Sequential Monte Carlo with Gaussian Mixture Approximation for Infinite-Dimensional Statistical Inverse Problems
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| Format: | Preprint |
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2025
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| author | Lu, Haoyu Jia, Junxiong Meng, Deyu |
| author_facet | Lu, Haoyu Jia, Junxiong Meng, Deyu |
| contents | 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 with Gaussian mixture approximation, together with an easy-to-use reduced version. 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 reduced version 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16028 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sequential Monte Carlo with Gaussian Mixture Approximation for Infinite-Dimensional Statistical Inverse Problems Lu, Haoyu Jia, Junxiong Meng, Deyu Numerical Analysis Statistics Theory 65L09, 49N45, 62F15 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 with Gaussian mixture approximation, together with an easy-to-use reduced version. 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 reduced version 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). |
| title | Sequential Monte Carlo with Gaussian Mixture Approximation for Infinite-Dimensional Statistical Inverse Problems |
| topic | Numerical Analysis Statistics Theory 65L09, 49N45, 62F15 |
| url | https://arxiv.org/abs/2503.16028 |