Abstract : In light of the great proliferation of data and increase in computational power, the importance of effectively combining observations with dynamical models for the purpose of prediction, parameter estimation, and modeling remains a fundamental challenge. Exciting recent works at the interface of machine learning, analysis of PDEs, and data assimilation have led to the development of new methods, analytic insights and unifying perspectives, as well as novel applications. This mini-symposium will bring together researchers that have made such contributions in analytical or computational capacities.
Organizer(s) : Jochen Broecker, Vincent Martinez, Sahani Pathiraja
00718 (1/3) : 1C @E817 A715 [Chair: Vincent Martinez]
[05063] Data-driven and model-driven techniques in DA: applications, numerics, rigorous results
Format : Online Talk on Zoom
Author(s) :
Jochen Broecker (Department of Mathematics and Statistics, University of Reading)
Abstract : Data Assimilation permeates all contributions in at least three ways: Firstly, novel approaches to data assimilation use machine learning and Bayesian inference to identify the current state as well as components of the system, two inextricably linked aims. Secondly, data assimilation has become an interesting application of (stochastic) PDE~theory. Thirdly, ergodic theory of infinite dimensional dynamical systems (asymptotic coupling) calls for sophisticated nudging or error feedback schemes. Finally, new venues for research will be sketched.
[02864] Consistency Results for some Bayesian PDE inverse problems
Format : Online Talk on Zoom
Author(s) :
Nathan Glatt-Holtz (Tulane)
Abstract : Frequently one would like to estimate functional parameters $u$ in a physical model defined by a partial differential equation from a collection of sparse and uncertain observations. Here a Bayesian methodology provides an attractive statistical approach for many such estimation problems, one which provides a comprehensive picture of uncertainties in the unknown. An important step in the validation of this Bayesian methodology is to establish conditions for posterior consistency. Specifically we would like to determine when $\mu_N \rightharpoonup \delta_{u_*}$ where $\mu_N$ is the Bayesian posterior conditioned on $N$ observations of the solution and $u_*$ is the true value of the unknown.
In this talk we describe some rigorous approaches that we have recently developed tailored to address consistency for PDE inverse problems involving the recovery of an infinite dimensional unknown. We describe how our approach applies to a gallery of model problems including the recovery of a divergence free velocity field from the measurement of a solute which is advecting and diffusing in the fluid medium. This is joint work with Jeff Borggaard, Christian Frederiksen and Justin Krometis.
[02753] Nonparametric Bayesian inference of discretely observed diffusions
Format : Talk at Waseda University
Author(s) :
Jean-Charles Croix (Amazon)
Masoumeh Dashti (University of Sussex)
Stylianos Katsarakis (University of Sussex)
Istvan Kiss (University of Sussex)
Tanja Zerenner (University of Bristol)
Abstract : We consider the inverse problem of recovering the diffusion and drift functions of a stochastic differential equation from discrete measurements of its solution. We show the stability of the posterior measure with respect to appropriate approximations of the underlying forward model allowing for priors with unbounded support. We then look at the approximated posterior obtained by Gaussian approximation of transition densities in the case where the diffusion coefficient is small.
[02850] A general involution framework for Metropolis-Hastings algorithms and applications to Bayesian inverse problems
Format : Talk at Waseda University
Author(s) :
Cecilia Mondaini (Drexel University)
Nathan Glatt-Holtz (Tulane University)
Abstract : We consider a general framework for Metropolis-Hastings algorithms used to sample from a given target distribution on a general state space. Our framework has at its core an involution structure, and is shown to encompass several popular algorithms as special cases, both in the finite- and infinite-dimensional settings. In particular, it includes random walk, preconditioned Crank-Nicolson (pCN), schemes based on a suitable Langevin dynamics such as the Metropolis Adjusted Langevin algorithm (MALA), and also ones based on Hamiltonian dynamics including several variants of the Hamiltonian Monte Carlo (HMC) algorithm. In addition, our framework comprises algorithms that generate multiple proposals at each iteration, which allow for greater efficiency through the use of modern parallel computing resources. Aside from encompassing existing algorithms, we also derive new schemes from this framework, including some multiproposal versions of the pCN algorithm. To illustrate effectiveness of these sampling procedures, we present applications in the context of certain Bayesian inverse problems in fluid dynamics. In particular, we consider the problem of recovering an incompressible background fluid flow from sparse and noisy measurements of the concentration of a passive solute advected by the flow. This talk is based on joint works with N. Glatt-Holtz (Tulane U), A. Holbrook (UCLA), and J. Krometis (Virginia Tech).
[02843] Insights from Nonlinear Continuous Data Assimilation for Turbulent Flows
Format : Talk at Waseda University
Author(s) :
Elizabeth Carlson (University of Victoria)
Adam Larios (University of Nebraska - Lincoln)
Edriss S Titi (University of Cambridge)
Abstract : One of the challenges of the accurate simulation of turbulent flows is that initial data is often incomplete. Data assimilation circumvents this issue by continually incorporating the observed data into the model. An emerging approach to data assimilation known as the Azouani-Olson-Titi (AOT) algorithm introduced a feedback control term to the 2D incompressible Navier-Stokes equations (NSE) in order to incorporate sparse measurements. The solution to the AOT algorithm applied to the 2D NSE was proven to converge exponentially to the true solution of the 2D NSE with respect to the given initial data. In this talk, we will focus on the insights of a nonlinear version of the AOT algorithm and distinguish the clear connections to the physics of the existing systems.
[02852] Linear response for nonlinear dissipative SPDEs
Format : Talk at Waseda University
Author(s) :
Giulia Carigi (University of L'Aquila)
Jochen Bröcker (University of Reading)
Tobias Kuna (University of L'Aquila)
Abstract : A framework suitable to establish response theory for a class of nonlinear stochastic partial differential equations is provided, exploiting coupling methods. The results are applied to the 2D stochastic Navier-Stokes equation and the stochastic two-layer quasi-geostrophic model. In particular, studying the response to perturbations in the forcings for models in geophysical fluid dynamics gives a mathematical insight into whether statistical properties derived under current conditions will be valid under different forcing scenarios.
[05078] Data assimilation of the 2D rotating NSE
Format : Talk at Waseda University
Author(s) :
Aseel Farhat (Florida State University)
Abstract : With sufficiently fast rotation, the solution of the 2D rotating NSE on the $\beta$ plane approaches a nearly zonal state. Additionally, the number of degrees of freedom of the system decrease with faster rotation. We validate this analytically and numerically in the context of a continuous data assimilation algorithm based on nudging.
[02847] Using machine learning in geophysical data assimilation (some of the issues and some ideas)
Format : Online Talk on Zoom
Author(s) :
Alberto Carrassi (Dept of Physics, University of Bologna)
Abstract : We show how ML can be included in the prediction and DA workflow in different ways. First, in “non-intrusive” ML, we show how supervised ML estimates the local Lyapunov exponents. ML is then combined with DA in an integrated fashion to learn a surrogate model from noisy and sparse data, and a parametrization of a physical’s model unresolved scales. DA is pivotal to extract information from the sparse, noisy, data that ML cannot handle alone.
00718 (3/3) : 1E @E817 A715 [Chair: Sahani Pathiraja]
[02853] Almost Sure Error Bounds for Data Assimilation in Dissipative Systems with Unbounded Observation Noise
Format : Online Talk on Zoom
Author(s) :
Tobias Kuna (Universita dell'Aquila)
Jochen Broecker (University of Reading)
Lea Oljaca (sustainable investment research)
Abstract : Data assimilation is widely used technique, in particular, in the geophysical community. It aims at inferring information of a model by combining incomplete and noisy observations with imperfect models even for very large models and often on the fly in real time. This methods have been extensively studied for a plethora of models, assimilation methods and error terms. In this talk, I will concentrate on how one can treat unbounded noise in the observations not only in expectation, but actually I will present a technique to obtain an a.s. bound. More specifically, we prove that the error is bounded by a finite and stationary processes. We use the simple replacement data assimilation scheme by Hayden, Olson and Titi, see [1] with observations discrete in time, including but not limited to 2D Navier-Stokes equation. The method should extend to more general algorithms like described in [2], as its estimates are based on absorbing and squeezing properties generalizing [3]. The content of the talk was published in [4].
[1] K. Hayden, E. Olson, and E. S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations, Phys. D, 240 (2011), pp. 1416–1425
[2] D. Sanz-Alonso and A. M. Stuart, Long-time asymptotics of the filtering distribution for partially observed chaotic dynamical systems, SIAM/ASA J. Uncertain. Quantif., 3 (2015), pp. 1200–1220,
[3] C. E. A. Brett, K. F. Lam, K. J. H. Law, D. S. McCormick, M. R. Scott, and A. M. Stuart, Accuracy and stability of filters for dissipative PDEs, Phys. D, 245 (2013), pp. 34–45,
[4] L. Oljača, J. Bröcker and T. Kuna, Almost sure error bounds for data assimilation in dissipative systems with unbounded observation noise, IAM J. Appl. Dyn. Syst., 17(4) (2018) pp. 2882--2914.
[04934] Challenges in high dimensional nonlinear filtering
Format : Talk at Waseda University
Author(s) :
Jana de Wiljes (Uni Potsdam)
Abstract : The seamless integration of large data sets into computational models is one of the central challenges for the mathematical sciences of the 21st century.
Despite the fact that the underlying assumptions do not hold for many applications, Gaussian approximative filters are considered state of the art as they have been successfully implemented for highly nonlinear settings with large dimensional state spaces. Moreover several recent studies have been devoted to showing accuracy of such filters in terms of tracking ability for nonlinear evolution models and we will present one of these results given in the form of distinct bounds for certain filter variants.
While the robustness of such Gaussian approximative filters is undeniable there has been considerable aspiration to design filters that can achieve even higher levels of accuracy while maintaining an appropriate level of robustness and stability.
Here we will discuss a family of such filters that do not require a parametrization of the posterior distribution and can be combined with traditional Gaussian filters via a likelihood split.
[02745] Particle Filters for Data Assimilation
Format : Online Talk on Zoom
Author(s) :
Dan Crisan (Imperial College London )
Abstract : I will present the latest developments of on-going work on the application of particle filters to develop high dimensional
data assimilation methodologies.
[04884] A novel regularity criterion for the 3D Navier-Stokes equations based on finitely many observations.
Format : Online Talk on Zoom
Author(s) :
Animikh Biswas (University of Maryland Baltimore County)
Abhishek Balakrishna (University of Maryland Baltimore County)
Abstract : We present a novel regularity criterion for the 3D Navier-Stokes equations (NSE) based on finitely many modal, nodal or volume element observations of the velocity field. The proof is based on a data assimilation algorithm utilizing a Newtonian relaxation scheme (nudging) motivated by feedback-control. The observations, which may be either modal, nodal or volume elements, are obtained from a weak solution of the 3D NSE and are collected almost everywhere in time over a finite grid. The regularity criterion we propose follows from our data assimilation algorithm and is hence intimately connected to the notion of determining functionals (modes, nodes and volumeelements). To the best of our knowledge, all existing regularity criteria require knowing the solutionof the 3D NSE almost everywhere in space. Our regularity criterion is fundamentally different fromany preexisting regularity criterion as it is based on finitely many observations (modes, nodes andvolume elements). We further prove that the regularity criterion we propose is both a necessaryand sufficient condition for regularity. Thus our result can be viewed as a natural generalizationof the notion of determining modes, nodes and volume elements as well as the asymptotic trackingproperty of the nudging algorithm for the 2D NSE to the 3D setting.
