TY - CHAP A1 - Kleefeld, Andreas ED - Constanda, Christian T1 - Numerical calculation of interior transmission eigenvalues with mixed boundary conditions T2 - Computational and Analytic Methods in Science and Engineering N2 - 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. Y1 - 2020 SN - 978-3-030-48185-8 (Hardcover) U6 - https://doi.org/10.1007/978-3-030-48186-5_9 SP - 173 EP - 195 PB - Birkhäuser CY - Cham ER - TY - JOUR A1 - Baringhaus, Ludwig A1 - Gaigall, Daniel A1 - Thiele, Jan Philipp T1 - Statistical inference for L²-distances to uniformity JF - Computational Statistics N2 - The paper deals with the asymptotic behaviour of estimators, statistical tests and confidence intervals for L²-distances to uniformity based on the empirical distribution function, the integrated empirical distribution function and the integrated empirical survival function. Approximations of power functions, confidence intervals for the L²-distances and statistical neighbourhood-of-uniformity validation tests are obtained as main applications. The finite sample behaviour of the procedures is illustrated by a simulation study. KW - Integrated empirical distribution (survival) function KW - Goodness-of-fit tests for uniformity KW - Numerical inversion of Laplace transforms KW - Coverage probability KW - Equivalence test Y1 - 2018 U6 - https://doi.org/10.1007/s00180-018-0820-0 SN - 1613-9658 VL - 2018 IS - 33 SP - 1863 EP - 1896 PB - Springer CY - Berlin ER - TY - JOUR A1 - Kleefeld, Andreas T1 - The hot spots conjecture can be false: some numerical examples JF - Advances in Computational Mathematics N2 - 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. KW - Numerics KW - Boundary integral equations KW - Potential theory KW - Helmholtz equation KW - Interior Neumann eigenvalues Y1 - 2021 U6 - https://doi.org/10.1007/s10444-021-09911-5 SN - 1019-7168 VL - 47 PB - Springer CY - Dordrecht ER - TY - JOUR A1 - Ditzhaus, Marc A1 - Gaigall, Daniel T1 - A consistent goodness-of-fit test for huge dimensional and functional data JF - Journal of Nonparametric Statistics N2 - A nonparametric goodness-of-fit test for random variables with values in a separable Hilbert space is investigated. To verify the null hypothesis that the data come from a specific distribution, an integral type test based on a Cramér-von-Mises statistic is suggested. The convergence in distribution of the test statistic under the null hypothesis is proved and the test's consistency is concluded. Moreover, properties under local alternatives are discussed. Applications are given for data of huge but finite dimension and for functional data in infinite dimensional spaces. A general approach enables the treatment of incomplete data. In simulation studies the test competes with alternative proposals. KW - Cramér-von-Mises statistic KW - separable Hilbert space KW - huge dimensional data KW - functional data Y1 - 2018 U6 - https://doi.org/10.1080/10485252.2018.1486402 SN - 1029-0311 VL - 30 IS - 4 SP - 834 EP - 859 PB - Taylor & Francis CY - Abingdon ER - TY - JOUR A1 - Harris, Isaac A1 - Kleefeld, Andreas T1 - Analysis and computation of the transmission eigenvalues with a conductive boundary condition JF - Applicable Analysis N2 - 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. KW - Boundary integral equations KW - Inverse spectral problem KW - Conductive boundary condition KW - Transmission eigenvalues Y1 - 2020 U6 - https://doi.org/10.1080/00036811.2020.1789598 SN - 1563-504X VL - 101 IS - 6 SP - 1880 EP - 1895 PB - Taylor & Francis CY - London ER - TY - JOUR A1 - Clausnitzer, Julian A1 - Kleefeld, Andreas T1 - A spectral Galerkin exponential Euler time-stepping scheme for parabolic SPDEs on two-dimensional domains with a C² boundary JF - Discrete and Continuous Dynamical Systems - Series B N2 - 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. KW - Nonlinear eigenvalue problems KW - Boundary integral equations, KW - Exponential Euler scheme, KW - Parabolic SPDEs Y1 - 2024 U6 - https://doi.org/10.3934/dcdsb.2023148 SN - 1531-3492 SN - 1553-524X (eISSN) VL - 29 IS - 4 SP - 1624 EP - 1651 PB - AIMS CY - Springfield ER - TY - JOUR A1 - Ayala, Rafael Ceja A1 - Harris, Isaac A1 - Kleefeld, Andreas T1 - Direct sampling method via Landweber iteration for an absorbing scatterer with a conductive boundary JF - Inverse Problems and Imaging N2 - 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. Y1 - 2024 U6 - https://doi.org/10.3934/ipi.2023051 SN - 1930-8337 SN - 1930-8345 (eISSN) VL - 18 IS - 3 SP - 708 EP - 729 PB - AIMS CY - Springfield ER - TY - JOUR A1 - Baringhaus, Ludwig A1 - Gaigall, Daniel T1 - Hotelling’s T² tests in paired and independent survey samples: An efficiency comparison JF - Journal of Multivariate Analysis N2 - Hotelling’s T² tests in paired and independent survey samples are compared using the traditional asymptotic efficiency concepts of Hodges–Lehmann, Bahadur and Pitman, as well as through criteria based on the volumes of corresponding confidence regions. Conditions characterizing the superiority of a procedure are given in terms of population canonical correlation type coefficients. Statistical tests for checking these conditions are developed. Test statistics based on the eigenvalues of a symmetrized sample cross-covariance matrix are suggested, as well as test statistics based on sample canonical correlation type coefficients. Y1 - 2017 U6 - https://doi.org/10.1016/j.jmva.2016.11.004 SN - 0047-259X VL - 2017 IS - 154 SP - 177 EP - 198 PB - Elsevier CY - Amsterdam ER - TY - JOUR A1 - Gaigall, Daniel A1 - Gerstenberg, Julian A1 - Trinh, Thi Thu Ha T1 - Empirical process of concomitants for partly categorial data and applications in statistics JF - Bernoulli N2 - On the basis of independent and identically distributed bivariate random vectors, where the components are categorial and continuous variables, respectively, the related concomitants, also called induced order statistic, are considered. The main theoretical result is a functional central limit theorem for the empirical process of the concomitants in a triangular array setting. A natural application is hypothesis testing. An independence test and a two-sample test are investigated in detail. The fairly general setting enables limit results under local alternatives and bootstrap samples. For the comparison with existing tests from the literature simulation studies are conducted. The empirical results obtained confirm the theoretical findings. KW - bootstrap KW - Categorial variable KW - Concomitant KW - Empirical process KW - Independence test Y1 - 2022 U6 - https://doi.org/10.3150/21-BEJ1367 SN - 1573-9759 VL - 28 IS - 2 SP - 803 EP - 829 PB - International Statistical Institute CY - Den Haag, NL ER - TY - JOUR A1 - Gaigall, Daniel T1 - Testing marginal homogeneity of a continuous bivariate distribution with possibly incomplete paired data JF - Metrika N2 - We discuss the testing problem of homogeneity of the marginal distributions of a continuous bivariate distribution based on a paired sample with possibly missing components (missing completely at random). Applying the well-known two-sample Crámer–von-Mises distance to the remaining data, we determine the limiting null distribution of our test statistic in this situation. It is seen that a new resampling approach is appropriate for the approximation of the unknown null distribution. We prove that the resulting test asymptotically reaches the significance level and is consistent. Properties of the test under local alternatives are pointed out as well. Simulations investigate the quality of the approximation and the power of the new approach in the finite sample case. As an illustration we apply the test to real data sets. KW - Marginal homogeneity test KW - Crámer–von-Mises distance KW - Paired sample KW - Incomplete data KW - Resampling test Y1 - 2019 U6 - https://doi.org/10.1007/s00184-019-00742-5 SN - 1435-926X VL - 2020 IS - 83 SP - 437 EP - 465 PB - Springer ER -