在无限维空间中概率测度的Kullback-Leibler近似算法
Algorithms for Kullback-Leibler Approximation of Probability Measures in Infinite Dimensions
AI总结：
本文研究了在无限维空间中通过Kullback-Leibler散度寻找高斯近似算法，提出两种参数化协方差的方法，并应用于非线性逆问题和条件扩散过程。
摘要：
本文研究了在无限维空间中寻找目标测度的高斯近似算法。目标测度通过其相对于参考高斯测度的密度定义。我们以Kullback-Leibler散度作为距离度量，通过最小化该距离找到最佳高斯近似。近似高斯测度必须与参考高斯测度等价，从而定义了底层变分问题的自然函数空间设置。我们引入了一种计算算法，该算法适用于所需最小化，通过将均值作为函数来寻找，并以两种不同方式参数化协方差：通过参考协方差的低秩扰动；以及通过Schrödinger势扰动的逆参考协方差。两个应用被展示：非线性逆问题在椭圆PDE中的应用，以及条件扩散过程。我们还展示了所获得的高斯近似如何用于生成改进的pCN-MCMC方法，这些方法不仅适合高维设置，而且在逆问题（小观测噪声）或条件扩散（小温度）中表现良好。
英文摘要：
In this paper we study algorithms to find a Gaussian approximation to a target measure defined on a Hilbert space of functions; the target measure itself is defined via its density with respect to a reference Gaussian measure. We employ the Kullback-Leibler divergence as a distance and find the best Gaussian approximation by minimizing this distance. It then follows that the approximate Gaussian must be equivalent to the Gaussian reference measure, defining a natural function space setting for the underlying calculus of variations problem. We introduce a computational algorithm which is well-adapted to the required minimization, seeking to find the mean as a function, and parameterizing the covariance in two different ways: through low rank perturbations of the reference covariance; and through Schrödinger potential perturbations of the inverse reference covariance. Two applications are shown: to a nonlinear inverse problem in elliptic PDEs, and to a conditioned diffusion process. We also show how the Gaussian approximations we obtain may be used to produce improved pCN-MCMC methods which are not only well-adapted to the high-dimensional setting, but also behave well with respect to small observational noise (resp. small temperatures) in the inverse problem (resp. conditioned diffusion).
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