arXiv subjects

Dan Crisan

Publications and source records attributed to Dan Crisan.

At least 19 recordsLinked to original sources

arXiv2026

An Operator-Theoretic Analysis of Nonlinear Filtering under Model Misspecification

We study Bayesian filtering under misspecified model dynamics. The optimal filter is governed by predictive operators $K=\{K_t\}_{t\ge 1}$ associated with the true dynamics, while the approximate filter uses perturbed operators $\widehat K=\{\widehat K_t\}_{t\ge 1}$. Both filters share the same update operators $U=\{U_t\}_{t\ge 1}$ induced by the observation model. We work in a fully probabilistic setting in which observations are random variables, rather than fixed realizations. This viewpoint yields a linear recursion for the mean measures of the optimal filter and makes the propagation of filtering errors depend on two ingredients: the contraction of the predictive operators, and the accumulated model discrepancy. Within this framework, we derive explicit bounds for the discrepancy between the optimal and misspecified filters. The bounds separate the effect of the initial-condition error from the perturbation introduced by the dynamical mismatch $\widehat K-K$. Under suitable contractivity assumptions, the cumulative error stabilizes at a steady-state level determined by the size of the misspecification and by the Dobrushin coefficients of the predictive operators. We also give analytical examples in which the assumptions can be verified directly. In particular, if the transition functions are bounded and the transition noise belongs to a family of elliptically symmetric distributions, including Gaussian, Student-$t$ and Laplace laws, then the filter error stabilizes as predicted by the theory. The results provide a theoretical basis for quantifying the robustness of nonlinear filters under random observations and misspecified dynamics.

math.ST↗

arXiv2026

Signature Kernel and Schwinger-Dyson Kernel Equations as Two-Parameter Rough Differential Equations

We develop a rough-path framework for two-parameter rough differential equations on rectangular and simplicial domains, motivated by the signature kernel and Schwinger--Dyson kernel equations. The theory is formulated in spaces of jointly controlled rough paths and is based on a robust two-parameter rough integration framework. In particular, we introduce a notion of rough integration over two-dimensional simplices at low regularity extending previous results in the literature. Within this setting, we show that the signature kernel equation arises naturally as a two-parameter rough differential equation and establish well-posedness and stability. We also extend the Schwinger--Dyson kernel equation, previously formulated for bounded-variation paths, to rough driving signals, proving existence and uniqueness in appropriate controlled rough path spaces. In the smooth rough path regime, we relate the resulting equations to PDE and integro-differential formulations. Finally, we derive and analyse a numerical scheme for the rough Schwinger--Dyson equation, including runtime and memory complexity estimates, and illustrate its performance with numerical experiments.

math.PR↗

arXiv2026

Well-posedness and Hurst parameter estimation for fluid equations driven by fractional transport noise

We study a two-dimensional incompressible vorticity equation on the torus driven by transport-type fractional Brownian noise with Hurst parameter $H \in (1/2,1)$. The model captures persistent, long-range correlated forcing consistent with inertial-range scaling laws and fractional Brownian approximations of turbulent fluctuations. A central ingredient of our approach is a version of the sewing lemma adapted to a class of integrands that includes, but is not limited to, transport-type structures. This result provides a flexible tool for constructing the Young integral and serves as a basis for analysing a wider class of stochastic partial differential equations. Using this approach, we establish existence and uniqueness of solutions via a fixed point argument and investigate statistical properties of the flow. In particular, we study quadratic functionals of the solution and derive an estimator for the Hurst parameter $H$.

math.PR↗

arXiv2026

Smoothness and other hyperparameter estimation for inverse problems related to data assimilation

We consider Bayesian inverse problems arising in data assimilation for dynamical systems governed by partial and stochastic partial differential equations. The space-time dependent field is inferred jointly with static parameters of the prior and likelihood densities. Particular emphasis is placed on the hyperparameter controlling the prior smoothness and regularity, which is critical in ensuring well-posedness, shaping posterior structure, and determining predictive uncertainty. Commonly it is assumed to be known and fixed a priori; however in this paper we will adopt a hierarchical Bayesian framework in which smoothness and other hyperparameters are treated as unknown and assigned hyperpriors. Posterior inference is performed using Metropolis-within-Gibbs sampling suitable to high dimensions, for which hyperparameter estimation involves little computational overhead. The methodology is demonstrated on inverse problems for the Navier-Stokes equations and the stochastic advection-diffusion equation, under sparse and dense observation regimes, using Gaussian priors with different covariance structure. Numerical results show that jointly estimating the smoothness substantially reduces the errors in uncertainty quantification and parameter estimation induced by smoothness misspecification, by achieving performance comparable to scenarios in which the true smoothness is known.

stat.CO↗

arXiv2026

Particle Filtering for a Class of State-Space Models with Low and Degenerate Observational Noise

We consider the discrete-time filtering problem in scenarios where the observation noise is low or degenerate. We focus on the case where the observation equation is a linear function of the state and the data involve additive noise. However, we place minimal assumptions on the hidden state process. For such a class of models we derive new particle filters (PFs) with the key property that their performance is robust to the size of the observation noise. As a consequence, the developed PFs are well-defined in the limiting case of degenerate observation noise. Indicatively, we prove (under assumptions) that the PF applied in this low noise setting inherits the properties of the PF used in the degenerate case. We extend our framework to the case where the hidden states are drawn from a diffusion process. In this scenario we develop new PFs which are robust to both low noise and fine levels of time discretization. We illustrate our algorithms numerically on several examples.

stat.CO↗

arXiv2025

Sequential Markov Chain Monte Carlo for Filtering of State-Space Models with Low or Degenerate Observation Noise

We consider the discrete-time filtering problem in scenarios where the observation noise is degenerate or low. More precisely, one is given access to a discrete time observation sequence which at any time $k$ depends only on the state of an unobserved Markov chain. We specifically assume that the functional relationship between observations and hidden Markov chain has either degenerate or low noise. In this article, under suitable assumptions, we derive the filtering density and its recursions for this class of problems on a specific sequence of manifolds defined through the observation function. We then design sequential Markov chain Monte Carlo methods to approximate the filter serially in time. For a certain linear observation model, we show that using sequential Markov chain Monte Carlo for low noise will converge as the noise disappears to that of using sequential Markov chain Monte Carlo for degenerate noise. We illustrate the performance of our methodology on several challenging stochastic models arising in statistics and applied mathematics.

stat.CO↗

arXiv2025

Data assimilation for energy-aware hybrid models

This work integrates ensemble-based data assimilation (DA) with the energy-aware hybrid modeling approach, applied to a three-layer quasi-geostrophic (QG) model of the Gulf Stream flow. Building on prior DA success in the QG channel regime, where stochastic corrections based on EOFs were effective, we show that this method fails to address persistent errors in the more complex, dynamically richer Gulf Stream setting.To overcome this, we employ a hybrid model that controls energy at selected scales, maintaining dynamic consistency and physical realism. We evaluate the combined effect of hybrid modeling and DA, using a particle filter which combines model reduction, tempering, jittering, and nudging. Numerical experiments show that the hybrid model reproduces both the large-scale jet and small-scale vortices seen in high-resolution reference simulations, but missing in the standard (non-hybrid) QG model. When DA is incorporated, the hybrid model further reduces tracking error and ensemble divergence. Moreover, targeted assimilation from the most energetic region matches tracking error and uncertainty reduction of full-domain networks, highlighting the critical importance of observation network design. These findings demonstrate that combining energy-aware hybrid modeling with ensemble-based DA enables high-fidelity, computationally efficient tracking of the reference solution even under sparse, noisy, localized observations.

physics.flu-dyn↗

arXiv2025

Data assimilation using a global Girsanov nudged particle filter

We present a particle filtering algorithm for stochastic models on infinite dimensional state space, making use of Girsanov perturbations to nudge the ensemble of particles into regions of higher likelihood. We argue that the optimal control problem needs to couple control variables for all of the particles to maintain an ensemble with good effective sample size (ESS). We provide an optimisation formulation that separates the problem into three stages, separating the nonlinearity in the ESS term in the functional with the nonlinearity due to the forward problem, and allowing independent parallel computation for each particle when calculations are performed over control variable space. The particle filter is applied to the stochastic Kuramoto-Sivashinsky equation, and compared with the temper-jitter particle filter approach. We observe that whilst the nudging filter is over spread compared to the temper-jitter filter, it responds to extreme events in the assimilated data more quickly and robustly.

math.NA↗

arXiv2025

A localized particle filter for geophysical data assimilation

Particle filters are computational techniques for estimating the state of dynamical systems by integrating observational data with model predictions. This work introduces a class of Localized Particle Filters (LPFs) that exploit spatial localization to reduce computational costs and mitigate particle degeneracy in high-dimensional systems. By partitioning the state space into smaller regions and performing particle weight updates and resampling separately within each region, these filters leverage assumptions of limited spatial correlation to achieve substantial computational gains. This approach proves particularly valuable for geophysical data assimilation applications, including weather forecasting and ocean modeling, where system dimensions are vast, and complex interactions and nonlinearities demand efficient yet accurate state estimation methods. We demonstrate the methodology on a partially observed rotating shallow water system, achieving favourable performance in terms of algorithm stability and error estimates.

stat.AP↗

arXiv2025

Nonlinear Stochastic Filtering with Volterra Gaussian noises

We develop a nonlinear filtering theory for signal-observation systems driven by Volterra Gaussian processes, covering both the Young and genuinely rough regimes. The dynamics are formulated as a rough differential equation in which the observation has a signal-dependent Volterra drift, a structure naturally induced by an equivalent change of measure. We establish global well-posedness of the coupled system and derive a Kallianpur-Striebel formula. We then obtain a robust pathwise representation of the filter. In the one-dimensional setting, we characterise the unnormalised conditional density through a rough Zakai equation and establish its well-posedness using an extension of the rough viscosity framework. Finally, under a partial H\"ormander-type condition, we prove that the conditional distribution of the signal admits a smooth density.

math.PR↗

arXiv2025

A uniform particle approximation to the Navier-Stokes-alpha models in three dimensions with advection noise

In this work, we investigate a system of interacting particles governed by a set of stochastic differential equations. Our main goal is to rigorously demonstrate that the empirical measure associated with the particle system converges uniformly, both in time and space, to the solution of the three dimensional Navier Stokes alpha model with advection noise. This convergence establishes a probabilistic framework for deriving macroscopic stochastic fluid equations from underlying microscopic dynamics. The analysis leverages semigroup techniques to address the nonlinear structure of the limiting equations, and we provide a detailed treatment of the well posedness of the limiting stochastic partial differential equation. This ensures that the particle approximation remains stable and controlled over time. Although similar convergence results have been obtained in two dimensional settings, our study presents the first proof of strong uniform convergence in three dimensions for a stochastic fluid model derived from an interacting particle system. Importantly, our results also yield new insights in the deterministic regime, namely, in the absence of advection noise, where this type of convergence had not been previously established.

math.PR↗

arXiv2024

Uniqueness of the solution of the filtering equations in spaces of measures

Nonlinear filtering is a pivotal problem that has attracted significant attention from mathematicians, statisticians, engineers, and various other scientific disciplines. The solution to this problem is governed by the so-called filtering equations. In this paper, we investigate the uniqueness of solutions to these equations within measure spaces and introduce a novel, generalized framework for this analysis. Our approach provides new insights and extends the applicability of existing theories in the study of nonlinear filtering.

math.PR↗

arXiv2024

Forecast error growth: A dynamic-stochastic model

There is a history of simple forecast error growth models designed to capture the key properties of error growth in operational numerical weather prediction (NWP) models. We propose here such a scalar model that relies on the previous ones and incorporates multiplicative noise in a nonlinear stochastic differential equation (SDE). We analyze the properties of this SDE, including the shape of the error growth curve for small times and its stationary distribution, and prove well-posedness and positivity of solutions. Next, we fit this model to operational NWP error growth curves, and show good agreement with both the mean and probabilistic features of the error growth. These results suggest that the dynamic-stochastic error growth model proposed herein and similar ones could play a role in many other areas of the sciences that involve prediction.

physics.ao-ph↗

arXiv2024

Pathwise Optimal Control and Rough Fractional Hamilton-Jacobi-Bellman Equations for Rough-Fractional Dynamics

In this work, we investigate the degeneracy problem in pathwise control, extending the framework developed in \cite{allan2020pathwise} to a more general class of driving signals and a broader set of admissible controls. Our approach consists in choosing admissible controls from a suitable class of H\"older-continuous paths. This leads naturally to the use of fractional derivatives and transforms the original control equation into a fractional dynamics system. Within this setting, we derive sufficient conditions ensuring that the control problem remains non-degenerate. We then build on the analysis developed in \cite{gomoyunov2020dynamic,gomoyunov2020theory,gomoyunov2021viscosity} to study the resulting value function and the associated Hamilton--Jacobi--Bellman equation.

math.OC↗

arXiv2024

Bayesian inference for geophysical fluid dynamics using generative models

Data assimilation plays a crucial role in numerical modeling, enabling the integration of real-world observations into mathematical models to enhance the accuracy and predictive capabilities of simulations. This approach is widely applied in fields such as meteorology, oceanography, and environmental science, where the dynamic nature of systems demands continuous updates to model states. However, the calibration of models in these high-dimensional, nonlinear systems poses significant challenges. In this paper, we explore a novel calibration methodology using diffusion generative models. We generate synthetic data that statistically aligns with a given set of observations (in this case the increments of the numerical approximation of a solution of a partial differential equation). This allows us to efficiently implement a model reduction and assimilate data from a reference system state modeled by a highly resolved numerical solution of the rotating shallow water equation of order 104 degrees of freedom into a stochastic system having two orders of magnitude less degrees of freedom. To do so, the new samples are incorporated into a particle filtering methodology augmented with tempering and jittering for dynamic state estimation, a method particularly suited for handling complex and multimodal distributions. This work demonstrates how generative models can be used to improve the predictive accuracy for particle filters, providing a more computationally efficient solution for data assimilation and model calibration.

math.NA↗

arXiv2024

Nudging state-space models for Bayesian filtering under misspecified dynamics

Nudging is a popular algorithmic strategy in numerical filtering to deal with the problem of inference in high-dimensional dynamical systems. We demonstrate in this paper that general nudging techniques can also tackle another crucial statistical problem in filtering, namely the misspecification of the transition kernel. Specifically, we rely on the formulation of nudging as a general operation increasing the likelihood and prove analytically that, when applied carefully, nudging techniques implicitly define state-space models that have higher marginal likelihoods for a given (fixed) sequence of observations. This provides a theoretical justification of nudging techniques as data-informed algorithmic modifications of state-space models to obtain robust models under misspecified dynamics. To demonstrate the use of nudging, we provide numerical experiments on linear Gaussian state-space models and a stochastic Lorenz 63 model with misspecified dynamics and show that nudging offers a robust filtering strategy for these cases.

stat.CO↗

arXiv2024

A uniform point vortex approximation for the solution of the two-dimensional Navier Stokes equation with transport noise

We study a model of interacting particles represented by a system of N stochastic differential equations. We establish that the mollified empirical distribution of the system converges uniformly with respect to both time and spatial variables to the solution of the two dimensional Navier Stokes equation with transport noise. The proofs are based on a semigroup approach.

math.PR↗

arXiv2024

The identification of diffusions from imperfect observations

This paper studies the identification of an $\mathbb{R}^d$-valued diffusion $X$ when a running function of it, say $h(X_t)$, is observed. A point-wise observation of the process (in other words, observing $h(X_t)$ in isolation) cannot identify $X_t$ unless the $h$ is injective. However observing $h(X_s)$ on a small interval $[t,t+\varepsilon]$ can be enough to determine $X_t$ exactly. The paper contain results that expand on this idea; in particular, a property of `fine total asymmetry' of twice continuously differentiable $h$ is introduced that depends on the fine topology of potential theory and that is both necessary and sufficient for $X$ to be adapted to a natural right-continuous filtration generated by the observations. This particular filtration, though augmented with null sets, does not depend on the distribution of $X_0$. For real-analytic $h$ the property reduces to simple asymmetry; that is, there is no nontrivial affine isometry $\kappa$ on $\mathbb{R}^d$ such that $h = h \circ \kappa$. A second result concerns the case where $X_0$ is given and $h$ is merely Borel; then $X$ is adapted to an augmented filtration generated by the observation process $(h(X_t))_{t\geq 0}$ if $h$ is `locally invertible' on a subset of $\mathbb{R}^d$ dense in the fine topology on $\mathbb{R}^d$.

math.PR↗