We address the problem of finding the closest matrix Ũ to a given U under the constraint that a prescribed second-moment matrix P̃ must be matched, i.e. Ũ^TŨ=P̃. We obtain a closed-form formula for the unique global optimizer Ũ for the full-rank case, that is related to U by an SPD (symmetric positive definite) linear transform. This result is generalized to rank-deficient cases as well as to infinite dimensions. We highlight the geometric intuition behind the theory and study the problem’s rich connections to minimum congruence transform, generalized polar decomposition, optimal transport, and rank-deficient data assimilation. In the special case of P̃=I, minimum-correction second-moment matching reduces to the well-studied optimal orthonormalization problem. We investigate the general strategies for numerically computing the optimizer and analyze existing polar decomposition and matrix square root algorithms. We modify and stabilize two Newton iterations previously deemed unstable for computing the matrix square root, such that they can now be used to efficiently compute both the orthogonal polar factor and the SPD square root. We then verify the higher performance of the various new algorithms using benchmark cases with randomly generated matrices. Lastly, we complete two applications for the stochastic Lorenz-96 dynamical system in a chaotic regime. In reduced subspace tracking using dynamically orthogonal equations, we maintain the numerical orthonormality and continuity of time-varying base vectors. In ensemble square root filtering for data assimilation, the prior samples are transformed into posterior ones by matching the covariance given by the Kalman update while also minimizing the corrections to the prior samples.

Though computing power continues to grow quickly, our appetite to solve larger and larger problems grows just as fast. As a consequence, reduced-order modeling has become an essential technique in the computational scientist’s toolbox. By reducing the dimensionality of a system, we are able to obtain approximate solutions to otherwise intractable problems. And because the methodology we develop is sufficiently general, we may agnostically apply it to a plethora of problems, whether the high dimensionality arises due to the sheer size of the computational domain, the fine resolution we require, or stochasticity of the dynamics. In this thesis, we develop time integration schemes, called retractions, to efficiently evolve the dynamics of a system’s low-rank approximation. Through the study of differential geometry, we are able to analyze the error incurred at each time step. A novel, explicit, computationally inexpensive set of algorithms, which we call perturbative retractions, are proposed that converge to an ideal retraction that projects exactly to the manifold of fixed-rank matrices. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. We show that these high-order retractions significantly reduce the numerical error incurred over time when compared to a naive Euler forward retraction. Through test cases, we demonstrate their efficacy in the cases of matrix addition, real-time data compression, and deterministic and stochastic differential equations.

MSEAS was selected as part of the team of Kanna Rajan for one of the 8 plenary presentations (out of 90) of the US Launch Meeting of the United Nations Ocean Decade that will take place on February 3-4, 2021, as well as for e-poster presentations as part of a few other teams.
The United Nations has proclaimed a Decade of Ocean Science for Sustainable Development (2021-2030) to support efforts to reverse the cycle of decline in ocean health and gather ocean stakeholders worldwide behind a common framework that will ensure ocean science can fully support countries in creating improved conditions for sustainable development of the Ocean. This event, hosted by the National Academies of Sciences, Engineering and Medicine, is the US launch of the UN Decade and it includes close to 90 e-posters of “ocean shots”, i.e. innovative propositions to the Decade from US scientists.
For more details about the UN Ocean Decade please visit: https://www.oceandecade.org/

To view the agenda please click here.

Transport of any material quantity due to background fields, i.e. advective transport, in fluid dynamical systems has been a widely studied problem. It is of crucial importance in classical fluid mechanics, geophysical flows, micro and nanofluidics, and biological flows. Even though mathematical models that thoroughly describe such transport exist, the inherent nonlinearities and the high dimensionality of complex fluid systems make it very challenging to develop the capabilities to accurately compute and characterize advective material transport. We systematically study the problems of predicting, uncovering, and learning the principal features of advective material transport in this work. The specific objectives of this thesis are to: (i) develop and apply new numerical methodologies to compute the solutions of advective transport equations with minimal errors and theoretical guarantees, (ii) propose and theoretically investigate novel criteria to detect sets of fluid parcels that remain the most coherent / incoherent throughout an extended time interval to quantify fluid mixing, and (iii) extend and develop new machine learning methods to infer and predict the transport features, given snapshot data about passive and active material transport.

The first part of this work deals with the development of the PDE-based ‘method of flow map composition’, which is a novel methodology to compute the solutions of the partial differential equation describing classical advective and advective-diffusive-reactive transport. The method of composition yields solutions almost devoid of numerical errors, and is readily parallelizable. It can compute more accurate solutions in less time than traditional numerical methods. We also complete a comprehensive theoretical analysis and analytically obtain the value of the numerical timestep that minimizes the net error. The method of flow map composition is extensively benchmarked and its applications are demonstrated in several analytical flow fields and realistic data-assimilative ocean plume simulations.

We then utilize the method of flow map composition to analyze Lagrangian material coherence in dynamic open domains. We develop new theory and schemes to efficiently predict the sets of fluid parcels that either remain the most or the least coherent over an extended amount of time. We also prove that these material sets are the ones to maximally resist advective stretching and diffusive transport. Thus, they are of significant importance in understanding the dynamics of fluid mixing and form the skeleton of material transport in unsteady fluid systems. The developed theory and numerical methods are utilized to analyze Lagrangian coherence in analytical and realistic scenarios. We emphasize realistic marine flows with multiple time-dependent inlets and outlets, and demonstrate applications in diverse dynamical regimes and several open ocean regions.

The final part of this work investigates the machine inference and prediction of the principal transport features from snapshot data about the transport of some material quantity. Our goals include machine learning the underlying advective transport features, coherent / incoherent sets, and attracting and repelling manifolds, given the snapshots of advective and advective-diffusive material fields. We also infer and predict high resolution transport features by optimally combining coarse resolution snapshot data with localized high resolution trajectory data. To achieve these goals, we use and extend recurrent neural networks, including a combination of long short-term memory networks with hypernetworks. We develop methods that leverage our knowledge of the physical system in the design and architecture of the neural network and enforce the known constraints that the results must satisfy (e.g. mass conservation) in the training loss function. This allows us to train the networks only with partial supervision, without samples of the expected output fields, and still infer and predict physically consistent quantities. The developed theory, methods, and computational software are analyzed, validated, and applied to a variety of analytical and realistic fluid flows, including high-resolution ocean transports in the Western Mediterranean Sea.

We develop an exact partial differential equation-based methodology that predicts time-energy optimal paths for autonomous vehicles navigating in dynamic environments. The differential equations solve the multi-objective optimization problem of navigating a vehicle autonomously in a dynamic flow field to any destination with the goal of minimizing travel time and energy use. Based on Hamilton-Jacobi theory for reachability and the level set method, the methodology computes the exact Pareto optimal solutions to the multi-objective path planning problem, numerically solving the equations governing time-energy reachability fronts and optimal paths. Our approach is applicable to path planning in various scenarios, however we primarily present examples of navigating in dynamic marine environments. First, we validate the methodology through a benchmark case of crossing a steady front (a highway flow) for which we compare our results to semi-analytical optimal path solutions. We then consider more complex unsteady environments and solve for time-energy optimal missions in a quasi-geostrophic double-gyre ocean flow field.