Probabilistic inverse problems arise in a variety of scientific and engineering applications. A particular type of probabilistic inverse problem, termed a generalized aggregate data inverse problem, involves the specification of expected value targets and/or probabilistic constraints at discrete times or in a discrete set of transformed domains. Because the transformed distributions themselves are not specified, these problems cannot be readily solved by typical Bayesian solution techniques. In this work, a novel solution technique using the Koopman operator is proposed. The Koopman operator is used to “pull-back” the cost and constraint functions acting on the transformed probability densities to the initial domain, forming a set of integral equations and integral inequalities over a common integration domain. A quadrature technique is employed to approximate this system of equations and inequalities, leading to the formulation of a linearly-constrained convex quadratic program that can be solved using a variety of well-known techniques. Results show that the method can be feasibly applied to solve several practical engineering problems of interest, illustrating tradeoffs in various aspects of the solution process.

概率逆问题出现在各种科学和工程应用中。一种特殊类型的概率反问题，称为广义集合数据反问题，涉及在离散时间或在一组离散的转换域中指定期望值目标和/或概率约束。由于未指定转换分布本身，因此无法通过典型的贝叶斯解决方案技术轻松解决这些问题。在这项工作中，提出了一种使用Koopman算子的新颖求解技术。使用Koopman运算符将作用在转换后的概率密度上的成本和约束函数“拉回”到初始域，从而在公共积分域上形成一组积分方程和积分不等式。采用正交技术来逼近该方程式和不等式系统，从而导致可以使用多种公知技术来求解线性约束凸二次方程序。结果表明，该方法可以切实可行地用于解决一些实际感兴趣的工程问题，从而说明了解决过程各个方面的权衡。