Adaptive Nonlinear Data Assimilation through P-Spline Triangular Measure Transport

TL;DR

This paper introduces an adaptive P-spline based measure transport method for nonlinear, non-Gaussian data assimilation that automatically balances model complexity and overfitting, demonstrated on high-dimensional problems.

Contribution

It develops an automatic, hyperparameter-free adaptation algorithm for measure transport in data assimilation using P-splines and an information criterion.

Findings

Effective in nonlinear, non-Gaussian problems

Handles high-dimensional systems efficiently

Requires no hyperparameter tuning

Abstract

Non-Gaussian statistics are a challenge for data assimilation. Linear methods oversimplify the problem, yet fully nonlinear methods are often too expensive to use in practice. The best solution usually lies between these extremes. Triangular measure transport offers a flexible framework for nonlinear data assimilation. Its success, however, depends on how the map is parametrized. Too much flexibility leads to overfitting; too little misses important structure. To address this balance, we develop an adaptation algorithm that selects a parsimonious parametrization automatically. Our method uses P-spline basis functions and an information criterion as a continuous measure of model complexity. This formulation enables gradient descent and allows efficient, fine-scale adaptation in high-dimensional settings. The resulting algorithm requires no hyperparameter tuning. It adjusts the transport map to the appropriate level of complexity based on the system statistics and ensemble size. We demonstrate its performance in nonlinear, non-Gaussian problems, including a high-dimensional distributed groundwater model.