Nonlocal models in computational Science and Engineering: treatment of interfaces in heterogeneous materials and media, image processing, and model learning.
01/10/2020
In this second talk I will address in details some of the challenges and applications mentioned in the first talk. Specifically, I will describe two techniques to tackle the unresolved treatment of nonlocal interfaces in the simulation of heterogeneous material and media. The first technique is based on the minimization of an energy principle and yields a wellposed and physically consistent nonlocal interface problem; the second is based on a new fractional model for anomalous diffusion with increased variability.
Then, I will describe a technique for optimal image denoising using nonlocal operators as filters. The optimal imaging problem is formulated as a bilevel optimization problem where the control variables are the denoising parameters. Several numerical results on benchmark images illustrate the applicability and improved accuracy of our approach.
If time allows, I will also present two recently developed machinelearning techniques for nonlocal model identification. These techniques are physicsinformed, datadriven, tools that allow us to reconstruct model parameters from sparse observations. I will also show one and twodimensional numerical tests that illustrate robustness and accuracy of our approaches

New Large Constellations of Low Earth Orbit Satellites  Astronomy and Space Debris Challenges
23/09/2020
Over the next decade plans have been advanced for 100,000 new satellites in Low Earth Orbit (LEO, altitude less than 2000 km). This will increase the total number of objects in this orbital regime by at least a factor of 5, including active satellites and debris larger than 10 cm. I'll review why these constellations of satellites are planned, why so many are needed, and what the basic design parameters of a satellite constellation are. The first of these constellations have been launched: SpaceX Starlink satellites and OneWeb satellites. For the appearance of the night sky to the unaided eye and ground and space based optical astronomy, the night sky will never be the same. These new satellites could be brighter than most of the objects in orbit today, producing contamination by satellite streaks in astronomical images. The growing spatial density of objects in LEO leads to an increased risk of collision between objects in LEO and the increase in the space debris population.

Nonlocal models in computational science and engineering: theory and challenges
21/09/2020
Nonlocal models such as peridynamics and fractional equations can capture effects that classical partial differential equations fail to capture. These effects include multiscale behavior, discontinuities in the solutions such as cracks, and anomalous behavior such as super and subdiffusion. For this reason, they provide an improved predictive capability for a large class of engineering and scientific applications including fracture mechanics, subsurface flow, turbulence, and image processing, to mention a few.
However, the improved accuracy of nonlocal formulations comes at the price of modeling and computational challenges that may hinder the usability of these models. Challenges include the nontrivial prescription of nonlocal boundary conditions, the unresolved treatment of nonlocal interfaces, the identification of model parameters, often sparse and subject to noise, and the incredibly high computational cost.
In this talk I will first introduce nonlocal models and describe a recently developed nonlocal calculus for their analysis. Then, I will discuss simulation challenges and describe in detail how we are addressing some of them at Sandia National Labs.

Multifidelity Strategies in Uncertainty Quantification: an overview on some recent trends in sampling based approaches
18/09/2020
In the last decades, the advancements in the areas of computer hardware/architectures and scientific computing algorithms enabled engineers and scientists to more rapidly study and design complex systems by heavily relaying on numerical simulations. The increased need for predictive numerical simulations exacerbated the requirement for an accurate quantification of the errors of the numerical simulations beyond the more classical algorithmic verification activities. As a consequence, Uncertainty Quantification (UQ) has been introduced as a task that allows for a formal characterization and propagation of the physical and numerical uncertainty through computational codes in order to obtain statistics of the system's response. Despite the recent efforts and successes in advancing the UQ algorithms’ efficiency, the simultaneous combination of a large number of uncertainty parameters (which often correlates to the complexity of the numerical/physical assumptions) and the lack of regularity of the system's response still represents a formidable challenge for UQ. One of the possible ways of circumventing these difficulties is to rely on samplingbased approaches which are generally robust, easy to implement and they possess a rate of convergence which is independent from the number of parameters. However, for many years the extreme computational cost of these methods prevented their widespread use for UQ in the context of highfidelity simulations. More recently several multilevel/multifidelity Monte Carlo strategies have been proposed to decrease the Monte Carlo cost without penalizing its accuracy. Several different versions of multifidelity methods exist, but they all share the main idea: whenever a set/cloud/sequence of system realizations with varying accuracy can be obtained, it is often more efficient to fuse data coming from all of them instead of relying to the higherfidelity model only. In this talk we summarize our recent efforts in investigating novel ways of increasing the efficiency of these multifidelity approaches. We will provide several theoretical and numerical results and we will discuss a collection of numerical examples ranging from simple analytical/verification test cases to more complex and realistic engineering systems.

The COVID19 epidemic in Italy  a modeling perspective (or What I did in the 2 Months Quarantine)
16/09/2020

Combustion Engines Are Not Dead Yet: Future of Power and Transportation
14/09/2020
Despite the widespread perception about batteryelectric and fuelcell vehicles as future transportation, there are many reasons to believe that “all electric vehicles” scenario is not only unrealistic but also undesirable. The presentation will attempt to present an objective assessment of the future transportation portfolio and the role of advanced internal combustion engines running on conventional and alternative fuels. In particular, objective welltowheel life cycle assessment for various competing vehicle technologies will be presented, through which it will become clear that advanced high efficiency internal combustion engines running on carbonneutral liquid fuels are the most feasible future direction for transportation at scale. Overviews will be given on relevant ongoing research activities and their opportunities and challenges will be addressed.

Development of reducedorder models based on dimensionality reduction, classification and regression for reacting flow applications.
10/09/2020
In this second part, the reduced representations are used to derive reducedorder models, in combination to typical MLbased tasks such as classification and regression. Examples of applications of these ROM are provided in the context of Large Eddy Simulations of turbulent reacting flows, as well as for the development of digital twins of combustion assets.

Simple Flows Using a Second Order Theory of Fluids
09/09/2020

Dimensionality reduction and feature extraction from highfidelity combustion data
07/09/2020
The use of machine learning algorithms to predict the behaviors of complex systems is booming. However, the key for an effective use of machine learning tools in multiphysics problems, including combustion, is to couple them to physical and computer models, to embody in them all the prior knowledge and physical constraints that can enhance their performances, and to improve them based on the feedback coming for the validation experiments. In other words, we need to adapt the scientific method to bring machine learning into the picture and make the best use of the massive amount of data we have produced thanks to the advances in numerical computing.
The webinars review some of the open opportunities for the application of datadriven, reducedorder modelling to combustion systems. In particular, the first webinar focuses on dimensionality reduction in the context of reacting flow applications. Different approaches (based on modal decomposition, neural networks, kernel methods, ...) are presented and compared, based on their ability to identify lowdimensional manifold and provide relevant features.

PhysicsInformed Neural Networks for Optimal Control
03/09/2020
PhysicsInformed Neural Networks (PINN) refer to recently defined a class of machine learning algorithms where the learning process for both regression and classification tasks is constrained to satisfy differential equations derived by the straightforward application of known physical laws. Indeed, Deep Neural Networks (DNN) have been successfully employed to solve a variety of ODEs and PDEs arising in fluid mechanics, quantum mechanics, just to mention a few. Optimal control problems, i.e. finding a feasible control that minimize a cost functional while satisfying physical, state and control constraints, are generally difficult to solve and one may nned to resort to specialized numerical methods. The application of Pontryagin minimum principle generates a complex twopoint boundary value problem that is very sensitive to the initial guess (“curse of complexity”). The application of dynamic programming principles generate a highdimensional PDE named HamiltonJacobiBellman (“Curse of Dimensionality”). In this talk we show the PINN can be employed to solve optimal control problems by tackling their solution using deep and/or shallow NNs. We show that such methods can be coupled with the Theory of Functional Connections (TFC, by Mortari et al.) to create numerical frameworks that generate efficient and accurate solutions with potential for realtime applications.

Lattice models arising from neural networks and their long term dynamics
27/07/2020
Lattice systems arising from neural networks, referred to as neural lattice models, have attracted much attention recently. They can be broadly classified into two types: one developed as the discretization of continuous neural field models, namely neural field lattice systems, and the other as the limit of finite dimensional discrete neural networks when their sizes become increasingly large. In the lecture we will introduce a few interesting neural lattice models and investigate their long term dynamics. These dynamics provide insight into the stability of large neural networks, as well as justification of finite dimensional approximations for numerical simulations of such networks.

From Art to Science: the Flower Constellations
24/07/2020
This year the Flower Constellations theory celebrates its 16th birthday. Many years were needed to fully understand the implications and to develop the theory. This new satellite constellations design tool is now ready for applications. The theory introduces a new class of space objects characterized by shape preserving configurations where the whole constellation behaves as a rigid object. By using minimal parameterization (Hermite normal form) the 2D Lattice Flower Constellations allows to include all spatial and temporal symmetric solutions, while the extension to 3D Lattice allows designers to use any inclination when selecting elliptical orbits under J2 perturbation. Recently, the Necklace theory applied to 2D and 3D Flower Constellations exponentially increases the space of potential solutions while keeping limited the number of satellites and launches (costs). The evolution of the mathematical theory is presented, showing some potential configurations to improve existing applications as well as configurations proposing new applications! The number of applications are many, including, positioning, communication, radio occultation, interferometric, and surveillance systems. In particular, the Flower Constellations theory allows to design conjunctionfree constellations with many thousands of satellites and a new class of orbits/constellations, called J2 propelled systems, where the Earth oblateness perturbation is used (rather than control) to cover spatial volumes around the Earth to measure or monitor physical quantities.

Deep Learning Models for Space Guidance and Control
23/07/2020
Autonomous exploration of small and large bodies of the solar system requires the development of a new class of intelligent systems capable of integrating in realtime stream of sensor data and autonomously take optimal decisions. Over the past few years, there has been an explosion of machine learning techniques involving the use of deep neural networks to solve a variety of problems ranging from object detection to image recognition and natural language processing. The recent success of deep learning is due to concurrent advancement of fundamental understanding on how to train deep architectures, the availability of large amount of data and critical advancements in computing power (use of GPUs). One can ask how such techniques can be employed to provide integrated and closed loop solutions for space autonomy as well as Guidance, Navigation and Control (GNC). In this talk we discuss the fundamentals of deep reinforcement learning and metalearning (“learntolearn) and their application to GNC in a variety of scenarios relevant to space exploration.

One lecture on data assimilation using some actual data of the Corona disease as an example.
22/07/2020
The lecture introduces adjoint methods for fluid dynamics. The
adjoint equations for the NavierStokes equations describe
sensitivities of a certain quantity of interest (think of drag of
an airfoil for example) to a certain input parameter (think of the
profile of the airfoil). Classical sensitivity analysis would
assume a profile and vary for example the chamber thickness to
calculate the sensitivity. One change results in one
sensitivity. Instead of changing the thickness and probing the
drag the adjoint equations provide an equation to directly
calculate the sensitivity for changing all surface points of the
profile. The method is capable of calculating the sensitivities
of millions of parameters in one step. This comes of course with
an effort, but immediately pays off, when more than a handful
parameters are to be investigated. In this lecture, the method is
introduced and an application to a particularly simple example of
an optimisation of an Epidemiology model for the COVID19 pandemic
presented, to illustrate the method. Then applications to several,
more complicated, fluid dynamical problems are discussed.

Higher order numerical schemes and their convergence for random differential equations
20/07/2020
Random ordinary differential equations (RODEs) are ordinary differential equations that include a stochastic process in their vector field. They seem to have had a shadow existence to Ito stochastic differential equations, but have been around for as long as if not longer and have many important applications. In the engineering and physics literature, a simpler kind of RODE is investigated with the vector field being chosen randomly rather than depending on a stochastic process that give rises to stochastic ordinary differential equations (SODEs). Unlike SODEs, RODEs can be analyzed pathwise with deterministic calculus but require further treatment beyond that of classical ODE theory. The sample paths of RODEs may be just Holder continuous and not even differentiable, and thus classical Taylor expansions do not apply. In this lecture we will introduce the special Taylor expansions for RODEs and use them to derive higher order numerical schemes. Convergence analysis will also be provided.

Direct Numerical Simulation of Engine Knock
13/07/2020
Knocking in internal combustion engines is a commonly encountered phenomenon but fundamental understanding of the underlying physical mechanism is lacking. In the present study, direct numerical simulations (DNS) of reactive mixtures with temperature and composition fluctuations are conducted in order to provide insights into the autoignition and subsequent development of knock and detonation. High order discretization schemes with shockcapturing algorithms allow an accurate realization of the flameacoustic interaction and the evolution of detonation accompanied by high pressure peaks. Parametric studies using one and twodimensional DNS at enginelike conditions allow a systematic characterization of the onset of knock in terms of key effects such as bulk mixture conditions and heat release rate. The original theory by Bradley and coworkers is revised to properly predict the onset of detonation and further validated by simulation and experimental data.

Applications of Functional Interpolation to Optimization
10/07/2020
This lecture summarizes what the Theory of Functional Connections (TFC) is and presents the most important applications to date. The TFC performs analytical functional interpolation. This allows to derive analytical expressions with embedded constraints, expressions describing all possible functions satisfying a set of constraints. These expressions are derived for a wide class of constraints, including points and derivatives constraints, relative constraints, linear combination of constraints, component constraints, and integral constraints.
An immediate impact of TFC is on constrained optimization problems as the whole search space is reduced to just the space of solutions fully satisfying the constraints. This way a large set of constrained optimization problems are turned into unconstrained problems, allowing more simple, fast, and accurate methods to solve them. For instance, TFC allows to obtain fast and machineerror accurate solutions of linear and nonlinear ordinary differential equations.
TFC has been extended to ndimensions (Multivariate TFC). This allows to derive efficient methods to solve partial and stochastic differential equations. This lecture also provides some examples in aerospace applications as, for instance, accurate perturbed orbit propagation, perturbed Lambert problem, energyefficient docking, and energy (or fuelefficient) optimal landing on large bodies.
