@article{KayaFreitag2022,
  author    = {Kaya, Adem and Freitag, Melina A.},
  title     = {Conditioning analysis for discrete Helmholtz problems},
  series = {Computers and mathematics with applications : an international journal},
  volume    = {118},
  journal   = {Computers and mathematics with applications : an international journal},
  publisher = {Elsevier Science},
  address   = {Amsterdam},
  issn      = {0898-1221},
  doi       = {10.1016/j.camwa.2022.05.016},
  pages     = {171 -- 182},
  year      = {2022},
  abstract  = {In this paper, we examine conditioning of the discretization of the Helmholtz problem. Although the discrete Helmholtz problem has been studied from different perspectives, to the best of our knowledge, there is no conditioning analysis for it. We aim to fill this gap in the literature. We propose a novel method in 1D to observe the near-zero eigenvalues of a symmetric indefinite matrix. Standard classification of ill-conditioning based on the matrix condition number is not true for the discrete Helmholtz problem. We relate the ill-conditioning of the discretization of the Helmholtz problem with the condition number of the matrix. We carry out analytical conditioning analysis in 1D and extend our observations to 2D with numerical observations. We examine several discretizations. We find different regions in which the condition number of the problem shows different characteristics. We also explain the general behavior of the solutions in these regions.},
  language  = {en}
}
@article{RedmannFreitag2021,
  author    = {Redmann, Martin and Freitag, Melina A.},
  title     = {Optimization based model order reduction for stochastic systems},
  series = {Applied mathematics and computation},
  volume    = {398},
  journal   = {Applied mathematics and computation},
  publisher = {Elsevier},
  address   = {New York},
  issn      = {0096-3003},
  doi       = {10.1016/j.amc.2020.125783},
  pages     = {18},
  year      = {2021},
  abstract  = {In this paper, we bring together the worlds of model order reduction for stochastic linear systems and H-2-optimal model order reduction for deterministic systems. In particular, we supplement and complete the theory of error bounds for model order reduction of stochastic differential equations. With these error bounds, we establish a link between the output error for stochastic systems (with additive and multiplicative noise) and modified versions of the H-2-norm for both linear and bilinear deterministic systems. When deriving the respective optimality conditions for minimizing the error bounds, we see that model order reduction techniques related to iterative rational Krylov algorithms (IRKA) are very natural and effective methods for reducing the dimension of large-scale stochastic systems with additive and/or multiplicative noise. We apply modified versions of (linear and bilinear) IRKA to stochastic linear systems and show their efficiency in numerical experiments.},
  language  = {en}
}
@article{KuerschnerFreitag2020,
  author    = {K{\"u}rschner, Patrick and Freitag, Melina A.},
  title     = {Inexact methods for the low rank solution to large scale Lyapunov equations},
  series = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  volume    = {60},
  journal   = {BIT : numerical mathematics ; the leading applied mathematics journal for all computational mathematicians},
  number    = {4},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {0006-3835},
  doi       = {10.1007/s10543-020-00813-4},
  pages     = {1221 -- 1259},
  year      = {2020},
  abstract  = {The rational Krylov subspace method (RKSM) and the low-rank alternating directions implicit (LR-ADI) iteration are established numerical tools for computing low-rank solution factors of large-scale Lyapunov equations. In order to generate the basis vectors for the RKSM, or extend the low-rank factors within the LR-ADI method, the repeated solution to a shifted linear system of equations is necessary. For very large systems this solve is usually implemented using iterative methods, leading to inexact solves within this inner iteration (and therefore to "inexact methods"). We will show that one can terminate this inner iteration before full precision has been reached and still obtain very good accuracy in the final solution to the Lyapunov equation. In particular, for both the RKSM and the LR-ADI method we derive theory for a relaxation strategy (e.g. increasing the solve tolerance of the inner iteration, as the outer iteration proceeds) within the iterative methods for solving the large linear systems. These theoretical choices involve unknown quantities, therefore practical criteria for relaxing the solution tolerance within the inner linear system are then provided. The theory is supported by several numerical examples, which show that the total amount of work for solving Lyapunov equations can be reduced significantly.},
  language  = {en}
}