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Inverse problems are common and important in many applications in computational physics but are inherently ill-posed with many possible model parameters resulting in satisfactory results in the observation space. When solving the inverse…

Computational Physics · Physics 2020-06-24 Xin-Lei Zhang , Carlos Michelén-Ströfer , Heng Xiao

Ensemble Kalman inversion (EKI) is a derivative-free, particle-based optimization method for solving inverse problems. It can be shown that EKI approximates a gradient flow, which allows the application of methods for accelerating gradient…

Optimization and Control · Mathematics 2025-07-04 Sydney Vernon , Eviatar Bach , Oliver R. A. Dunbar

This work proposes a novel Alternating Direction Method of Multipliers (ADMM)-based Ensemble Kalman Inversion (EKI) algorithm for solving constrained nonlinear model predictive control (NMPC) problems. First, stage-wise nonlinear inequality…

Optimization and Control · Mathematics 2026-03-26 Ahmed Khalil , Mohamed Safwat , Efstathios Bakolas

The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the Kalman filter for high dimensional linear Gaussian state space models. EnKF methods have also been developed for parameter inference of static Bayesian models with a…

Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are popular sampling methods for obtaining a target posterior distribution. They can be seem as one step (the analysis step) in the data assimilation method Ensemble…

Numerical Analysis · Mathematics 2025-03-07 Zhiyan Ding , Qin Li , Jianfeng Lu

In the process of reproducing the state dynamics of parameter dependent distributed systems, data from physical measurements can be incorporated into the mathematical model to reduce the parameter uncertainty and, consequently, improve the…

Numerical Analysis · Mathematics 2022-10-06 Francesco A. B. Silva , Cecilia Pagliantini , Martin Grepl , Karen Veroy

Numerical models of geothermal reservoirs typically depend on hundreds or thousands of unknown parameters, which must be estimated using sparse, noisy data. However, these models capture complex physical processes, which frequently results…

This work presents new results and understanding of the Ensemble Kalman filter (EnKF) for inverse problems. In particular, using a Lagrangian dual perspective we show that EnKF can be derived from the sample average approximation (SAA) of…

Numerical Analysis · Mathematics 2026-01-27 C G Krishnanunni , Jonathan Wittmer , Tan Bui-Thanh , Quoc P. Nguyen

We consider the inverse problem of determining the Winkler subgrade reaction coefficient of a slab foundation modelled as a thin elastic plate clamped at the boundary. The plate is loaded by a concentrated force and its transversal…

Analysis of PDEs · Mathematics 2016-02-17 Giovanni Alessandrini , Antonino Morassi , Edi Rosset , Sergio Vessella

Although the governing equations of many systems, when derived from first principles, may be viewed as known, it is often too expensive to numerically simulate all the interactions they describe. Therefore researchers often seek simpler…

Computation · Statistics 2021-05-03 Tapio Schneider , Andrew M. Stuart , Jin-Long Wu

Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper, we propose a new version of such a methodology, a framework…

Dynamical Systems · Mathematics 2019-10-17 Alfredo Garbuno-Inigo , Franca Hoffmann , Wuchen Li , Andrew M. Stuart

Ensemble Kalman Inversion (EKI) has been proposed as an efficient method for the approximate solution of Bayesian inverse problems with expensive forward models. However, when applied to the Bayesian inverse problem EKI is only exact in the…

Computation · Statistics 2024-11-19 Richard D. P. Grumitt , Minas Karamanis , Uroš Seljak

In recent years, uncertainty-aware full waveform inversion (FWI) has received increasing attention, with a growing emphasis on producing informative uncertainty estimates alongside inversion results. Bayesian inference methods--particularly…

Geophysics · Physics 2025-05-14 Yunduo Li , Yijie Zhang , Xueyu Zhu , Jinghuai Gao

The use of ensemble methods to solve inverse problems is attractive because it is a derivative-free methodology which is also well-adapted to parallelization. In its basic iterative form the method produces an ensemble of solutions which…

Numerical Analysis · Mathematics 2018-05-09 Neil K. Chada , Marco A. Iglesias , Lassi Roininen , Andrew M. Stuart

The Ensemble Kalman filter (EnKF) was introduced by Evensen in 1994 [10] as a novel method for data assimilation: state estimation for noisily observed time-dependent problems. Since that time it has had enormous impact in many application…

Optimization and Control · Mathematics 2013-04-08 Marco A. Iglesias , Kody J. H. Law , Andrew M. Stuart

Elastic full-waveform inversion (EFWI) is a process used to estimate subsurface properties by fitting seismic data while satisfying wave propagation physics. The problem is formulated as a least-squares data fitting minimization problem…

Numerical Analysis · Mathematics 2024-04-12 Kamal Aghazade , Ali Gholami , Hossein S. Aghamiry , Hamid Reza Siahkoohi

In this paper we discuss a deterministic form of ensemble Kalman inversion as a regularization method for linear inverse problems. By interpreting ensemble Kalman inversion as a low-rank approximation of Tikhonov regularization, we are able…

Numerical Analysis · Mathematics 2023-10-31 Fabian Parzer , Otmar Scherzer

The unscented Kalman inversion (UKI) presented in [1] is a general derivative-free approach to solving the inverse problem. UKI is particularly suitable for inverse problems where the forward model is given as a black box and may not be…

Numerical Analysis · Mathematics 2021-02-24 Daniel Z. Huang , Jiaoyang Huang

We propose the application of iterative regularization for the development of ensemble methods for solving Bayesian inverse problems. In concrete, we construct (i) a variational iterative regularizing ensemble Levenberg-Marquardt method…

Numerical Analysis · Mathematics 2014-06-25 Marco A. Iglesias

Analytic and optimization methods for solving inverse kinematics (IK) problems have been deeply studied throughout the history of robotics. The two strategies have complementary strengths and weaknesses, but developing a unified approach to…

Robotics · Computer Science 2026-02-06 Thomas Cohn , Lihan Tang , Alexandre Amice , Russ Tedrake