arXiv subjects

Andrey A Popov

Publications and source records attributed to Andrey A Popov.

12 recordsLinked to original sources

arXiv2025

A geometric ensemble method for Bayesian inference

Conventional approximations to Bayesian inference rely on either approximations by statistics such as mean and covariance or by point particles. Recent advances such as the ensemble Gaussian mixture filter have generalized these notions to sums of parameterized distributions. This work presents a new methodology for approximating Bayesian inference by sums of uniform distributions on convex polytopes. The methodology presented herein is developed from the simplest convex polytope filter that takes advantage of uniform prior and measurement uncertainty, to an operationally viable ensemble filter with Kalmanized approximations to updating convex polytopes. Numerical results on the Ikeda map show the viability of this methodology in the low-dimensional setting, and numerical results on the Lorenz '96 equations similarly show viability in the high-dimensional setting.

math.OC↗

arXiv2025

Deterministic Optimal Transport-based Gaussian Mixture Particle Filtering for Verifiable Applications

Mixture-model particle filters such as the ensemble Gaussian mixture filter require a resampling procedure in order to converge to exact Bayesian inference. Canonically, stochastic resampling is performed, which provides useful samples with no guarantee of usefulness for a finite ensemble. We propose a new resampling procedure based on optimal transport that deterministically selects optimal resampling points. We show on a toy 3-variable problem that it significantly reduces the amount of particles required for useful state estimation. Finally, we show that this filter improves the state estimation of a seldomly-observed space object in an NRHO around the moon.

stat.CO↗

arXiv2024

Ensemble Variational Fokker-Planck Methods for Data Assimilation

Particle flow filters solve Bayesian inference problems by smoothly transforming a set of particles into samples from the posterior distribution. Particles move in state space under the flow of an McKean-Vlasov-Ito process. This work introduces the Variational Fokker-Planck (VFP) framework for data assimilation, a general approach that includes previously known particle flow filters as special cases. The McKean-Vlasov-Ito process that transforms particles is defined via an optimal drift that depends on the selected diffusion term. It is established that the underlying probability density - sampled by the ensemble of particles - converges to the Bayesian posterior probability density. For a finite number of particles the optimal drift contains a regularization term that nudges particles toward becoming independent random variables. Based on this analysis, we derive computationally-feasible approximate regularization approaches that penalize the mutual information between pairs of particles, and avoid particle collapse. Moreover, the diffusion plays a role akin to a particle rejuvenation approach that aims to alleviate particle collapse. The VFP framework is very flexible. Different assumptions on prior and intermediate probability distributions can be used to implement the optimal drift, and localization and covariance shrinkage can be applied to alleviate the curse of dimensionality. A robust implicit-explicit method is discussed for the efficient integration of stiff McKean-Vlasov-Ito processes. The effectiveness of the VFP framework is demonstrated on three progressively more challenging test problems, namely the Lorenz '63, Lorenz '96 and the quasi-geostrophic equations.

math.OC↗

arXiv2022

An Adaptive Covariance Parameterization Technique for the Ensemble Gaussian Mixture Filter

The ensemble Gaussian mixture filter combines the simplicity and power of Gaussian mixture models with the provable convergence and power of particle filters. The quality of the ensemble Gaussian mixture filter heavily depends on the choice of covariance matrix in each Gaussian mixture. This work extends the ensemble Gaussian mixture filter to an adaptive choice of covariance based on the parameterized estimates of the sample covariance matrix. Through the use of the expectation maximization algorithm, optimal choices of the covariance matrix parameters are computed in an online fashion. Numerical experiments on the Lorenz '63 equations show that the proposed methodology converges to classical results known in particle filtering. Further numerical results with more advances choices of covariance parameterization and the medium-size Lorenz '96 equations show that the proposed approach can perform significantly better than the standard EnGMF, and other classical data assimilation algorithms.

math.OC↗

arXiv2022

The Model Forest Ensemble Kalman Filter

Traditional data assimilation uses information obtained from the propagation of one physics-driven model and combines it with information derived from real-world observations in order to obtain a better estimate of the truth of some natural process. However, in many situations multiple simulation models that describe the same physical phenomenon are available. Such models can have different sources. On one hand there are theory-guided models are constructed from first physical principles, while on the other there are data-driven models that are constructed from snapshots of high fidelity information. In this work we provide a possible way to make use of this collection of models in data assimilation by generalizing the idea of model hierarchies into model forests -- collections of high fidelity and low fidelity models organized in a groping of model trees such as to capture various relationships between different models. We generalize the multifidelity ensemble Kalman filter that previously operated on model hierarchies into the model forest ensemble Kalman filter through a generalized theory of linear control variates. This new filter allows for much more freedom when treading the line between accuracy and speed. Numerical experiments with a high fidelity quasi-geostrophic model and two of its low fidelity reduced order models validate the accuracy of our approach.

cs.CE↗

arXiv2022

A Stochastic Covariance Shrinkage Approach in Ensemble Transform Kalman Filtering

The Ensemble Kalman Filters (EnKF) employ a Monte-Carlo approach to represent covariance information, and are affected by sampling errors in operational settings where the number of model realizations is much smaller than the model state dimension. To alleviate the effects of these errors EnKF relies on model-specific heuristics such as covariance localization, which takes advantage of the spatial locality of correlations among the model variables. This work proposes an approach to alleviate sampling errors that utilizes a locally averaged-in-time dynamics of the model, described in terms of a climatological covariance of the dynamical system. We use this covariance as the target matrix in covariance shrinkage methods, and develop a stochastic covariance shrinkage approach where synthetic ensemble members are drawn to enrich both the ensemble subspace and the ensemble transformation. We additionally provide for a way in which this methodology can be localized similar to the state-of-the-art LETKF method, and that for a certain model setup, our methodology significantly outperforms it.

stat.ME↗

arXiv2022

A fast time-stepping strategy for dynamical systems equipped with a surrogate model

Simulation of complex dynamical systems arising in many applications is computationally challenging due to their size and complexity. Model order reduction, machine learning, and other types of surrogate modeling techniques offer cheaper and simpler ways to describe the dynamics of these systems but are inexact and introduce additional approximation errors. In order to overcome the computational difficulties of the full complex models, on one hand, and the limitations of surrogate models, on the other, this work proposes a new accelerated time-stepping strategy that combines information from both. This approach is based on the multirate infinitesimal general-structure additive Runge-Kutta (MRI-GARK) framework. The inexpensive surrogate model is integrated with a small timestep to guide the solution trajectory, and the full model is treated with a large timestep to occasionally correct for the surrogate model error and ensure convergence. We provide a theoretical error analysis, and several numerical experiments, to show that this approach can be significantly more efficient than using only the full or only the surrogate model for the integration.

math.NA↗

arXiv2021

A Stochastic Covariance Shrinkage Approach to Particle Rejuvenation in the Ensemble Transform Particle Filter

Rejuvenation in particle filters is necessary to prevent the collapse of the weights when the number of particles is insufficient to sample the high probability regions of the state space. Rejuvenation is often implemented in a heuristic manner by the addition of stochastic samples that widen the support of the ensemble. This work aims at improving canonical rejuvenation methodology by the introduction of additional prior information obtained from climatological samples; the dynamical particles used for importance sampling are augmented with samples obtained from stochastic covariance shrinkage. The ensemble transport particle filter, and its second order variant, are extended with the proposed rejuvenation approach. Numerical experiments show that modified filters significantly improve the analyses for low dynamical ensemble sizes.

math.ST↗

arXiv2021

Multifidelity Ensemble Kalman Filtering Using Surrogate Models Defined by Physics-Informed Autoencoders

Data assimilation is a Bayesian inference process that obtains an enhanced understanding of a physical system of interest by fusing information from an inexact physics-based model, and from noisy sparse observations of reality. The multifidelity ensemble Kalman filter (MFEnKF) recently developed by the authors combines a full-order physical model and a hierarchy of reduced order surrogate models in order to increase the computational efficiency of data assimilation. The standard MFEnKF uses linear couplings between models, and is statistically optimal in case of Gaussian probability densities. This work extends MFEnKF to work with non-linear couplings between the models. Optimal nonlinear projection and interpolation operators are obtained by appropriately trained physics-informed autoencoders, and this approach allows to construct reduced order surrogate models with less error than conventional linear methods. Numerical experiments with the canonical Lorenz '96 model illustrate that nonlinear surrogates perform better than linear projection-based ones in the context of multifidelity filtering.

math.OC↗

arXiv2020

A Multifidelity Ensemble Kalman Filter with Reduced Order Control Variates

This work develops a new multifidelity ensemble Kalman filter (MFEnKF) algorithm based on linear control variate framework. The approach allows for rigorous multifidelity extensions of the EnKF, where the uncertainty in coarser fidelities in the hierarchy of models represent control variates for the uncertainty in finer fidelities. Small ensembles of high fidelity model runs are complemented by larger ensembles of cheaper, lower fidelity runs, to obtain much improved analyses at only small additional computational costs. We investigate the use of reduced order models as coarse fidelity control variates in the MFEnKF, and provide analyses to quantify the improvements over the traditional ensemble Kalman filters. We apply these ideas to perform data assimilation with a quasi-geostrophic test problem, using direct numerical simulation and a corresponding POD-Galerkin reduced order model. Numerical results show that the two-fidelity MFEnKF provides better analyses than existing EnKF algorithms at comparable or reduced computational costs.

math.NA↗

arXiv2020

An Explicit Probabilistic Derivation of Inflation in a Scalar Ensemble Kalman Filter for Finite Step, Finite Ensemble Convergence

This paper uses a probabilistic approach to analyze the converge of an ensemble Kalman filter solution to an exact Kalman filter solution in the simplest possible setting, the scalar case, as it allows us to build upon a rich literature of scalar probability distributions and non-elementary functions. To this end we introduce the bare-bones Scalar Pedagogical Ensemble Kalman Filter (SPEnKF). We show that in the asymptotic case of ensemble size, the expected value of both the analysis mean and variance estimate of the SPEnKF converges to that of the true Kalman filter, and that the variances of both tend towards zero, at each time moment. We also show that the ensemble converges in probability in the complementary case, when the ensemble is finite, and time is taken to infinity. Moreover, we show that in the finite-ensemble, finite-time case, variance inflation and mean correction can be leveraged to coerce the SPEnKF converge to its scalar Kalman filter counterpart. We then apply this framework to analyze perturbed observations and explain why perturbed observations ensemble Kalman filters underperform their deterministic counterparts.

math.OC↗

arXiv2018

A Bayesian Approach to Multivariate Adaptive Localization in Ensemble-Based Data Assimilation with Time-Dependent Extensions

Ever since its inception, the Ensemble Kalman Filter has elicited many heuristic methods that sought to correct it. One such method is localization---the thought that `nearby' variables should be highly correlated with `far away' variable not. Recognizing that correlation is a time-dependent property, adaptive localization is a natural extension to these heuristics. We propose a Bayesian approach to adaptive Schur-product localization for the DEnKF, and extend it to support multiple radii of influence. We test both the empirical validity of (multivariate) adaptive localization, and of our approach. We test a simple toy problem (Lorenz'96), extending it to a multivariate model, and a more realistic geophysical problem (1.5 Layer Quasi-Geostrophic). We show that the multivariate approach has great promise on the toy problem, and that the univariate approach leads to improved filter performance for the realistic geophysical problem.

math.NA↗