The automatic projection filter is a recently developed numerical method for projection filtering that leverages sparse-grid integration and automatic differentiation. However, its accuracy is highly sensitive to the accuracy of the cumulant-generating function computed via the sparse-grid integration, which in turn is also sensitive to the choice of the bijection from the canonical hypercube to the state space. In this paper, we propose two new adaptive parametric bijections for the automatic projection filter. The first bijection relies on the minimization of Kullback--Leibler divergence, whereas the second method employs the sparse-grid Gauss--Hermite quadrature. The two new bijections allow the sparse-grid nodes to adaptively move within the high-density region of the state space, resulting in a substantially improved approximation while using only a small number of quadrature nodes. The practical applicability of the methodology is illustrated in three simulated nonlinear filtering problems.

翻译：自动投影滤波是一种近期发展起来的数值方法，用于投影滤波，它利用了稀疏网格积分和自动微分。然而，其精度高度依赖于通过稀疏网格积分计算的累积生成函数的精度，而该精度又对从标准超立方体到状态空间的映射选择敏感。在本文中，我们提出了两种适用于自动投影滤波的新自适应参数化双射。第一种双射依赖于最小化Kullback-Leibler散度，而第二种方法采用稀疏网格Gauss-Hermite求积。这两种新的双射允许稀疏网格节点自适应地在状态空间的高密度区域内移动，从而在使用少量求积节点的情况下显著提高近似精度。该方法的实际适用性通过三个模拟非线性滤波问题进行了说明。