Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems

• URL: http://arxiv.org/abs/2508.09623v2
• Date: Fri, 15 Aug 2025 02:50:23 GMT
• Title: Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems
• Authors: Akshay Thakur, Sawan Kumar, Matthew Zahr, Souvik Chakraborty,
• Abstract summary: We develop an $h$-adaptive probabilistic solver that can scale to a large number of collocation points.<n>We demonstrate the efficacy of the proposed solver on benchmark PDEs, as well as a time-dependent parabolic PDE formulated in a space-time setting.
• Score: 7.807210884802377
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and imposing the governing PDE as a constraint at a finite set of collocation points, probabilistic numerics delivers mesh-free solutions at arbitrary locations. However, the high computational cost, which scales cubically with the number of collocation points, remains a critical bottleneck, particularly for large-scale or high-dimensional problems. We propose a scalable enhancement to this paradigm through two key innovations. First, we develop a stochastic dual descent algorithm that reduces the per-iteration complexity from cubic to linear in the number of collocation points, enabling tractable inference. Second, we exploit a clustering-based active learning strategy that adaptively selects collocation points to maximize information gain while minimizing computational expense. Together, these contributions result in an $h$-adaptive probabilistic solver that can scale to a large number of collocation points. We demonstrate the efficacy of the proposed solver on benchmark PDEs, including two- and three-dimensional steady-state elliptic problems, as well as a time-dependent parabolic PDE formulated in a space-time setting.

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