• 主题
高级搜索  >
历史搜索 清空历史
首页> 外文学位 >Algorithms for stochastic Galerkin projections: Solvers, basis adaptation and multiscale modeling and reduction.
【24h】

Algorithms for stochastic Galerkin projections: Solvers, basis adaptation and multiscale modeling and reduction.

机译：随机Galerkin投影的算法：求解器，基础自适应以及多尺度建模和归约。

获取原文
获取原文并翻译 | 示例
页面导航
  • 摘要
  • 著录项
  • 相似文献
  • 相关主题

摘要

This dissertation focuses on facilitating the analysis of probabilistic models for physical systems. To that end, novel contributions are made to various aspects of the problem, namely, 1) development of efficient algorithms for solving stochastic system of equations governing the physical problem, 2) stochastic basis adaptation methods to compute the solution in reduced stochastic dimensional space and 3) stochastic upscaling methods to find coarse-scale models from the fine-scale stochastic solutions. In particular, algorithms are developed for stochastic systems that are governed by partial differential equations (PDEs) with random coefficients. Polynomial chaos-based stochastic Galerkin and stochastic collocation methods are employed for solving these equations. Solvers and preconditioners based on Gauss-Seidel and Jacobi algorithms are explored for solving system of linear equations arising from stochastic Galerkin discretization of PDEs with random input data. Gauss-Seidel and Jacobi algorithms are formulated such that the existing software is leveraged in the computational effort. These algorithms are also used to develop preconditioners to Krylov iterative methods. These solvers and preconditioners are tested by solving a steady state diffusion equation and a steady state advection-diffusion equation. Upon discretization, the former PDE results in a symmetric positive definite matrix on left-hand-side, whereas the latter results in a non-symmetric positive definite matrix. The stochastic systems face significant computational challenge due the curse of dimensionality as the solution often lives in very high dimensional space. This challenge is addressed in the present work by recognizing the low dimensional structure of many quantities of interest (QoI) even in problems that have been embedded, via parameterization, in very high-dimensional settings. A new method for the characterization of subspaces associated with low-dimensional QoI is presented here. The probability density function of these QoI is found to be concentrated around one-dimensional subspaces for which projection operators are developed. This approach builds on the properties of Gaussian Hilbert spaces and associated tensor product spaces.;For many physical problems, the solution lives in multiple scales, and it is important to capture the physics at all scales. To address this issue, a stochastic upscaling methodology is developed in which the above developed algorithms and basis adaptation methods are used. In particular upscaling methodology is demonstrated by developing a coarse scale stochastic porous medium model that replaces a fine-scale which consists of flow past fixed solid inclusions. The inclusions have stochastic spatially varying thermal conductivities and generate heat that is transported by the fluid. The permeability and conductivity of the effective porous medium are constructed as statistically dependent stochastic processes that are both explicitly dependent on the fine scale random conductivity.;Another contribution of this thesis is development of a probabilistic framework for synthesizing high resolution micrographs from low resolution ones using a parametric texture model and a particle filter. Information contained in high resolution micrographs is relevant to the accurate prediction of microstructural behavior and the nucleation of instabilities. As these micrographs may be tedious and uneconomical to obtain over an extended spatial domain, A statistical approach is proposed for interpolating fine details over a whole computational domain starting with a low resolution prior and high resolution micrographs available only at a few spatial locations. As a first step, a small set of high resolution micrographs are decomposed into a set of multi-scale and multi-orientation subbands using a complex wavelet transform. Parameters of a texture model are computed as the joint statistics of the decomposed subbands. The synthesis algorithm then generates random micrographs satisfying the parameters of the texture model by recursively updating the gray level values of the pixels in the input micrograph. A density-based Monte Carlo filter is used at each step of the recursion to update the generated micrograph, using a low resolution micrograph at that location as a measurement. The process is continued until the synthesized micrograph has the same statistics as those from the high resolution micrographs. The proposed method combines a texture synthesis procedure with a particle filter and produces good quality high resolution micrographs.
机译：本文致力于促进物理系统概率模型的分析。为此，对问题的各个方面做出了新的贡献，即：1）开发用于求解控制物理问题的方程组的有效算法的有效算法； 2）在减小的随机维空间中计算解的随机基础适应方法；以及3）随机放大方法，从精细尺度随机解中找到粗糙尺度模型。特别是，为随机系统开发了算法，该算法由具有随机系数的偏微分方程（PDE）控制。基于多项式混沌的随机Galerkin和随机搭配方法用于求解这些方程。探索了基于高斯-赛德尔（Gauss-Seidel）和雅可比（Jacobi）算法的求解器和预处理器，用于求解随机输入数据对随机偏微分方程的随机Galerkin离散化产生的线性方程组。制定了高斯-塞德尔（Gauss-Seidel）算法和雅可比（Jacobi）算法，以便在计算工作中利用现有软件。这些算法还用于开发Krylov迭代方法的预处理器。通过求解稳态扩散方程和稳态对流扩散方程来测试这些求解器和预处理器。离散化后，前者的PDE会在左侧生成一个对称的正定矩阵，而后者会导致一个非对称的正定矩阵。随机系统由于维数的诅咒而面临巨大的计算挑战，因为解决方案通常生活在非常高的维数空间中。通过识别许多关注量（QoI）的低维结构，即使在已经通过参数化将其嵌入到非常高维的设置中的问题中，也可以解决本挑战。这里提出了一种新的表征与低维QoI相关的子空间的方法。发现这些QoI的概率密度函数集中在为其开发投影算子的一维子空间周围。这种方法建立在高斯希尔伯特空间和相关的张量积空间的属性上。对于许多物理问题，解决方案存在于多个尺度上，并且捕获所有尺度的物理学很重要。为了解决这个问题，开发了一种随机放大方法，其中使用了上面开发的算法和基础自适应方法。特别是，通过开发一种粗尺度的随机多孔介质模型来证明按比例放大的方法，该模型可以代替由经过固定固体包裹体的流动组成的细尺度。夹杂物在空间上具有随机变化的热导率，并产生通过流体传输的热量。有效的多孔介质的渗透率和电导率被构造为统计相关的随机过程，都明确地依赖于小尺度的随机电导率。本论文的另一贡献是开发了概率框架，该框架用于从低分辨率的显微照片中合成低分辨率的显微照片。参数纹理模型和粒子过滤器。高分辨率显微照片中包含的信息与微观结构行为的精确预测和不稳定性的成核有关。由于在扩展的空间域上获取这些显微照片可能是乏味且不经济的，因此提出了一种统计方法，用于在整个计算域上内插精细的细节，从只能在几个空间位置获得的低分辨率先驱和高分辨率显微照片开始。第一步，使用复杂的小波变换将一小组高分辨率的显微照片分解为一组多尺度和多方向的子带。计算纹理模型的参数作为分解子带的联合统计量。然后，合成算法通过递归更新输入显微照片中像素的灰度值，生成满足纹理模型参数的随机显微照片。在递归的每个步骤中都使用基于密度的蒙特卡洛滤波器，以使用该位置的低分辨率显微照片作为度量来更新生成的显微照片。继续该过程，直到合成的显微照片具有与高分辨率显微照片相同的统计量为止。所提出的方法结合了纹理合成程序和粒子过滤器，并产生了高质量的高分辨率显微照片。

著录项

  • 作者

    Tipireddy, Ramakrishna.;

  • 作者单位

    University of Southern California.;

  • 授予单位 University of Southern California.;
  • 学科 Engineering Civil.
  • 学位 Ph.D.
  • 年度 2013
  • 页码 128 p.
  • 总页数 128
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

相似文献

  • 外文文献
  • 中文文献
  • 专利
获取原文

客服邮箱：kefu@zhangqiaokeyan.com

京公网安备：11010802029741号 ICP备案号：京ICP备15016152号-6 六维联合信息科技 (北京) 有限公司©版权所有

  • 客服微信

  • 服务号