Inverse Models for Estimating the Initial Condition of Spatio-Temporal Advection-Diffusion Processes

TL;DR

This paper develops inverse modeling techniques to estimate the initial state of a spatio-temporal advection-diffusion process from sparse data, employing spectral methods and convex optimization for different sampling schemes.

Contribution

It introduces a unified convex optimization framework for inverse problems under various sampling schemes, with efficient spectral solutions for non-uniform and shifted uniform sampling.

Findings

Spectral domain solutions improve computational efficiency.

The approach effectively estimates initial conditions from sparse data.

Code availability facilitates reproducibility and application.

Abstract

Inverse problems involve making inference about unknown parameters of a physical process using observational data. This paper investigates an important class of inverse problems -- the estimation of the initial condition of a spatio-temporal advection-diffusion process using spatially sparse data streams. Three spatial sampling schemes are considered, including irregular, non-uniform and shifted uniform sampling. The irregular sampling scheme is the general scenario, while computationally efficient solutions are available in the spectral domain for non-uniform and shifted uniform sampling. For each sampling scheme, the inverse problem is formulated as a regularized convex optimization problem that minimizes the distance between forward model outputs and observations. The optimization problem is solved by the Alternating Direction Method of Multipliers algorithm, which also handles the situation when a linear inequality constraint (e.g., non-negativity) is imposed on the model output. Numerical examples are presented, code is made available on GitHub, and discussions are provided to generate some useful insights of the proposed inverse modeling approaches.