Anne Gelb
Dartmouth College
Title
The Residual Prior Transform: From Signal Recovery to Data Assimilation
Abstract
Recovering signals, images, and physical states from noisy or incomplete data is typically framed as a regularization problem: enforce that some transform Lx of the unknown x is sparse. Classical choices such as total variation and its higher-order variants fix a single assumed order of smoothness, which forces a trade-off between staircasing and ringing whenever the true variability changes across the domain – exactly the situation for piecewise smooth signals and for physical states governed by hyperbolic conservation laws, whose solutions develop shocks even from smooth initial data.
We introduce the residual transform operator R = T − S, built from two distinct operators, T ̸= S, that nevertheless satisfy Tx ≈ Sx to the same order of approximation in the transform domain. Because T and S are matched in this way regardless of the underlying signal’s local smoothness, R acts as an annihilating operator: it is small wherever the underlying signal is smooth and large only at genuine discontinuities, without ever needing to know the signal’s variability in advance. Placed inside a generalized sparse Bayesian learning framework, R yields not only a robust point estimate but full uncertainty quantification, with the regularization strength learned automatically from the data rather than hand-tuned.
We then carry R into a new setting: variational data assimilation (3D-Var) for hyperbolic conservation laws, where the usual Gaussian background term is ill-suited to a shock’s genuinely multimodal position uncertainty. Replacing this with residual-based regularization yields analyses that remain accurate even when the background-error covariance is misspecified, with the advantage over standard baselines growing precisely as observations become sparser – the regime where this structure-preserving approach earns its
keep.
This research is done in collaboration with Yao Xiao and Adityvikram Viswanathan, and is partially supported by supported by DOD (ONR MURI) #N00014-20-1-2595 and DOE ASCR #DE-SC0025555
In addition, there will be a pre-talk for graduate students in the same location at 2:15 pm
