arXiv subjects

Matthias Morzfeld

Publications and source records attributed to Matthias Morzfeld.

At least 19 recordsLinked to original sources

arXiv2026

Why we should condition denoising diffusion generative models on windows of past observations

Data assimilation (DA) is, traditionally, a cycling process that relies on time-dependent priors to propagate information from past observations to future cycles. Using denoising diffusion generative modeling for DA is challenging because standard approaches use a fixed training data set, which in turn leads to a static prior that ignores information from past observations. Because past observations are ignored, DA systems with static priors lead to larger posterior errors than cycling DA systems. Incorporating time-dependent priors into generative models, however, requires expensive and frequent retraining. Motivated by linear systems theory - where the dependence of a prediction of a Kalman filter on past observations decays exponentially - we condition diffusion models on short windows of past observations. Specifically, we describe training procedures for two frameworks: a diffusion DA system predicting the current state given a set of past observations, and a diffusion ``direct observation prediction'' (DOP) system, predicting future observations given a set of past observations. Using a canonical linear system, we show that both systems can achieve the minimal posterior error characteristic of a fully-cycled DA/DOP system, without re-training, provided the time windows are long enough. The linear setup ensures analytical tractability, avoids confounding neural network training errors, and confirms that conditioning on windows of past observations is required for efficient and accurate diffusion-based DA or DOP.

math-ph↗

arXiv2026

Neural ensemble Kalman filter: Data assimilation for compressible flows with shocks

Data assimilation (DA) for compressible flows with shocks is challenging because many classical DA methods generate spurious oscillations and nonphysical features near uncertain shocks. We focus here on the ensemble Kalman filter (EnKF). We show that the poor performance of the EnKF may be attributed to the bimodal forecast distribution that can arise in the vicinity of an uncertain shock location; this violates the assumptions underpinning the EnKF, which assume a forecast which is close to Gaussian. To address this issue we introduce the new neural EnKF. The basic idea is to systematically embed neural function approximations within ensemble DA by mapping the forecast ensemble of shocked flows to the parameter space (weights and biases) of a deep neural network (NN) and to subsequently perform DA in that space. The nonlinear mapping encodes sharp and smooth flow features in an ensemble of NN parameters. Neural EnKF updates are therefore well-behaved only if the NN parameters vary smoothly within the neural representation of the forecast ensemble. We show that such a smooth variation of network parameters can be enforced via physics-informed transfer learning, and demonstrate that in so-doing the neural EnKF avoids the spurious oscillations and nonphysical features that plague the EnKF. The applicability of the neural EnKF is demonstrated through a series of systematic numerical experiments with the inviscid Burgers' equation, the Sod shock tube, and a two-dimensional blast wave.

physics.flu-dyn↗

arXiv2025

The Ensemble Kalman Inversion Race

Ensemble Kalman methods were initially developed to solve nonlinear data assimilation problems in oceanography but are now popular in applications far beyond their original use cases. Of particular interest is climate model calibration. As hybrid physics and machine-learning models advance, the number of parameters and complexity of parameterizations in climate models will continue to grow. To fully realize these advances, we must move from laborious hand-tuning to calibration-driven model development in rapid iteration cycles. Thus, robust calibration of these parameters plays an increasingly important role. We focus on learning climate model parameters by minimizing the misfit between modeled and observed climate statistics in an idealized setting. Ensemble Kalman methods are a natural choice for this problem because they are derivative-free, scalable to high dimensions, and robust to noise caused by statistical observations. Given the many variants of ensemble methods proposed, an important question is: Which ensemble Kalman method should be used for climate model calibration? To answer this question, we perform systematic numerical experiments to explore the relative computational efficiencies of several ensemble Kalman methods. The numerical experiments involve statistical observations of Lorenz-type models of increasing complexity, frequently used to represent simplified atmospheric systems, with some featuring neural network parameterizations. For each test problem, several ensemble Kalman methods and a derivative-based method "race" to reach a specified accuracy, and we measure the computational cost required to achieve the desired accuracy. We investigate how prior information and the parameter or data dimensions define the computational costs of the various methods.

physics.data-an↗

arXiv2025

Numerical study of high-dimensional covariance estimation and localization for data assimilation

Covariance localization is a critical component of ensemble-based data assimilation (DA) and many current localization schemes simply dampen correlations as a function of distance. Increases in computational resources, broadening scope of application for DA, and advances in general statistical methodology raise the question as to whether alternative localization methods may improve ensemble DA relative to current schemes. We carefully explore this issue by comparing distance based localization with alternative covariance localization techniques, partially those taken from the statistical literature. The comparison is done on test problems that we designed to challenge distance-based localization, including joint state-parameter estimation in a modified Lorenz '96 model and state estimation in a two-layer quasi-geostrophic model. Across all sets of experiments, we find that while localization of any kind (with rare exceptions) can lead to significant reductions in error, traditional, distance-based localization generally leads to the largest error reduction. More general localization schemes can sometimes lead to greater error reduction, though the impacts may only be marginal and may require more tuning and/or prior information.

physics.data-an↗

arXiv2025

Using Diffusion Models to do Data Assimilation

The recent surge in machine learning (ML) methods for geophysical modeling has raised the question of how these methods might be applied to data assimilation (DA). We focus on diffusion modeling (a form of generative artificial intelligence) for systems that can perform the entire DA process, rather than on ML-based tools used within a conventional DA system. We identify at least three distinct types of diffusion-based DA systems and show that they differ in the posterior distribution they target for sampling. These posterior distributions correspond to different priors and/or likelihoods, which in turn result in unique training datasets, computational requirements, and state estimate qualities. Our analysis further shows that a diffusion DA system designed to target the same posterior distribution as current ensemble DA algorithms requires re-training at each DA cycle, which is computationally costly. We discuss the implications of these findings for the use of diffusion modeling in DA.

physics.ao-ph↗

arXiv2023

Localized covariance estimation: A Bayesian perspective

A major problem in numerical weather prediction (NWP) is the estimation of high-dimensional covariance matrices from a small number of samples. Maximum likelihood estimators cannot provide reliable estimates when the overall dimension is much larger than the number of samples. Fortunately, NWP practitioners have found ingenious ways to boost the accuracy of their covariance estimators by leveraging the assumption that the correlations decay with spatial distance. In this work, Bayesian statistics is used to provide a new justification and analysis of the practical NWP covariance estimators. The Bayesian framework involves manipulating distributions over symmetric positive definite matrices, and it leads to two main findings: (i) the commonly used "hybrid estimator" for the covariance matrix has a naturally Bayesian interpretation; (ii) the very commonly used "Schur product estimator" is not Bayesian, but it can be studied and understood within the Bayesian framework. As practical implications, the Bayesian framework shows how to reduce the amount of tuning required for covariance estimation, and it suggests that efficient covariance estimation should be rooted in understanding and penalizing conditional correlations, rather than correlations.

stat.ME↗

arXiv2022

Localization in Ensemble Kalman inversion

Ensemble Kalman inversion (EKI) is a technique for the numerical solution of inverse problems. A great advantage of the EKI's ensemble approach is that derivatives are not required in its implementation. But theoretically speaking, EKI's ensemble size needs to surpass the dimension of the problem. This is because of EKI's "subspace property", i.e., that the EKI solution is a linear combination of the initial ensemble it starts off with. We show that the ensemble can break out of this initial subspace when "localization" is applied. In essence, localization enforces an assumed correlation structure onto the problem, and is heavily used in ensemble Kalman filtering and data assimilation. We describe and analyze how to apply localization to the EKI, and how localization helps the EKI ensemble break out of the initial subspace. Specifically, we show that the localized EKI (LEKI) ensemble will collapse to a single point (as intended) and that the LEKI ensemble mean will converge to the global optimum at a sublinear rate. Under strict assumptions on the localization procedure and observation process, we further show that the data misfit decays uniformly. We illustrate our ideas and theoretical developments with numerical examples with simplified toy problems, a Lorenz model, and an inversion of electromagnetic data, where some of our mathematical assumptions may only be approximately valid.

math.NA↗

arXiv2020

Can one use Earth's magnetic axial dipole field intensity to predict reversals?

We study predictions of reversals of Earth's axial magnetic dipole field that are based solely on the dipole's intensity. The prediction strategy is, roughly, that once the dipole intensity drops below a threshold, then the field will continue to decrease and a reversal (or a major excursion) will occur. We first present a rigorous definition of an intensity threshold-based prediction strategy and then describe a mathematical and numerical framework to investigate its validity and robustness in view of the data being limited. We apply threshold-based predictions to a hierarchy of numerical models, ranging from simple scalar models to 3D geodynamos. We find that the skill of threshold-based predictions varies across the model hierarchy. The differences in skill can be explained by differences in how reversals occur: if the field decreases towards a reversal slowly (in a sense made precise in this paper), the skill is high, and if the field decreases quickly, the skill is low. Such a property could be used as an additional criterion to identify which models qualify as Earth-like. Applying threshold-based predictions to Virtual Axial Dipole Moment (VADM) paleomagnetic reconstructions (PADM2M and Sint-2000) covering the last two million years, reveals a moderate skill of threshold-based predictions for Earth's dynamo. Besides all of their limitations, threshold-based predictions suggest that no reversal is to be expected within the next 10 kyr. Most importantly, however, we show that considering an intensity threshold for identifying upcoming reversals is intrinsically limited by the dynamic behavior of Earth's magnetic field.

physics.geo-ph↗

arXiv2018

A Bayesian approach to calibrating hydrogen flame kinetics using many experiments and parameters

First-principles Markov Chain Monte Carlo sampling is used to investigate uncertainty quantification and uncertainty propagation in parameters describing hydrogen kinetics. Specifically, we sample the posterior distribution of thirty-one parameters focusing on the H2O2 and HO2 reactions resulting from conditioning on ninety-one experiments. Established literature values are used for the remaining parameters in the mechanism. The samples are computed using an affine invariant sampler starting with broad, noninformative priors. Autocorrelation analysis shows that O(1M) samples are sufficient to obtain a reasonable sampling of the posterior. The resulting distribution identifies strong positive and negative correlations and several non-Gaussian characteristics. Using samples drawn from the posterior, we investigate the impact of parameter uncertainty on the prediction of two more complex flames: a 2D premixed flame kernel and the ignition of a hydrogen jet issuing into a heated chamber. The former represents a combustion regime similar to the target experiments used to calibrate the mechanism and the latter represents a different combustion regime. For the premixed flame, the net amount of product after a given time interval has a standard deviation of less than 2% whereas the standard deviation of the ignition time for the jet is more than 10%. The samples used for these studies are posted online. These results indicate the degree to which parameters consistent with the target experiments constrain predicted behavior in different combustion regimes. This process provides a framework for both identifying reactions for further study from candidate mechanisms as well as combining uncertainty quantification and propagation to, ultimately, tie uncertainty in laboratory flame experiments to uncertainty in end-use numerical predictions of more complicated scenarios.

math.NA↗

arXiv2017

Localization for MCMC: sampling high-dimensional posterior distributions with local structure

We investigate how ideas from covariance localization in numerical weather prediction can be used in Markov chain Monte Carlo (MCMC) sampling of high-dimensional posterior distributions arising in Bayesian inverse problems. To localize an inverse problem is to enforce an anticipated "local" structure by (i) neglecting small off-diagonal elements of the prior precision and covariance matrices; and (ii) restricting the influence of observations to their neighborhood. For linear problems we can specify the conditions under which posterior moments of the localized problem are close to those of the original problem. We explain physical interpretations of our assumptions about local structure and discuss the notion of high dimensionality in local problems, which is different from the usual notion of high dimensionality in function space MCMC. The Gibbs sampler is a natural choice of MCMC algorithm for localized inverse problems and we demonstrate that its convergence rate is independent of dimension for localized linear problems. Nonlinear problems can also be tackled efficiently by localization and, as a simple illustration of these ideas, we present a localized Metropolis-within-Gibbs sampler. Several linear and nonlinear numerical examples illustrate localization in the context of MCMC samplers for inverse problems.

stat.ME↗

arXiv2017

Symmetrized importance samplers for stochastic differential equations

We study a class of importance sampling methods for stochastic differential equations (SDEs). A small-noise analysis is performed, and the results suggest that a simple symmetrization procedure can significantly improve the performance of our importance sampling schemes when the noise is not too large. We demonstrate that this is indeed the case for a number of linear and nonlinear examples. Potential applications, e.g., data assimilation, are discussed.

math.NA↗

arXiv2016

Analysis of the ensemble Kalman filter for marginal and joint posteriors

The ensemble Kalman filter (EnKF) is widely used to sample a probability density function (pdf) generated by a stochastic model conditioned by noisy data. This pdf can be either a joint posterior that describes the evolution of the state of the system in time, conditioned on all the data up to the present, or a particular marginal of this posterior. We show that the EnKF collapses in the same way and under even broader conditions as a particle filter when it samples the joint posterior. However, this does not imply that EnKF collapses when it samples the marginal posterior. We we show that a localized and inflated EnKF can efficiently sample this marginal, and argue that the marginal posterior is often the more useful pdf in geophysics. This explains the wide applicability of EnKF in this field. We further investigate the typical tuning of EnKF, in which one attempts to match the mean square error (MSE) to the marginal posterior variance, and show that sampling error may be huge, even if the MSE is moderate.

physics.data-an↗

arXiv2016

Iterative importance sampling algorithms for parameter estimation

In parameter estimation problems one computes a posterior distribution over uncertain parameters defined jointly by a prior distribution, a model, and noisy data. Markov Chain Monte Carlo (MCMC) is often used for the numerical solution of such problems. An alternative to MCMC is importance sampling, which can exhibit near perfect scaling with the number of cores on high performance computing systems because samples are drawn independently. However, finding a suitable proposal distribution is a challenging task. Several sampling algorithms have been proposed over the past years that take an iterative approach to constructing a proposal distribution. We investigate the applicability of such algorithms by applying them to two realistic and challenging test problems, one in subsurface flow, and one in combustion modeling. More specifically, we implement importance sampling algorithms that iterate over the mean and covariance matrix of Gaussian or multivariate t-proposal distributions. Our implementation leverages massively parallel computers, and we present strategies to initialize the iterations using "coarse" MCMC runs or Gaussian mixture models.

math.NA↗

arXiv2015

What the collapse of the ensemble Kalman filter tells us about particle filters

The ensemble Kalman filter (EnKF) is a reliable data assimilation tool for high-dimensional meteorological problems. On the other hand, the EnKF can be interpreted as a particle filter, and particle filters collapse in high-dimensional problems. We explain that these seemingly contradictory statements offer insights about how particle filters function in certain high-dimensional problems, and in particular support recent efforts in meteorology to "localize" particle filters, i.e., to restrict the influence of an observation to its neighborhood.

math.NA↗

arXiv2015

Sampling, feasibility, and priors in Bayesian estimation

Importance sampling algorithms are discussed in detail, with an emphasis on implicit sampling, and applied to data assimilation via particle filters. Implicit sampling makes it possible to use the data to find high-probability samples at relatively low cost, making the assimilation more efficient. A new analysis of the feasibility of data assimilation is presented, showing in detail why feasibility depends on the Frobenius norm of the covariance matrix of the noise and not on the number of variables. A discussion of the convergence of particular particle filters follows. A major open problem in numerical data assimilation is the determination of appropriate priors, a progress report on recent work on this problem is given. The analysis highlights the need for a careful attention both to the data and to the physics in data assimilation problems.

stat.CO↗

arXiv2014

Implicit sampling for path integral control, Monte Carlo localization, and SLAM

The applicability and usefulness of implicit sampling in stochastic optimal control, stochastic localization, and simultaneous localization and mapping (SLAM), is explored; implicit sampling is a recently-developed variationally-enhanced sampling method. The theory is illustrated with examples, and it is found that implicit sampling is significantly more efficient than current Monte Carlo methods in test problems for all three applications.

math.OC↗

arXiv2014

Small-noise analysis and symmetrization of implicit Monte Carlo samplers

Implicit samplers are algorithms for producing independent, weighted samples from multi-variate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis suggests a symmetrization of the algo- rithms that leads to improved (implicit) sampling schemes at a rel- atively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems.

math.NA↗

arXiv2014

Conditions for successful data assimilation

We show, using idealized models, that numerical data assimilation can be successful only if an effective dimension of the problem is not excessive. This effective dimension depends on the noise in the model and the data, and in physically reasonable problems it can be moderate even when the number of variables is huge. We then analyze several data assimilation algorithms, including particle filters and variational methods. We show that well-designed particle filters can solve most of those data assimilation problems that can be solved in principle, and compare the conditions under which variational methods can succeed to the conditions required of particle filters. We also discuss the limitations of our analysis.

math-ph↗