A FAST AND SCALABLE METHOD FOR A-OPTIMAL DESIGN OF EXPERIMENTS FOR INFINITE-DIMENSIONAL BAYESIAN NONLINEAR INVERSE PROBLEMS
A FAST AND SCALABLE METHOD FOR A-OPTIMAL DESIGN OF EXPERIMENTS FOR INFINITE-DIMENSIONAL BAYESIAN NONLINEAR INVERSE PROBLEMS
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DOI：
10.1137/140992564
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发表时间：
2016-01-01
影响因子：
3.1
通讯作者：
Ghattas, Omar
中科院分区：
文献类型：
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作者：
Alexanderian, Alen;Petra, Noemi;Ghattas, Omar
我们解决了由部分微分方程（PDE）控制的贝叶斯非线性反问题的最佳实验设计（OED）问题。逆问题试图从在一组传感器位置和管理PDES上观察到的实验数据中推断出无限维度参数。 OED问题的目的是找到最佳的传感器位置，以最大程度地减少推断参数字段中的不确定性。具体而言，我们从固定的候选传感器位置中寻求最佳的传感器子集。我们通过使用后协方差痕迹的期望值概括经典的A-最佳实验设计标准来制定OED目标函数。该预期值是通过在可能的一组实验数据上取平均值来计算的。为了应对参数场的无限维特征，我们以最大a后验概率（MAP）点构建高斯近似值，并使用所得的协方差算子来定义OED目标函数。我们使用随机痕量估计来计算此协方差算子的轨迹，该轨迹仅被隐式定义。由此产生的OED问题包括限制了表征地图点的PDE系统，以及描述了协方差（高斯近似与后部）对向量的协方差的作用的PDE。我们使用稀疏惩罚功能控制传感器配置的稀疏性。变分的伴随方法用于有效计算PDE约束OED目标函数的梯度。我们详细阐述了OED方法，以确定最佳传感器配置以最好地推断椭圆PDE的系数。此外，我们为在多孔介质流问题中推断对数渗透性场的推断提供了数值结果。数值结果表明，评估OED目标函数所需的PDE数量及其梯度基本上与参数维度和传感器维度（即候选传感器位置的数量）无关。用于计算OED的准Newton迭代的数量也表现出相同的维度不变性属性。
We address the problem of optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs). The inverse problem seeks to infer an infinite-dimensional parameter from experimental data observed at a set of sensor locations and from the governing PDEs. The goal of the OED problem is to find an optimal placement of sensors so as to minimize the uncertainty in the inferred parameter field. Specifically, we seek an optimal subset of sensors from among a fixed set of candidate sensor locations. We formulate the OED objective function by generalizing the classical A-optimal experimental design criterion using the expected value of the trace of the posterior covariance. This expected value is computed through sample averaging over the set of likely experimental data. To cope with the infinite-dimensional character of the parameter field, we construct a Gaussian approximation to the posterior at the maximum a posteriori probability (MAP) point, and use the resulting covariance operator to define the OED objective function. We use randomized trace estimation to compute the trace of this covariance operator, which is defined only implicitly. The resulting OED problem includes as constraints the system of PDEs characterizing the MAP point, and the PDEs describing the action of the covariance (of the Gaussian approximation to the posterior) to vectors. We control the sparsity of the sensor configurations using sparsifying penalty functions. Variational adjoint methods are used to efficiently compute the gradient of the PDE-constrained OED objective function. We elaborate our OED method for the problem of determining the optimal sensor configuration to best infer the coefficient of an elliptic PDE. Furthermore, we provide numerical results for inference of the log permeability field in a porous medium flow problem. Numerical results show that the number of PDE solves required for the evaluation of the OED objective function and its gradient is essentially independent of both the parameter dimension and the sensor dimension (i.e., the number of candidate sensor locations). The number of quasi-Newton iterations for computing an OED also exhibits the same dimension invariance properties.