Research map / Probabilistic and scientific modeling / Variational and generative sampling
Christoffel-DPS: Optimal sensor placement in diffusion posterior sampling for arbitrary distributions
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Proposes a distribution-free optimal sensor placement framework based on Christoffel functions for posterior sampling, with non-asymptotic recovery bounds, and instantiates Christoffel-DPS (offline and online ensemble variants) for diffusion/flow-matching reconstruction on Pinball, Darcy and Kolmogorov flow.
Problem
Choosing sensor locations for state reconstruction from sparse measurements when the prior is complex and non-Gaussian, where classical Gaussian-based optimal sensor placement (OSP) methods do not apply and generative-model placement lacks non-asymptotic guarantees.
Approach
Derives a Christoffel-function-based sampling measure over sensor locations (secant-set supremum) that minimizes a sample-complexity bound for posterior sampling with arbitrary priors and linear sensing; instantiates it with empirical secant sets (offline) and ensemble Tweedie estimates during DPS (online), plus a greedy column-pivoted QR variant.
Key results
- On Pinball, online ensemble Christoffel-DPS reaches the error of random and classical baselines at roughly half the sensor budget, per the paper's text.
- Offline Christoffel-DPS variants match or exceed the strongest classical baseline across all budgets on Pinball.
- On Darcy flow, greedy Christoffel-DPS reaches a lower error plateau and outperforms classical methods at low sensor budgets.
Surprising findings
- Non-greedy (random i.i.d. Christoffel) sampling tends to underperform, attributed to the cost of estimating the empirical secant set.
- D- and E-optimal POD collapse on Darcy flow due to rank deficiency of the low-frequency pressure field, requiring Tikhonov regularization.
Limitations
- Experiments only use point evaluations for state reconstruction; other linear measurements (gradient-augmented, inverse PDE) not tested.
- Estimating the empirical secant set and the online ensemble with autodiff guidance is computationally expensive.
- Gains over classical OED are milder on approximately Gaussian, linear datasets.
Experimental setup
Datasets: Pinball advection-diffusion snapshots (FEM mesh, ~7,500 nodes) Darcy flow snapshots (128x128) Kolmogorov flow vorticity snapshots
Benchmarks: Pinball Darcy flow Kolmogorov flow
Metrics: relative L2 error
Baselines: random placement A-optimal POD D-optimal POD E-optimal POD SSPOR PhySense
Builds on: diffusion posterior sampling PhySense GRIFDIR DiffusionPDE POD-based A/D/E-optimal design
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