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32
2010: Quantifying uncertainty in climate change science through empirical information theory
 Proc
"... Information theory provides a concise systematic framework for measuring climate consistency and sensitivity for imperfect models. A suite of increasingly complex physically relevant linear Gaussian models with time periodic features mimicking the seasonal cycle is utilized to elucidate central iss ..."
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Cited by 20 (6 self)
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Information theory provides a concise systematic framework for measuring climate consistency and sensitivity for imperfect models. A suite of increasingly complex physically relevant linear Gaussian models with time periodic features mimicking the seasonal cycle is utilized to elucidate central issues that arise in contemporary climate science. These include the role of model error, thememory of initial conditions, and effects of coarse graining in producing short, medium, and longrange forecasts. In particular, this study demonstrates how relative entropy can be used to improve climate consistency of an overdamped imperfect model by inflating stochastic forcing. Moreover, the authors show that, in the considered models, by improving climate consistency, this simultaneously increases the predictive skill of an imperfect model in response to external perturbation, a property of crucial importance in the context of climate change. The three models range in complexity from a scalar time periodic model mimicking seasonal fluctuations in a mean jet to a spatially extended system of turbulent Rossby waves to, finally, the behavior of a turbulent tracer with a mean gradient with the background turbulent field velocity generated by the first two models. This last model mimics the global and regional behavior of turbulent passive tracers under various climate change scenarios. This detailed study provides important guidelines for extending these strategies to more complicated and nonGaussian physical systems. 1.
Lessons in Uncertainty Quantification for Turbulent Dynamical System
 DCDSA
"... Abstract. The modus operandi of modern applied mathematics in developing very recent mathematical strategies for uncertainty quantification in partially observed highdimensional turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines wit ..."
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Cited by 15 (4 self)
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Abstract. The modus operandi of modern applied mathematics in developing very recent mathematical strategies for uncertainty quantification in partially observed highdimensional turbulent dynamical systems is emphasized here. The approach involves the synergy of rigorous mathematical guidelines with a suite of physically relevant and progressively more complex test models which are mathematically tractable while possessing such important features as the twoway coupling between the resolved dynamics and the turbulent fluxes, intermittency and positive Lyapunov exponents, eddy diffusivity parameterization and turbulent spectra. A large number of new theoretical and computational phenomena which arise in the emerging statisticalstochastic framework for quantifying and mitigating model error in imperfect predictions, such as the existence of information barriers to model improvement, are developed and reviewed here with the intention to introduce mathematicians, applied mathematicians, and scientists to these remarkable emerging topics with increasing practical importance. 1. Introduction. The ‘inevitable reality
FUNDAMENTAL LIMITATIONS OF AD HOC LINEAR AND QUADRATIC MULTILEVEL REGRESSION MODELS FOR PHYSICAL SYSTEMS
"... Abstract. A central issue in contemporary applied mathematics is the development of simpler dynamical models for a reduced subset of variables in complex high dimensional dynamical systems with many spatiotemporal scales. Recently, ad hoc quadratic multilevel regression models have been proposed t ..."
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Cited by 9 (4 self)
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Abstract. A central issue in contemporary applied mathematics is the development of simpler dynamical models for a reduced subset of variables in complex high dimensional dynamical systems with many spatiotemporal scales. Recently, ad hoc quadratic multilevel regression models have been proposed to provide suitable reduced nonlinear models directly from data. The main results developed here are rigorous theorems demonstrating the nonphysical finite time blowup and large time instability in statistical solutions of general scalar multilevel quadratic regression models with corresponding unphysical features of the invariant measure. Surprising intrinsic model errors due to discrete sampling errors are also shown to occur rigorously even for linear multilevel regression dynamic models. all of these theoretical results are corroborated by numerical experiments with simple models. Single level nonlinear regression strategies with physical cubic damping are shown to have significant skill on the same test problems. 1. Introduction. A
Accuracy and Stability of The ContinuousTime 3DVAR Filter for The NavierStokes Equation
, 2012
"... The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understan ..."
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Cited by 8 (3 self)
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The 3DVAR filter is prototypical of methods used to combine observed data with a dynamical system, online, in order to improve estimation of the state of the system. Such methods are used for high dimensional data assimilation problems, such as those arising in weather forecasting. To gain understanding of filters in applications such as these, it is hence of interest to study their behaviour when applied to infinite dimensional dynamical systems. This motivates study of the problem of accuracy and stability of 3DVAR filters for the NavierStokes equation. We work in the limit of high frequency observations and derive continuous time filters. This leads to a stochastic partial differential equation (SPDE) for state estimation, in the form of a dampeddriven NavierStokes equation, with meanreversion to the signal, and spatiallycorrelated timewhite noise. Both forward and pullback accuracy and stability results are proved for this SPDE, showing in particular that when enough low Fourier modes are observed, and when the model uncertainty is larger than the data uncertainty in these modes (variance inflation), then the filter can lock on to a small neighbourhood of the true signal, recovering from order one initial error, if the error in the observations modes is small. Numerical examples are given to illustrate the theory.
NonGaussian Test Models for Prediction and State Estimation with Model Errors
"... Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where statistical ensemble prediction and realtime filtering/state estimation ..."
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Cited by 6 (5 self)
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Turbulent dynamical systems involve dynamics with both a large dimensional phase space and a large number of positive Lyapunov exponents. Such systems are ubiquitous in applications in contemporary science and engineering where statistical ensemble prediction and realtime filtering/state estimation are needed despite the underlying complexity of the system. Statistically exactly solvable test models have a crucial role to provide firm mathematical underpinning or new algorithms for vastly more complex scientific phenomena. Here a class of statistically exactly solvable nonGaussian test models are introduced where a generalized FeynmanKac formulation reduces the exact behavior of conditional statistical moments to the solution of inhomogeneous FokkerPlanck equations modified by linear lower order coupling and source terms. This procedure is applied to a test model with hidden instabilities and combined with information theory to address two important issues in contemporary statistical prediction of turbulent dynamical systems: coarsegrained ensemble prediction in a perfect model and improving long range forecasting in imperfect models. The models discussed here should be useful for many other applications and algorithms for real time prediction and state estimation. 1
Dynamic Stochastic Superresolution of sparseley observed turbulent systems
 Journal of Computational Physics
"... Realtime capture of the relevant features of the unresolved turbulent dynamics of complex natural systems from sparse noisy observations and imperfect models is a notoriously difficult problem. The resulting lack of resolution and statistical accuracy in estimating the important turbulent processes ..."
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Cited by 5 (1 self)
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Realtime capture of the relevant features of the unresolved turbulent dynamics of complex natural systems from sparse noisy observations and imperfect models is a notoriously difficult problem. The resulting lack of resolution and statistical accuracy in estimating the important turbulent processes which intermittently send significant energy to the largescale fluctuations hinders efficient parameterization and realtime prediction using discretized PDE models. This issue is particularly subtle and important when dealing with turbulent geophysical systems with an enormous range of interacting spatiotemporal scales and rough energy spectra near the mesh scale. Here, we introduce and study a suite of general Dynamic Stochastic Superresolution (DSS) algorithms and show that, by appropriately filtering sparse regular observations with the help of cheap stochastic exactly solvable models, one can derive stochastically ‘superresolved’ velocity fields and gain insight into the important characteristics of the unresolved dynamics. The DSS algorithms operate in Fourier domain and exploit the fact that the coarse observation network aliases highwavenumber information into the resolved waveband. It is shown that these cheap algorithms are robust and have significant skill on a test bed of turbulent solutions from realistic nonlinear turbulent spatially extended systems in the presence of a significant model error. In particular, the DSS algorithms are
2011: Diagnosing lateral mixing in the upper ocean with virtual tracers: Spatial and temporal resolution dependence
 J. Phys. Oceanogr
"... Several recent studies diagnose lateral stirring and mixing in the upper ocean using altimetryderived velocity fields to advect ‘‘virtual’ ’ particles and fields offline. However, the limited spatiotemporal resolution of altimetric maps leads to errors in the inferred diagnostics, because unresolv ..."
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Cited by 5 (1 self)
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Several recent studies diagnose lateral stirring and mixing in the upper ocean using altimetryderived velocity fields to advect ‘‘virtual’ ’ particles and fields offline. However, the limited spatiotemporal resolution of altimetric maps leads to errors in the inferred diagnostics, because unresolved scales are necessarily imperfectly modeled. The authors examine a range of tracer diagnostics in two models of baroclinic turbulence: the standard Phillips model, in which dispersion is controlled by largescale eddies, and the Eady model, where dispersion is determined by local scales of motion. These models serve as a useful best and worstcase comparison and a valuable test of the resolution sensitivity of tracer diagnostics. The effect of unresolved scales is studied by advecting tracers using model velocity fields subsampled in space and time and comparing the derived tracer diagnostics with their ‘‘true’ ’ value obtained from the fully resolved flow. The authors find that eddy diffusivity and absolute dispersion, which are governed by largescale dynamics, are insensitive to spatial sampling error in either flow.Measures that depend strongly on small scales, such as relative dispersion and finitetime Lyapunov exponents, are highly sensitive to spatial sampling in the Eady model. Temporal sampling error is found to have a more complicated behavior because of the onset of particle overshoot leading to scrambling of Lagrangian diagnostics. This leads to a potential restriction on the utility of raw altimetrymaps for studyingmixing in the upper ocean. The authors conclude that offline diagnostics of mixing in ocean flows with an energized submesoscale should be viewed with some caution. 1.
ANALYSIS OF THE 3DVAR FILTER FOR THE PARTIALLY OBSERVED LORENZ’63 MODEL
"... Abstract. The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for highdimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an ..."
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Cited by 5 (2 self)
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Abstract. The problem of effectively combining data with a mathematical model constitutes a major challenge in applied mathematics. It is particular challenging for highdimensional dynamical systems where data is received sequentially in time and the objective is to estimate the system state in an online fashion; this situation arises, for example, in weather forecasting. The sequential particle filter is then impractical and ad hoc filters, which employ some form of Gaussian approximation, are widely used. Prototypical of these ad hoc filters is the 3DVAR method. The goal of this paper is to analyze the 3DVAR method, using the Lorenz ’63 model to exemplify the key ideas. The situation where the data is partial and noisy is studied, and both discrete time and continuous time data streams are considered. The theory demonstrates how the widely used technique of variance inflation acts to stabilize the filter, and hence leads to asymptotic accuracy. 1. Introduction. Data
Quantifying bayesian filter performance for turbulent dynamical systems through information theory
 Comm. Math. Sci
"... Incomplete knowledge of the true dynamics and its partial observations pose a notoriously difficult problem in many contemporary scientific applications which require predictions of highdimensional dynamical systems with physical instabilities and energy fluxes across a wide range of scales. In suc ..."
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Cited by 4 (1 self)
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Incomplete knowledge of the true dynamics and its partial observations pose a notoriously difficult problem in many contemporary scientific applications which require predictions of highdimensional dynamical systems with physical instabilities and energy fluxes across a wide range of scales. In such cases assimilation of real data into the modeled dynamics is necessary for mitigating model error and for improving the stability and predictive skill of imperfect models. However, the practically implementable data assimilation/filtering strategies are also imperfect and not optimal due to the formidably complex nature of the underlying dynamics. Here, the connections between information theory and the filtering problem are exploited in order to establish bounds on the filter error statistics, and to systematically study the statistical accuracy of various Kalman filters with model error for estimating the dynamics of spatially extended, partially observed turbulent systems. The effects of model error on filter stability and accuracy in this highdimensional setting are analyzed through appropriate information measures which naturally extend the common pathwise estimates of filter performance, like the meansquare error or pattern correlation, to the statistical superensemble setting that involves all possible initial conditions and all realizations of noisy observations of the truth signal. This informationtheoretic framework for an offline assessment of filter performance is an important complement to the pathwise approach, and it has natural generalizations to Kalman filtering with nonGaussian statistically exactly solvable forecast models. Here, this framework is utilized to study the performance of imperfect, reducedorder filters with Gaussian models which use various spatiotemporal discretizations to approximate the dynamics of the stochastically forced advection