@article{PieronekKleefeld2024, author = {Pieronek, Lukas and Kleefeld, Andreas}, title = {On trajectories of complex-valued interior transmission eigenvalues}, series = {Inverse problems and imaging : IPI}, volume = {18}, journal = {Inverse problems and imaging : IPI}, number = {2}, publisher = {AIMS}, address = {Springfield, Mo}, issn = {1930-8337 (Print)}, doi = {10.3934/ipi.2023041}, pages = {480 -- 516}, year = {2024}, abstract = {This paper investigates the interior transmission problem for homogeneous media via eigenvalue trajectories parameterized by the magnitude of the refractive index. In the case that the scatterer is the unit disk, we prove that there is a one-to-one correspondence between complex-valued interior transmission eigenvalue trajectories and Dirichlet eigenvalues of the Laplacian which turn out to be exactly the trajectorial limit points as the refractive index tends to infinity. For general simply-connected scatterers in two or three dimensions, a corresponding relation is still open, but further theoretical results and numerical studies indicate a similar connection.}, 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} } @article{HarrisKleefeld2018, author = {Harris, Isaac and Kleefeld, Andreas}, title = {The inverse scattering problem for a conductive boundary condition and transmission eigenvalues}, series = {Applicable Analysis}, volume = {99}, journal = {Applicable Analysis}, number = {3}, publisher = {Taylor \& Francis}, address = {London}, issn = {1563-504X}, doi = {10.1080/00036811.2018.1504028}, pages = {508 -- 529}, year = {2018}, abstract = {In this paper, we consider the inverse scattering problem associated with an inhomogeneous media with a conductive boundary. In particular, we are interested in two problems that arise from this inverse problem: the inverse conductivity problem and the corresponding interior transmission eigenvalue problem. The inverse conductivity problem is to recover the conductive boundary parameter from the measured scattering data. We prove that the measured scatted data uniquely determine the conductivity parameter as well as describe a direct algorithm to recover the conductivity. The interior transmission eigenvalue problem is an eigenvalue problem associated with the inverse scattering of such materials. We investigate the convergence of the eigenvalues as the conductivity parameter tends to zero as well as prove existence and discreteness for the case of an absorbing media. Lastly, several numerical and analytical results support the theory and we show that the inside-outside duality method can be used to reconstruct the interior conductive eigenvalues.}, language = {en} } @article{KleefeldPieronek2020, author = {Kleefeld, Andreas and Pieronek, J.}, title = {Elastic transmission eigenvalues and their computation via the method of fundamental solutions}, series = {Applicable Analysis}, volume = {100}, journal = {Applicable Analysis}, number = {16}, publisher = {Taylore \& Francis}, address = {London}, issn = {1563-504X}, doi = {10.1080/00036811.2020.1721473}, pages = {3445 -- 3462}, year = {2020}, abstract = {A stabilized version of the fundamental solution method to catch ill-conditioning effects is investigated with focus on the computation of complex-valued elastic interior transmission eigenvalues in two dimensions for homogeneous and isotropic media. Its algorithm can be implemented very shortly and adopts to many similar partial differential equation-based eigenproblems as long as the underlying fundamental solution function can be easily generated. We develop a corroborative approximation analysis which also implicates new basic results for transmission eigenfunctions and present some numerical examples which together prove successful feasibility of our eigenvalue recovery approach.}, language = {en} } @article{BreussKleefeld2020, author = {Breuß, Michael and Kleefeld, Andreas}, title = {Implicit monotone difference methods for scalar conservation laws with source terms}, series = {Acta Mathematica Vietnamica}, volume = {45}, journal = {Acta Mathematica Vietnamica}, publisher = {Springer Singapore}, address = {Singapore}, issn = {2315-4144}, doi = {10.1007/s40306-019-00354-1}, pages = {709 -- 738}, year = {2020}, abstract = {In this article, a concept of implicit methods for scalar conservation laws in one or more spatial dimensions allowing also for source terms of various types is presented. This material is a significant extension of previous work of the first author (Breuß SIAM J. Numer. Anal. 43(3), 970-986 2005). Implicit notions are developed that are centered around a monotonicity criterion. We demonstrate a connection between a numerical scheme and a discrete entropy inequality, which is based on a classical approach by Crandall and Majda. Additionally, three implicit methods are investigated using the developed notions. Next, we conduct a convergence proof which is not based on a classical compactness argument. Finally, the theoretical results are confirmed by various numerical tests.}, language = {en} } @article{AsanteAsamaniKleefeldWade2020, author = {Asante-Asamani, E.O. and Kleefeld, Andreas and Wade, B.A.}, title = {A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting}, series = {Journal of Computational Physics}, volume = {415}, journal = {Journal of Computational Physics}, publisher = {Elsevier}, address = {Amsterdam}, issn = {0021-9991}, doi = {10.1016/j.jcp.2020.109490}, year = {2020}, abstract = {A second-order L-stable exponential time-differencing (ETD) method is developed by combining an ETD scheme with approximating the matrix exponentials by rational functions having real distinct poles (RDP), together with a dimensional splitting integrating factor technique. A variety of non-linear reaction-diffusion equations in two and three dimensions with either Dirichlet, Neumann, or periodic boundary conditions are solved with this scheme and shown to outperform a variety of other second-order implicit-explicit schemes. An additional performance boost is gained through further use of basic parallelization techniques.}, language = {en} } @incollection{Kleefeld2020, author = {Kleefeld, Andreas}, title = {Numerical calculation of interior transmission eigenvalues with mixed boundary conditions}, 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 (Hardcover)}, doi = {10.1007/978-3-030-48186-5_9}, pages = {173 -- 195}, year = {2020}, abstract = {Interior transmission eigenvalue problems for the Helmholtz equation play an important role in inverse wave scattering. Some distribution properties of those eigenvalues in the complex plane are reviewed. Further, a new scattering model for the interior transmission eigenvalue problem with mixed boundary conditions is described and an efficient algorithm for computing the interior transmission eigenvalues is proposed. Finally, extensive numerical results for a variety of two-dimensional scatterers are presented to show the validity of the proposed scheme.}, language = {en} }