# The IV AMMCS International Conference

## Waterloo, Ontario, Canada | August 20-25, 2017

# AMMSCS 2017 Plenary Talk

## NEW STRATEGIES FOR REDUCED-ORDER MODELS FOR PREDICTING THE STATISTICAL RESPONSES AND UNCERTAINTY QUANTIFICATION IN COMPLEX TURBULENT DYNAMICAL SYSTEMS

### Andrew Majda (New York University)

MORSE PROFESSOR OF ARTS AND SCIENCE

DEPARTMENT OF MATHEMATICS

FOUNDER OF CENTER FOR ATMOSPHERE OCEAN SCIENCE (CAOS)

COURANT INSTITUTE, NEW YORK UNIVERSITY

AND

PRINCIPAL INVESTIGATOR, CENTER FOR PROTOTYPE CLIMATE MODELING (CPCM)

NYU – ABU DHABI (NYUAD)

Turbulent dynamical systems characterized by both a
high-dimensional phase space and a large number of instabilities
are ubiquitous among many complex systems in science and engineering including
climate, material, and neural science. The existence of a strange attractor in the turbulent systems containing a large number of
positive Lyapunov exponents results in a rapid growth of small uncertainties from imperfect modeling equations or perturbations in initial
values, requiring naturally a probabilistic characterization for the evolution of the turbulent system. Uncertainty quantification (UQ) in
turbulent dynamical systems is a grand challenge where the goal is to obtain statistical estimates such as the change in mean and
variance for key physical quantities in their nonlinear
responses to changes in external forcing parameters or uncertain initial data. In the
development of a proper UQ scheme for systems of high or infinite dimensionality with instabilities, significant model errors compared
with the true natural signal are always unavoidable due to both the imperfect understanding of the underlying physical processes and the
limited computational resources available through direct Monte-Carlo integration. One central issue in contemporary research is the
development of a systematic methodology that can recover the crucial features of the natural system in statistical equilibrium (model
fidelity) and improve the imperfect model prediction skill in response to various external perturbations (model sensitivity).

Here we discuss a general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems. There are three stages in the modeling strategy, imperfect model selection; calibration of the imperfect model in a training phase using only data in the more complex perfect model statistics; and prediction of the responses with UQ to a wide class of forcing and perturbation scenarios. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. Several mathematical ideas will be introduced to improve the prediction skill of the imperfect reduced-order models. Most importantly, empirical information theory and statistical linear response theory are applied in the training phase for calibrating model errors to achieve optimal imperfect model parameters; and total statistical energy dynamics are introduced to improve the model sensitivity in the prediction phase especially when strong external perturbations are exerted. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed. The uncertainty quantification and statistical response for these complex models are accurately captured by the reduced-order models with only 2×10

Here we discuss a general mathematical framework to construct statistically accurate reduced-order models that have skill in capturing the statistical variability in the principal directions with largest energy of a general class of damped and forced complex turbulent dynamical systems. There are three stages in the modeling strategy, imperfect model selection; calibration of the imperfect model in a training phase using only data in the more complex perfect model statistics; and prediction of the responses with UQ to a wide class of forcing and perturbation scenarios. The methods are developed under a universal class of turbulent dynamical systems with quadratic nonlinearity that is representative in many applications in applied mathematics and engineering. Several mathematical ideas will be introduced to improve the prediction skill of the imperfect reduced-order models. Most importantly, empirical information theory and statistical linear response theory are applied in the training phase for calibrating model errors to achieve optimal imperfect model parameters; and total statistical energy dynamics are introduced to improve the model sensitivity in the prediction phase especially when strong external perturbations are exerted. The validity of general framework of reduced-order models is demonstrated on instructive stochastic triad models. Recent applications to two-layer baroclinic turbulence in the atmosphere and ocean with combinations of turbulent jets and vortices are also surveyed. The uncertainty quantification and statistical response for these complex models are accurately captured by the reduced-order models with only 2×10

^{2}modes in a highly turbulent system with 1×10^{5}degrees of freedom. Less than 0.15% of the total spectral modes are needed in the reduced-order models.
Andrew J. Majda is the Morse Professor of Arts and Sciences at the Courant
Institute of New York University. Majda's primary research interests are
modern applied mathematics in the broadest possible sense merging asymptotic
methods, numerical methods, physical reasoning, and rigorous mathematical
analysis.
Majda is a member of the National Academy of Sciences and has received
numerous honors and awards including the National Academy of Science Prize in
Applied Mathematics, the John von Neumann Prize of the Society of Industrial
and Applied Mathematics, and the Gibbs Prize of the American Mathematical
Society. He is also a member of the American Academy of Arts and Science. He
has been awarded the Medal of the College de France, twice, and is a Fellow of
the Japan Society for the Promotion of Science. He has received an honorary
doctorate from his undergraduate alma mater, Purdue University.
In the past several years at the Courant Institute, Majda has created the
Center for Atmosphere Ocean Science with a multi-disciplinary faculty to
promote cross-disciplinary research with modern applied mathematics in climate
modeling and prediction. Majda's current research interests include
multi-scale multi-cloud modeling for the tropics, reduced stochastic and
statistical modeling for climate, and novel mathematical strategies for
prediction and data assimilation in complex multi-scale systems.
Professor Majda’s recent awards include the 2015 ICIAM Lagrange Prize and the
2016 Steel Prize for Seminal Research Contributions.