arXiv subjects

Alexander Ostermann

Publications and source records attributed to Alexander Ostermann.

66 records · Page 4Linked to original sources

arXiv2015

Fast algorithms for morphological operations using run-length encoded binary images

This paper presents innovative algorithms to efficiently compute erosions and dilations of run-length encoded (RLE) binary images with arbitrary shaped structuring elements. An RLE image is given by a set of runs, where a run is a horizontal concatenation of foreground pixels. The proposed algorithms extract the skeleton of the structuring element and build distance tables of the input image, which are storing the distance to the next background pixel on the left and right hand sides. This information is then used to speed up the calculations of the erosion and dilation operator by enabling the use of techniques which allow to skip the analysis of certain pixels whenever a hit or miss occurs. Additionally the input image gets trimmed during the preprocessing steps on the base of two primitive criteria. Experimental results show the advantages over other algorithms. The source code of our algorithms is available in C++.

cs.CV↗

arXiv2014

Overcoming order reduction in diffusion-reaction splitting. Part 1: Dirichlet boundary conditions

For diffusion-reaction equations employing a splitting procedure is attractive as it reduces the computational demand and facilitates a parallel implementation. Moreover, it opens up the possibility to construct second-order integrators that preserve positivity. However, for boundary conditions that are neither periodic nor of homogeneous Dirichlet type order reduction limits its usefulness. In the situation described the Strang splitting procedure is not more accurate than Lie splitting. In this paper, we propose a splitting procedure that, while retaining all the favorable properties of the original method, does not suffer from order reduction. We demonstrate our results by conducting numerical simulations in one and two space dimensions with inhomogeneous and time dependent Dirichlet boundary conditions. In addition, a mathematical rigorous convergence analysis is conducted that confirms the results observed in the numerical simulations.

math.NA↗

arXiv2014

On the error propagation of semi-Lagrange and Fourier methods for advection problems

In this paper we study the error propagation of numerical schemes for the advection equation in the case where high precision is desired. The numerical methods considered are based on the fast Fourier transform, polynomial interpolation (semi-Lagrangian methods using a Lagrange or spline interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is conservative and has to store more than a single value per cell). We demonstrate, by carrying out numerical experiments, that the worst case error estimates given in the literature provide a good explanation for the error propagation of the interpolation-based semi-Lagrangian methods. For the discontinuous Galerkin semi-Lagrangian method, however, we find that the characteristic property of semi-Lagrangian error estimates (namely the fact that the error increases proportionally to the number of time steps) is not observed. We provide an explanation for this behavior and conduct numerical simulations that corroborate the different qualitative features of the error in the two respective types of semi-Lagrangian methods. The method based on the fast Fourier transform is exact but, due to round-off errors, susceptible to a linear increase of the error in the number of time steps. We show how to modify the Cooley--Tukey algorithm in order to obtain an error growth that is proportional to the square root of the number of time steps. Finally, we show, for a simple model, that our conclusions hold true if the advection solver is used as part of a splitting scheme.

math.NA↗

arXiv2014

A strategy to suppress recurrence in grid-based Vlasov solvers

In this paper we propose a strategy to suppress the recurrence effect present in grid-based Vlasov solvers. This method is formulated by introducing a cutoff frequency in Fourier space. Since this cutoff only has to be performed after a number of time steps, the scheme can be implemented efficiently and can relatively easily be incorporated into existing Vlasov solvers. Furthermore, the scheme proposed retains the advantage of grid-based methods in that high accuracy can be achieved. This is due to the fact that in contrast to the scheme proposed by Abbasi et al. no statistical noise is introduced into the simulation. We will illustrate the utility of the method proposed by performing a number of numerical simulations, including the plasma echo phenomenon, using a discontinuous Galerkin approximation in space and a Strang splitting based time integration.

physics.comp-ph↗

arXiv2013

An almost symmetric Strang splitting scheme for the construction of high order composition methods

In this paper we consider splitting methods for nonlinear ordinary differential equations in which one of the (partial) flows that results from the splitting procedure can not be computed exactly. Instead, we insert a well-chosen state $y_{\star}$ into the corresponding nonlinearity $b(y)y$, which results in a linear term $b(y_{\star})y$ whose exact flow can be determined efficiently. Therefore, in the spirit of splitting methods, it is still possible for the numerical simulation to satisfy certain properties of the exact flow. However, Strang splitting is no longer symmetric (even though it is still a second order method) and thus high order composition methods are not easily attainable. We will show that an iterated Strang splitting scheme can be constructed which yields a method that is symmetric up to a given order. This method can then be used to attain high order composition schemes. We will illustrate our theoretical results, up to order six, by conducting numerical experiments for a charged particle in an inhomogeneous electric field, a post-Newtonian computation in celestial mechanics, and a nonlinear population model and show that the methods constructed yield superior efficiency as compared to Strang splitting. For the first example we also perform a comparison with the standard fourth order Runge--Kutta methods and find significant gains in efficiency as well better conservation properties.

math.NA↗

arXiv2013

Exponential Integrators on Graphic Processing Units

In this paper we revisit stencil methods on GPUs in the context of exponential integrators. We further discuss boundary conditions, in the same context, and show that simple boundary conditions (for example, homogeneous Dirichlet or homogeneous Neumann boundary conditions) do not affect the performance if implemented directly into the CUDA kernel. In addition, we show that stencil methods with position-dependent coefficients can be implemented efficiently as well. As an application, we discuss the implementation of exponential integrators for different classes of problems in a single and multi GPU setup (up to 4 GPUs). We further show that for stencil based methods such parallelization can be done very efficiently, while for some unstructured matrices the parallelization to multiple GPUs is severely limited by the throughput of the PCIe bus.

math.NA↗

arXiv2013

An almost symmetric Strang splitting scheme for nonlinear evolution equations

In this paper we consider splitting methods for the time integration of parabolic and certain classes of hyperbolic partial differential equations, where one partial flow can not be computed exactly. Instead, we use a numerical approximation based on the linearization of the vector field. This is of interest in applications as it allows us to apply splitting methods to a wider class of problems from the sciences. However, in the situation described the classic Strang splitting scheme, while still a method of second order, is not longer symmetric. This, in turn, implies that the construction of higher order methods by composition is limited to order three only. To remedy this situation, based on previous work in the context of ordinary differential equations, we construct a class of Strang splitting schemes that are symmetric up to a desired order. We show rigorously that, under suitable assumptions on the nonlinearity, these methods are of second order and can then be used to construct higher order methods by composition. In addition, we illustrate the theoretical results by conducting numerical experiments for the Brusselator system and the KdV equation.

math.NA↗

arXiv2013

Explicit exponential Runge-Kutta methods of high order for parabolic problems

Exponential Runge-Kutta methods constitute efficient integrators for semilinear stiff problems. So far, however, explicit exponential Runge-Kutta methods are available in the literature up to order 4 only. The aim of this paper is to construct a fifth-order method. For this purpose, we make use of a novel approach to derive the stiff order conditions for high-order exponential methods. This allows us to obtain the conditions for a method of order 5 in an elegant way. After stating the conditions, we first show that there does not exist an explicit exponential Runge-Kutta method of order 5 with less than or equal to 6 stages. Then, we construct a fifth-order method with 8 stages and prove its convergence for semilinear parabolic problems. Finally, a numerical example is given that illustrates our convergence bound.

math.CA↗

arXiv2013

Convergence analysis of Strang splitting for Vlasov-type equations

A rigorous convergence analysis of the Strang splitting algorithm for Vlasov-type equations in the setting of abstract evolution equations is provided. It is shown that under suitable assumptions the convergence is of second order in the time step τ. As an example, it is verified that the Vlasov-Poisson equation in 1+1 dimensions fits into the framework of this analysis. Also, numerical experiments for the latter case are presented.

math.NA↗

arXiv2012

Analysis of exponential splitting methods for inhomogeneous parabolic equations

We analyze the convergence of the exponential Lie and exponential Strang splitting applied to inhomogeneous second-order parabolic equations with Dirichlet boundary conditions. A recent result on the smoothing properties of these methods allows us to prove sharp convergence results in the case of homogeneous Dirichlet boundary conditions. When no source term is present and natural regularity assumptions are imposed on the initial value, we show full-order convergence of both methods. For inhomogeneous equations, we prove full-order convergence for the exponential Lie splitting, whereas order reduction to 1.25 for the exponential Strang splitting. Furthermore, we give sufficient conditions on the inhomogeneity for full-order convergence of both methods. Moreover our theoretical convergence results explain the severe order reduction to 0.25 of splitting methods applied to problems involving inhomogeneous Dirichlet boundary conditions. We include numerical experiments to underline the sharpness of our theoretical convergence results.

math.NA↗

arXiv2012

Convergence analysis of a discontinuous Galerkin/Strang splitting approximation for the Vlasov--Poisson equations

A rigorous convergence analysis of the Strang splitting algorithm with a discontinuous Galerkin approximation in space for the Vlasov--Poisson equations is provided. It is shown that under suitable assumptions the error is of order $\mathcal{O}(τ^2+h^q +h^q / τ)$, where $τ$ is the size of a time step, $h$ is the cell size, and $q$ the order of the discontinuous Galerkin approximation. In order to investigate the recurrence phenomena for approximations of higher order as well as to compare the algorithm with numerical results already available in the literature a number of numerical simulations are performed.

math.NA↗

arXiv2011

Investigation of the recombination of the retarded shell of "born-again" CSPNe by time-dependent radiative transfer models

A standard planetary nebula stays more than 10 000 years in the state of a photoionized nebula. As long as the timescales of the most important ionizing processes are much smaller, the ionization state can be characterized by a static photoionization model and simulated with codes like CLOUDY (Ferland et al. 1998). When the star exhibits a late Helium flash, however, its ionizing flux stops within a very short period. The star then re-appears from itsopaque shell after a few years (or centuries) as a cold giant star without any hard ionizing photons. Describing the physics of such behavior requires a fully time-dependent radiative transfer model. Pollacco (1999), Kerber et al. (1999) and Lechner & Kimeswenger (2004) used data of the old nebulae around V605 Aql and V4334 Sgr to derive a model of the pre-outburst state of the CSPN in a static model. Their argument was the long recombination time scale for such thin media. With regard to these models Schoenberner (2008) critically raised the question whether a significant change in the ionization state (and thus the spectrum) has to be expected after a time of up to 80 years, and whether static models are applicable at all.

astro-ph.GA↗