In the first part of the thesis, we develop a generalized polynomial chaos approach based stochastic Galerkin (gPC-SG) method for the linear semi-conductor Boltzmann equation with random inputs and diffusive scalings. The random inputs are due to uncertainties in the collision kernel or initial data. We study the regularity (uniform in the Knudsen number) of the solution in the random space, and prove the spectral accuracy of the gPC-SG method. We then use the asymptotic-preserving framework for the deterministic counterpart to come up with the stochastic asymptotic-preserving (sAP) gPC-SG method for the problem under study which is efficient in the diffusive regime. Numerical experiments are conducted to validate the accuracy and asymptotic properties of the method. In the second part, we study the linear transport equation under diffusive scaling and with random inputs. The method is based on the gPC-SG framework. Several theoretical aspects will be addressed. A uniform numerical stability with respect to the Knudsen number and a uniform error estimate is given. For temporal and spatial discretizations, we apply the implicit-explicit (IMEX) scheme under the micro-macro decomposition framework and the discontinuous Galerkin (DG) method. A rigorous proof of the sAP property is given. Extensive numerical experiments that validate the accuracy and sAP of the method are shown. In the last part, we study a class of highly oscillatory transport equations that arise in semiclassical modeling of non-adiabatic quantum dynamics. These models contain uncertainties, particularly in coefficients that correspond to the potentials of the molecular system. We first focus on a highly oscillatory scalar model with random uncertainty. Our method is built upon the nonlinear geometrical optics (NGO) based method for numerical approximations of deterministic equations, which can obtain accurate pointwise solution even without numerically resolving spatially and temporally the oscillations. With the random uncertainty, we show that such a method has oscillatory higher order derivatives in the random space, thus requires a frequency dependent discretization in the random space. We modify this method by introducing a new "time" variable based on the phase, which is shown to be non-oscillatory in the random space, based on which we develop a gPC-SG method that can capture oscillations with the frequency-independent time step, mesh size as well as the degree of polynomial chaos. A similar approach is then extended to a semiclassical surface hopping model system with a similar numerical conclusion. Various numerical examples attest that these methods indeed capture accurately the solution statistics pointwisely even though none of the numerical parameters resolve the high frequencies of the solution.

This book explores recent advances in uncertainty quantification for hyperbolic, kinetic, and related problems. The contributions address a range of different aspects, including: polynomial chaos expansions, perturbation methods, multi-level Monte Carlo methods, importance sampling, and moment methods. The interest in these topics is rapidly growing, as their applications have now expanded to many areas in engineering, physics, biology and the social sciences. Accordingly, the book provides the scientific community with a topical overview of the latest research efforts.

Uncertainty Quantification in Multiscale Materials Modeling provides a complete overview of uncertainty quantification (UQ) in computational materials science. It provides practical tools and methods along with examples of their application to problems in materials modeling. UQ methods are applied to various multiscale models ranging from the nanoscale to macroscale. This book presents a thorough synthesis of the state-of-the-art in UQ methods for materials modeling, including Bayesian inference, surrogate modeling, random fields, interval analysis, and sensitivity analysis, providing insight into the unique characteristics of models framed at each scale, as well as common issues in modeling across scales. Synthesizes available UQ methods for materials modeling Provides practical tools and examples for problem solving in modeling material behavior across various length scales Demonstrates UQ in density functional theory, molecular dynamics, kinetic Monte Carlo, phase field, finite element method, multiscale modeling, and to support decision making in materials design Covers quantum, atomistic, mesoscale, and engineering structure-level modeling and simulation

This thesis gives an overview of the current results on uncertainty quantification and sensitivity analysis for multiscale kinetic equations with random inputs, with an emphasis on the author's contribution to this field. In the first part of this thesis we consider a kinetic-fluid model for disperse two-phase flows with uncertainty in the fine particle regime. We propose a stochastic asymptotic-preserving (s-AP) scheme in the generalized polynomial chaos stochastic Galerkin (gPC-sG) framework, which allows the efficient computation of the problem in both kinetic and hydrodynamic regimes. The s-AP property is proved by deriving the equilibrium of the gPC version of the Fokker-Planck operator. The coefficient matrices that arise in a Helmholtz equation and a Poisson equation, essential ingredients of the algorithms, are proved to be positive definite under reasonable and mild assumptions. The computation of the gPC version of a translation operator that arises in the inversion of the Fokker-Planck operator is accelerated by a spectrally accurate splitting method. Numerical examples illustrate the s-AP property and the efficiency of the gPC-sG method in various asymptotic regimes. In the second part of this thesis we consider the same kinetic-fluid model with random initial inputs in the light particle regime. Using energy estimates, we prove the uniform regularity in the random space of the model for random initial data near the global equilibrium in some suitable Sobolev spaces, with the randomness in the initial particle distribution and fluid velocity. By hypocoercivity arguments, we prove that the energy decays exponentially in time, which means that the long time behavior of the solution is insensitive to such randomness in the initial data. Then we consider the gPC-sG method for the same model. For initial data near the global equilibrium and smooth enough in the physical and random spaces, we prove that the gPC-sG method has spectral accuracy, uniformly in time and the Knudsen number, and the error decays exponentially in time. In the third part of this thesis we propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. The method uses locally supported piecewise polynomials as an orthonormal basis of the random space. By a sparse approach, only a moderate number of basis functions is required to achieve good accuracy in multi-dimensional random spaces. We discover a sparse structure of a set of basis-related coefficients, which allows us to accelerate the computation of the collision operator. Regularity of the solution of the Boltzmann equation in the random space and an accuracy result of the stochastic Galerkin method are proved in multi-dimensional cases. The efficiency of the method is illustrated by numerical examples with uncertainties from the initial data, boundary data and collision kernel. In the fourth part of this thesis we explore the possibility of using Generalized polynomial chaos (gPC) for uncertainty quantification in hyperbolic problems. GPC has been extensively used in uncertainty quantification problems to handle random variables. For gPC to be valid, one requires high regularity on the random space that hyperbolic type problems usually cannot provide, and thus it is believed to behave poorly in those systems. We provide a counter-argument, and show that despite the solution profile itself develops singularities in the random space, which prevents the use of gPC, the physical quantities such as shock emergence time, shock location, and shock width are all smooth functions of random variables in the initial data: with proper shifting, the solution's polynomial interpolation approximates with high accuracy. The studies were inspired by the stability results from hyperbolic systems. We use the Burgers' equation as an example for thorough analysis, and the analysis could be extended to general conservation laws with convex fluxes.

This dissertation consists of two parts: quantum transitions (Part 1) and hydrodynamic limits of kinetic equations (Part 2). In both parts, we investigate the inner mathematical connections between equations for different physics at different scales, and use these connections to design efficient computational methods for multi-scale problems. Despite its numerous applications in chemistry and physics, the mathematics of quantum transition is not well understood. Using the Wigner transformation, we derive semi-classical models in phase space for two problems: the dynamics of electrons in crystals near band- crossing points; surface hopping of quantum molecules when the Born-Oppenheimer approximation breaks down. In both cases, particles may jump between states with comparable energies. Our models can capture the transition rates for such processes. We provide analytic analysis of and numerical methods for our models, demonstrated by explicit examples. The second part is to construct numerical methods for kinetic equation that are efficient in the hydrodynamic regime. Asymptotically, the kinetic equations reduce to fluid dynamics described by the Euler or Navier-Stokes equations in the fluid regime. Numerically the Boltzmann equation is still hard to handle in the hydrodynamic regime due to the stiff collision term. We review the theoretical work that links the two sets of equations, and present our asymptotic-preserving numerical solvers for the Boltzmann equation that naturally capture the asymptotic limits in the hydrodynamic regime. We also extend our methods to the case of multi-species systems.

This book presents the fundamental notions and advanced mathematical tools in the stochastic modeling of uncertainties and their quantification for large-scale computational models in sciences and engineering. In particular, it focuses in parametric uncertainties, and non-parametric uncertainties with applications from the structural dynamics and vibroacoustics of complex mechanical systems, from micromechanics and multiscale mechanics of heterogeneous materials. Resulting from a course developed by the author, the book begins with a description of the fundamental mathematical tools of probability and statistics that are directly useful for uncertainty quantification. It proceeds with a well carried out description of some basic and advanced methods for constructing stochastic models of uncertainties, paying particular attention to the problem of calibrating and identifying a stochastic model of uncertainty when experimental data is available. This book is intended to be a graduate-level textbook for students as well as professionals interested in the theory, computation, and applications of risk and prediction in science and engineering fields.

This book deals with the application of spectral methods to problems of uncertainty propagation and quanti?cation in model-based computations. It speci?cally focuses on computational and algorithmic features of these methods which are most useful in dealing with models based on partial differential equations, with special att- tion to models arising in simulations of ?uid ?ows. Implementations are illustrated through applications to elementary problems, as well as more elaborate examples selected from the authors’ interests in incompressible vortex-dominated ?ows and compressible ?ows at low Mach numbers. Spectral stochastic methods are probabilistic in nature, and are consequently rooted in the rich mathematical foundation associated with probability and measure spaces. Despite the authors’ fascination with this foundation, the discussion only - ludes to those theoretical aspects needed to set the stage for subsequent applications. The book is authored by practitioners, and is primarily intended for researchers or graduate students in computational mathematics, physics, or ?uid dynamics. The book assumes familiarity with elementary methods for the numerical solution of time-dependent, partial differential equations; prior experience with spectral me- ods is naturally helpful though not essential. Full appreciation of elaborate examples in computational ?uid dynamics (CFD) would require familiarity with key, and in some cases delicate, features of the associated numerical methods. Besides these shortcomings, our aim is to treat algorithmic and computational aspects of spectral stochastic methods with details suf?cient to address and reconstruct all but those highly elaborate examples.