Path-OED for infinite-dimensional Bayesian linear inverse problems governed by PDEs
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908780006998016 |
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| author | Neuberger, J. Nicholas Alexanderian, Alen Waanders, Bart van Bloemen Attia, Ahmed |
| author_facet | Neuberger, J. Nicholas Alexanderian, Alen Waanders, Bart van Bloemen Attia, Ahmed |
| contents | We consider infinite-dimensional Bayesian linear inverse problems governed by time-dependent partial differential equations (PDEs) and develop a mathematical and computational framework for optimal design of mobile sensor paths in this setting. The proposed path optimal experimental design (path-OED) framework is established rigorously in a function space setting and elaborated for the case of Bayesian c-optimality, which quantifies the posterior variance in a linear functional of the inverse parameter. The latter is motivated by goal-oriented formulations, where we seek to minimize the uncertainty in a scalar prediction of interest. To facilitate computations, we complement the proposed infinite-dimensional framework with discretized formulations, in suitably weighted finite-dimensional inner product spaces, and derive efficient methods for finding optimal sensor paths. The resulting computational framework is flexible, scalable, and can be adapted to a broad range of linear inverse problems and design criteria. We also present extensive computational experiments, for a model inverse problem constrained by an advection-diffusion equation, to demonstrate the effectiveness of the proposed approach. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_15168 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Path-OED for infinite-dimensional Bayesian linear inverse problems governed by PDEs Neuberger, J. Nicholas Alexanderian, Alen Waanders, Bart van Bloemen Attia, Ahmed Optimization and Control We consider infinite-dimensional Bayesian linear inverse problems governed by time-dependent partial differential equations (PDEs) and develop a mathematical and computational framework for optimal design of mobile sensor paths in this setting. The proposed path optimal experimental design (path-OED) framework is established rigorously in a function space setting and elaborated for the case of Bayesian c-optimality, which quantifies the posterior variance in a linear functional of the inverse parameter. The latter is motivated by goal-oriented formulations, where we seek to minimize the uncertainty in a scalar prediction of interest. To facilitate computations, we complement the proposed infinite-dimensional framework with discretized formulations, in suitably weighted finite-dimensional inner product spaces, and derive efficient methods for finding optimal sensor paths. The resulting computational framework is flexible, scalable, and can be adapted to a broad range of linear inverse problems and design criteria. We also present extensive computational experiments, for a model inverse problem constrained by an advection-diffusion equation, to demonstrate the effectiveness of the proposed approach. |
| title | Path-OED for infinite-dimensional Bayesian linear inverse problems governed by PDEs |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2601.15168 |