A High Order Stochastic Asymptotic Preserving Scheme for Chemotaxis Kinetic Models with Random Inputs

In this paper, we develop a stochastic Asymptotic-Preserving (sAP) scheme for the kinetic chemotaxis system with random inputs, which will converge to the modified Keller-Segel model with random inputs in the diffusive regime. Based on the generalized Polynomial Chaos (gPC) approach, we design a high order stochastic Galerkin method using implicit-explicit (IMEX) Runge-Kutta (RK) time discretization with a macroscopic penalty term. The new schemes improve the parabolic CFL condition to a hyperbolic type when the mean free path is small, which shows significant efficiency especially in uncertainty quantification (UQ) with multi-scale problems. The stochastic Asymptotic-Preserving property will be shown asymptotically and verified numerically in several tests. Many other numerical tests are conducted to explore the effect of the randomness in the kinetic system, in the aim of providing more intuitions for the theoretic study of the chemotaxis models

Element learning: a systematic approach of accelerating finite element-type methods via machine learning, with applications to radiative transfer

In this paper, we propose a systematic approach for accelerating finite element-type methods by machine learning for the numerical solution of partial differential equations (PDEs). The main idea is to use a neural network to learn the solution map of the PDEs and to do so in an element-wise fashion. This map takes input of the element geometry and the PDEs' parameters on that element, and gives output of two operators -- (1) the in2out operator for inter-element communication, and (2) the in2sol operator (Green's function) for element-wise solution recovery. A significant advantage of this approach is that, once trained, this network can be used for the numerical solution of the PDE for any domain geometry and any parameter distribution without retraining. Also, the training is significantly simpler since it is done on the element level instead on the entire domain. We call this approach element learning. This method is closely related to hybridizbale discontinuous Galerkin (HDG) methods in the sense that the local solvers of HDG are replaced by machine learning approaches. Numerical tests are presented for an example PDE, the radiative transfer equation, in a variety of scenarios with idealized or realistic cloud fields, with smooth or sharp gradient in the cloud boundary transition. Under a fixed accuracy level of $10^{-3}$ in the relative $L^2$ error, and polynomial degree $p=6$ in each element, we observe an approximately 5 to 10 times speed-up by element learning compared to a classical finite element-type method

Mini-Workshop: Innovative Trends in the Numerical Analysis and Simulation of Kinetic Equations

In multiscale modeling hierarchy, kinetic theory plays a vital role in connecting microscopic Newtonian mechanics and macroscopic continuum mechanics. As computing power grows, numerical simulation of kinetic equations has become possible and undergone rapid development over the past decade. Yet the unique challenges arising in these equations, such as highdimensionality, multiple scales, random inputs, positivity, entropy dissipation, etc., call for new advances of numerical methods. This mini-workshop brought together both senior and junior researchers working on various fastpaced growing numerical aspects of kinetic equations. The topics include, but were not limited to, uncertainty quantification, structure-preserving methods, phase transitions, asymptotic-preserving schemes, and fast methods for kinetic equations

Monte Carlo gPC methods for diffusive kinetic flocking models with uncertainties

In this paper we introduce and discuss numerical schemes for the approximation of kinetic equations for flocking behavior with phase transitions that incorporate uncertain quantities. This class of schemes here considered make use of a Monte Carlo approach in the phase space coupled with a stochastic Galerkin expansion in the random space. The proposed methods naturally preserve the positivity of the statistical moments of the solution and are capable to achieve high accuracy in the random space. Several tests on a kinetic alignment model with self propulsion validate the proposed methods both in the homogeneous and inhomogeneous setting, shading light on the influence of uncertainties in phase transition phenomena driven by noise such as their smoothing and confidence band

Adaptive Methods for Modeling Chemical Transport in the Earth's atmosphere

This work is devoted to analysis and development of efficient adaptive algorithms for problems related to the transport of chemical species in the Earth’s atmosphere from data of remote-sensing instruments. Focal point of the thesis is the assessment of different types of errors by a posteriori error analysis. On the basis of these a posteriori error estimates the algebraic iteration can be adjusted to discretization within a succesive mesh adaptation process. The presented adaptive algorithms are applicable for a wide range of problems

Model atmospheres of sub-stellar mass objects

We present an outline of basic assumptions and governing structural equations describing atmospheres of substellar mass objects, in particular the extrasolar giant planets and brown dwarfs. Although most of the presentation of the physical and numerical background is generic, details of the implementation pertain mostly to the code CoolTlusty. We also present a review of numerical approaches and computer codes devised to solve the structural equations, and make a critical evaluation of their efficiency and accuracy.Comment: 31 pages, 10 figure

Uncertainty quantification for problems in radionuclide transport

The field of radionuclide transport has long recognised the stochastic nature of the problems encountered. Many parameters that are used in computational models are very difficult, if not impossible, to measure with any great degree of confidence. For example, bedrock properties can only be measured at a few discrete points, the properties between these points may be inferred or estimated using experiments but it is difficult to achieve any high levels of confidence. This is a major problem when many countries around the world are considering deep geologic repositories as a disposal option for long-lived nuclear waste but require a high degree of confidence that any release of radioactive material will not pose a risk to future populations. In this thesis we apply Polynomial Chaos methods to a model of the biosphere that is similar to those used to assess exposure pathways for humans and associated dose rates by many countries worldwide. We also apply the Spectral-Stochastic Finite Element Method to the problem of contaminated fluid flow in a porous medium. For this problem we use the Multi-Element generalized Polynomial Chaos method to discretise the random dimensions in a manner similar to the well known Finite Element Method. The stochastic discretisation is then refined adaptively to mitigate the build up errors over the solution times. It was found that these methods have the potential to provide much improved estimates for radionuclide transport problems. However, further development is needed in order to obtain the necessary efficiency that would be required to solve industrial problems