords = {Prony's method, parallel algorithm, efficient GPU-CPU implementation, numerical analysis},
volume = {42},
number = {2},
issn = {0895-4798},
}

$[DOI]$ Z. Bujanović and D. Kressner, “Norm and trace estimation with random rank-one vectors,” SIAM journal on matrix analysis and applications, vol. 42, iss. 1, pp. 202-223, 2021.
[Bibtex]

@article{article,
author = {Bujanovi\'{c}, Zvonimir and Kressner, Daniel},
title = {Norm and trace estimation with random rank-one vectors},
journal = {SIAM journal on matrix analysis and applications},
year = {2021},
pages = {202-223},
DOI = {10.1137/20M1331718},
keywords = {norm of a matrix, trace of a matrix, stochastic estimator, rank-one vectors},
volume = {42},
number = {1},
issn = {0895-4798},
}

$[DOI]$ R. Mohr, M. Fonoberova, Z. Drmač, I. Manojlović, and I. Mezić, “Predicting the Critical Number of Layers for Hierarchical Support Vector Regression,” Entropy (Basel. Online), vol. 23, iss. 1, p. 16, 2021.
[Bibtex]

@article{article,
author = {Mohr, Ryan and Fonoberova, Maria and Drma\v{c}, Zlatko and Manojlovi\'{c}, Iva and Mezi\'{c}, Igor},
year = {2021},
pages = {16},
DOI = {10.3390/e23010037},
chapter = {37},
keywords = {support vector regression, Fourier transform, dynamic mode decomposition, Koopman operator},
journal = {Entropy (Basel. Online)},
doi = {10.3390/e23010037},
volume = {23},
number = {1},
title = {Predicting the Critical Number of Layers for Hierarchical Support Vector Regression},
keyword = {support vector regression, Fourier transform, dynamic mode decomposition, Koopman operator},
chapternumber = {37}
}

$[DOI]$ L. Grubišić, M. Hajba, and D. Lacmanović, “Deep Neural Network Model for Approximating Eigenmodes Localized by a Confining Potential,” Entropy (Basel. Online), vol. 23, iss. 1, p. 19, 2021.
[Bibtex]

@article{article,
author = {Grubi\v{s}i\'{c}, Luka and Hajba, Marko and Lacmanovi\'{c}, Domagoj},
year = {2021},
pages = {19},
DOI = {10.3390/e23010095},
chapter = {95},
keywords = {Anderson localization, deep neural networks, residual error estimates, physics informed neural networks},
journal = {Entropy (Basel. Online)},
doi = {10.3390/e23010095},
volume = {23},
number = {1},
title = {Deep Neural Network Model for Approximating
Eigenmodes Localized by a Confining Potential},
keyword = {Anderson localization, deep neural networks, residual error estimates, physics informed neural networks},
chapternumber = {95}
}

2020

$[DOI]$ B. Peherstorfer, Z. Drmač, and S. Gugercin, “Stability of Discrete Empirical Interpolation and Gappy Proper Orthogonal Decomposition with Randomized and Deterministic Sampling Points,” SIAM Journal on Scientific Computing, vol. 42, iss. 5, p. A2837-A2864, 2020.
[Bibtex]

@article{article,
author = {Peherstorfer, Benjamin and Drma\v{c}, Zlatko and Gugercin, Serkan},
year = {2020},
pages = {A2837-A2864},
DOI = {10.1137/19m1307391},
keywords = {model reduction, empirical interpolation, gappy proper orthogonal decomposition, noisy observations, nonlinear model reduction, randomized model reduction},
journal = {SIAM Journal on Scientific Computing},
doi = {10.1137/19m1307391},
volume = {42},
number = {5},
title = {Stability of Discrete Empirical Interpolation and
Gappy Proper Orthogonal Decomposition with Randomized
and Deterministic Sampling Points},
keyword = {model reduction, empirical interpolation, gappy proper orthogonal decomposition, noisy observations, nonlinear model reduction, randomized model reduction}
}

$[DOI]$ C. Beattie, Z. Drmač, and S. Gugercin, “Revisiting IRKA: Connections with Pole Placement and Backward Stability,” Vietnam journal of mathematics, vol. 48, pp. 963-985, 2020.
[Bibtex]

@article{article,
author = {Beattie, Christopher and Drma\v{c}, Zlatko and Gugercin, Serkan},
year = {2020},
pages = {963-985},
DOI = {10.1007/s10013-020-00424-0},
keywords = {Interpolation, Model reduction, H2-optimality, Pole placement, Backward stability},
journal = {Vietnam journal of mathematics},
doi = {10.1007/s10013-020-00424-0},
volume = {48},
title = {Revisiting IRKA: Connections with Pole Placement
and Backward Stability},
keyword = {Interpolation, Model reduction, H2-optimality, Pole placement, Backward stability}
}

Submitted manuscripts / In preparation

2024

N. Bosner, “Parallel implementations of Riemannian conjugate gradient methods for joint approximate diagonalization,” , 2024.
[Bibtex]

@article{Bos24,
author = {Bosner, Nela},
title = {Parallel implementations of Riemannian conjugate gradient methods for joint approximate diagonalization},
year = {2024},
vol = {(submitted)},
}