@article{ClausnitzerKleefeld2024,
  author    = {Clausnitzer, Julian and Kleefeld, Andreas},
  title     = {A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a C² boundary},
  series = {Discrete and Continuous Dynamical Systems - Series B},
  volume    = {29},
  journal   = {Discrete and Continuous Dynamical Systems - Series B},
  number    = {4},
  publisher = {AIMS},
  address   = {Springfield},
  issn      = {1531-3492},
  doi       = {10.3934/dcdsb.2023148},
  pages     = {1624 -- 1651},
  year      = {2024},
  abstract  = {We consider the numerical approximation of second-order semi-linear parabolic stochastic partial differential equations interpreted in the mild sense which we solve on general two-dimensional domains with a C² boundary with homogeneous Dirichlet boundary conditions. The equations are driven by Gaussian additive noise, and several Lipschitz-like conditions are imposed on the nonlinear function. We discretize in space with a spectral Galerkin method and in time using an explicit Euler-like scheme. For irregular shapes, the necessary Dirichlet eigenvalues and eigenfunctions are obtained from a boundary integral equation method. This yields a nonlinear eigenvalue problem, which is discretized using a boundary element collocation method and is solved with the Beyn contour integral algorithm. We present an error analysis as well as numerical results on an exemplary asymmetric shape, and point out limitations of the approach.},
  language  = {en}
}
@article{AyalaHarrisKleefeld2024,
  author    = {Ayala, Rafael Ceja and Harris, Isaac and Kleefeld, Andreas},
  title     = {Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary},
  series = {Inverse Problems and Imaging},
  volume    = {18},
  journal   = {Inverse Problems and Imaging},
  number    = {3},
  publisher = {AIMS},
  address   = {Springfield},
  issn      = {1930-8337},
  doi       = {10.3934/ipi.2023051},
  pages     = {708 -- 729},
  year      = {2024},
  abstract  = {In this paper, we consider the inverse shape problem of recovering isotropic scatterers with a conductive boundary condition. Here, we assume that the measured far-field data is known at a fixed wave number. Motivated by recent work, we study a new direct sampling indicator based on the Landweber iteration and the factorization method. Therefore, we prove the connection between these reconstruction methods. The method studied here falls under the category of qualitative reconstruction methods where an imaging function is used to recover the absorbing scatterer. We prove stability of our new imaging function as well as derive a discrepancy principle for recovering the regularization parameter. The theoretical results are verified with numerical examples to show how the reconstruction performs by the new Landweber direct sampling method.},
  language  = {en}
}
@inproceedings{KahraBreussKleefeldetal.2024,
  author    = {Kahra, Marvin and Breuß, Michael and Kleefeld, Andreas and Welk, Martin},
  title     = {An Approach to Colour Morphological Supremum Formation Using the LogSumExp Approximation},
  series = {Discrete Geometry and Mathematical Morphology},
  booktitle = {Discrete Geometry and Mathematical Morphology},
  editor    = {Brunetti, Sara and Frosini, Andrea and Rinaldi, Simone},
  publisher = {Springer},
  address   = {Cham},
  isbn      = {978-3-031-57793-2},
  doi       = {10.1007/978-3-031-57793-2_25},
  pages     = {325 -- 337},
  year      = {2024},
  abstract  = {Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations. In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric 2x2 matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.},
  language  = {en}
}