skip to main content
US FlagAn official website of the United States government
dot gov icon
Official websites use .gov
A .gov website belongs to an official government organization in the United States.
https lock icon
Secure .gov websites use HTTPS
A lock ( lock ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites.

Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


Title: A Stochastic Galerkin Method for the Boltzmann Equation with Multi-Dimensional Random Inputs Using Sparse Wavelet Bases
Abstract We propose a stochastic Galerkin method using sparse wavelet bases for the Boltzmann equation with multi-dimensional random inputs. Themethod 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.  more » « less
Award ID(s):
1654152
PAR ID:
10062086
Author(s) / Creator(s):
; ;
Date Published:
Journal Name:
Numerical Mathematics: Theory, Methods and Applications
Volume:
10
Issue:
02
ISSN:
1004-8979
Page Range / eLocation ID:
465 to 488
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. (, Journal of Functional Analysis)
    Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann by analogy with the isotropic Landau equation introduced by Krieger and Strain (2012) [35]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau. Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the “very soft potentials” regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani and Guillen (2022) [22] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities. 
    more » « less
  2. ; (, International Journal for Numerical Methods in Engineering)
    Summary This paper presents an approach for efficient uncertainty analysis (UA) using an intrusive generalized polynomial chaos (gPC) expansion. The key step of the gPC‐based uncertainty quantification(UQ) is the stochastic Galerkin (SG) projection, which can convert a stochastic model into a set of coupled deterministic models. The SG projection generally yields a high‐dimensional integration problem with respect to the number of random variables used to describe the parametric uncertainties in a model. However, when the number of uncertainties is large and when the governing equation of the system is highly nonlinear, the SG approach‐based gPC can be challenging to derive explicit expressions for the gPC coefficients because of the low convergence in the SG projection. To tackle this challenge, we propose to use a bivariate dimension reduction method (BiDRM) in this work to approximate a high‐dimensional integral in SG projection with a few one‐ and two‐dimensional integrations. The efficiency of the proposed method is demonstrated with three different examples, including chemical reactions and cell signaling. As compared to other UA methods, such as the Monte Carlo simulations and nonintrusive stochastic collocation (SC), the proposed method shows its superior performance in terms of computational efficiency and UA accuracy. 
    more » « less
  3. ; ; ; (, BIT numerical mathematics)
    In uncertainty quantification, it is commonly required to solve a forward model consisting of a partial differential equation (PDE) with a spatially varying uncertain coefficient that is represented as an affine function of a set of random variables, or parameters. Discretizing such models using stochastic Galerkin finite element methods (SGFEMs) leads to very high-dimensional discrete problems that can be cast as linear multi-term matrix equations (LMTMEs). We develop efficient computational methods for approximating solutions of such matrix equations in low rank. To do this, we follow an alternating energy minimization (AEM) framework, wherein the solution is represented as a product of two matrices, and approximations to each component are sought by solving certain minimization problems repeatedly. Inspired by proper generalized decomposition methods, the iterative solution algorithms we present are based on a rank-adaptive variant of AEM methods that successively computes a rank-one solution component at each step. We introduce and evaluate new enhancement procedures to improve the accuracy of the approximations these algorithms deliver. The efficiency and accuracy of the enhanced AEM methods is demonstrated through numerical experiments with LMTMEs associated with SGFEM discretizations of parameterized linear elliptic PDEs. 
    more » « less
  4. ; ; ; ; (, Mathematics of Climate and Weather Forecasting)
    Abstract We develop a stochastic Galerkin method for a coupled Navier-Stokes-cloud system that models dynamics of warm clouds. Our goal is to explicitly describe the evolution of uncertainties that arise due to unknown input data, such as model parameters and initial or boundary conditions. The developed stochastic Galerkin method combines the space-time approximation obtained by a suitable finite volume method with a spectral-type approximation based on the generalized polynomial chaos expansion in the stochastic space. The resulting numerical scheme yields a second-order accurate approximation in both space and time and exponential convergence in the stochastic space. Our numerical results demonstrate the reliability and robustness of the stochastic Galerkin method. We also use the proposed method to study the behavior of clouds in certain perturbed scenarios, for examples, the ones leading to changes in macroscopic cloud pattern as a shift from hexagonal to rectangular structures. 
    more » « less
  5. ; ; ; (, XXXV IUPAP. Comp. Physics (CCP2024))
    Fast solution of wave propagation in periodic structures usually relies on simplified approaches, such as analytical methods, transmission line models, scattering matrix approaches, plane wave methods, etc. For complex multi-dimensional problems, computationally intensive direct numerical simulation (DNS) is always needed. This study demonstrates a fast and accurate simulation methodology enabled by a physics-based learning methodology, derived from proper orthogonal decomposition (POD) and Galerkin projection, for periodic quantum nanostructure and photonic crystals. POD is a projection-based method that generates optimal basis functions (or POD modes) via solution data collected from DNSs. This process trains the POD modes to adapt parametric variations of the system and offers the best least squares (LS) fit to the solution using the smallest number of modes. This is very different from other projection approaches, e.g., Fourier, Legendre, Bessel, Airy functions, etc., that adopt assumed basis functions selected for the problem based on the solution form. After generating the optimal POD modes, Galerkin projection of the wave equation onto each of the POD modes is performed to close the model and incorporate physical principles guided by the wave equation. Such a rigorous approach offers efficient simulations with high accuracy and exhibits the extrapolation ability in cases reasonably beyond the training bounds. The POD-Galerkin methodology is applied in this study to predict band structures and wave solutions for 2D periodic quantum-dot and photonic-lattice structures. The plane-wave approach is also included in a periodic quantum-dot structure to illustrate the superior performance of the POD-Galerkin methodology. The POD-Galerkin approach offers a 2-order computing speedup for both nanostructure and optical superlattices, compared to DNS, when solving both the wave solution and band structure. If the band structure is the only concern, a 4-order improvement in computational efficiency can be achieved. Fig. 1(a) shows the optical superlattice in a demonstration, where a unit cell includes 22 discs with diagonally symmetrical refractive indices and the background index n = 1. The POD modes for this case are trained by TE mode electric field data collected from DNSs with variation of diagonally symmetrical refractive indices. The LS error of the predicted electric field wave solution from the POD-Galerkin approach, shown in Fig. 1(b) compared to DNS, is below 1% with just 8 POD modes that offer a more than 4-order reduction in the degrees of freedom, compared to DNS. In addition, an extremely accurate prediction of band structure is illustrated in Fig. 1(c) with a maximum error below 0.1% in the entire Brillouin zone. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Save / Share this Record
  • Send to Email