680 research outputs found
An introduction to uncertainty quantification for kinetic equations and related problems
We overview some recent results in the field of uncertainty quantification
for kinetic equations and related problems with random inputs. Uncertainties
may be due to various reasons, such as lack of knowledge on the microscopic
interaction details or incomplete information at the boundaries or on the
initial data. These uncertainties contribute to the curse of dimensionality and
the development of efficient numerical methods is a challenge. After a brief
introduction on the main numerical techniques for uncertainty quantification in
partial differential equations, we focus our survey on some of the recent
progress on multi-fidelity methods and stochastic Galerkin methods for kinetic
equations
Structure preserving stochastic Galerkin methods for Fokker-Planck equations with background interactions
This paper is devoted to the construction of structure preserving stochastic
Galerkin schemes for Fokker-Planck type equations with uncertainties and
interacting with an external distribution, that we refer to as a background
distribution. The proposed methods are capable to preserve physical properties
in the approximation of statistical moments of the problem like nonnegativity,
entropy dissipation and asymptotic behaviour of the expected solution. The
introduced methods are second order accurate in the transient regimes and high
order for large times. We present applications of the developed schemes to the
case of fixed and dynamic background distribution for models of collective
behaviour
Multi-scale control variate methods for uncertainty quantification in kinetic equations
Kinetic equations play a major rule in modeling large systems of interacting
particles. Uncertainties may be due to various reasons, like lack of knowledge
on the microscopic interaction details or incomplete informations at the
boundaries. These uncertainties, however, contribute to the curse of
dimensionality and the development of efficient numerical methods is a
challenge. In this paper we consider the construction of novel multi-scale
methods for such problems which, thanks to a control variate approach, are
capable to reduce the variance of standard Monte Carlo techniques
Monte Carlo stochastic Galerkin methods for the Boltzmann equation with uncertainties: space-homogeneous case
In this paper we propose a novel numerical approach for the Boltzmann
equation with uncertainties. The method combines the efficiency of classical
direct simulation Monte Carlo (DSMC) schemes in the phase space together with
the accuracy of stochastic Galerkin (sG) methods in the random space. This
hybrid formulation makes it possible to construct methods that preserve the
main physical properties of the solution along with spectral accuracy in the
random space. The schemes are developed and analyzed in the case of space
homogeneous problems as these contain the main numerical difficulties. Several
test cases are reported, both in the Maxwell and in the variable hard sphere
(VHS) framework, and confirm the properties and performance of the new methods
Atomic radius and charge parameter uncertainty in biomolecular solvation energy calculations
Atomic radii and charges are two major parameters used in implicit solvent
electrostatics and energy calculations. The optimization problem for charges
and radii is under-determined, leading to uncertainty in the values of these
parameters and in the results of solvation energy calculations using these
parameters. This paper presents a new method for quantifying this uncertainty
in implicit solvation calculations of small molecules using surrogate models
based on generalized polynomial chaos (gPC) expansions. There are relatively
few atom types used to specify radii parameters in implicit solvation
calculations; therefore, surrogate models for these low-dimensional spaces
could be constructed using least-squares fitting. However, there are many more
types of atomic charges; therefore, construction of surrogate models for the
charge parameter space requires compressed sensing combined with an iterative
rotation method to enhance problem sparsity. We demonstrate the application of
the method by presenting results for the uncertainties in small molecule
solvation energies based on these approaches. The method presented in this
paper is a promising approach for efficiently quantifying uncertainty in a wide
range of force field parameterization problems, including those beyond
continuum solvation calculations.The intent of this study is to provide a way
for developers of implicit solvent model parameter sets to understand the
sensitivity of their target properties (solvation energy) on underlying choices
for solute radius and charge parameters
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
Advancements in uncertainty quantification with stochastic expansions applied to supersonic and hypersonic flows
The primary objective of this study was to develop improved methodologies for efficient and accurate uncertainty quantification with stochastic expansions and apply them to problems in supersonic and hypersonic flows. Methods introduced included approaches for efficient dimension reduction, sensitivity analysis, and sparse approximations. These methods and procedures were demonstrated on multiple stochastic models of hypersonic, planetary entry flows, which included high-fidelity, computational fluid dynamics models of radiative heating on the surface of hypersonic inflatable aerodynamic decelerators during Mars and Titan entry. For these stochastic problems, construction of an accurate surrogate model was achieved with as few as 10% of the number of model evaluations needed to construct a full dimension, total order expansion. Another objective of this work was to introduce methodologies used for further advancement of a quantification of margins and uncertainties framework. First, the use of stochastic expansions was introduced to efficiently quantify the uncertainty in system design performance metrics and performance boundaries. Then, procedures were defined to measure margin and uncertainty metrics for systems subject to multiple types of uncertainty in operating conditions and physical models. To demonstrate the new quantification of margins and uncertainties methodologies, two multi-system, multi-physics stochastic models were investigated: (1) a model for reentry dynamics, control, and convective heating and (2) a model of ground noise prediction of low-boom, supersonic aircraft configurations. Overall the methods and results of this work have outlined many effective approaches to uncertainty quantification of large-scale, high-dimension, aerospace problems containing both epistemic and inherent uncertainty. The methods presented showed significant improvement in the efficiency and accuracy of uncertainty analysis capability when stochastic expansions were used for uncertainty quantification. --Abstract, page iii
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