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.
We establish asymptotic diffusion limits of the non-classical transport equation derived in . 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.
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.
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.
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.