The coolest path problem
Martin Frank Armin Fügenschuh Michael Herty Lars Schewe
Networks & Heterogeneous Media 2010, 5(1): 143-162 doi: 10.3934/nhm.2010.5.143
We introduce the coolest path problem, which is a mixture of two well-known problems from distinct mathematical fields. One of them is the shortest path problem from combinatorial optimization. The other is the heat conduction problem from the field of partial differential equations. Together, they make up a control problem, where some geometrical object traverses a digraph in an optimal way, with constraints on intermediate or the final state. We discuss some properties of the problem and present numerical solution techniques. We demonstrate that the problem can be formulated as a linear mixed-integer program. Numerical solutions can thus be achieved within one hour for instances with up to 70 nodes in the graph.
keywords: Shortest Path. Integer Programming Heat Equation
Fractional diffusion limits of non-classical transport equations
Martin Frank Weiran Sun
Kinetic & Related Models 2018, 11(6): 1503-1526 doi: 10.3934/krm.2018059

We establish asymptotic diffusion limits of the non-classical transport equation derived in [12]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis is based on a combination of the Fourier transform and a moment method. We put special focus on dealing with anisotropic scattering, which compared to the isotropic case makes the analysis significantly more involved.

keywords: Fractional diffusion limit non-classical transport
Optimal prediction for radiative transfer: A new perspective on moment closure
Martin Frank Benjamin Seibold
Kinetic & Related Models 2011, 4(3): 717-733 doi: 10.3934/krm.2011.4.717
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.
keywords: diffusion approximation. method of moments Radiative transfer optimal prediction measure
On a generalized Boltzmann equation for non-classical particle transport
Martin Frank Thierry Goudon
Kinetic & Related Models 2010, 3(3): 395-407 doi: 10.3934/krm.2010.3.395
We are interested in non-standard transport equations where the description of the scattering events involves an additional "memory variable''. We establish the well posedness and investigate the diffusion asymptotics of such models. While the questions we address are quite classical the analysis is original since the usual dissipative properties of collisional transport equations is broken by the introduction of the memory terms.
keywords: Transport equations Diffusion asymptotics Radiative transfer.
Perturbed, entropy-based closure for radiative transfer
Martin Frank Cory D. Hauck Edgar Olbrant
Kinetic & Related Models 2013, 6(3): 557-587 doi: 10.3934/krm.2013.6.557
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 $M_1$ or $PM_1$ 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 $M_1$ model are observed in cases where unphysical shocks develop in the $M_1$ model.
keywords: method of moments Radiative transfer entropy closure.

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