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Necessary and sufficient conditions are found under which operators of the form Sigma (m, j=1) x (2) sub j + X sub O can be made constant coefficient. In addition, necessary and sufficient conditions are derived which classify those linear partial differential operators that can be moved to the Kolmogorov type.
The NAVSTAR Global Positioning System (GPS) is a satellite based navigation system that can be operated in a differential mode. The basic concept of differential GPS is to have a local ground station at a known location. The ground station continuously determines bias corrections based on the received satellite signals and transmits these corrections to local users. This allows the user to improve significantly their estimate of position. The architecture of a real time differential GPS ground station based on two STI (Stanford Telecommunications, Inc.) single-channel GPS receivers is presented. Receiver requirements are described, together with the receiver architectures. Real time operation results are presented for recent data collected from six satellites.
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The command and control model for spacecraft operations, as well as the distribution of tasks between ground assets and in space assets, whether with a crew or solely robotic, is fundamentally constrained by the round trip light time between the space asset and the control facility (presumably on Earth, though not required). For an asset on Mars, the round trip light time varies, from roughly fourteen minutes to up to forty minutes. For a Lunar asset the round-trip light time is measured in only a few seconds, but current communications systems may more than double the latency with system overhead. For a Lunar Asset the total command latency may range from six seconds to more than forty, depending on communications overhead and data rates. Further, these variables are not always predictable, thus complicating operations. There are several differentiating factors for Lunar vs. Mars operations, Round trip light time/Atmosphere/Lighting and ShadowsTerrain type and knowledge/Round trip light time has implications for the distribution of tasks between ground and in space assets. Even at Lunar Distances, the combination of round trip light time plus communications systems overhead does not enable joy stick driving of a rover. The best that can be done, if driving from Earth, is near real time command and control. By 2030, driving from in space may be possible. Productivity on Mars requires either long operational sequences of commands, as is done for current rovers such as Curiosity, significant autonomous capability or, as may be possible by 2030, command and control support from space. Another implication of the long round trip light time from Earth to Mars, is that flight software functions must be resident on the in space asset. On the Moon, there is considerably more flexibility, enabling processing functions, to be resident on Earth or in space. This provides the opportunity to take advantage of the considerable processing power available on the ground, but may be constrained by data rates. On the Moon, for practical operational purposes, there is no atmosphere. Hence there is no scattering of light in the shadows. This has implications for image interpretation and driving near the poles. The Moon has permanently shadowed regions (PSR), unique terrain with unknown surface properties. With no scattering of light in shadows, driving on the Moon, particularly at the poles, where we have strong evidence of water, may prove to be hazardous and complex, requiring non-optical sensors, such as LIDAR.
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New coarse grid multigrid operators for problems with highly oscillatory coefficients are developed. These types of operators are necessary when the characters of the differential equations on coarser grids or longer wavelengths are different from that on the fine grid. Elliptic problems for composite materials and different classes of hyperbolic problems are practical examples. The new coarse grid operators can be constructed directly based on the homogenized differential operators or hierarchically computed from the finest grid. Convergence analysis based on the homogenization theory is given for elliptic problems with periodic coefficients and some hyperbolic problems. These are classes of equations for which there exists a fairly complete theory for the interaction between shorter and longer wavelengths in the problems. Numerical examples are presented.
There has been considerable research investigating the Ferroelectric Field-Effect Transistor (FeFET) in memory circuits. However, very little research has been performed in applying the FeFET to analog circuits. This paper investigates the use of FeFETs in a common analog circuit, the differential amplifier. The two input Metal-Oxide-Semiconductor (MOS) transistors in a general MOS differential amplifier circuit are replaced with FeFETs. Resistors are used in place of the other three MOS transistors. The FeFET model used in the analysis has been previously reported and was based on experimental device data. Because of the FeFET hysteresis, the FeFET differential amplifier has four different operating modes depending on whether the FeFETs are positively or negatively polarized. The FeFET differential amplifier operation in the different modes was analyzed by calculating the amplifier voltage transfer and gain characteristics shown in figures 2 through 5. Comparisons were made between the FeFET differential amplifier and the standard MOS differential amplifier. Possible applications and benefits of the FeFET differential amplifier are discussed.
This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.
We propose a bilevel optimization approach for the determination of parameters in nonlocal image denoising. We consider both spatial weights in front of the fidelity term, as well as weights within the kernel of the nonlocal operator. In both cases we investigate the differentiability of the solution operator in function spaces and derive a first order optimality system that characterizes local minima. For the numerical solution of the problems, we propose a second-order optimization algorithm in combination with a finite element discretization of the nonlocal denoising models and a computational strategy for the solution of the resulting dense linear systems. Several experiments are run in order to show the suitability of our approach.
We propose a bilevel optimization approach for the estimation of parameters in nonlocal image denoising models. The parameters we consider are both the space-dependent fidelity weight and weights within the kernel of the nonlocal operator. In both cases we investigate the differentiability of the solution operator in function spaces and derive a first order optimality system that characterizes local minima. For the numerical solution of the problems, we propose a second-order trust-region algorithm in combination with a finite element discretization of the nonlocal denoising models and we introduce a computational strategy for the solution of the resulting dense linear systems. Several experiments illustrate the applicability and effectiveness of our approach.
Differentiating matrices allow the numerical differentiation of functions defined at points of a discrete grid. A type of differentiating matrix based on local approximation on a sequence of sliding subgrids is considered. Previous derivations of this type of matrix have been restricted to grids with uniformly spaced points, and the resulting derivative approximations have lacked precision, especially at endpoints. The new formulation allows grids which have arbitrarily space points. It is shown that high accuracy can be achieved through use of differentiating matrices on non-uniform grids which include near-boundary points. Use of the differentiating matrix as an operator to solve eigenvalue problems involving ordinary differential equations is also considered.
In this paper we present a non-intrusive data-driven approach for model reduction of nonlinear systems. The approach considers the particular case of nonlinear partial differential equations (PDEs) that form systems of partial differential–algebraic equations (PDAEs) when lifted to polynomial form. Such systems arise, for example, when the governing equations include Arrhenius reaction terms (e.g., in reacting flow models) and thermodynamic terms (e.g., the Helmholtz free energy terms in a phase-field solidification model). Using the known structured form of the lifted algebraic equations, the approach computes the reduced operators for the algebraic equations explicitly, using straightforward linear algebra operations on the basis matrices. The reduced operators for the differential equations are inferred from lifted snapshot data using operator inference, which solves a linear least squares regression problem. The approach is illustrated for the nonlinear model of solidification of a pure material. The lifting transformations reformulate the solidification PDEs as a system of PDAEs that have cubic structure. The operators of the lifted system for this solidification example have affine dependence on key process parameters, permitting us to learn a parametric reduced model with operator inference. Numerical experiments show the effectiveness of the resulting reduced models in capturing key aspects of the solidification dynamics.
Solid-state 2-microns laser has been receiving considerable interest because of its eye-safe property and efficient diode pump operation, It has potential for multiple lidar applications to detect water vapor. carbon dioxide and winds. In this paper, we describe a 2-microns double pulsed Ho:Tm:YLF laser and end-pumped amplifier system. A comprehensive theoretical model has been developed to aid the design and optimization of the laser performance. In a single Q-switched pulse operation the residual energy stored in the Tm atoms will be wasted. However, in a double pulses operation mode, the residual energy stored in the Tm atoms will repopulate the Ho atoms that were depleted by the extraction of the first Q-switched pulse. Thus. the Tin sensitized Ho:YLF laser provides a unique advantage in applications that require double pulse operation, such as Differential Absorption Lidar (DIAL). A total output energy of 146 mJ per pulse pair under Q-switch operation is achieved with as high as 4.8% optical to optical efficiency. Compared to a single pulse laser, 70% higher laser efficiency is realized. To obtain high energy while maintaining the high beam quality, a master-oscillator-power-amplifier 2-microns system is designed. We developed an end-pumped Ho:Tm:YLF disk amplifier. This amplifier uses two diode arrays as pump source. A non-imaging lens duct is used to couple the radiation from the laser diode arrays to the laser disk. Preliminary result shows that the efficiency of this laser can be as high as 3%, a factor of three increases over side-pump configuration. This high energy, highly efficient and high beam quality laser is a promising candidate for use in an efficient, multiple lidar applications.
Operator learning techniques have recently emerged as a powerful tool for learning maps between infinite-dimensional Banach spaces. Trained under appropriate constraints, they can also be effective in learning the solution operator of partial differential equations (PDEs) in an entirely self-supervised manner. In this work we analyze the training dynamics of deep operator networks (DeepONets) through the lens of Neural Tangent Kernel theory, and reveal a bias that favors the approximation of functions with larger magnitudes. To correct this bias we propose to adaptively re-weight the importance of each training example, and demonstrate how this procedure can effectively balance the magnitude of back-propagated gradients during training via gradient descent. We also propose a novel network architecture that is more resilient to vanishing gradient pathologies. Taken together, our developments provide new insights into the training of DeepONets and consistently improve their predictive accuracy by a factor of 10-50x, demonstrated in the challenging setting of learning PDE solution operators in the absence of paired input-output observations.
A control system to be used with a plasma arc welding apparatus is disclosed. The plasma arc welding apparatus includes a plasma arc power supply, a contactor, and an electrode assembly for moving the electrode relative to a work piece. The electrode assembly is raised or lowered by a drive motor. The present apparatus includes a plasma arc adapter connected across the power supply to measure the voltage across the plasma arc. The plasma arc adapter forms a dc output signal input to a differential amplifier. A second input is defined by an adjustable resistor connected to a dc voltage supply to permit operator control. The differential amplifier forms an output difference signal provided to an adder circuit. The adder circuit then connects with a power amplifier which forms the driving signal for the motor. In addition, the motor connects to a tachometor which forms a feedback signal delivered to the adder to provide damping, therby avoiding servo loop overshoot.
Capacitive strain gages designed for measurements in wind tunnels to 2000 F were built and evaluated. Two design approaches were followed. One approach was based on fixed capacitor plates with a movable ground plane inserted between the plates to effect differential capacitive output with strain. The second approach was based on movable capacitor plates suspended between sapphire bearings, housed in a rugged body, and arranged to operate as a differential capacitor. A sapphire bearing gage (1/4 in. diameter x 1 in. in size) was built with a range of 50,000 and a resolution of 200 microstrain. Apparent strain on Rene' 41 was less than + or - 1000 microstrain from room temperature to 2000 F. Three gage models were built from the Ground Plane Differential concept. The first was 1/4 in. square by 1/32 in. high and useable to 700 F. The second was 1/2 in. square by 1/16 in. high and useable to 1440 F. The third, also 1/2 in. square by 1/16 in. high was expected to operate in the 1600 to 2000 F range, but was not tested because time and funding ended.