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NASA NTRS2020

Assessment of Transition Modeling Capability in OVERFLOW with Emphasis on Swept-Wing Configurations

In preparation for comparisons with data obtained from the recently concluded experiments in the National Transonic Facility at the NASA Langley Research Center on the new common research model with natural laminar flow (CRM-NLF), an assessment of the current transition modeling capability in the NASA OVERFLOW 2.2o code has been carried out. A combination of the available experimental data and linear stability analysis is used to evaluate the accuracy and robustness of these models for selected swept-wing type configurations, with significant crossflow. An additional goal for this work involves providing a comparative assessment of the relevant transition models in the context of a single flow solver and identifying model limitations as well as the potential for future improvements that would help strengthen the physical basis of such transition models. Included in this investigation is an assessment of the sensitivities of the underlying transition models to grid resolution (wall-normal, as well as streamwise and spanwise) and the values of extra input parameters such as the level of surface roughness and freestream turbulence variables. The flow configurations targeted in this assessment include the NASA NLF(2)-0415 swept-wing configuration, the sickle-shaped wing introduced by the Technical University of Braunschweig, and the wing-body configuration of the CRM model from the fourth and fifth AIAA CFD Drag Prediction Workshops.

Freestream velocity↗

NASA NTRS2001

Design of a LQR Controller of Reduced Inputs for Multiple Spacecraft Formation Flying

Regarding multiple spacecraft formation flying, the observation is made that control thrust need only be applied coplanar to the local horizon to achieve complete controllability of a two-satellite formation. Without the need for zenith-nadir (radial) thrust, simplifications and reduction of the weight of the propulsion system may be accomplished. This work focuses on the validation of this radial-excluding control system on its own merits, and in comparison to a related system which does provide thrust parallel to the orbital radius. Simulations are performed using commercial ODE solvers to propagate the Keplerian dynamics of a controlled satellite relative to an uncontrolled, leader satellite. The conclusion is drawn that, despite the exclusion of the radial thrust axis, the remaining control thrust available still provides enough control to design a gain matrix of adequate performance using linear-quadratic regulator (LQR) techniques.

Starin, Scott R.↗

NASA NTRS2002

A Nonlinear, Human-Centered Approach to Motion Cueing with a Neurocomputing Solver

This paper discusses the continuation of research into the development of new motion cueing algorithms first reported in 1999. In this earlier work, two viable approaches to motion cueing were identified: the coordinated adaptive washout algorithm or 'adaptive algorithm', and the 'optimal algorithm'. In this study, a novel approach to motion cueing is discussed that would combine features of both algorithms. The new algorithm is formulated as a linear optimal control problem, incorporating improved vestibular models and an integrated visual-vestibular motion perception model previously reported. A control law is generated from the motion platform states, resulting in a set of nonlinear cueing filters. The time-varying control law requires the matrix Riccati equation to be solved in real time. Therefore, in order to meet the real time requirement, a neurocomputing approach is used to solve this computationally challenging problem. Single degree-of-freedom responses for the nonlinear algorithm were generated and compared to the adaptive and optimal algorithms. Results for the heave mode show the nonlinear algorithm producing a motion cue with a time-varying washout, sustaining small cues for a longer duration and washing out larger cues more quickly. The addition of the optokinetic influence from the integrated perception model was shown to improve the response to a surge input, producing a specific force response with no steady-state washout. Improved cues are also observed for responses to a sway input. Yaw mode responses reveal that the nonlinear algorithm improves the motion cues by reducing the magnitude of negative cues. The effectiveness of the nonlinear algorithm as compared to the adaptive and linear optimal algorithms will be evaluated on a motion platform, the NASA Langley Research Center Visual Motion Simulator (VMS), and ultimately the Cockpit Motion Facility (CMF) with a series of pilot controlled maneuvers. A proposed experimental procedure is discussed. The results of this evaluation will be used to assess motion cueing performance.

Telban, Robert J.↗

DOE OSTI2020

A low-rank solver for the stochastic unsteady Navier–Stokes problem

Here we study a low-rank iterative solver for the unsteady Navier–Stokes equations for incompressible flows with a stochastic viscosity. The equations are discretized using the stochastic Galerkin method, and we consider an all-at-once formulation where the algebraic systems at all the time steps are collected and solved simultaneously. The problem is linearized with Picard’s method. To efficiently solve the linear systems at each step, we use low-rank tensor representations within the Krylov subspace method, which leads to significant reductions in storage requirements and computational costs. Combined with effective mean-based preconditioners and the idea of inexact solve, we show that only a small number of linear iterations are needed at each Picard step. The proposed algorithm is tested with a model of flow in a two-dimensional symmetric step domain with different settings to demonstrate the computational efficiency.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2020

Multidimensional Tests of a Finite-Volume Solver for MHD With a Real-Gas Equation of State

This work considers two algorithms of a finite-volume solver for the MHD equations with a real-gas equation of state (EOS). Both algorithms use a multistate form of Harten-Lax-Van Leer approximate Riemann solver as formulated for MHD discontinuities. This solver is modified to use the generalized sound speed from the real-gas EOS. Two methods are tested: EOS evaluation at cell centers and at flux interfaces where the former is more computationally efficient. A battery of 1D and 2D tests are employed: convergence of 1D and 2D linearized waves, shock tube Riemann problems, a 2D nonlinear circularly polarized Alfvén wave, and a 2D magneto-Rayleigh-Taylor instability test. The cell-centered EOS evaluation algorithm produces unresolvable thermodynamic inconsistencies in the intermediate states leading to spurious solutions while the flux-interface EOS evaluation algorithm robustly produces the correct solution. The linearized wave tests show this inconsistency is associated with the magnetosonic waves and the magneto-Rayleigh-Taylor instability test demonstrates simulation findings where the spurious solution leads to an unphysical simulation.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1998

Euler Technology Assessment for Preliminary Aircraft Design-Unstructured/Structured Grid NASTD Application for Aerodynamic Analysis of an Advanced Fighter/Tailless Configuration

This study supports the NASA Langley sponsored project aimed at determining the viability of using Euler technology for preliminary design use. The primary objective of this study was to assess the accuracy and efficiency of the Boeing, St. Louis unstructured grid flow field analysis system, consisting of the MACGS grid generation and NASTD flow solver codes. Euler solutions about the Aero Configuration/Weapons Fighter Technology (ACWFT) 1204 aircraft configuration were generated. Several variations of the geometry were investigated including a standard wing, cambered wing, deflected elevon, and deflected body flap. A wide range of flow conditions, most of which were in the non-linear regimes of the flight envelope, including variations in speed (subsonic, transonic, supersonic), angles of attack, and sideslip were investigated. Several flowfield non-linearities were present in these solutions including shock waves, vortical flows and the resulting interactions. The accuracy of this method was evaluated by comparing solutions with test data and Navier-Stokes solutions. The ability to accurately predict lateral-directional characteristics and control effectiveness was investigated by computing solutions with sideslip, and with deflected control surfaces. Problem set up times and computational resource requirements were documented and used to evaluate the efficiency of this approach for use in the fast paced preliminary design environment.

Michal, Todd R.↗

NASA NTRS2017

Geometrically Flexible and Efficient Flow Analysis of High Speed Vehicles Via Domain Decomposition, Part 1: Unstructured-Grid Solver for High Speed Flows

The ability to solve the equations governing the hypersonic turbulent flow of a real gas on unstructured grids using a spatially-elliptic, 2nd-order accurate, cell-centered, finite-volume method has been recently implemented in the VULCAN-CFD code. This paper describes the key numerical methods and techniques that were found to be required to robustly obtain accurate solutions to hypersonic flows on non-hex-dominant unstructured grids. The methods and techniques described include: an augmented stencil, weighted linear least squares, cell-average gradient method, a robust multidimensional cell-average gradient-limiter process that is consistent with the augmented stencil of the cell-average gradient method and a cell-face gradient method that contains a cell skewness sensitive damping term derived using hyperbolic diffusion based concepts. A data-parallel matrix-based symmetric Gauss-Seidel point-implicit scheme, used to solve the governing equations, is described and shown to be more robust and efficient than a matrix-free alternative. In addition, a y+ adaptive turbulent wall boundary condition methodology is presented. This boundary condition methodology is deigned to automatically switch between a solve-to-the-wall and a wall-matching-function boundary condition based on the local y+ of the 1st cell center off the wall. The aforementioned methods and techniques are then applied to a series of hypersonic and supersonic turbulent flat plate unit tests to examine the efficiency, robustness and convergence behavior of the implicit scheme and to determine the ability of the solve-to-the-wall and y+ adaptive turbulent wall boundary conditions to reproduce the turbulent law-of-the-wall. Finally, the thermally perfect, chemically frozen, Mach 7.8 turbulent flow of air through a scramjet flow-path is computed and compared with experimental data to demonstrate the robustness, accuracy and convergence behavior of the unstructured-grid solver for a realistic 3-D geometry on a non-hex-dominant grid.

White, Jeffery A.↗

DOE OSTI2024

Supporting ARPA-E Power Grid Optimization (Final Report)

Pacific Northwest National Laboratory (PNNL), Arizona State University (ASU), Georgia Institute of Technology (Georgia Tech), Los Alamos National Laboratory (LANL), National Renewable Energy Laboratory (NREL), Texas A&M University (TAMU), The University of Texas at Austin (UT), and the University of Wisconsin-Madison (UW-M) supported the ARPA-E Grid Optimization (GO) Competition by providing a common problem formulation, data format, datasets, evaluation mechanism, scoring, rules, and results that resulted in the awarding of $\$9.24$ million dollars to teams from academia, industry, and national labs for solving three sets of increasingly difficult non-linear, security- constrained AC Optimal Powerflow (AC-OPF) optimization problems in order to increase the efficiency of the US Electric Grid. It is estimated that a 1% increase in efficiency can save $\$1$ billion. Current industry practices typically use a linear DC model (DC-OPF) in order solve the OPF problem within the time constraints of the operation schedule. The GO Competition challenges the best power engineers, mathematicians, and computer scientists to make possible operational decisions based on accurate physical models. To accomplish this, the GO Competition created a series of Challenges and funded teams to produce the best solver. Challenge 1 was to solve the security constrained Alternating Current Optimal Power Flow (ACOPF) problem. Challenge 2 extended that to by adding adjustable transformer tap ratios, phase shifting transformers, switchable shunts, price-responsive demand, ramp rate constrained generators and loads, and fast-start unit commitment (UC). Furthermore, Challenge 2 was a maximization problem while Challenge 1 was a minimization problem. While Challenge 3 was being developed, the entrants were invited to find better solutions to the Challenge 2 synthetic datasets with no restrictions on time, hardware, or algorithms. The Challenge 2 solutions turned out to be very good. Challenge 3 expanded the Challenge 2 problem further by using multiperiod dynamic markets, including advisory models for extreme weather events, day-ahead markets, and the real-time markets with an extended look-ahead. These problems included active bid-in demand and topology optimization. Together the Challenges used nearly 30 million CPU hours. Since each team was working on the same problem, using the same data, and running on the same hardware, fair comparisons could be drawn as to the best solver. The datasets were varied enough, however, that the best solver for one dataset was not necessarily the best at another, so cumulative scores were used. The process was managed by the PNNL maintained website https://GOCompetition.energy.gov, where Entrants could find information about the problem, the data, the rules, submit their solver for evaluation, and see the scores of all the competing teams on a Leaderboard. Interest was world-wide but only American teams were eligible for prizes. The Competition has produced 34 journal articles 115 papers and been cited over 500 times in the literature, including 12 dissertations (4 from foreign countries; Columbia (2), Germany, and Italy) and 3 from the DOE ExaScale project. Software developed by Pearl Street Technologies for Challenges 1 and 2 is now deployed by Southwest Power Pool (SPP) and Midcontinent Independent Service Operator (MISO). Other teams have received inquiries from venture capitalists. Google DeepMind has thanked the Competition for making the datasets developed for the Competition public. They are using it to train machine learning models. The larger datasets have billions of unknowns to be solved for, but only a small percent matter in the final solution. Knowing what unknowns are important can dramatically speedup the solution.

24 POWER TRANSMISSION AND DISTRIBUTION↗

DOE OSTI2026

Extreme-scale EV charging infrastructure planning for last-mile delivery using high-performance parallel computing

Here, this paper addresses stochastic charger location and allocation problems under queue congestion for last-mile delivery using electric vehicles (EVs). The objective is to decide where to open charging stations and how many chargers of each type to install, subject to budgetary and waiting-time constraints. We formulate the problem as a mixed-integer non-linear program, where each station-charger pair is modeled as a multiserver queue with stochastic arrivals and service times to capture the notion of waiting in fleet operations. The model is extremely large, with billions of variables and constraints for a typical metropolitan area; even loading the model in solver memory is difficult, let alone solving it. To address this challenge, we develop a Lagrangian-based dual decomposition framework that decomposes the problem by station and leverages parallelization on high-performance computing systems, where the subproblems are solved by using a cutting plane method and their solutions are collected at the master level. We also develop a three-step rounding heuristic to transform the fractional subproblem solutions into feasible integral solutions. Computational experiments on data from the Chicago metropolitan area with hundreds of thousands of households and thousands of candidate stations show that our approach produces high-quality solutions in cases where existing exact methods cannot even load the model in memory. We also analyze various policy scenarios, demonstrating that combining existing depots with newly built stations under multiagency collaboration substantially reduces costs and congestion. These findings offer a scalable and efficient framework for developing sustainable large-scale EV charging networks.

Capacity allocation↗

DOE OSTI2020

A Scalable Multigrid Reduction Framework for Multiphase Poromechanics of Heterogeneous Media

Simulation of multiphase poromechanics involves solving a multiphysics problem in which multiphase flow and transport are tightly coupled with the porous medium deformation. To capture this dynamic interplay, fully implicit methods, also known as monolithic approaches, are usually preferred. The main bottleneck of a monolithic approach is that it requires solution of large linear systems that result from the discretization and linearization of the governing balance equations. Because such systems are nonsymmetric, indefinite, and highly ill-conditioned, preconditioning is critical for fast convergence. Recently, most efforts in designing efficient preconditioners for multiphase poromechanics have been dominated by physics-based strategies. Current state-of-the-art “black-box” solvers such as algebraic multigrid (AMG) are ineffective because they cannot effectively capture the strong coupling between the mechanics and the flow subproblems, as well as the coupling inherent in the multiphase flow and transport process. In this work, we develop an algebraic framework based on multigrid reduction (MGR) that is suited for tightly coupled systems of PDEs. Using this framework, the decoupling between the equations is done algebraically through defining appropriate interpolation and restriction operators. One can then employ existing solvers for each of the decoupled blocks or design a new solver based on knowledge of the physics. We demonstrate the applicability of our framework when used as a “black-box” solver for multiphase poromechanics. Here, we show that the framework is flexible to accommodate a wide range of scenarios, as well as efficient and scalable for large problems.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2021

Sensor enabled data-driven predictive analytics for modeling and control with high penetration of DERs in distribution systems

The electric power grid is undergoing a tremendous transformation due to the increasing penetration of renewable energy resources beginning with wind and more recently with the distributed energy resources (DERs) such as solar and battery storage. DERs have dramatically changed the role of the distribution systems in the overall power grid, and they are expected to contribute a significant portion of power generation in the future. If current trends for DERs continue, system operation and control will need to change dramatically for improved grid reliability and resiliency. As renewable resources increase in penetration, new and challenging operational, planning, and design problems are expected to emerge. Some of the key challenges that arise in the planning and operation of the future grid are: 1) Quantifying the impact of high DER penetration in distribution systems on bulk grid behavior over multiple time scales. 2) Identifying whether a particular DER configuration/settings have a large impact on the overall grid behavior. These challenges can be addressed in an offline manner using detailed T&D grid models and they can also be addressed in an online manner using sensor measurements. In particular, the advancement and planned growth in sensor technology in power grid over various voltage levels provide us with a unique opportunity to tackle these challenges from a data analytic perspective without needing detailed T&D grid models. A few questions that naturally arise when addressing the challenges from DERs using sensor data are: 1) How can we use limited sensor measurements to monitor & control voltage stability and small signal stability of the bulk system? 2) How can we ensure that the developed data analytic methods are robust to data availability and quality issues? 3) How can we compute the developed analytics in a scalable manner using streaming measurements? In this project, we addressed the aforementioned challenges arising from DERs and answered the questions raised above on how to effectively use the sensor measurements to enhance the reliability and performance of the electric grid. Thus, the overarching goal of this project is to develop effective reduced/representative system models from data that make the computational complexity sufficiently manageable so as to be useful to simulate, analyze, and even control complex non-linear power systems dynamics with large penetrations of DERs. In order to achieve the objective, the project team established a four-fold technical approach 1) Formulated a combined transmission-distribution co-simulation framework for data generation and validation, 2) Derived reduced/representative models of power systems based on data-driven methods for efficient computation and appropriate representation of system behavior, 3) Developed data driven characterization of power system behavior based on transfer operator theory, machine learning and optimization for model estimation, 4) Incorporated a scalable data management and processing architecture using distributed Kafka streaming applications that coordinate input data streams to the developed data analytics. The key accomplishments of the project are: 1) Development of a scalable multi-timescale T&D co-simulation framework (both for steady state and for dynamic co-simulation) using commercial solvers (PSSE and GridLAB-D). The steady-state T&D co-simulation interface is shared with our industry partner (PJM). 2) A structured reduced order dynamic model of distribution systems that can represent partial motor stalling along with a systematic procedure to derive the model parameters. 3) A PMU based online method to monitor, localize and mitigate fault-induced delayed voltage recovery using DER reactive support and load control in distribution systems. 4) Development of linear operator based robust methodologies for dynamic state estimation, uncertainty quantification, system identification and trajectory prediction for power system dynamics. 5) An adaptive damping control for utilizing wind energy resources to provide oscillation damping and system stability. 6) Implementation of Kafka-based framework for efficient processing of streaming data using Linux-based local virtual environment.

DER integration↗

DOE OSTI2026

Active operator learning with predictive uncertainty quantification for partial differential equations

With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying reliable surrogate models in scientific applications. Existing uncertainty quantification (UQ) frameworks employ ensembles or Bayesian methods, which can incur substantial computational costs during both training and inference. Here, we propose a lightweight predictive UQ method tailored for Deep operator networks (DeepONets) that also generalizes to other operator networks. Numerical experiments on linear and nonlinear PDEs demonstrate that the framework’s uncertainty estimates are unbiased and provide accurate out-of-distribution uncertainty predictions with a sufficiently large training dataset. Our framework provides fast inference and uncertainty estimates that can efficiently drive outer-loop analyses that would be prohibitively expensive with conventional solvers. We demonstrate how predictive uncertainties can be used in the context of Bayesian optimization and active learning problems to yield improvements in accuracy and data-efficiency for outer-loop optimization procedures. In the active learning setup, we extend the framework to Fourier Neural Operators (FNO) and describe a generalized method for other operator networks. To enable real-time deployment, we introduce an inference strategy based on precomputed trunk outputs and a sparse placement matrix, reducing evaluation time by more than a factor of five. Our method provides a practical route to uncertainty-aware operator learning in time-sensitive settings.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS2018

Numerical Simulations of a Quiet SuperSonic Technology (QueSST) Aircraft Preliminary Design

Reynolds Averaged Navier-Stokes (RANS) simulations were performed on a Lockheed Martin Quiet SuperSonic Technology (QueSST) aircraft preliminary design to assess inlet performance. The FUN3D flow solver and its adjoint-based grid refinement capability were used for the simulations in hopes of determining internal "best practices" for predicting inlet performance on top-aft-mounted inlets. Several parameters were explored including tetrahedral vs. pentahedral cells in/around the boundary-layer regions, an engine axis-aligned linear pressure sensor vs. a pressure box objective as the grid adaptation metric, and the number of grid adaptation cycles performed. Additional simulations were performed on manually refined grids for comparison with the adjoint-based adapted grids. Results showed poor agreement in predicted inlet performance on the refined grids compared to experimental data. This was true regardless of whether the refinement was adjoint-based or manual, the cell type in/near the boundary-layer regions, or the grid adaptation metric used. In addition, the 40-probe total pressure recovery was shown to decrease asymptotically as the number of adaptation cycles is increased. Solutions on the unadapted grids generally had better agreement with experimental data than their refined grid counterparts.

Inlet Performance↗

NASA NTRS2018

Numerical Simulations of a Quiet SuperSonic Technology (QueSST) Aircraft Preliminary Design

Reynolds Averaged Navier-Stokes (RANS) simulations were performed on the Lockheed Martin Quiet SuperSonic Technology (QueSST) aircraft preliminary design to assess inlet performance. The FUN3D flow solver and its adjoint-based grid refinement capability was used for the simulations in hopes of determining internal "best practices" for predicting inlet performance on top-aft-mounted inlets. Several parameters were explored including tetrahedral vs. pentahedral cells in/around the boundary-layer regions, an engine axis-aligned linear pressure sensor vs. a pressure box objective as the grid adaptation metric, and the number of grid adaptation cycles performed. Additional simulations were performed on manually refined grids for comparison with the adjoint-based adapted grids

Adjoint-based Grid Adaptation↗

NASA NTRS2018

Numerical Simulations of a Quiet SuperSonic Technology (QueSST) Aircraft Preliminary Design

Adjoint-based Grid Adaptation↗

NASA NTRSDate not supplied

Aeroelastic Analysis of Highly Flexible Wings with Linearized Frequency-Domain Aerodynamics

Flutter analysis of configurations with geometric structural nonlinearities typically is done with time-domain analysis. The results from this process are computationally expensive and can yield cumbersome results that may be difficult to manage/interpret. Compared to time-domain methods, frequency-domain flutter analysis can provide additional insight into the characteristics of a flutter stability problem. By linearizing the aeroelastic problem about the nonlinear equilibrium state, this work applies frequency-domain aeroelastic analysis to the Pazy wing, the subject of the Large Deformation Working Group in the Aeroelastic Prediction Workshop. Generalized aerodynamic forces (GAFs) are computed with both a doublet-lattice method and a computational fluid dynamics solver at a range of reduced frequencies as well as a range of dynamic pressures to account for the dependence of the mode shapes on the nonlinear equilibrium state. These GAFs are used in a p-k flutter solver, which is modified to handle the dependence of the stiffness matrix and GAFs on the dynamic pressure.

Bret K Stanford↗

NASA NTRSDate not supplied

Automatic Boundary-Layer Adaptation of Structured Grids in VULCAN-CFD

In supersonic and hypersonic flow computations, well-resolved boundary layers are essential for accurate quantification of surface heating and transition prediction, particularly via linear stability analysis. Grid design for hypersonic flows with shocks, boundary-layer separation, and/or complex mean flow features incorporating spanwise/azimuthal inhomogeneities is a difficult issue. In comparison to a manual grid adaptation procedure, an autonomous grid adaptation technique offers significant improvements in computing time and solution quality. The VULCAN-CFD solver already includes a validated procedure for automatic adaption of structured grids to the bow shock. The present focus is on implementing an automatic boundary-layer adaptation capability in VULCAN-CFD that adapts structured, multiblock grids to both the bow shock and the boundary layer at the same time. The boundary-layer adaptation algorithm allows the user to specify the number of cells within the boundary layer, along with the input parameters used for detecting the edge of the boundary layer, namely, the variable used in the edge detection criterion,the edge detection method, the detection direction, and the relaxation factor used during the morphing of the grid. The algorithm automatically distributes grid points along the wall-normal direction to achieve a smooth variation in grid spacing from the edge of the boundary layer to a"junction" location within the outer part of the grid. Illustrative results are presented for three different high-speed configurations: the two-dimensional flow over a cylinder at Mach 17.6 and unit Reynolds number of Re=0.38x10^6 m^-1, the axisymmetric flow over a cone-cylinder-flare model at Mach 6.0 and Re = 10.5×10^6 m^-1, and the three-dimensional flow over a blunt, 7-degree half-angle cone at 5-degree angle of attack in a Mach 9.79 flow with Re = 17.1×10^6 m^-1. The automated boundary-layer adaptation is shown to provide an adequate grid topology that is aligned with the bow shock in the outer part of the grid and also resolves the viscous boundary-layer region close to the surface.

boundary layer transition↗

DOE OSTI2025

Randomized Preconditioned Solvers for Strong Constraint 4D-Var Data Assimilation

The Strong Constraint 4D Variational (SC-4DVAR) data assimilation method is widely used in climate and weather applications. SC-4DVAR involves solving a minimization problem to compute the maximum a posteriori estimate, which we tackle using the Gauss-Newton method. The computation of the descent direction is expensive since it involves the solution of a large-scale and potentially ill-conditioned linear system, solved using the preconditioned conjugate gradient (PCG) method. Here, to address this cost, we efficiently construct scalable preconditioners using three different randomization techniques, which all rely on a certain low-rank structure involving the Gauss-Newton Hessian. The proposed techniques come with theoretical guarantees on the condition number, and at the same time, are amenable to parallelization. We also develop an adaptive approach to estimate the sketch size and choose between the reuse or recomputation of the preconditioner. We demonstrate the performance and effectiveness of our methodology on two representative model problems—the Burgers and barotropic vorticity equation—showing a drastic reduction in both the number of PCG iterations and the number of Gauss-Newton Hessian products after including the preconditioner construction cost.

Gauss-Newton↗