@article{Kleefeld2021,
  author    = {Kleefeld, Andreas},
  title     = {The hot spots conjecture can be false: some numerical examples},
  series = {Advances in Computational Mathematics},
  volume    = {47},
  journal   = {Advances in Computational Mathematics},
  publisher = {Springer},
  address   = {Dordrecht},
  issn      = {1019-7168},
  doi       = {10.1007/s10444-021-09911-5},
  year      = {2021},
  abstract  = {The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10- 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.},
  language  = {en}
}
@misc{BurgethKleefeldNaegeletal.2020,
  author    = {Burgeth, Bernhard and Kleefeld, Andreas and Naegel, Beno{\^i}t and Perret, Benjamin},
  title     = {Editorial — Special Issue: ISMM 2019},
  series = {Mathematical Morphology - Theory and Applications},
  volume    = {4},
  journal   = {Mathematical Morphology - Theory and Applications},
  number    = {1},
  publisher = {De Gruyter},
  address   = {Warschau},
  issn      = {2353-3390},
  doi       = {10.1515/mathm-2020-0200},
  pages     = {159 -- 161},
  year      = {2020},
  abstract  = {This editorial presents the Special Issue dedicated to the conference ISMM 2019 and summarizes the articles published in this Special Issue.},
  language  = {en}
}
@article{KleefeldZimmermann2022,
  author    = {Kleefeld, Andreas and Zimmermann, M.},
  title     = {Computing Elastic Interior Transmission Eigenvalues},
  series = {Integral Methods in Science and Engineering},
  journal   = {Integral Methods in Science and Engineering},
  editor    = {Constanda, Christian and Bodmann, Bardo E.J. and Harris, Paul J.},
  publisher = {Birkh{\"a}user},
  address   = {Cham},
  isbn      = {978-3-031-07171-3},
  doi       = {10.1007/978-3-031-07171-3_10},
  pages     = {139 -- 155},
  year      = {2022},
  abstract  = {An alternative method is presented to numerically compute interior elastic transmission eigenvalues for various domains in two dimensions. This is achieved by discretizing the resulting system of boundary integral equations in combination with a nonlinear eigenvalue solver. Numerical results are given to show that this new approach can provide better results than the finite element method when dealing with general domains.},
  language  = {en}
}
@inproceedings{BurgethKleefeldZhangetal.2022,
  author    = {Burgeth, Bernhard and Kleefeld, Andreas and Zhang, Eugene and Zhang, Yue},
  title     = {Towards Topological Analysis of Non-symmetric Tensor Fields via Complexification},
  series = {Discrete Geometry and Mathematical Morphology},
  booktitle = {Discrete Geometry and Mathematical Morphology},
  editor    = {Baudrier, {\´E}tienne and Naegel, Beno{\^i}t and Kr{\"a}henb{\"u}hl, Adrien and Tajine, Mohamed},
  publisher = {Springer},
  address   = {Cham},
  isbn      = {978-3-031-19897-7},
  doi       = {10.1007/978-3-031-19897-7_5},
  pages     = {48 -- 59},
  year      = {2022},
  abstract  = {Fields of asymmetric tensors play an important role in many applications such as medical imaging (diffusion tensor magnetic resonance imaging), physics, and civil engineering (for example Cauchy-Green-deformation tensor, strain tensor with local rotations, etc.). However, such asymmetric tensors are usually symmetrized and then further processed. Using this procedure results in a loss of information. A new method for the processing of asymmetric tensor fields is proposed restricting our attention to tensors of second-order given by a 2x2 array or matrix with real entries. This is achieved by a transformation resulting in Hermitian matrices that have an eigendecomposition similar to symmetric matrices. With this new idea numerical results for real-world data arising from a deformation of an object by external forces are given. It is shown that the asymmetric part indeed contains valuable information.},
  language  = {en}
}
@article{HarrisKleefeld2022,
  author    = {Harris, Isaac and Kleefeld, Andreas},
  title     = {Analysis and computation of the transmission eigenvalues with a conductive boundary condition},
  series = {Applicable Analysis},
  volume    = {101},
  journal   = {Applicable Analysis},
  number    = {6},
  publisher = {Taylor \& Francis},
  address   = {London},
  issn      = {1563-504X},
  doi       = {10.1080/00036811.2020.1789598},
  pages     = {1880 -- 1895},
  year      = {2022},
  abstract  = {We provide a new analytical and computational study of the transmission eigenvalues with a conductive boundary condition. These eigenvalues are derived from the scalar inverse scattering problem for an inhomogeneous material with a conductive boundary condition. The goal is to study how these eigenvalues depend on the material parameters in order to estimate the refractive index. The analytical questions we study are: deriving Faber-Krahn type lower bounds, the discreteness and limiting behavior of the transmission eigenvalues as the conductivity tends to infinity for a sign changing contrast. We also provide a numerical study of a new boundary integral equation for computing the eigenvalues. Lastly, using the limiting behavior we will numerically estimate the refractive index from the eigenvalues provided the conductivity is sufficiently large but unknown.},
  language  = {en}
}
@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}
}
@incollection{PieronekKleefeld2019,
  author    = {Pieronek, Lukas and Kleefeld, Andreas},
  title     = {The Method of Fundamental Solutions for Computing Interior Transmission Eigenvalues of Inhomogeneous Media},
  series = {Integral Methods in Science and Engineering: Analytic Treatment and Numerical Approximations},
  booktitle = {Integral Methods in Science and Engineering: Analytic Treatment and Numerical Approximations},
  editor    = {Constanda, Christian and Harris, Paul},
  publisher = {Birkh{\"a}user},
  address   = {Cham},
  isbn      = {978-3-030-16077-7},
  doi       = {10.1007/978-3-030-16077-7_28},
  pages     = {353 -- 365},
  year      = {2019},
  abstract  = {The method of fundamental solutions is applied to the approximate computation of interior transmission eigenvalues for a special class of inhomogeneous media in two dimensions. We give a short approximation analysis accompanied with numerical results that clearly prove practical convenience of our alternative approach.},
  language  = {en}
}
@incollection{AbeleKleefeld2020,
  author    = {Abele, Daniel and Kleefeld, Andreas},
  title     = {New Numerical Results for the Optimization of Neumann Eigenvalues},
  series = {Computational and Analytic Methods in Science and Engineering},
  booktitle = {Computational and Analytic Methods in Science and Engineering},
  editor    = {Constanda, Christian},
  publisher = {Birkh{\"a}user},
  address   = {Cham},
  isbn      = {978-3-030-48185-8 (Print)},
  doi       = {10.1007/978-3-030-48186-5_1},
  pages     = {1 -- 20},
  year      = {2020},
  abstract  = {We present new numerical results for shape optimization problems of interior Neumann eigenvalues. This field is not well understood from a theoretical standpoint. The existence of shape maximizers is not proven beyond the first two eigenvalues, so we study the problem numerically. We describe a method to compute the eigenvalues for a given shape that combines the boundary element method with an algorithm for nonlinear eigenvalues. As numerical optimization requires many such evaluations, we put a focus on the efficiency of the method and the implemented routine. The method is well suited for parallelization. Using the resulting fast routines and a specialized parametrization of the shapes, we found improved maxima for several eigenvalues.},
  language  = {en}
}