A variational Bayesian method for inverse problems with impulsive noise

We propose a novel numerical method for solving inverse problems subject to impulsive noises which possibly contain a large number of outliers. The approach is of Bayesian type, and it exploits a heavy-tailed t distribution for data noise to achieve robustness with respect to outliers. A hierarchical model with all hyper-parameters automatically determined from the given data is described. An algorithm of variational type by minimizing the Kullback-Leibler divergence between the true posteriori distribution and a separable approximation is developed. The numerical method is illustrated on several one- and two-dimensional linear and nonlinear inverse problems arising from heat conduction, including estimating boundary temperature, heat flux and heat transfer coefficient. The results show its robustness to outliers and the fast and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy

Review of Summation-by-parts schemes for initial-boundary-value problems

High-order finite difference methods are efficient, easy to program, scales well in multiple dimensions and can be modified locally for various reasons (such as shock treatment for example). The main drawback have been the complicated and sometimes even mysterious stability treatment at boundaries and interfaces required for a stable scheme. The research on summation-by-parts operators and weak boundary conditions during the last 20 years have removed this drawback and now reached a mature state. It is now possible to construct stable and high order accurate multi-block finite difference schemes in a systematic building-block-like manner. In this paper we will review this development, point out the main contributions and speculate about the next lines of research in this area

Coupling problem in thermal systems simulations

Building energy simulation is playing a key role in building design in order to reduce the energy consumption and, consequently, the CO2 emissions. An object-oriented tool called NEST is used to simulate all the phenomena that appear in a building. In the case of energy and momentum conservation and species transport, the current solver behaves well, but in the case of mass conservation it takes a lot of time to reach a solution. For this reason, in this work, instead of solving the continuity equations explicitly, an implicit method based on the Trust Region algorithm is proposed. Previously, a study of the properties of the model used by NEST-Building software has been done in order to simplify the requirements of the solver. For a building with only 9 rooms the new solver is a thousand times faster than the current method

Linear instability of Poiseuille flows with highly non-ideal fluids

The objective of this work is to investigate linear modal and algebraic instability in Poiseuille flows with fluids close to their vapour-liquid critical point. Close to this critical point, the ideal gas assumption does not hold and large non-ideal fluid behaviours occur. As a representative non-ideal fluid, we consider supercritical carbon dioxide (CO$_2$) at pressure of 80 bar, which is above its critical pressure of 73.9 bar. The Poiseuille flow is characterized by the Reynolds number ($Re=\rho_{w}^{*}u_{r}^{*}h^{*}/\mu_{w}^{*}$), the product of Prandtl ($Pr=\mu_{w}^{*}C_{pw}^{*}/\kappa_{w}^{*}$) and Eckert number ($Ec=u_{r}^{*2}/C_{pw}^{*}T_{w}^{*}$), and the wall temperature that in addition to pressure determines the thermodynamic reference condition. For low Eckert numbers, the flow is essentially isothermal and no difference with the well-known stability behaviour of incompressible flows is observed. However, if the Eckert number increases, the viscous heating causes gradients of thermodynamic and transport properties, and non-ideal gas effects become significant. Three regimes of the laminar base flow can be considered, subcritical (temperature in the channel is entirely below its pseudo-critical value), transcritical, and supercritical temperature regime. If compared to the linear stability of an ideal gas Poiseuille flow, we show that the base flow is more unstable in the subcritical regime, inviscid unstable in the transcritical regime, while significantly more stable in the supercritical regime. Following the corresponding states principle, we expect that qualitatively similar results will be obtained for other fluids at equivalent thermodynamic states.Comment: 34 pages, 22 figure

The Gravothermal Instability at all scales: from Turnaround Radius to Supernovae

The gravitational instability, responsible for the formation of the structure of the Universe, occurs below energy thresholds and above spatial scales of a self-gravitating expanding region, when thermal energy can no longer counterbalance self-gravity. I argue that at sufficiently-large scales, dark energy may restore thermal stability. This stability re-entrance of an isothermal sphere defines a turnaround radius, which dictates the maximum allowed size of any structure generated by gravitational instability. On the opposite limit of high energies and small scales, I will show that an ideal, quantum or classical, self-gravitating gas is subject to a high-energy relativistic gravothermal instability. It occurs at sufficiently-high energy and small radii, when thermal energy cannot support its own gravitational attraction. Applications of the phenomenon include neutron stars and core-collapse supernovae. I also extend the original Oppenheimer--Volkov calculation of the maximum mass limit of ideal neutron cores to the non-zero temperature regime, relevant to the whole cooling stage from a hot proto-neutron star down to the final cold state.Comment: Minor amendments to match published versio

Standing and travelling waves in cylindrical Rayleigh-Benard convection

The Boussinesq equations for Rayleigh-Benard convection are simulated for a cylindrical container with an aspect ratio near 1.5. The transition from an axisymmetric stationary flow to time-dependent flows is studied using nonlinear simulations, linear stability analysis and bifurcation theory. At a Rayleigh number near 25,000, the axisymmetric flow becomes unstable to standing or travelling azimuthal waves. The standing waves are slightly unstable to travelling waves. This scenario is identified as a Hopf bifurcation in a system with O(2) symmetry