Andreas Kleefeld

A numerical method to compute interior transmission eigenvalues

In this paper the numerical calculation of eigenvalues of the interior transmission problem arising in acoustic scattering for constant contrast in three dimensions is considered. From the computational point of view existing methods are very expensive, and are only able to show the existence of such transmission eigenvalues. Furthermore, they have trouble finding them if two or more eigenvalues are situated closely together. We present a new method based on complex-valued contour integrals and the boundary integral equation method which is able to calculate highly accurate transmission eigenvalues. So far, this is the first paper providing such accurate values for various surfaces different from a sphere in three dimensions. Additionally, the computational cost is even lower than those of existing methods. Furthermore, the algorithm is capable of finding complex-valued eigenvalues for which no numerical results have been reported yet. Until now, the proof of existence of such eigenvalues is still open. Finally, highly accurate eigenvalues of the interior Dirichlet problem are provided and might serve as test cases to check newly derived Faber–Krahn type inequalities for larger transmission eigenvalues that are not yet available.

Folded- and Log-Folded-t Distributions as Models for Insurance Loss Data

A rich variety of probability distributions has been proposed in the actuarial literature for fitting of insurance loss data. Examples include: lognormal, log-t, various versions of Pareto, loglogistic, Weibull, gamma and its variants, and generalized beta of the second kind distributions, among others. In this paper, we supplement the literature by adding the log-folded-normal and log-folded-t families. Shapes of the density function and key distributional properties of the ‘folded’ distributions are presented along with three methods for the estimation of parameters: method of maximum likelihood; method of moments; and method of trimmed moments. Further, large and small-sample properties of these estimators are studied in detail. Finally, we fit the newly proposed distributions to data which represent the total damage done by 827 fires in Norway for the year 1988. The fitted models are then employed in a few quantitative risk management examples, where point and interval estimates for several value-at-risk measures are calculated.

An approach to color-morphology based on Einstein addition and Loewner order

In this article, a new method to process color images via mathematical morphology is presented. Precisely, each pixel of an image contained in the rgb-space is converted into a 2x2-matrix representing a point in a color bi-cone. The supremum and infimum needed for dilation and erosion, respectively, are calculated with respect to the Loewner order. Since the result can be outside the bi-cone, a map is proposed to ensure the algebraic closure in the bi-cone. A new addition and subtraction based on Einsteins Relativity Theory is used to define morphological operation such as top-hats, gradients, and morphological Laplacian. Since the addition and subtraction is defined in the unit ball, a suitable mapping between those two spaces is constructed. A comparison with the component-wise approach and the full ordering using lexicographical cascades approach demonstrate the feasibility and capabilities of the proposed approach.

Morphology for Color Images via Loewner Order for Matrix Fields

Mathematical morphology is a very successful branch of image processing with a history of more than four decades. Its fundamental operations are dilation and erosion, which are based on the notion of a maximum and a minimum with respect to an order. Many operators constructed from dilation and erosion are available for grey value images, and recently useful analogs of these processes for matrix-valued images have been introduced by taking advantage of the so-called Loewner order. There has been a number of approaches to morphology for vector-valued images, that is, colour images based on various orders, however, each with its merits and shortcomings. In this article we propose an approach to (elementary) morphology for colour images that relies on the existing order based morphology for matrix fields of symmetric 2×2-matrices. An RGB-image is embedded into a field of those 2×2-matrices by exploiting the geometrical properties of the order cone associated with the Loewner order. To this end a modification of the HSL-colour model and a relativistic addition of matrices is introduced.

The Levenberg–Marquardt method applied to a parameter estimation problem arising from electrical resistivity tomography

Abstract An efficient and robust electrical resistivity tomographic inversion algorithm based on the Levenberg–Marquardt method is considered to obtain quantities like grain size, spatial scale and particle size distribution of mineralized rocks. The corresponding model in two dimensions is based on the Maxwell equations and leads to a partial differential equation with mixed Dirichlet–Neumann boundary conditions. The forward problem is solved numerically with the finite-difference method. However, the inverse problem at hand is a classical nonlinear and ill-posed parameter estimation problem. Linearizing and applying the Tikhonov regularization method yields an iterative scheme, the Levenberg–Marquardt method. Several large systems of equations have to be solved efficiently in each iteration step which is accomplished by the conjugate gradient method without setting up the corresponding matrix. Instead fast matrix–vector multiplications are performed directly. Therefore, the derivative and its adjoint for the parameter-to-solution map are needed. Numerical results demonstrate the performance of our method as well as the possibility to reconstruct some of the desired parameters.

Modeling Severity and Measuring Tail Risk of Norwegian Fire Claims

The probabilistic behavior of the claim severity variable plays a fundamental role in calculation of deductibles, layers, loss elimination ratios, effects of inflation, and other quantities arising in insurance. Among several alternatives for modeling severity, the parametric approach continues to maintain the leading position, which is primarily due to its parsimony and flexibility. In this article, several parametric families are employed to model severity of Norwegian fire claims for the years 1981 through 1992. The probability distributions we consider include generalized Pareto, lognormal-Pareto (two versions), Weibull-Pareto (two versions), and folded-t. Except for the generalized Pareto distribution, the other five models are fairly new proposals that recently appeared in the actuarial literature. We use the maximum likelihood procedure to fit the models and assess the quality of their fits using basic graphical tools (quantile-quantile plots), two goodness-of-fit statistics (Kolmogorov-Smirnov and Anderson-Darling), and two information criteria (AIC and BIC). In addition, we estimate the tail risk of “ground up” Norwegian fire claims using the value-at-risk and tail-conditional median measures. We monitor the tail risk levels over time, for the period 1981 to 1992, and analyze predictive performances of the six probability models. In particular, we compute the next-year probability for a few upper tail events using the fitted models and compare them with the actual probabilities.

Authors’ Reply to ‘Letter to the Editor regarding folded models and the paper by Brazauskas and Kleefeld (2011)’

We thank Dr. Scollnik for reading our paper carefully and for pointing out an important issue in the numerical example of the paper. Yes, we agree that our comparison of the newly proposed model with its closest competitors was “quick” and a bit unfair to the other distributions. Indeed, using the logtransformed data instead of original data for comparing the fits of various distributions gives a homecourt advantage to the folded-t7 (FT7) model. As one can see from Tables 1 and 2 in Scollnik (2012), the logarithmic transformation changes the values of statistical performance measures for the truncated generalized Pareto distribution (GPD), as it should, and makes the GPD a much more competitive model for the data under consideration. Moreover, we see that the fit of the truncated lognormal model is borderline and that of the truncated composite lognormal-Pareto (LNPa) is excellent. Using the fminsearch function in MATLAB for finding maximum likelihood estimators, we were able to replicate (within a small margin of rounding error) all numbers in Table 2 of the discussion paper. The direct fit of the GPD to the Norwegian fire claims now clearly passes the χ2 test and the values of its (appropriately transformed) negative log-likelihood, NLL, and the Akaike information criterion (AIC) are substantially smaller. However, while the GPD looks more competitive now, it is still uniformly outperformed by the FT7 model, according to the NLL, AIC and the χ 2 criteria. Consequently, since the truncated lognormal model yields inferior fit when compared to that of the GPD, it is also uniformly outperformed by the FT7 model. Further, since the LNPa model has three parameters (all other distributions under consideration have at most two parameters), it was not viewed in our paper as one of the “closest competitors”. Nonetheless, it certainly fits the Norwegian data very well and thus merits further investigation. To this end, we first note that, under reasonable circumstances, one would expect the model with more parameters to fit the given data set better and to have a smaller NLL than a more parsimonious model. Therefore, in such situations information-based decision rules, such as the AIC and Schwarz Bayesian criterion (SBC), come in handy. According to the AIC measure, the penalty to the LNPa model for having an additional parameter is relatively small and thus the LNPa outperforms the FT7 model (AICLNPa = 1688.521 < 1690.834 = AICFT7). On the other hand, according to the SBC measure, the conclusion is opposite: the FT7 outperforms the LNPa model (SBCFT7 = 1700.270 < 1702.674 = SBCLNPa). Note that in all these comparisons the FT7 was treated as a two-parameter model, although its degrees of freedom were fixed (ν = 7). When both parameters are estimated using

The factorization method for the acoustic transmission problem

In this work, the shape reconstruction problem of acoustically penetrable bodies from the far-field data corresponding to time-harmonic plane wave incidence is investigated within the framework of the factorization method. Although the latter technique has received considerable attention in inverse scattering problems dealing with impenetrable scatterers and it has not been elaborated for inverse transmission problems with the only exception being a work by the first two authors and co-workers. We aim to bridge this gap in the field of acoustic scattering; the paper on one hand focuses on establishing rigorously the necessary theoretical framework for the application of the factorization method to the inverse acoustic transmission problem. The main outcome of the investigation undertaken is the derivation of an explicit formula for the scatterers characteristic function, which depends solely on the far-field data feeding the inverse scattering scheme. Extended numerical examples in three dimensions are also presented, where a variety of different surfaces are successfully reconstructed by the factorization method, thus, complementing the methods validation from the computational point of view.

The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions

In this article, a new integral equation is derived to solve the exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions, and existence and uniqueness is proven for all wave numbers. We apply the boundary element collocation method to solve the system of Fredholm integral equations of the second kind, where we use constant interpolation. We observe superconvergence at the collocation nodes and illustrate it with numerical results for several smooth surfaces.

Adaptive Filters for Color Images: Median Filtering and Its Extensions

In this paper we are concerned with robust structure-preserving denoising filters for color images. We build on a recently proposed transformation from the RGB color space to the space of symmetric \(2\times 2\) matrices that has already been used to transfer morphological dilation and erosion concepts from matrix-valued data to color images. We investigate the applicability of this framework to the construction of color-valued median filters. Additionally, we introduce spatial adaptivity into our approach by morphological amoebas that offer excellent capabilities for structure-preserving filtering. Furthermore, we define color-valued amoeba M-smoothers as a generalization of the median-based concepts. Our experiments confirm that all these methods work well with color images. They demonstrate the potential of our approach to define color processing tools based on matrix field techniques.