Sampling-free linear Bayesian update of polynomial chaos representations
Sampling-free linear Bayesian update of polynomial chaos representations
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DOI：
10.1016/j.jcp.2012.04.044
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发表时间：
2012-07
期刊：
影响因子：
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通讯作者：
B. Rosic;A. Litvinenko;O. Pajonk;H. Matthies
中科院分区：
文献类型：
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作者：
B. Rosic;A. Litvinenko;O. Pajonk;H. Matthies
我们提出了一个完全确定性的方法来概率解释的反问题，其中未知量表示的随机场或过程，可能的非高斯分布。在“白色噪声”的框架下给出了随机场的描述，这使得我们能够通过Galerkin投影到多项式混沌上来解决随机正问题。借助于这种表示，概率识别问题被转换成多项式混沌展开设置和贝叶斯线性形式的更新。通过引入埃尔米特代数，这成为一个直接的，纯代数的方式计算后，这是比较便宜的评估。此外，我们表明，著名的卡尔曼滤波器是低阶部分的更新。所提出的方法是在这里测试的固定扩散方程与规定的源项，其特征在于由一个不确定的电导率参数，然后确定从有限的和嘈杂的数据通过测量的扩散量。
We present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes, described by possibly non-Gaussian distributions. The description of the introduced random fields is given in a “white noise” framework, which enables us to solve the stochastic forward problem through Galerkin projection onto polynomial chaos. With the help of such a representation the probabilistic identification problem is cast in a polynomial chaos expansion setting and the Baye’s linear form of updating. By introducing the Hermite algebra this becomes a direct, purely algebraic way of computing the posterior, which is comparatively inexpensive to evaluate. In addition, we show that the well-known Kalman filter is the low order part of this update. The proposed method is here tested on a stationary diffusion equation with prescribed source terms, characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity.