arXiv subjects

Martin Frank

Publications and source records attributed to Martin Frank.

68 records · Page 4Linked to original sources

arXiv2014

StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent $P_N$ and $SP_N$ equations, of radiative transfer. The method, which works for arbitrary moment order $N$, makes use of the specific coupling between the moments in the $P_N$ equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.

physics.comp-ph↗

arXiv2014

Higher order mixed moment approximations for the Fokker-Planck equation in one space dimension

We study mixed-moment models (full zeroth moment, half higher moments) for a Fokker-Planck equation in one space dimension. Mixed-moment minimum-entropy models are known to overcome the zero net-flux problem of full-moment minimum entropy Mn models. Realizability theory for these mixed moments of arbitrary order is derived, as well as a new closure, which we refer to as Kershaw closures. They provide non-negative distribution functions combined with an analytical closure. Numerical tests are performed with standard first-order finite volume schemes and compared with a finite-difference Fokker-Planck scheme.

math-ph↗

arXiv2012

Perturbed, Entropy-Based Closure for Radiative Transfer

We derive a hierarchy of closures based on perturbations of well-known entropy-based closures; we therefore refer to them as perturbed entropy-based models. Our derivation reveals final equations containing an additional convective and diffusive term which are added to the flux term of the standard closure. We present numerical simulations for the simplest member of the hierarchy, the perturbed M1 or PM1 model, in one spatial dimension. Simulations are performed using a Runge-Kutta discontinuous Galerkin method with special limiters that guarantee the realizability of the moment variables and the positivity of the material temperature. Improvements to the standard M1 model are observed in cases where unphysical shocks develop in the M1 model.

physics.comp-ph↗

arXiv2011

Optimal radiotherapy treatment planning using minimum entropy models

We study the problem of finding an optimal radiotherapy treatment plan. A time-dependent Boltzmann particle transport model is used to model the interaction between radiative particles with tissue. This model allows for the modeling of inhomogeneities in the body and allows for anisotropic sources modeling distributed radiation---as in brachytherapy---and external beam sources---as in teletherapy. We study two optimization problems: minimizing the deviation from a spatially-dependent prescribed dose through a quadratic tracking functional; and minimizing the survival of tumor cells through the use of the linear-quadratic model of radiobiological cell response. For each problem, we derive the optimality systems. In order to solve the state and adjoint equations, we use the minimum entropy approximation; the advantages of this method are discussed. Numerical results are then presented.

math.OC↗

arXiv2011

Optimal prediction for radiative transfer: A new perspective on moment closure

Moment methods are classical approaches that approximate the mesoscopic radiative transfer equation by a system of macroscopic moment equations. An expansion in the angular variables transforms the original equation into a system of infinitely many moments. The truncation of this infinite system is the moment closure problem. Many types of closures have been presented in the literature. In this note, we demonstrate that optimal prediction, an approach originally developed to approximate the mean solution of systems of nonlinear ordinary differential equations, can be used to derive moment closures. To that end, the formalism is generalized to systems of partial differential equations. Using Gaussian measures, existing linear closures can be re-derived, such as $P_N$, diffusion, and diffusion correction closures. This provides a new perspective on several approximations done in the process and gives rise to ideas for modifications to existing closures.

math-ph↗

arXiv2009

Application of Generalized Fokker-Planck Theory To Electron And Photon Transport In Tissue

We study a deterministic method for particle transport in tissue in selected medical applications. Generalized Fokker-Planck (GFP) theory has been developed to improve the Fokker-Planck (FP) equation in cases where scattering is forward-peaked and where there is a sufficient amount of large-angle scattering. We compare grid-based numerical solutions to Fokker-Planck and Generalized Fokker-Planck (GFP) in realistic applications. Electron dose calculations in heterogeneous parts of the human body are performed. Accurate electron scattering cross sections are therefore included and their incorporation in our model is extensively described. Moreover, we solve GFP approximations of the radiative transport equation to investigate reflectance and transmittance of light in tissue. All results are compared with either Monte Carlo or discrete-ordinates transport solutions.

physics.med-ph↗

arXiv2009

Deterministic Partial Differential Equation Model for Dose Calculation in Electron Radiotherapy

Treatment with high energy ionizing radiation is one of the main methods in modern cancer therapy that is in clinical use. During the last decades, two main approaches to dose calculation were used, Monte Carlo simulations and semi-empirical models based on Fermi-Eyges theory. A third way to dose calculation has only recently attracted attention in the medical physics community. This approach is based on the deterministic kinetic equations of radiative transfer. Starting from these, we derive a macroscopic partial differential equation model for electron transport in tissue. This model involves an angular closure in the phase space. It is exact for the free-streaming and the isotropic regime. We solve it numerically by a newly developed HLLC scheme based on [BerCharDub], that exactly preserves key properties of the analytical solution on the discrete level. Several numerical results for test cases from the medical physics literature are presented.

physics.med-ph↗

arXiv2009

Diffusive Corrections to Pn Approximations

In this paper, we investigate moment methods from a general point of view using an operator notation. This theoretical approach lets us explore the moment closure problem in more detail. This gives rise to a new idea, proposed in [Levermore2005, Levermore2009], of how to improve the well-known Pn approximations. We systematically develop a diffusive correction to the Pn equations from the operator formulation - the so-called Dn approximation. We validate the new approach with numerical examples in one and two dimensions.

physics.comp-ph↗

arXiv2009

Optimal prediction for moment models: Crescendo diffusion and reordered equations

A direct numerical solution of the radiative transfer equation or any kinetic equation is typically expensive, since the radiative intensity depends on time, space and direction. An expansion in the direction variables yields an equivalent system of infinitely many moments. A fundamental problem is how to truncate the system. Various closures have been presented in the literature. We want to study moment closure generally within the framework of optimal prediction, a strategy to approximate the mean solution of a large system by a smaller system, for radiation moment systems. We apply this strategy to radiative transfer and show that several closures can be re-derived within this framework, e.g. $P_N$, diffusion, and diffusion correction closures. In addition, the formalism gives rise to new parabolic systems, the reordered $P_N$ equations, that are similar to the simplified $P_N$ equations. Furthermore, we propose a modification to existing closures. Although simple and with no extra cost, this newly derived crescendo diffusion yields better approximations in numerical tests.

math-ph↗

arXiv2009

Optimal treatment planning governed by kinetic equations

In this paper we study a problem in radiotherapy treatment planning, where the evolution of the radiation field is governed by a deterministic Boltzmann transport equation. We show existence, uniqueness and regularity of solutions to an optimal dose distribution problem constrained by the Boltzmann Continuous Slowing-Down equation in an appropriate function space. The main new difficulty is the treatment of the stopping power term. Furthermore, we characterize optimal controls for problems governed by this transport equation.

math.OC↗

arXiv2009

Adaptive Finite Element Simulation of the Time-dependent Simplified PN Equations

The steady-state simplified $P_N$ approximation to the radiative transport equation has been successfully applied to many problems involving radiation. Recently, time-dependent simplified $P_N$ equations have been derived by an asymptotic analysis similar to the asymptotic derivation of the steady-state $SP_N$ equations \cite{FraKlaLarYas07}. In this paper, we present computational results for the time-dependent $SP_N$ equations in two dimensions, obtained by using an adaptive finite element approach. Several numerical comparisons with other existing models are shown.

physics.comp-ph↗

arXiv2008

Higher order minimum entropy approximations in radiative transfer

In this paper we approximate the radiative transfer equations by the method of moments, constructing mesoscopic approximations of arbitrary order of the otherwise microscopic system. To define the necessary closure a minimum entropy approach is utilized. While in radiative transfer, the minimum entropy closure for moment systems up to the first-order moment is well known, higher-order minimum entropy closures have not been implemented. This is probably due to the fact that the closure cannot be expressed in analytical form. Our focus thus lies in developing some general results about the minimum entropy system and in deriving a numerical closure. By extending to higher order, among increasing the precision, we are able to overcome difficulties that arise for the first order minimum entropy method. Numerical experiments in a 1-dimensional domain irradiated by two beams or with internal source show the accuracy of this approach.

physics.comp-ph↗

arXiv2005

Optimizing at the Ergodic Edge

Using a simple, annealed model, some of the key features of the recently introduced extremal optimization heuristic are demonstrated. In particular, it is shown that the dynamics of local search possesses a generic critical point under the variation of its sole parameter, separating phases of too greedy (non-ergodic, jammed) and too random (ergodic) exploration. Comparison of various local search methods within this model suggests that the existence of the critical point is essential for the optimal performance of the heuristic.

physics.comp-ph↗

arXiv2001

Brillouin Scattering Study of Propylene Carbonate: An Evaluation of Phenomenological and Mode Coupling Analyses

Brillouin scattering spectra of the molecular glassformer propylene carbonate (PC) in the temperature range 140 K to 350 K were analyzed using both the phenomenological Cole-Davidson memory function and a hybrid memory function consisting of the Cole-Davidson function plus a power-law term representing the critical decay part of the fast beta relaxation. The spectra were also analyzed using the extended two-correlator schematic MCT model recently employed by Gotze and Voigtmann to analyze depolarized light backscattering,dielectric, and neutron-scattering spectra of PC [W.Gotze and Th. Voigtman, Phys. Rev. E61, 4133 (2000). We assess the ability of the phenomenological and MCT fits, each with three free fitting parameters, to simultaneously describe the spectra and give reasonable values for the alpha-relaxation time tau-alpha

cond-mat.soft↗