@misc{KleefeldKirsch, author = {Kleefeld, Andreas and Kirsch, Andreas}, title = {The factorization method for a conductive boundary condition}, series = {Journal of Integral Equations and Applications}, volume = {24}, journal = {Journal of Integral Equations and Applications}, number = {4}, issn = {0897-3962}, doi = {10.1216/JIE-2012-24-4-575}, pages = {575 -- 601}, abstract = {In this article, the inverse scattering problem for obstacles with conductive boundary conditions is considered. The problem at hand is solved with the Factorization Method; that is, the operator and its factorization is derived. To obtain generic far field data for several obstacles the boundary element method is employed to solve the integral equation numerically. The far field data are highly accurate due to superconvergence. Finally, we show that we are able to reconstruct several obstacles and thus confirm the validity of the new operator factorization.}, language = {en} } @misc{KleefeldReissel, author = {Kleefeld, Andreas and Reißel, Martin}, title = {The Levenberg-Marquardt method applied to a parameter estimation problem arising from electrical resistivity tomography}, series = {Journal of Applied Mathematics and Computation}, volume = {217}, journal = {Journal of Applied Mathematics and Computation}, number = {9}, issn = {0096-3003}, pages = {4490 -- 4501}, abstract = {An efficient and robust electrical resistivity tomographic inversion algorithm based on the Levenberg-Marquardt method is considered to obtain quantities like grain size, source discrimination and particle size distribution. 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 classic nonlinear and ill-posed parameter estimation problem. Linearizing it and applying the Tikhonov regularization method yields an iterative scheme, the Levenberg-Marquardt method. Several systems of equations of large size 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.}, language = {en} } @misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {Folded and log-folded-t distributions as models for insurance loss data}, doi = {http://www.tandfonline.com/doi/abs/10.1080/03461230903424199}, abstract = {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.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {Linear Fredholm integral equations of the first kind over surfaces in 3D}, abstract = {Linear Fredholm integral equations of the first kind over surfaces are less familiar than those of the second kind, although they arise in many applications like computer tomography, heat conduction and inverse scattering. This article emphasizes their numerical treatment, since discretization usually leads to ill-conditioned linear systems. Strictly speaking, the matrix is nearly singular and ordinary numerical methods fail. However, there exists a numerical regularization method - the Tikhonov method - to deal with this ill-conditioning and to obtain accurate numerical results.}, language = {en} } @misc{KleefeldLin, author = {Kleefeld, Andreas and Lin, Tzu-Chu}, title = {The nonlinear Landweber method applied to an inverse scattering problem for sound-soft obstacles in 3D}, abstract = {The inverse scattering problem for sound-soft obstacles is considered for both smooth and piecewise smooth surfaces in 3D. The nonlinear and ill-posed integral equation of the first kind is solved by the nonlinear Landweber method. It is an iterative regularization scheme to obtain approximations for the unknown boundary of the obstacle. It is stable with respect to noise and essentially no extra work is required to incorporate several incident waves. So far, it has only been applied to the two dimensional case. Two different integral equations are presented to obtain far-field data. Furthermore, the domain derivative and its adjoint are characterized. The integral equations of the second kind are approximated by a boundary element collocation method. The two-grid method is used to solve the large and dense linear systems. Numerical examples are illustrated to show that both smooth and piecewise smooth obstacles can be reconstructed with this method, where the latter case has not yet been reported.}, language = {en} } @misc{KleefeldLin, author = {Kleefeld, Andreas and Lin, Tzu-Chu}, title = {Boundary element collocation method for solving the exterior Neumann problem for Helmholtz's equation in three dimensions}, series = {Electronic transactions on numerical analysis}, volume = {39}, journal = {Electronic transactions on numerical analysis}, issn = {1068-9613}, pages = {113 -- 143}, abstract = {We describe a boundary integral equation that solves the exterior Neumann problem for the Helmholtz equation in three dimensions. The unique solution is found by approximating a Fredholm integral equation of the second kind with the boundary element collocation method. We prove uperconvergence at the collocation points, distinguishing the cases of even and odd interpolation. Numerical examples demonstrate the performance of the method solving the integral equation and confirm the superconvergence.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {The transmission problem for the Helmholtz equation in R³}, series = {Computational Methods in Applied Mathematics}, volume = {12}, journal = {Computational Methods in Applied Mathematics}, number = {3}, issn = {1609-4840}, pages = {330 -- 350}, abstract = {The problem under consideration is the three-dimensional transmission problem for time harmonic acoustic waves for two homogeneous media. A simply-connected and bounded region with sufficiently smooth boundary is immersed in an infinite medium. Each medium is c haracterized by the space independent wave number κ and the density μ. The system of boundary integral equations is reviewed as well as an existence and uniqueness result. The system is approximated by the boundary element collocation method and consistency, stability, and convergence is proved. In addition, superconvergence is proved and numerical results illustrate the agreement with these theoretical results. No numerical results seem to be reported for this method yet.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {A modified boundary integral equation for solving the exterior Robin problem for the Helmholtz equation in three dimensions}, series = {Applied Mathematics and Computation}, volume = {2012}, journal = {Applied Mathematics and Computation}, number = {219}, issn = {0096-3003}, doi = {10.1016/j.amc.2012.08.055}, pages = {2114 -- 2123}, abstract = {A boundary integral equation with modified fundamental solution to solve the exterior Robin problem for Helmholtz's equation is considered. A specific choice of the coefficients of these added terms ensure the unique solvability and, in addition, they can be chosen to ensure the minimization of the least-squares difference of the modified and the exact Green's function or the minimization of the condition number. Numerical results are reported showing the robustness and the superconvergence of the new method.}, language = {en} } @misc{KleefeldBrazauskas, author = {Kleefeld, Andreas and Brazauskas, Vytaras}, title = {A statistical application of the quantile mechanics approach: MTM estimators for the parameters of t and gamma distribution}, series = {European Journal of Applied Mathematics}, volume = {23}, journal = {European Journal of Applied Mathematics}, number = {5}, issn = {0956-7925}, doi = {10.1017/S0956792512000137}, pages = {593 -- 610}, abstract = {In this paper, we revisit the quantile mechanics approach, which was introduced by Steinbrecher and Shaw (Steinbrecher, G. \& Shaw, W. T. (2008) Quantile mechanics. European. J. Appl. Math. 19, 87-112). Our objectives are (i) to derive the method of trimmed moments (MTM) estimators for the parameters of gamma and Student's t distributions, and (ii) to examine their large- and small-sample statistical properties. Since trimmed moments are defined through the quantile function of the distribution, quantile mechanics seems like a natural approach for achieving objective (i). To accomplish the second goal, we rely on the general large sample results for MTMs, which were established by Brazauskas et al. (Brazauskas, V., Jones, B. L. \& Zitikis, R. (2009) Robust fitting of claim severity distributions and the method of trimmed moments. J. Stat. Plan. Inference 139, 2028-2043), and then use Monte Carlo simulations to investigate small-sample behaviour of the newly derived estimators. We find that, unlike the maximum likelihood method, which usually yields fully efficient but non-robust estimators, the MTM estimators are robust and offer competitive trade-offs between robustness and efficiency. These properties are essential when one employs gamma or Student's t distributions in such outlier-prone areas as insurance and finance.}, language = {en} } @misc{Kleefeld, author = {Kleefeld, Andreas}, title = {The exterior problem for the Helmholtz equation with mixed boundary conditions in three dimensions}, series = {International Journal of Computer Mathematics}, volume = {89}, journal = {International Journal of Computer Mathematics}, number = {17}, issn = {0020-7160}, doi = {10.1080/00207160.2012.709932}, pages = {2392 -- 2409}, abstract = {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.}, language = {en} }