Lessons in uncertainty quantification for turbulent dynamical systems

Abstract

The modus operandi of modern applied mathematics in developing very recent mathematical strategies for uncertainty quantification in partially observed high-dimensional 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 two-way 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 statistical-stochastic 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.

Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems

“…An information-theoretic framework has recently been developed and applied to quantify model error, model sensitivity and prediction skill [10,[26][27][28][29][30][31][32][33][34]. The natural way to measure the lack of information in one probability density q(u) compared with the true probability density p(u) is through the relative entropy P (p, q) [26,32,40],…”

“…They are characterized by a large dimensional state space and a large dimension of strong instabilities which transfer energy throughout the system. Key mathematical issues are their basic mathematical structural properties and qualitative features [2,3,8,9], statistical prediction and uncertainty quantification (UQ) [10][11][12], state estimation or data assimilation [13][14][15][16][17], and coping with the inevitable model errors that arise in approximating such complex systems [10,[18][19][20][21]. Understanding and predicting complex multiscale turbulent dynamical systems involve the blending of rigorous mathematical theory, qualitative and quantitative modelling, and novel numerical procedures [2,22].…”

“…Recently, an information-theoretic framework has been developed and is applied together with other mathematical tools to address all the issues mentioned above [10,[26][27][28][29][30][31][32][33][34]. This information-theoretic framework provides an unbiased way to quantify the model error and model fidelity [18,[35][36][37] in complex nonlinear dynamical systems, which in turn offers a systematic procedure for model selection and parameter estimation within a given class of imperfect models [1,[26][27][28].…”

A fast stochastic solution method for the Blade Element Momentum equations for long‐term load assessment

“…Because realizations of large sets of random variables can be generated very quickly, this is not a problem for deterministic load analyses. However, the computational cost of stochastic tools increase dramatically with the number of random variables, a phenomenon commonly known as the “curse of dimensionality.” To keep the number of random variables manageable a reduced‐order wind model is used, cf Fluck and Crawford: where S is the (symmetric) wind speed power spectrum (eg, a Kaimal spectrum, cf Equation 2.24), ω m are selected spectral frequencies, ξ =[ ξ m ] is a random vector, and Δ θ =[Δ θ m k ] is a matrix of (deterministic, known, and usually constant) phase increments implicitly containing the spatial structure (ie, coherence) of the wind field. This brings the count of necessary random numbers ξ m down to N R = N F , independently of how many data points are contained in the wind data grid.…”

Stochastic reduced-order models for stable nonlinear ordinary differential equations

“…Other well-tested methodologies used for the calculation of response statistics of nonlinear systems involve the use of the Fokker Plank equation [21][22][23], moments and cumulant closures [24,25], stochastic modelling using polynomial chaos [26,27], or reduced orders of the nonlinear system by using linear or nonlinear normal modes [28][29][30]. This enumeration of methods used for solving stochastic nonlinear equations is not exhaustive, but so far using these approaches proved to make capturing statistics of extreme responses of nonlinear systems difficult or even impossible due to their built-in limitations [31], because they have solutions only for special cases [32], or are too computationally expensive [33].…”