Adaptive multi-element generalized polynomial chaos: algorithms and applications
Adaptive multi-element generalized polynomial chaos: algorithms and applications
复制标题
自适应多元素广义多项式混沌：算法与应用
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
X. Wan
中科院分区：
文献类型：
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作者：
G. Karniadakis;X. Wan
随着微分方程确定性解的方法不断成熟，人们对更现实的数学模型的不确定性量化产生了浓厚的兴趣，其中物理应用受到输入数据中的不确定性的影响，例如模型参数，边界/初始条件，强迫项等。提出了一种非采样的多元广义多项式混沌方法（ME-gPC），用于研究不确定性量化问题。特别是，自适应耦合与这种方法来控制仿真误差。
Karhuen-Loeve（K-L）展开是一种有效的随机输入建模技术，本文首先介绍了该展开的快速数值算法，K-L展开的关键部分快速高斯变换的一个尖锐的误差估计，并提出了一个快速的特征值求解器的高斯型协方差核，这产生了节省高达三个数量级的时间和内存。
接下来介绍ME-gPC算法。ME-gPC基于随机空间的分解和谱展开。在每个随机元素中，定义了服从条件概率密度函数（PDF）的新随机变量。正交多项式加权的条件PDF数值构造快速，准确。证明了当函数在参数空间中具有足够的正则性时，该算法是hp收敛的。
随后开发了用于具有随机系数的常微分方程和偏微分方程的自适应ME-gPC算法。对于常微分方程，基于局部多项式混沌展开式系数的衰减率，提出了一种启发式自适应判据。对于偏微分方程，基于后验误差估计，给出了一个严格的自适应准则。特别是，一个减少的空间被构造，以减少求解误差方程的成本。
接下来研究了多项式混沌方法的两个基本问题：长期积分和非高斯随机输入。对于具有随机输运速度的双曲型方程，比较了gPC和ME-gPC的性能。结果表明，ME-gPC可以通过采用h收敛来延长给定精度的有效积分时间。提出了一种多项式混沌方法，可以更有效地处理非高斯随机输入。
开发的算法，然后应用到流体力学中的物理应用，包括在一个3D的电子芯片的热传导，热对流在一个槽道和嘈杂的流动通过一个固定的圆柱。
除了ME-gPC方法外，本文还研究了谱/hp有限元方法的通量型后验误差估计。给出了二维情况下的显式通量。提出了一种更一般的方法，通过最小化加权目标函数来获得局部纯Neumann问题的平衡通量。
As methods for the deterministic solutions of differential equations continue to mature, there is intensive interest in uncertainty quantification for more realistic mathematical models, where the physical applications are affected by uncertainty in the input data, such as model parameters, boundary/initial conditions, forcing terms, etc. In this work, a non-sampling method called "multi-element generalized polynomial chaos (ME-gPC)" is developed to study uncertainty quantification. In particular, adaptivity is coupled with this method to control simulation errors.
The Karhuen-Loeve (K-L) expansion, an efficient technique for modeling of random inputs, is first presented, where we focus on fast numerical algorithms for the associated eigenvalue problem, i.e., the key part of the K-L expansion. A sharp error estimate for the fast Gauss transform is developed and used to propose a fast eigenvalue solver for Gaussian-type covariance kernels, which yields a saving up to three orders of magnitude in time and memory.
The algorithm of ME-gPC is next presented. ME-gPC is based on the decomposition of random space and spectral expansions. In each random element, a new random variable subject to a conditional probability density function (PDF) is defined. Orthogonal polynomials weighted by the conditional PDF are numerically constructed fast and accurately. The hp-convergence ME-gPC is proved if the function has enough regularity in the parametric space.
Adaptive ME-gPC algorithms for ODEs and PDEs with random coefficients are subsequently developed. For ODEs, a heuristic adaptivity criterion is developed based on the decay rate of coefficients of local polynomial chaos expansions. For PDEs, a rigorous adaptivity criterion is developed based on a posteriori error estimators. In particular, a reduced space is constructed to reduce the cost of solving the error equations.
Two fundamental problems of polynomial chaos methods are next studied: long-term integration and non-Gaussian random inputs. The performance of gPC and ME-gPC is compared for a hyperbolic equation with a random transport velocity. It is shown that ME-gPC can extend the valid integration time for a given accuracy by employing h-convergence. A methodology is developed for polynomial chaos methods to deal with non-Gaussian random inputs more efficiently.
The developed algorithms are then applied to physical applications in fluid mechanics including heat conduction in a 3D electric chip, heat convection in a grooved channel and noisy flow past a stationary circular cylinder.
In addition to the ME-gPC method, a study on the flux-type a posteriori error estimation for spectral/hp finite element methods is included in this work. Explicit fluxes for the two-dimensional case are given. A more general approach is proposed to obtain equilibrated fluxes for the local pure Neumann problems by minimizing a weighted target function.
《极端条件下的XMCD研究——巡回电子系统中的磁相变——》（特邀）
DOI：
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发表时间：
2007
期刊：
影响因子：
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作者：
H.;Maruyama
通讯作者：
Maruyama