Results for “inverse methods” — Page 63

NASA NTRS1994

An Efficient Spectral Method for Ordinary Differential Equations with Rational Function Coefficients

We present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients. We employ expansions in terms of a large class of orthogonal polynomial families, including all the classical orthogonal polynomials. These families obey a simple three-term recurrence relation for differentiation, which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse. The inverse is an integration operator for the family, and it is simply the tridiagonal coefficient matrix for the recurrence. Since in these families convolution operators (i.e. matrix representations of multiplication by a function) are banded for polynomials, we are able to obtain a banded representation for linear differential operators with rational coefficients. This leads to a method of solution of initial or boundary value problems that, besides having an operation count that scales linearly with the order of truncation N, is computationally well conditioned. Among the applications considered is the use of rational maps for the resolution of sharp interior layers.

Coutsias, Evangelos A.↗

DOE OSTI2020

On the Derivation of Quasi-Newton Formulas for Optimization in Function Spaces

Newton’s method is usually preferred when solving optimization problems due to its superior convergence properties compared to gradient-based or derivative-free optimization algorithms. However, deriving and computing second-order derivatives needed by Newton’s method often is not trivial and, in some cases, not possible. In such cases quasi-Newton algorithms are a great alternative. In this paper, we provide a new derivation of well-known quasi-Newton formulas in an infinite-dimensional Hilbert space setting. Furthermore, it is known that quasi-Newton update formulas are solutions to certain variational problems over the space of symmetric matrices. In this paper, we formulate similar variational problems over the space of bounded symmetric operators in Hilbert spaces. By changing the constraints of the variational problem we obtain updates (for the Hessian and Hessian inverse) not only for the Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method but also for Davidon–Fletcher–Powell (DFP), Symmetric Rank One (SR1), and Powell-Symmetric-Broyden (PSB). In addition, for an inverse problem governed by a partial differential equation (PDE), we derive DFP and BFGS “structured” secant formulas that explicitly use the derivative of the regularization and only approximates the second derivative of the misfit term. We show numerical results that demonstrate the desired mesh-independence property and superior performance of the resulting quasi-Newton methods.

97 MATHEMATICS AND COMPUTING↗

NASA NTRS1993

Precision slew/settle technologies for flexible spacecraft

Many spacecraft missions in the next decade will require both a high degree of agility and precision pointing. Agility includes both rotational maneuvering for retargeting and translational motion for orbit adjustment and threat avoidance. The major challenge associated with such missions is the need for control over a wide range of amplitudes and frequencies, ranging from tens of degrees at less than 1 Hz to a few micron radians at hundreds of Hz. TRW's internally funded Precision Control of Agile Spacecraft (PCAS) project is concerned with developing and validating in hardware the tools necessary to successfully complete the combined agile maneuvering/precision pointing missions. Development has been undertaken on a number of fronts for quietly slewing flexible structures. Various methods for designing slew torque profiles have been investigated. Prime candidates for slew/settle scenarios include Inverse Dynamics and Parameterized Function Space. Joint work with Processor Bayo at the University of California, Santa Barbara and Professor Flashner at the University of Southern California has led to promising torque profile design methods. Active and passive vibration suppression techniques also play a key role for rapid slew/settle mission scenarios. Active members with local control loops and passive members with high loss factor viscoelastic material have been selected for hardware verification. Progress in each of these areas produces large gains in the quiet slewing of flexible spacecraft. The main thrust of the effort to date has been the development of a modular testbed for hardware validation of the precision control concepts. The testbed is a slewing eighteen foot long flexible truss. Active and passive members can be interchanged with the baseline aluminum members to augment the inherent damping in the system. For precision control the active members utilize control laws running on a high speed digital structural control processor. Tip and midspan motions of the truss are determined using optical sensors while accelerometers can be used to monitor the motions of other points of interest. Preliminary results indicate that a mix of technologies produces the greatest benefit. For example, shaping the torque profile produces large improvements in slew/settle performance, but without added damping settling times may still be excessive. With the introduction of moderate amounts of damping, slew/settle performance is vastly improved. On the other hand, introducing damping without shaping the torque profile may not yield the desired level of performance.

Manning, R. A.↗

NASA NTRS1987

Three dimensional numerical scattering from ocean-like surfaces

A new method of calculating the electric field scattered from two-dimensional conducting rough surfaces is presented. This method uses the operator conjugate gradients technique and the extended boundary condition equation. This avoids the build-up of round-off errors encountered in the inversion of large matrices. It also avoids the singularities of the integral kernels usually encountered in the method of moments. Techniques for improving the rate of convergence by applying physical constraints are discussed. This method is presently being used to calculate the polarization signatures from simulated ocean-like surfaces.

Rodriguez, Ernesto↗

NASA NTRS2002

Turbomachinery Airfoil Design Optimization Using Differential Evolution

An aerodynamic design optimization procedure that is based on a evolutionary algorithm known at Differential Evolution is described. Differential Evolution is a simple, fast, and robust evolutionary strategy that has been proven effective in determining the global optimum for several difficult optimization problems, including highly nonlinear systems with discontinuities and multiple local optima. The method is combined with a Navier-Stokes solver that evaluates the various intermediate designs and provides inputs to the optimization procedure. An efficient constraint handling mechanism is also incorporated. Results are presented for the inverse design of a turbine airfoil from a modern jet engine and compared to earlier methods. The capability of the method to search large design spaces and obtain the optimal airfoils in an automatic fashion is demonstrated. Substantial reductions in the overall computing time requirements are achieved by using the algorithm in conjunction with neural networks.

Madavan, Nateri K.↗

NASA NTRS2002

Turbomachinery Airfoil Design Optimization Using Differential Evolution

An aerodynamic design optimization procedure that is based on a evolutionary algorithm known at Differential Evolution is described. Differential Evolution is a simple, fast, and robust evolutionary strategy that has been proven effective in determining the global optimum for several difficult optimization problems, including highly nonlinear systems with discontinuities and multiple local optima. The method is combined with a Navier-Stokes solver that evaluates the various intermediate designs and provides inputs to the optimization procedure. An efficient constraint handling mechanism is also incorporated. Results are presented for the inverse design of a turbine airfoil from a modern jet engine. The capability of the method to search large design spaces and obtain the optimal airfoils in an automatic fashion is demonstrated. Substantial reductions in the overall computing time requirements are achieved by using the algorithm in conjunction with neural networks.

Madavan, Nateri K.↗

DOE OSTI2022

Probabilistic inversion of circular phase spectra: application to two-station phase-velocity dispersion estimation in western Canada

SUMMARY Periodic directional and temporal measurements are common in seismology, and necessitate specific statistical analyses that are appropriate for circular quantities. In this work, we explore the use of a von Mises distribution as a representation of errors on circular seismological observations. Specifically, we automate the estimation of surface-wave phase-velocity dispersion for the teleseismic two-station method, which generally suffers from a 2π phase ambiguity. The use of Bayesian inverse techniques, which aim to rigorously quantify model parameter uncertainty, have become widespread throughout seismology over the last decade. Here, we apply Bayesian inversion to measurements of surface-wave phase spectra in order to estimate 1-D, path-averaged Earth structure between station pairs. The dispersion curve and associated uncertainties are additional results of the inversion, which can then be used as input for subsequent analyses (e.g. tomography). We demonstrate this technique through application to surface-wave recordings from long-running seismic stations throughout western Canada. Our results for over 10 000 station pairs reveal first-order tectonic features consistent with previous studies, which provides confidence in our approach as well as an initial step towards resolving a full 3-D seismic velocity model for the region. This work also presents a foundation for the inversion of surface-wave phase spectra to estimate 3-D Earth structure directly. Finally, the ideas presented in this work are not limited to the inversion of surface-wave phase spectra, but can also be considered for Bayesian geophysical inversion of any circular quantities.

Gosselin, Jeremy M. (ORCID:0000000203754102)↗

DOE OSTI2021

Scalable ab initio fragmentation methods based on a truncated expansion of the non-orthogonal molecular orbital model

An alternative formulation of the non-orthogonal molecular orbital model of electronic structure theory is developed based on the expansion of the inverse molecular orbital overlap matrix. From this model, a hierarchy of ab initio fragment-based quantum chemistry methods, referred to as the nth-order expanded non-orthogonal molecular orbital methods, are developed using a minimal number of approximations, each of which is frequently employed in intermolecular interaction theory. These novel methods are compared to existing fragment-based quantum chemistry methods, and the implications of those significant differences, where they exist, between the methods developed herein and those already existing methods are examined in detail. Computational complexities and theoretical scaling are also analyzed and discussed. Future extensions for the hierarchy of methods, to account for additional intrafragment and interfragment interactions, are outlined.

Westheimer, Bryce M. (ORCID:0000000267645714)↗

NASA NTRS1985

Microwave inversion of leaf area and inclination angle distributions from backscattered data

The backscattering coefficient from a slab of thin randomly oriented dielectric disks over a flat lossy ground is used to reconstruct the inclination angle and area distributions of the disks. The disks are employed to model a leafy agricultural crop, such as soybeans, in the L-band microwave region of the spectrum. The distorted Born approximation, along with a thin disk approximation, is used to obtain a relationship between the horizontal-like polarized backscattering coefficient and the joint probability density of disk inclination angle and disk radius. Assuming large skin depth reduces the relationship to a linear Fredholm integral equation of the first kind. Due to the ill-posed nature of this equation, a Phillips-Twomey regularization method with a second difference smoothing condition is used to find the inversion. Results are obtained in the presence of 1 and 10 percent noise for both leaf inclination angle and leaf radius densities.

Lang, R. H.↗

DOE OSTI2023

A Modified Maximum Entropy Inverse Reinforcement Learning Approach for Microgrid Energy Scheduling

Increasing popularity of integrating distributed energy resources (DERs) into the power system brings a challenge to optimize the microgrid dispatch policy. The reinforcement learning methods suffer from a long-time problem with the theoretical assumption of the objective/reward function for the microgrid system. Although the traditional inverse reinforcement learning (IRL) approaches can solve this problem to some extent, they encounter a limitation of complex computations for state visitation frequency in the large and continuous state space. To alleviate this limitation, we propose a modified maximum entropy IRL (MMIRL) method to extract the reward function from the expert demonstrations for solving the microgrid energy scheduling problem. The proposed MMIRL algorithm is promising in recovering the reward function and learning the dispatch policy compared to conventional approaches. Case studies are performed in an energy arbitrage problem and a microgrid system with DERs. Results substantiate that the proposed MMIRL approach can learn the dispatch policy with more than 99% efficiency and outperforms other comparative methods.

artificial intelligence, reinforcement learning, m↗

NASA NTRS1984

Conservative streamtube solution of steady-state Euler equations

This paper presents a new method for solving the steady state Euler equations. The method is similar to streamline curvature methods but has a conserative finite volume formulation which ensures correct shock capturing. Either wall position or wall pressure may be prescribed as boundary conditions, permitting both direct and inverse calculations. In supersonic applications the solution is obtained by space-marching while in subsonic and transonic applications iterative relaxation methods are used. Numerical results are given for: (1) supersonic diffuser with oblique shocks (direct calculation); (2) supersonic jet entering still reservoir (inverse calculation); (3) subsonic bump in a channel with 25 percent blockage (direct and inverse); (4) subsonic high-work turbine cascade (direct); and (5) transonic bump in a channel with 12 percent blockage (direct calculation).

Drela, M.↗

DOE OSTI2025

Enhancing ACPF Analysis: Integrating Newton-Raphson Method with Gradient Descent and Computational Graphs

This paper presents a new method for enhancing Alternating Current Power Flow (ACPF) analysis. The method integrates the Newton-Raphson (NR) method with Enhanced-Gradient Descent (GD) and computational graphs. The integration of renewable energy sources in power systems introduces variability and unpredictability, and this method addresses these challenges. It leverages the robustness of NR for accurate approximations and the flexibility of GD for handling variable conditions, all without requiring Jacobian matrix inversion. Furthermore, computational graphs provide a structured and visual framework that simplifies and systematizes the application of these methods. The goal of this fusion is to overcome the limitations of traditional ACPF methods and improve the resilience, adaptability, and efficiency of modern power grid analyses. We validate the effectiveness of our advanced algorithm through comprehensive testing on established IEEE benchmark systems. Furthermore, our findings demonstrate that our approach not only speeds up the convergence process but also ensures consistent performance across diverse system states, representing a significant advancement in power flow computation.

24 POWER TRANSMISSION AND DISTRIBUTION↗

NASA NTRS1998

Numerical Procedures for Inlet/Diffuser/Nozzle Flows

Two primitive variable, pressure based, flux-split, RNS/NS solution procedures for viscous flows are presented. Both methods are uniformly valid across the full Mach number range, Le., from the incompressible limit to high supersonic speeds. The first method is an 'optimized' version of a previously developed global pressure relaxation RNS procedure. Considerable reduction in the number of relatively expensive matrix inversion, and thereby in the computational time, has been achieved with this procedure. CPU times are reduced by a factor of 15 for predominantly elliptic flows (incompressible and low subsonic). The second method is a time-marching, 'linearized' convection RNS/NS procedure. The key to the efficiency of this procedure is the reduction to a single LU inversion at the inflow cross-plane. The remainder of the algorithm simply requires back-substitution with this LU and the corresponding residual vector at any cross-plane location. This method is not time-consistent, but has a convective-type CFL stability limitation. Both formulations are robust and provide accurate solutions for a variety of internal viscous flows to be provided herein.

Rubin, Stanley G.↗

NASA NTRS2001

A forward model-based validation of cardiovascular system identification

We present a theoretical evaluation of a cardiovascular system identification method that we previously developed for the analysis of beat-to-beat fluctuations in noninvasively measured heart rate, arterial blood pressure, and instantaneous lung volume. The method provides a dynamical characterization of the important autonomic and mechanical mechanisms responsible for coupling the fluctuations (inverse modeling). To carry out the evaluation, we developed a computational model of the cardiovascular system capable of generating realistic beat-to-beat variability (forward modeling). We applied the method to data generated from the forward model and compared the resulting estimated dynamics with the actual dynamics of the forward model, which were either precisely known or easily determined. We found that the estimated dynamics corresponded to the actual dynamics and that this correspondence was robust to forward model uncertainty. We also demonstrated the sensitivity of the method in detecting small changes in parameters characterizing autonomic function in the forward model. These results provide confidence in the performance of the cardiovascular system identification method when applied to experimental data.

NASA Discipline Cardiopulmonary↗

NASA NTRS1992

Kinematics of an in-parallel actuated manipulator based on the Stewart platform mechanism

This paper presents kinematic equations and solutions for an in-parallel actuated robotic mechanism based on Stewart's platform. These equations are required for inverse position and resolved rate (inverse velocity) platform control. NASA LaRC has a Vehicle Emulator System (VES) platform designed by MIT which is based on Stewart's platform. The inverse position solution is straight-forward and computationally inexpensive. Given the desired position and orientation of the moving platform with respect to the base, the lengths of the prismatic leg actuators are calculated. The forward position solution is more complicated and theoretically has 16 solutions. The position and orientation of the moving platform with respect to the base is calculated given the leg actuator lengths. Two methods are pursued in this paper to solve this problem. The resolved rate (inverse velocity) solution is derived. Given the desired Cartesian velocity of the end-effector, the required leg actuator rates are calculated. The Newton-Raphson Jacobian matrix resulting from the second forward position kinematics solution is a modified inverse Jacobian matrix. Examples and simulations are given for the VES.

Williams, Robert L., II↗

NASA NTRS2023

Modeling and Simulation of the Angel Upper Limb Offload Device: Branching Into New Methods

BACKGROUND: The Active Response Gravity Offload System (ARGOS) provides an analog environment for extravehicular activity (EVA) testing and training. Discomfort has been observed during longer suited test sessions. While the subject’s core is offloaded during surface EVA evaluations, his/her arms experience full Earth gravity and can become overly fatigued, especially during suited tests which involve reaching and prolonged arm extensions. A device (ARGOS Negation of Gravitational Effects on the Limbs: ANGEL) to offload the weight of the arms and suit sleeves is being developed by JSC’s Flight Systems Branch of the Software, Robotics, and Simulation Division. Previously we have shared preliminary modeling of that device and kinematics based on motion capture data. Here we present an alternative approach to determine device kinematics by calculating ANGEL component angles with an OpenSim plugin. We compare calculated angles to inverse kinematics (IK) derived ones with the goal of validating the model. This new method can be further informative for device design and analytically testing different configurations to achieve desired reduced gravity conditions (e.g., lunar gravity (Lg) or Martian gravity (Mg)). We have compared calculated angles with IK-derived angles in tests with a shirt-sleeve subject positioned in a test stand with a Mark-III Hard Upper Torso (HUT) and Portable Life Support System (PLSS) mockup and arm weights to emulate the weight of the suit sleeve as well as a suited subject in ARGOS with a Mark-III suit. A variety of upper body tasks were completed in the former and full-body tasks in the latter. METHODS AND RESULTS: To model the offload device, we augment the OpenSim human model topology with the offload mechanism components and joints, using CAD models to represent the mechanism graphically. The joint angles of the device are calculated in the OpenSim plugin by modeling how the components configure themselves under the offloading spring tension given a particular IK-derived arm position. There are four ANGEL components with a total of 5 degrees of freedom (DOFs), each component has a single DOF except for the cuff which is modeled as 2 DOFs. The sickle/yaw bracket and cuff rotation angles are determined statically based on the assumptions that the sickle will track the attachment point of the cuff and that the cuff will rotate such that the attachment point is at its highest point. The cuff tilt, linker and V-bracket angles are then determined by optimizing their positions to approach a mechanical equilibrium. The calculated linker angle is compared to three different methods of determining the linker line-of-force kinematically (from V-bracket to center-cuff, cuff highest point or marker-derived position). Given the joint angles of the device, the spring force and resulting force on the arm is computed by the plugin and applied as an external load in inverse dynamics (ID) to enable study of overall shoulder joint torques as well as offload achieved. We verify the calculated joint angles by using the inverse kinematic data. The average difference in angles is the smallest for the V-bracket and linker, around 1 to 5 degrees for most trials. The resulting offload and shoulder torque are comparable between calculated and IK-derived angles. In summary, we have developed a method to calculate the joint angles of an exoskeleton-like upper limb offloading device currently in development. We have also developed a custom plugin which will be a valuable tool to optimize device configurations for a desired gravitational environment, probe the offload achieved for motions recorded outside of our test suite, and inform future design improvements.

L B Nilsson↗

NASA NTRS2023

Modeling and Simulation of The Angel Upper Limb Offload Device: Branching into New Methods

BACKGROUND: The Active Response Gravity Offload System (ARGOS) provides an analog environment for extravehicular activity (EVA) testing and training. Discomfort has been observed during longer suited test sessions. While the subject’s core is offloaded during surface EVA evaluations, his/her arms experience full Earth gravity and can become overly fatigued, especially during suited tests which involve reaching and prolonged arm extensions. A device (ARGOS Negation of Gravitational Effects on the Limbs: ANGEL) to offload the weight of the arms and suit sleeves is being developed by JSC’s Flight Systems Branch of the Software, Robotics, and Simulation Division. Previously we have shared preliminary modeling of that device and kinematics based on motion capture data. Here we present an alternative approach to determine device kinematics by calculating ANGEL component angles with an OpenSim plugin. We compare calculated angles to inverse kinematics (IK) derived ones with the goal of validating the model. This new method can be further informative for device design and analytically testing different configurations to achieve desired reduced gravity conditions (e.g., lunar gravity (Lg) or Martian gravity (Mg)). We have compared calculated angles with IK-derived angles in tests with a shirt-sleeve subject positioned in a test stand with a Mark-III Hard Upper Torso (HUT) and Portable Life Support System (PLSS) mockup and arm weights to emulate the weight of the suit sleeve as well as a suited subject in ARGOS with a Mark-III suit. A variety of upper body tasks were completed in the former and full-body tasks in the latter. METHODS AND RESULTS: To model the offload device, we augment the OpenSim human model topology with the offload mechanism components and joints, using CAD models to represent the mechanism graphically. The joint angles of the device are calculated in the OpenSim plugin by modeling how the components configure themselves under the offloading spring tension given a particular IK-derived arm position. There are four ANGEL components with a total of 5 degrees of freedom (DOFs), each component has a single DOF except for the cuff which is modeled as 2 DOFs. The sickle/yaw bracket and cuff rotation angles are determined statically based on the assumptions that the sickle will track the attachment point of the cuff and that the cuff will rotate such that the attachment point is at its highest point. The cuff tilt, linker and V-bracket angles are then determined by optimizing their positions to approach a mechanical equilibrium. The calculated linker angle is compared to three different methods of determining the linker line-of-force kinematically (from V-bracket to center-cuff, cuff highest point or marker-derived position). Given the joint angles of the device, the spring force and resulting force on the arm is computed by the plugin and applied as an external load in inverse dynamics (ID) to enable study of overall shoulder joint torques as well as offload achieved. We verify the calculated joint angles by using the inverse kinematic data. The average difference in angles is the smallest for the V-bracket and linker, around 1 to 5 degrees for most trials. The resulting offload and shoulder torque are comparable between calculated and IK-derived angles. In summary, we have developed a method to calculate the joint angles of an exoskeleton-like upper limb offloading device currently in development. We have also developed a custom plugin which will be a valuable tool to optimize device configurations for a desired gravitational environment, probe the offload achieved for motions recorded outside of our test suite, and inform future design improvements.

L B Nilsson↗

NASA NTRS1993

Measurement methods and algorithms for comparison of local and remote clocks

Several methods for characterizing the performance of clocks with special emphasis on using calibration information that is acquired via an unreliable or noisy channel is discussed. Time-domain variance estimators and frequency-domain techniques such as cross-spectral analysis are discussed. Each of these methods has advantages and limitations that will be illustrated using data obtained via GPS, ACTS, and other methods. No one technique will be optimum for all of these analyses, and some of these problems cannot be completely characterized by any of the techniques discussed. The inverse problem of communicating frequency and time corrections to a real-time steered clock are also discussed. Methods were developed to mitigate the disastrous problems of data corruption and loss of computer control.

Levine, Judah↗