Quantifying the uncertainty of Lagrangian motion can be performed by solving a large number of ordinary differential equations with random velocities, or equivalently a stochastic transport partial differential equation (PDE) for the ensemble of flow-maps. The Dynamically Orthogonal (DO) decomposition is applied as an efficient dynamical model order reduction to solve for such stochastic advection and Lagrangian transport. Its interpretation as the method that applies instantaneously the truncated SVD on the matrix discretization of the original stochastic PDE is used to obtain new numerical schemes. Fully linear, explicit central advection schemes stabilized with numerical filters are selected to ensure efficiency, accuracy, stability, and direct consistency between the original deterministic and stochastic DO advections and flow-maps. Various strategies are presented for selecting a time-stepping that accounts for the curvature of the fixed rank manifold and the error related to closely singular coefficient matrices. Efficient schemes are developed to dynamically evolve the rank of the reduced solution and to ensure the orthogonality of the basis matrix while preserving its smooth evolution over time. Finally, the new schemes are applied to quantify the uncertain Lagrangian motions of a 2D double gyre flow with random frequency and of a stochastic flow past a cylinder.

Iterated pressure-correction projection schemes for the unsteady incompressible Navier-Stokes equations are developed, analyzed and exemplified, in relation to preconditioned iterative methods and the pressure-Schur complement equation. Typical pressure-correction schemes perform only one iteration per stage or time step, and suffer from splitting errors that result in spurious numerical boundary layers and a limited order of convergence in time. We show that performing iterations not only reduces the effects of the splitting errors, but can also be more efficient computationally than merely reducing the time step. We devise stopping criteria to recover the desired order of temporal convergence, and to drive the splitting error below the time-integration error. We also develop and implement the iterated pressure corrections with both multi-step and multi-stage time integration schemes. Finally, to reduce further the computational cost of the iterated approach, we combine it with an Aitken acceleration scheme. Our theoretical results are validated and illustrated by numerical test cases for the Stokes and Navier-Stokes equations, using implicit-explicit (IMEX) backwards differences and Runge-Kutta time-integration solvers. The test cases comprise a now classical manufactured solution in the projection method community and a modified version of a more recently proposed manufactured solution. The different error types, stopping criterion, recovered orders of convergence, and acceleration rates are illustrated, as well as the effects of the rotational corrections and time-integration schemes. It is found that iterated pressure-correction schemes can retrieve the accuracy and temporal convergence order of fully-coupled schemes and are computationally more efficient than classic pressure-correction schemes.

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.  The curvature of the manifold is characterized and related to the smallest singular value through the study of the Weingarten map.  Differentiability results for the orthogonal projection onto embedded manifolds are reviewed and used to derive an explicit dynamical system for tracking the truncated Singular Value Decomposition (SVD)  of a time-dependent matrix. 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, which translates into an explicit dependence of this error on the gap between singular values.  The DO approximation is also justified as the dynamical system that applies instantaneously the SVD truncation to optimally constrain the rank of the reduced solution.  Riemannian matrix optimization is investigated in this extrinsic framework to provide algorithms that adaptively update the best low rank approximation of a smoothly varying matrix.  The related gradient flow provides a dynamical system that converges to the truncated SVD of an input matrix for almost every initial data.

Accounting for uncertainty in optimal path planning is essential for many applications. We present and apply stochastic level-set partial differential equations that govern the stochastic time-optimal reachability fronts and time-optimal paths for vehicles navigating in uncertain, strong, and dynamic flow fields. To solve these equations efficiently, we obtain and employ their dynamically orthogonal reduced-order projections, maintaining accuracy while achieving several orders of magnitude in computational speed-up when compared to classic Monte Carlo methods. We utilize the new equations to complete stochastic reachability and time-optimal path planning in three test cases: (i) a canonical stochastic steady-front with uncertain flow strength, (ii) a stochastic barotropic quasi-geostrophic double-gyre circulation, and (iii) a stochastic flow past a circular island. For all the three test cases, we analyze the results with a focus on studying the effect of flow uncertainty on the reachability fronts and time-optimal paths, and their probabilistic properties. With the first test case, we demonstrate the approach and verify the accuracy of our solutions by comparing them with the Monte Carlo solutions.With the second, we show that different flow field realizations can result in paths with high spatial dissimilarity but with similar arrival times. With the third, we provide an example where time-optimal path variability can be very high and sensitive to uncertainty in eddy shedding direction downstream of the island. Keywords: Stochastic Path Planning, Level Set Equations, Dynamically Orthogonal, Ocean Modeling, AUV, Uncertainty Quantification

Deepak Subramani, a sixth year graduate student, has been awarded the SNAME Travel Award in Ocean Engineering by MIT-MechE to present his work at the 2017 American Geophysical Union Fall Meeting, to be held from December 11 to 15, 2017, in New Orleans, Louisiana, U.S.A.