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Any model order reduced dynamical system that evolves a modal decomposition to approximate the discretized solution of a stochastic PDE can be related to a vector field tangent to the manifold of fixed rank matrices. The Dynamically Orthogonal (DO) approximation is the canonical reduced order model for which the corresponding vector field is the orthogonal projection of the original system dynamics onto the tangent spaces of this manifold. The embedded geometry of the fixed rank matrix manifold is thoroughly analyzed. Geodesic equations are derived and extrinsic curvatures are characterized through the study of the Weingarten map. Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit formula for the differential of the truncated Singular Value Decomposition (SVD). A similar analysis applied to the group of orthogonal matrices yields the differential of the polar decomposition. It is demonstrated that the error made by the DO approximation remains controlled under the minimal condition that the original solution stays close to the low rank manifold. Numerically, the DO approximation is also the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution. The geometric analysis is used to provide improved numerical time-integration schemes. Riemannian matrix optimization including gradient and Newton methods allows to adaptively track the best low rank approximation of dynamical matrices.

The quantification of uncertainties is critical when systems are nonlinear and have uncertain terms in their governing equations or are constrained by limited knowledge of initial
and boundary conditions. Such situations are common in multiscale, intermittent and non-
homogeneous fluid and ocean flows. The Dynamically Orthogonal (DO) field equations
provide an efficient time-dependent adaptive methodology to predict the probability density functions of such flows. The present work derives efficient computational schemes for
the DO methodology applied to unsteady stochastic Navier-Stokes and Boussinesq equations, and illustrates and studies the numerical aspects of these schemes. Semi-implicit
projection methods are developed for the mean and for the orthonormal modes that define
a basis for the evolving DO subspace, and time-marching schemes of first to fourth order
are used for the stochastic coefficients. Conservative second-order finite-volumes are employed in physical space with Total Variation Diminishing schemes for the advection terms.
Other results specific to the DO equations include: (i) the definition of pseudo-stochastic
pressures to obtain a number of pressure equations that is linear in the subspace size in-
stead of quadratic; (ii) symmetric Total Variation Diminishing-based advection schemes
for the stochastic velocities; (iii) the use of generalized inversion to deal with singular
subspace covariances or deterministic modes; and (iv) schemes to maintain orthonormal
modes at the numerical level. To verify the correctness of our implementation and study
the properties of our schemes and their variations, a set of stochastic flow benchmarks are
defined including asymmetric Dirac and symmetric lock-exchange flows, lid-driven cavity flows, and flows past objects in a confined channel. Different Reynolds number and
Grashof number regimes are employed to illustrate robustness. Optimal convergence under both time and space refinements is shown as well as the convergence of the probability
density functions with the number of stochastic realizations.

Rixen, M., P.F.J. Lermusiaux and J. Osler, 2012. *Quantifying, Predicting and Exploiting Uncertainties in Marine Environments*, Ocean Dynamics, 62(3):495â€“499, doi: 10.1007/s10236-012-0526-8.

Following the scientific, technical and field trial initiatives ongoing since the Maritime Rapid Environmental Assessment (MREA) conferences in 2003, 2004 and 2007, the MREA10 conference provided a timely opportunity to review the progress on various aspects of MREA, with a particular emphasis on marine environmental uncertainty management. A key objective of the conference was to review the present state-of-the art in Quantifying, Predicting and Exploiting (QPE) marine environmental uncertainties. The integration of emerging environmental monitoring and modeling techniques into data assimilation streams and their subsequent exploitation at an operational level involves a complex chain of non-linear uncertainty transfers, including human factors. Accordingly the themes for the MREA10 conference were selected to develop a better understanding of uncertainty, from its inception in the properties being measured and instrumentation employed, to its eventual impact in the applications that rely upon environmental information.

Contributions from the scientific community were encouraged on all aspects of environmental uncertainties: their quantification, prediction, understanding and exploitation. Contributions from operational communities, the consumers of environmental information who have to cope with uncertainty, were also encouraged. All temporal and spatial scales were relevant: tactical, operational, and strategic, including uncertainty studies for topics with long-term implications. Manuscripts reporting new technical and theoretical developments in MREA, but acknowledging effects of uncertainties to be accounted for in future research, were also included.

The response was excellent with 87 oral presentations (11 of which were invited keynote speakers) and 24 poster presentations during the conference. A subset of these presentations was submitted to this topical issue and 22 manuscripts have been published by Ocean Dynamics.

We estimate and study the evolution of the dominant dimensionality of
dynamical systems with uncertainty governed by stochastic partial differential
equations, within the context of dynamically orthogonal (DO) field equations.
Transient nonlinear dynamics, irregular data and non-stationary statistics are
typical in a large range of applications such as oceanic and atmospheric flow
estimation. To efficiently quantify uncertainties in such systems, it is
essential to vary the dimensionality of the stochastic subspace with time. An
objective here is to provide criteria to do so, working directly with the
original equations of the dynamical system under study and its DO
representation. We first analyze the scaling of the computational cost of
these DO equations with the stochastic dimensionality and show that unlike
many other stochastic methods the DO equations do not suffer from the curse of
dimensionality. Subsequently, we present the new adaptive criteria for the
variation of the stochastic dimensionality based on instantaneous i) stability
arguments and ii) Bayesian data updates. We then illustrate the capabilities
of the derived criteria to resolve the transient dynamics of two 2D stochastic
fluid flows, specifically a double-gyre wind-driven circulation and a
lid-driven cavity flow in a basin. In these two applications, we focus on the
growth of uncertainty due to internal instabilities in deterministic flows. We
consider a range of flow conditions described by varied Reynolds numbers and
we study and compare the evolution of the uncertainty estimates under these
varied conditions.

In this work we derive an exact, closed set of evolution equations for general continuous stochastic fields
described by a Stochastic Partial Differential Equation (SPDE). By hypothesizing a decomposition of the
solution field into a mean and stochastic dynamical component, we derive a system of field equations
consisting of a Partial Differential Equation (PDE) for the mean field, a family of PDEs for the orthonormal
basis that describe the stochastic subspace where the stochasticity `lives’ as well as a system of Stochastic
Differential Equations that defines how the stochasticity evolves in the time varying stochastic subspace.
These new evolution equations are derived directly from the original SPDE, using nothing more than
a dynamically orthogonal condition on the representation of the solution. If additional restrictions are
assumed on the form of the representation, we recover both the Proper Orthogonal Decomposition
equations and the generalized Polynomial Chaos equations. We apply this novel methodology to two
cases of two-dimensional viscous fluid flows described by the NavierStokes equations and we compare
our results with Monte Carlo simulations.

Scientific computations for the quantification, estimation and prediction of uncertainties for ocean dynamics are developed
and exemplified. Primary characteristics of ocean data, models and uncertainties are reviewed and quantitative data
assimilation concepts defined. Challenges involved in realistic data-driven simulations of uncertainties for four-dimensional
interdisciplinary ocean processes are emphasized. Equations governing uncertainties in the Bayesian probabilistic
sense are summarized. Stochastic forcing formulations are introduced and a new stochastic-deterministic ocean model
is presented. The computational methodology and numerical system, Error Subspace Statistical Estimation, that is used
for the efficient estimation and prediction of oceanic uncertainties based on these equations is then outlined. Capabilities
of the ESSE system are illustrated in three data-assimilative applications: estimation of uncertainties for physical-biogeochemical
fields, transfers of ocean physics uncertainties to acoustics, and real-time stochastic ensemble predictions with
assimilation of a wide range of data types. Relationships with other modern uncertainty quantification schemes and promising
research directions are discussed.

Lermusiaux, P.F.J., C.-S. Chiu, G.G. Gawarkiewicz, P. Abbot, A.R. Robinson, R.N. Miller, P.J. Haley, W.G. Leslie, S.J. Majumdar, A. Pang and F. Lekien, 2006. *Quantifying Uncertainties in Ocean Predictions.* Refereed invited manuscript. Oceanography, Special issue on "Advances in Computational Oceanography", T. Paluszkiewicz and S. Harper (Office of Naval Research), Eds., 19, 1, 92-105, doi: 10.5670/oceanog.2006.93.

An interdisciplinary team of scientists is collaborating to enhance the understanding of the uncertainty in the ocean environment, including the sea bottom, and characterize its impact on tactical system performance. To accomplish these goals quantitatively an end-to-end system approach is necessary. The conceptual basis of this approach and the framework of the end-to-end system, including its components, is the subject of this presentation. Specifically, we present a generic approach to characterize variabilities and uncertainties arising from regional scales and processes, construct uncertainty models for a generic sonar system, and transfer uncertainties from the acoustic environment to the sonar and its signal processing. Illustrative examples are presented to highlight recent progress toward the development of the methodology and components of the system.

Lermusiaux, P.F.J., 2001. *Evolving the subspace of the three-dimensional multiscale ocean variability: Massachusetts Bay.* Journal of Marine Systems, Special issue on "Three-dimensional ocean circulation: Lagrangian measurements and diagnostic analyses", 29/1-4, 385-422, doi: 10.1016/S0924-7963(01)00025-2.