# Inverse Problems and Data Assimilation

arXiv: https://arxiv.org/abs/1810.06191 · HTML: https://mlsanity.com/paper/1810.06191/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Oct 2018.

## Built on by (most cited first, 18 of 18)

- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Auto-differentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2107.07687/) (2021)
- [Reduced-Order Autodifferentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2301.11961/) (2023)
- [Learning About Structural Errors in Models of Complex Dynamical Systems](https://mlsanity.com/paper/2401.00035/) (2023)
- [LD-EnSF: Synergizing Latent Dynamics with Ensemble Score Filters for Fast Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2411.19305/) (2024)
- [Learning Optimal Filters Using Variational Inference](https://mlsanity.com/paper/2406.18066/) (2024)
- [Enhancing Gaussian Process Surrogates for Optimization and Posterior Approximation via Random Exploration](https://mlsanity.com/paper/2401.17037/) (2024)
- [Non-Asymptotic Analysis of Ensemble Kalman Updates: Effective Dimension and Localization](https://mlsanity.com/paper/2208.03246/) (2022)
- [Hyperparameter Optimization for Randomized Algorithms: A Case Study on Random Features](https://mlsanity.com/paper/2407.00584/) (2024)
- [Optimization on Manifolds via Graph Gaussian Processes](https://mlsanity.com/paper/2210.10962/) (2022)
- [Ensemble Kalman filter in latent space using a variational autoencoder pair](https://mlsanity.com/paper/2502.12987/) (2025)
- [A Primer on Variational Inference for Physics-Informed Deep Generative Modelling](https://mlsanity.com/paper/2409.06560/) (2024)
- [A Variational Inference Approach to Inverse Problems with Gamma Hyperpriors](https://mlsanity.com/paper/2111.13329/) (2021)
- [DC-LA: Difference-of-Convex Langevin Algorithm](https://mlsanity.com/paper/2601.22932/) (2026)
- [Control-Augmented Autoregressive Diffusion for Data Assimilation](https://mlsanity.com/paper/2510.06637/) (2025)
- [Filtered Neural Galerkin model reduction schemes for efficient propagation of initial condition uncertainties in digital twins](https://mlsanity.com/paper/2511.00670/) (2025)
- [Preserving linear invariants in ensemble filtering methods](https://mlsanity.com/paper/2404.14328/) (2024)

## Abstract (arXiv)

We provide a clear and concise introduction to the subjects of inverse problems and data assimilation, and their inter-relations. The first part of our notes covers inverse problems; this refers to the study of how to estimate unknown model parameters from data. The second part of our notes covers data assimilation; this refers to a particular class of inverse problems in which the unknown parameter is the initial condition (and/or state) of a dynamical system, and the data comprises partial and noisy observations of the state. The third and final part of our notes describes the use of data assimilation methods to solve generic inverse problems by introducing an artificial algorithmic time. Our notes cover, among other topics, maximum a posteriori estimation, (stochastic) gradient descent, variational Bayes, Monte Carlo, importance sampling and Markov chain Monte Carlo for inverse problems; and 3DVAR, 4DVAR, extended and ensemble Kalman filters, and particle filters for data assimilation. Each of parts one and two starts with a chapter on the Bayesian formulation, in which the problem solution is given by a posterior distribution on the unknown parameter. Then the following chapter specializes the Bayesian formulation to a linear-Gaussian setting where explicit characterization of the posterior is possible and insightful. The next two chapters explore methods to extract information from the posterior in nonlinear and non-Gaussian settings using optimization and Gaussian approximations. The final two chapters describe sampling methods that can reproduce the full posterior in the large sample limit. Each chapter closes with a bibliography containing citations to alternative pedagogical literature and to relevant research literature. We also include a set of exercises at the end of parts one and two. Our notes are thus useful for both classroom teaching and self-guided study.