Enhanced goal-oriented error assessment and computational strategies in adaptive reduced basis solver for stochastic problems

This work focuses on providing accurate low-cost approximations of stochastic ¿nite elements simulations in the framework of linear elasticity. In a previous work, an adaptive strategy was introduced as an improved Monte-Carlo method for multi-dimensional large stochastic problems. We provide here a complete analysis of the method including a new enhanced goal-oriented error estimator and estimates of CPU (computational processing unit) cost gain. Technical insights of these two topics are presented in details, and numerical examples show the interest of these new developments.Postprint (author's final draft

Unified Gas-kinetic Wave-Particle Methods III: Multiscale Photon Transport

In this paper, we extend the unified gas-kinetic wave-particle (UGKWP) method to the multiscale photon transport. In this method, the photon free streaming and scattering processes are treated in an un-splitting way. The duality descriptions, namely the simulation particle and distribution function, are utilized to describe the photon. By accurately recovering the governing equations of the unified gas-kinetic scheme (UGKS), the UGKWP preserves the multiscale dynamics of photon transport from optically thin to optically thick regime. In the optically thin regime, the UGKWP becomes a Monte Carlo type particle tracking method, while in the optically thick regime, the UGKWP becomes a diffusion equation solver. The local photon dynamics of the UGKWP, as well as the proportion of wave-described and particle-described photons are automatically adapted according to the numerical resolution and transport regime. Compared to the $S_n$ -type UGKS, the UGKWP requires less memory cost and does not suffer ray effect. Compared to the implicit Monte Carlo (IMC) method, the statistical noise of UGKWP is greatly reduced and computational efficiency is significantly improved in the optically thick regime. Several numerical examples covering all transport regimes from the optically thin to optically thick are computed to validate the accuracy and efficiency of the UGKWP method. In comparison to the $S_n$ -type UGKS and IMC method, the UGKWP method may have several-order-of-magnitude reduction in computational cost and memory requirement in solving some multsicale transport problems.Comment: 27 pages, 15 figures. arXiv admin note: text overlap with arXiv:1810.0598

Hybrid PDE solver for data-driven problems and modern branching

The numerical solution of large-scale PDEs, such as those occurring in data-driven applications, unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgement in the modelling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD. We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods. We close this work with an invitation to the fully nonlinear case and open research questions.Comment: 23 pages, 7 figures; Final SMUR version; To appear in the European Journal of Applied Mathematics (EJAM

Towards a unified linear kinetic transport model with the trace ion module for EIRENE

Linear kinetic Monte Carlo particle transport models are frequently employed in fusion plasma simulations to quantify atomic and surface effects on the main plasma flow dynamics. Separate codes are used for transport of neutral particles (incl. radiation) and charged particles (trace impurity ions). Integration of both modules into main plasma fluid solvers provides then self consistent solutions, in principle. The required interfaces are far from trivial, because rapid atomic processes in particular in the edge region of fusion plasmas require either smoothing and resampling, or frequent transfer of particles from one into the other Monte Carlo code. We propose a different scheme here, in which despite the inherently different mathematical form of kinetic equations for ions and neutrals (e.g. Fokker-Planck vs. Boltzmann collision integrals) both types of particle orbits can be integrated into one single code. We show that the approximations and shortcomings of this "single sourcing" concept (e.g., restriction to explicit ion drift orbit integration) can be fully tolerable in a wide range of typical fusion edge plasma conditions, and be overcompensated by the code-system simplicity, as well as by inherently ensured consistency in geometry (one single numerical grid only) and (the common) atomic and surface process modulesComment: 15 pages, 7 figure

Energy Harvesting From Bistable Systems Under Random Excitation

The transformation of otherwise unused vibrational energy into electric energy through the use of piezoelectric energy harvesting devices has been the subject of numerous investigations. The mechanical part of such a device is often constructed as a cantilever beam with applied piezo patches. If the harvester is designed as a linear resonator the power output relies strongly on the matching of the natural frequency of the beam and the frequency of the harvested vibration which restricts the applicability since most vibrations which are found in built environments are broad-banded or stochastic in nature. A possible approach to overcome this restriction is the use of permanent magnets to impose a nonlinear restoring force on the beam that leads to a broader operating range due to large amplitude motions over a large range of excitation frequencies. In this paper such a system is considered introducing a refined modeling with a modal expansion that incorporates two modal functions and a refined modeling of the magnet beam interaction. The corresponding probability density function in case of random excitation is calculated by the solution of the corresponding Fokker-Planck equation and compared with results from Monte Carlo simulations. Finally some measurements of ambient excitations are discussed.DFG, 253161314, Untersuchung des nichtlinearen dynamischen Verhaltens von stochastisch erregten Energy Harvesting Systemen mittels Lösung der Fokker-Planck-Gleichun