Semi-groups, groups and Lyapunov stability of partial differential equations.
Semigroups, groups and Liapunov stability of partial differential equation, obtaining operator differential equations in Hilbert spaces
Buis, G. R.↗
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Flight-test evaluation of civil helicopter terminal approach operations using differential GPS
A civil code differential Global Positioning System (DGPS) has been developed and flight-tested by the NASA Ames Research Center. The system was used to evaluate the performance of the DGPS for support of helicopter terminal approach operations. The airborne component of the DGPS was installed in a NASA helicopter. The ground-reference component was installed in a mobile van and equipped with a real-time VHF telemetry data link to transmit correction information to the aircraft system. An extensive series of tests was conducted to evaluate the performance of the system for several different configurations of the airborne navigation filter. This paper will describe the systems, the results of the flight tests, and the results of the posttest analysis.
Edwards, F. G.↗
Civil helicopter flight operations using differential GPS
The results of NASA flight trials of a dual-receiver differential global positioning system (DGPS) for civilian helicopter navigation applications, are presented. The three principal components of the DGPS system are described, including the GPS ground-reference system, a range tracking system, and an on-board sequential GPS receiver. The ground-based receiver in the DGPS operates at a known fixed location and receives C/A code signals from NAVSTAR satellites. System bias errors in the ground receiver are subtracted from the airborne solution for the navigational fix. Calculations of the differential bias error are carried out using an on-board PDP-11/34 M research computer. The ground-reference differential corrections for satellites are given in a table. It is shown that the differential correction signal of the DGPS contains only a small (0.1 rad/sec) high-frequency component which can be attributed to system error. A schematic diagram of the DGPS postflight data processing routine is provided.
Edwards, F. G.↗
Challenges in Training PINNs: A Loss Landscape Perspective
This paper explores challenges in training Physics Informed Neural Networks (PINNs), emphasizing the role of the loss landscape in the training process. We examine difficulties in minimizing the PINN loss function, particularly due to ill conditioning caused by differential operators in the residual term. We compare gradient-based optimizers Adam, L-BFGS, and their combination Adam+L-FGS, showing the superiority of Adam+L-BFGS, and introduce a novel secondorder optimizer, NysNewton-CG (NNCG), which significantly improves PINN performance. Theoretically, our work elucidates the connection between ill-conditioned differential operators and ill-conditioning in the PINN loss and shows the benefits of combining first- and second-order optimization methods. Our work presents valuable insights and more powerful optimization strategies for training PINNs, which could improve the utility of PINNs for solving difficult partial differential equations.
Rathore, Pratik↗
Modified Chebyshev pseudospectral method with O(N exp -1) time step restriction
The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N exp 2) where N is the number of grid points. As a result of this, the allowable time step in an explicit time marching algorithm is O(N exp -2) which, in many cases, is much below the time step dictated by the physics of the partial differential equation. A new set of interpolating points is introduced such that the eigenvalues of the differentiation operator are O(N) and the allowable time step is O(N exp -1). The properties of the new algorithm are similar to those of the Fourier method. The new algorithm also provides a highly accurate solution for non-periodic boundary value problems.
Kosloff, Dan↗
A modified Chebyshev pseudospectral method with an O(N exp -1) time step restriction
The extreme eigenvalues of the Chebyshev pseudospectral differentiation operator are O(N exp 2) where N is the number of grid points. As a result of this, the allowable time step in an explicit time marching algorithm is O(N exp -2) which, in many cases, is much below the time step dictated by the physics of the partial differential equation. A new set of interpolating points is introduced such that the eigenvalues of the differentiation operator are O(N) and the allowable time step is O(N exp -1). The properties of the new algorithm are similar to those of the Fourier method. The new algorithm also provides a highly accurate solution for non-periodic boundary value problems.
Kosloff, Dan↗
PLZT block data composers operated in differential phase mode
The use of a PLTZ block data composer as a matrix type phase modulator to record and process digital data by the differential phase mode in a holographic recording/processing system was investigated. The system has readout contrast ratios of between 10 and 15 to 1. The differential phase mode has the advantage that strain bias is not required and thickness and strain variations in the PLZT are cancelled out.
Drake, M. D.↗
Thermal Vacuum and Vibration Testing of the Differential Thermal Expansion Thermal Switch
The Extended Stroke, Reverse Operation, Differential Expansion Thermal Switch for Extreme Variable Environments uses materials with both positive and negative coefficients of thermal expansion (CTE) arranged in concentric cylinders so that the thermal switch (TS) operates passively, opening a gap between interfaces to create a low thermal contact when cold, and closing the gap between interfaces to create a high thermal contact when hot. This paper details the testing of the TS at Marshall Space Flight Center including the pre-vibe thermal vacuum cycle test, vibration test, post-vibe thermal vacuum cycle test, and long duration thermal vacuum cycle test. The pre- and post-vibe thermal cycle and the long duration cycle test tested to hot and cold plateau initial temperatures of 40 °C, and -173 °C, respectively. The pre- and post-vibe tests required plateau holds of at least 2 hours, whereas the long duration cycle test required plateau holds of at least 20 hours. The tests each used a rate of temperature change of < 2 °C/min when transitioning between plateaus. The data collected from each thermal test is compared to show the operation of the TS both before and after the vibration test to GEVS levels, and throughout the long duration cycle test. From the data collected, the total conductance through the TS interfaces is calculated for both on (TS closed – during hot plateau) and off (TS open – during cold plateau) configurations, and the turndown ratio is also calculated. Challenges and lessons learned from the testing are also discussed.
thermal control technology↗
Thermal Vacuum and Vibration Testing of the Differential Thermal Expansion Thermal Switch
The Extended Stroke, Reverse Operation, Differential Thermal Expansion (DTE) Thermal Switch (TS) for Extreme Variable Environments uses materials with both positive and negative coefficients of thermal expansion (CTE) arranged in concentric cylinders so that the TS operates passively, opening a gap between interfaces to create a low thermal contact when cold, and closing the gap between interfaces to create a high thermal contact when hot. This paper details the testing of the TS at Marshall Space Flight Center (MSFC) including the vibration (vibe) test, pre-vibe thermal vacuum (t-vac) cycle test, post-vibe t-vac cycle test, and long duration t-vac cycle test. The data collected from the pre- and post-vibe tests are compared to show the impact of the vibe test on performance. The data collected from the long duration test is used to calculate the total thermal contact conductance through the TS interfaces for both ‘on’ (TS closed, during hot plateau) and ‘off’ (TS open, during cold plateau) configurations, and the turndown ratio is also calculated. Challenges experienced throughout testing are also discussed.
Thermal control technology↗
Thermal Vacuum and Vibration Testing of the Differential Thermal Expansion Thermal Switch
The Extended Stroke, Reverse Operation, Differential Thermal Expansion (DTE) Thermal Switch (TS) for Extreme Variable Environments uses materials with both positive and negative coefficients of thermal expansion (CTE) arranged in concentric cylinders so that the TS operates passively, opening a gap between interfaces to create a low thermal contact when cold, and closing the gap between interfaces to create a high thermal contact when hot. This paper details the testing of the TS at Marshall Space Flight Center (MSFC) including the vibration (vibe) test, pre-vibe thermal vacuum (t-vac) cycle test, post-vibe t-vac cycle test, and long duration t-vac cycle test. The data collected from the pre- and post-vibe tests are compared to show the impact of the vibe test on performance. The data collected from the long duration test is used to calculate the total thermal contact conductance through the TS interfaces for both ‘on’ (TS closed, during hot plateau) and ‘off’ (TS open, during cold plateau) configurations, and the turndown ratio is also calculated. Challenges experienced throughout testing are also discussed.
Thermal control technology↗
A higher-order finite-element implementation of the nonlinear Fokker–Planck collision operator for charged particle collisions in a low density plasma
Collisions between particles in a low density plasma are described by the Fokker–Planck collision operator. In applications, this nonlinear integro-differential operator is often approximated by linearised or ad-hoc model operators due to computational cost and complexity. In this work, we present an implementation of the nonlinear Fokker–Planck collision operator written in terms of Rosenbluth potentials in the Rosenbluth–MacDonald–Judd (RMJ) form. The Rosenbluth potentials may be obtained either by direct integration or by solving partial differential equations (PDEs) similar to Poisson's equation: we optimise for performance and scalability by using sparse matrices to solve the relevant PDEs. We represent the distribution function using a tensor-product continuous-Galerkin finite-element representation and we derive and describe the implementation of the weak form of the collision operator. We present tests demonstrating a successful implementation using an explicit time integrator and we comment on the speed and accuracy of the operator. Finally, we speculate on the potential for applications in the current and next generation of kinetic plasma models.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Worldwide differential GPS for Space Shuttle landing operations
Worldwide differential Global Positioning System (WWDGPS) is viewed as an effective method of offering continuous high-quality navigation worldwide. The concept utilizes a network with as few as 33 ground stations to observe most of the error sources of GPS and provide error corrections to users on a worldwide basis. The WWDGPS real-time GPS tracking concept promises a threefold or fourfold improvement in accuracy for authorized dual-frequency users, and in addition maintains an accurate and current ionosphere model for single-frequency users. A real-time global tracking network also has the potential to reverse declarations of poor health on marginal satellites, increasing the number of satellites in the constellation and lessening the probability of GPS navigation outage. For Space Shuttle operations, the use of WWDGPS-aided P-code equipment promises performance equal to or better than other current landing guidance systems in terms of accuracy and reliability. This performance comes at significantly less cost to NASA, which will participate as a customer in a system designed as a commercial operation serving the global civil navigation community.
Loomis, Peter V. W.↗
On the convergence of local approximations to pseudodifferential operators with applications
We consider the approximation of a class pseudodifferential operators by sequences of operators which can be expressed as compositions of differential operators and their inverses. We show that the error in such approximations can be bounded in terms of L(1) error in approximating a convolution kernel, and use this fact to develop convergence results. Our main result is a finite time convergence analysis of the Engquist-Majda Pade approximants to the square root of the d'Alembertian. We also show that no spatially local approximation to this operator can be convergent uniformly in time. We propose some temporally local but spatially nonlocal operators with better long time behavior. These are based on Laguerre and exponential series.
Hagstrom, Thomas↗
Explicit block encodings of boundary value problems for many-body elliptic operators
Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.
Kharazi, Tyler [University of California, Berkeley↗
Special Component Designs for Differential-Amplifier MMICs
Special designs of two types of electronic components transistors and transmission lines have been conceived to optimize the performances of these components as parts of waveguide-embedded differential-amplifier monolithic microwave integrated circuits (MMICs) of the type described in the immediately preceding article. These designs address the following two issues, the combination of which is unique to these particular MMICs: Each MMIC includes a differential double-strip transmission line that typically has an impedance between 60 and 100 W. However, for purposes of matching of impedances, transmission lines having lower impedances are also needed. The transistors in each MMIC are, more specifically, one or more pair(s) of InP-based high-electron-mobility transistors (HEMTs). Heretofore, it has been common practice to fabricate each such pair as a single device configured in the side-to-side electrode sequence source/gate/drain/gate/source. This configuration enables low-inductance source grounding from the sides of the device. However, this configuration is not suitable for differential operation, in which it is necessary to drive the gates differentially and to feed the output from the drain electrodes differentially. The special transmission-line design provides for three conductors, instead of two, in places where lower impedance is needed. The third conductor is a metal strip placed underneath the differential double-strip transmission line. The third conductor increases the capacitance per unit length of the transmission line by such an amount as to reduce the impedance to between 5 and 15 W. In the special HEMT-pair design, the side-to-side electrode sequence is changed to drain/gate/source/gate/ drain. In addition, the size of the source is reduced significantly, relative to corresponding sizes in prior designs. This reduction is justified by the fact that, by virtue of the differential configuration, the device has an internal virtual ground, and therefore there is no need for a low-resistance contact between the source and the radio-frequency circuitry. The source contact is needed only for DC biasing. These designs were implemented in a single-stage-amplifier MMIC. In a test at a frequency of 305 GHz, the amplifier embedded in a waveguide exhibited a gain of 0 dB; after correcting for the loss in the waveguide, the amplifier was found to afford a gain of 0.9 dB. In a test at 220 GHz, the overall gain of the amplifier- and-waveguide assembly was found to be 3.5 dB.
Kangaslahti, Pekka↗
Simulation and analysis of differential global positioning system for civil helicopter operations
A Differential Global Positioning System (DGPS) computer simulation was developed, to provide a versatile tool for assessing DGPS referenced civil helicopter navigation. The civil helicopter community will probably be an early user of the GPS capability because of the unique mission requirements which include offshore exploration and low altitude transport into remote areas not currently served by ground based Navaids. The Monte Carlo simulation provided a sufficiently high fidelity dynamic motion and propagation environment to enable accurate comparisons of alternative differential GPS implementations and navigation filter tradeoffs. The analyst has provided the capability to adjust most aspects of the system, the helicopter flight profile, the receiver Kalman filter, and the signal propagation environment to assess differential GPS performance and parameter sensitivities. Preliminary analysis was conducted to evaluate alternative implementations of the differential navigation algorithm in both the position and measurement domain. Results are presented to show that significant performance gains are achieved when compared with conventional GPS but that differences due to DGPS implementation techniques were small. System performance was relatively insensitive to the update rates of the error correction information.
Denaro, R. P.↗
Absorbing boundary conditions for second-order hyperbolic equations
A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.
Jiang, Hong↗
Absorbing boundary conditions for second-order hyperbolic equations
A uniform approach to construct absorbing artificial boundary conditions for second-order linear hyperbolic equations is proposed. The nonlocal boundary condition is given by a pseudodifferential operator that annihilates travelling waves. It is obtained through the dispersion relation of the differential equation by requiring that the initial-boundary value problem admits the wave solutions travelling in one direction only. Local approximation of this global boundary condition yields an nth-order differential operator. It is shown that the best approximations must be in the canonical forms which can be factorized into first-order operators. These boundary conditions are perfectly absorbing for wave packets propagating at certain group velocities. A hierarchy of absorbing boundary conditions is derived for transonic small perturbation equations of unsteady flows. These examples illustrate that the absorbing boundary conditions are easy to derive, and the effectiveness is demonstrated by the numerical experiments.
Jiang, Hong↗