| Abstract: |
| Diffusion models have emerged as powerful tools for solving nonlinear inverse problems, offering high-quality reconstructions through conditional reverse sampling. However, challenges persist in handling partial differential equation (PDE)-constrained inverse problems, including computational inefficiency, discretization errors, and the inherent ill-posedness of nonlinear systems such as travel-time tomography and ultrasound computed tomography (USCT). In this talk, we present two synergistic frameworks that address these challenges by integrating diffusion priors with PDE-aware numerical and learned solvers.
First, we propose a subspace diffusion approach for PDE-based travel-time tomography, introducing (1) a plug-and-play posterior sampling process that leverages adjoint-state equations for gradient updates and (2) a coarse-to-fine subspace acceleration technique to reduce sampling time while preserving reconstruction quality. Second, we develop Diff-ANO (Diffusion-based Models with Adjoint Neural Operators), a unified framework for Helmholtz equation-constrained USCT that combines (1) a conditional consistency model for few-step, measurement-conditional sampling and (2) neural operator surrogates to replace traditional PDE solvers, enabling memory-efficient adjoint gradient computation via our Batch-based Convergent Born Series (BCBS) strategy. Our experiments demonstrate significant improvements in both reconstruction accuracy and computational efficiency across sparse-view and partial-view measurement scenarios. The proposed methods bridge the gap between generative priors and PDE constraints, offering scalable solutions for nonlinear inverse problems in imaging. |
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