By formulating the inverse problem of partial differential equations (PDEs) as a statistical inference problem, the Bayesian approach provides a general framework for quantifying uncertainties. In the inverse problem of PDEs, parameters are defined on an infinite-dimensional function space, and the PDEs induce a computationally intensive likelihood function. Additionally, sparse data tends to lead to a multi-modal posterior. These features make it difficult to apply existing sequential Monte Carlo (SMC) algorithms. To overcome these difficulties, we propose new conditions for the likelihood functions, construct a Gaussian mixture based preconditioned Crank-Nicolson transition kernel, and demonstrate the universal approximation property of the infinite-dimensional Gaussian mixture probability measure. By combining these three novel tools, we propose a new SMC algorithm, named SMC-GM. For this new algorithm, we obtain a convergence theorem that allows Gaussian priors, illustrating that the sequential particle filter actually reproduces the true posterior distribution. Furthermore, the proposed new algorithm is rigorously defined on the infinite-dimensional function space, naturally exhibiting the discretization-invariant property. Numerical experiments demonstrate that the new approach has a strong ability to probe the multi-modality of the posterior, significantly reduces the computational burden, and numerically exhibits the discretization-invariant property (important for large-scale problems).

翻译：通过将偏微分方程（PDE）反问题表述为统计推断问题，贝叶斯方法为量化不确定性提供了一个通用框架。在PDE反问题中，参数定义在无限维函数空间上，且PDE导出了一个计算密集的似然函数。此外，稀疏数据往往导致后验分布呈现多模态特性。这些特征使得现有的序贯蒙特卡洛（SMC）算法难以应用。为克服这些困难，我们提出了似然函数的新条件，构建了一种基于高斯混合的预处理Crank-Nicolson转移核，并证明了无限维高斯混合概率测度的通用逼近性质。通过结合这三种新工具，我们提出了一种新的SMC算法，命名为SMC-GM。针对该新算法，我们获得了允许高斯先验的收敛定理，证明序贯粒子滤波器实际上能够重现真实后验分布。此外，所提出的新算法在无限维函数空间上被严格定义，自然展现出离散化不变性。数值实验表明，新方法具有强大的探测后验多模态的能力，显著降低了计算负担，并在数值上展现了离散化不变性（这对于大规模问题至关重要）。