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Entirely rigorous proofs of the symmetries induced by the phase matrix into the reflection and transmission operators used in the matrix operator theory are given. Results are obtained for multiple scattering in both homogeneous and inhomogeneous atmospheres. These results will be useful to researchers using the method since large savings in computer time and storage are obtainable.
A microcomputer-based real-time programmable optical signal processing system utilizing a magneto-optic spatial light modulator (MOSLM) and a liquid crystal light valve (LCLV) is described. This system can perform a myriad of complicated optical operations, such as image correlation, image subtraction, and matrix multiplication. Its important asserts are its programmability and the capability of real-time addressing. These are important for telerobotic vision in space automation applications. Design specifications and suggestions for practical implementation of the system are discussed. Some preliminary experimental demonstrations are conducted to demonstrate applications of the proposed system to image correlation for optical pattern recognition, image subtraction for lC chip inspection, and matrix multiplication for optical computing.
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In a communications channel, the space environment between a spacecraft and an Earth ground station can potentially cause the loss of a data link or at least degrade its performance due to atmospheric effects, shadowing, multipath, or other impairments. In adaptive and coded modulation, the signal power level at the receiver can be used in order to choose a modulation-coding technique that maximizes throughput while meeting bit error rate (BER) and other performance requirements. It is the goal of this research to implement a generalized interacting multiple model (IMM) filter based on Kalman filters for improved received power estimation on software-dened radio (SDR) technology for satellite communications applications. The IMM filter has been implemented in Verilog consisting of a customizable bank of Kalman filters for choosing between performance and resource utilization. Each Kalman filter can be implemented using either solely a Schur complement module (for high area efficiency) or with Schur complement, matrix multiplication, and matrix addition modules (for high performance). These modules were simulated and synthesized for the Virtex II platform on the JPL Radio Experimenter Development System (EDS) at NASA Glenn Research Center. The results for simulation, synthesis, and hardware testing are presented.
The paper describes an advanced technology development which utilizes a high speed analog/binary CCD correlator to perform the matrix multiplications necessary to implement onboard feature classification. The matrix manipulation module uses the maximum likelihood classification algorithm assuming a Gaussian probability density function. The module will process 16 element multispectral vectors at rates in excess of 500 thousand multispectral vector elements per second. System design considerations for the optimum use of this module are discussed, test results from initial device fabrication runs are presented, and the performance in typical processing applications is described
Two analog very large scale integration (VLSI) vector matrix multiplier integrated circuit chips were designed, fabricated, and partially tested. They can perform both vector-matrix and matrix-matrix multiplication operations at high speeds. The 32 by 32 vector-matrix multiplier chip and the 128 by 64 vector-matrix multiplier chip were designed to perform 300 million and 3 billion multiplications per second, respectively. An additional circuit that has been developed is a continuous-time adaptive learning circuit. The performance achieved thus far for this circuit is an adaptivity of 28 dB at 300 KHz and 11 dB at 15 MHz. This circuit has demonstrated greater than two orders of magnitude higher frequency of operation than any previous adaptive learning circuit.
Fatigue tests of the SCS-6/Timetal 21S composite system were performed to characterize the fatigue behavior for unnotched conditions. The stress-life behavior of the unnotched (9/90)2s laminates was investigated for stress ratios of R = 0.1 and R = 0.3. The occurrence of matrix cracking was also examined in these specimens. This revealed multiple matrix crack initiation sites throughout the composite, as well as evenly spaced surface cracks along the length of the specimens. No difference in fatigue lives were observed for stress ratios of R = 0.1 and R = 0.3 when compared on a stress range basis. The unnotched SCS-6/Timetal 21S composites had shorter fatigue lives than the SCS-6/Ti-15-3 composites, however the neat Timetal 21S matrix material had a longer fatigue life than the neat Ti-15-3.
A compact differentiation technique (without using indexes) is developed for scalar functions that depend on complex matrix arguments which are combined by operations of complex conjugation, transposition, addition, multiplication, matrix inversion and taking the direct product. The differentiation apparatus is developed in order to simplify the solution of extremum problems of scalar functions of matrix arguments.
In this paper, we extend the theory and practice regarding algorithmic fault-tolerant matrix-matrix multiplication, C = AB, in a number of ways.
In this work, we describe methods for selecting the neural network architectures and input spaces to implement belief state inference on generic commercial transport aircraft. First, we highlight a case study on the planning, execution, and analysis of a set of experiments to determine the configurations of a conditional variational autoencoder (CVAE). We present a structured method that can be used in a number of aerospace applications, to optimize the structure and training parameters of the CVAE for belief state inference, using Design of Experiments (DOE) statistical methodologies. The motivation for this specific DOE was to identify the appropriate hyperparameters for measuring the CVAE reconstruction probability and latent space, such that the measurements can be used to infer qualitative state changes for the aircraft. We demonstrate that this process yields information about a trained neural network’s utility for this specific application, along with a quantifiable range of certainty. We execute 84 experiments using loss-of-control flight maneuver data from a NASA T-2 aircraft, demonstrating that this empirical process allows us to construct cheap and simple models with specific attributes amenable to belief state inference in aerospace applications. While theoretically, we could create a single CVAE with an input space the size of all measurable flight variables and environmental dynamics, it becomes intractable to use such a neural network in an in-situ intelligent multi-agent system. Using the recommendations from our case study, we introduce a technical approach for feasibly describing the belief space by (1) identifying significant statistical relationships among flight variables using rule induction, (2) using a set of rules that cover all features to define the input space of multiple CVAEs, and (3) forming a belief space based on the joint probability density of their collective latent spaces. This results in a series of relatively small matrix multiplications that can be performed in real time, as opposed to large matrix computations in a single CVAE. We demonstrate the application of this approach on the T-2 flight loss-of control experiments, using the architecture and hyperparameter recommendations from the case study. We compare the utilities of an individual CVAE trained on all flight variables and multiple CVAEs defined on subsets of flight variables for detecting qualitative changes in flight. We demonstrate that the use of multiple CVAEs with smaller input spaces permits the CVAE to capture more granular relationships in the latent space, permitting better state space characterization and loss-of-control detection.
A simple algorithm is presented for matrix multiplication in just over half the number of operations entailed by the conventional algorithm, in cases where the matrix possesses the degree of symmetry widely exhibited by derivative matrices. The algorithm is used to multiply the Chebyshev derivative matrix by a vector. For the larger values of n, the ratio of execution times approached the expected value of 2.
Damage development in woven SiC/SiNC ceramic matrix composites (CMC's) under tensile and cyclic loading both at room and elevated temperatures have been investigated for the exhaust nozzle of high-efficient turbine engines. The ultimate strength, failure strain, proportional limit and modulus data at a temperature range of 23 to 1250 C are generated. The tensile strength of SiC/SiNC woven composites have been observed to increase with increased temperatures up to 1000 C. The stress/strain plot shows a pseudo-yield point at 25 percent of the failure strain (epsilon(sub r)) which indicates damage initiation in the form of matrix cracking. The evolution of damage beyond 0.25 epsilon(sub f), both at room and elevated temperature comprises multiple matrix cracking, interfacial debonding, and fiber pullout. Although the nature of the stress/strain plot shows damage-tolerant behavior under static loading both at room and elevated temperature, the life expectancy of SiC/SiNC composites degrades significantly under cyclic loading at elevated temperature. This is mostly due to the interactions of fatigue damage caused by the mechanically induced plastic strain and the damage developed by the creep strain. The in situ damage evolutions are monitored by acoustic event parameters, ultrasonic C-scan and stiffness degradation. Rate equations for modulus degradation and fatigue life prediction of ceramic matrix composites both at room and elevated temperatures are developed. These rate equations are observed to show reasonable agreement with experimental results.
A product Q=P1 ... P(sub r) of m x m Householder matrices can be written in the form Q = I + WY(sup T), where W and Y are each m x r. This is called the WY representation of Q. It is of interest when implementing Householder techniques in high-performance computing environments that are especially good at matrix-matrix multiplication. In this note a storage-efficient way to implement the WY representation is described. In particular, it is shown how the matrix Q can be expressed in the form Q = I + YTY(sup T). Usually r much less than m and so this 'compact' WY representation requires less storage. When compared with the recent block-reflector strategy the new technique still has a storage advantage and involves a comparable amount of work.
Program termination analysis is an important task in logic and computer science. While determining if a program terminates is known to be undecidable in general, there has been a significant amount of attention given to finding sufficient and computationally practical conditions to prove termination. One such method takes a program and builds from it a matrix weighted digraph. These are directed graphs whose edges are labeled by square matrices with entries in {-1,0,1}, equipped with a nonstandard matrix multiplication. Certain properties of this digraph are known to imply the termination of the related program. In particular, termination of the program can be determined from the weights of the circuits in the digraph. In this talk, the motivation for addressing termination and how matrix weighted digraphs arise will be briefly discussed. The remainder of the talk will describe an efficient method for bounding the weights of a finite set of the circuits in a matrix weighted digraph, which allows termination of the related program to be deduced.
Strassen's algorithm for fast matrix-matrix multiplication has been implemented for matrices of arbitrary shapes on the CRAY-2 and CRAY Y-MP supercomputers. Several techniques have been used to reduce the scratch space requirement for this algorithm while simultaneously preserving a high level of performance. When the resulting Strassen-based matrix multiply routine is combined with some routines from the new LAPACK library, LU decomposition can be performed with rates significantly higher than those achieved by conventional means. We succeeded in factoring a 2048 x 2048 matrix on the CRAY Y-MP at a rate equivalent to 325 MFLOPS.
Walsh functions form an orthonormal basis set consisting of square waves. The discontinuous nature of square waves make the system well suited for representing functions with discontinuities. The product of any two Walsh functions is another Walsh function - a feature that can radically change an algorithm for solving non-linear partial differential equations (PDEs). The solution algorithm of non-linear differential equations using Walsh function series is unique in that integrals and derivatives may be computed using simple matrix multiplication of series representations of functions. Solutions to PDEs are derived as functions of wave component amplitude. Three sample problems are presented to illustrate the Walsh function series approach to solving unsteady PDEs. These include an advection equation, a Burgers equation, and a Riemann problem. The sample problems demonstrate the use of the Walsh function solution algorithms, exploiting Fast Walsh Transforms in multi-dimensions (O(Nlog(N))). Details of a Fast Walsh Reciprocal, defined here for the first time, enable inversion of aWalsh Symmetric Matrix in O(Nlog(N)) operations. Walsh functions have been derived using a fractal recursion algorithm and these fractal patterns are observed in the progression of pairs of wave number amplitudes in the solutions. These patterns are most easily observed in a remapping defined as a fractal fingerprint (FFP). A prolongation of existing solutions to the next highest order exploits these patterns. The algorithms presented here are considered a work in progress that provide new alternatives and new insights into the solution of non-linear PDEs.
This paper discusses the design and implementation of an analog optical vecto-rmatrix coprocessor with a throughput of 128 Mops for a personal computer. Vector matrix calculations are inherently parallel, providing a promising domain for the use of optical calculators. However, to date, digital optical systems have proven too cumbersome to replace electronics, and analog processors have not demonstrated sufficient accuracy in large scale systems. The goal of the work described in this paper is to demonstrate a viable optical coprocessor for linear operations. The analog optical processor presented has been integrated with a personal computer to provide full functionality and is the first demonstration of an optical linear algebra processor with a throughput greater than 100 Mops. The optical vector matrix processor consists of a laser diode source, an acoustooptical modulator array to input the vector information, a liquid crystal spatial light modulator to input the matrix information, an avalanche photodiode array to read out the result vector of the vector matrix multiplication, as well as transport optics and the electronics necessary to drive the optical modulators and interface to the computer. The intent of this research is to provide a low cost, highly energy efficient coprocessor for linear operations. Measurements of the analog accuracy of the processor performing 128 Mops are presented along with an assessment of the implications for future systems. A range of noise sources, including cross-talk, source amplitude fluctuations, shot noise at the detector, and non-linearities of the optoelectronic components are measured and compared to determine the most significant source of error. The possibilities for reducing these sources of error are discussed. Also, the total error is compared with that expected from a statistical analysis of the individual components and their relation to the vector-matrix operation. The sufficiency of the measured accuracy of the processor is compared with that required for a range of typical problems. Calculations resolving alloy concentrations from spectral plume data of rocket engines are implemented on the optical processor, demonstrating its sufficiency for this problem. We also show how this technology can be easily extended to a 100 x 100 10 MHz (200 Cops) processor.
The potential uses of Charge Transfer Devices (CTDs) in pattern classification operations are explored. The needs for a hardware-based pattern classifier are established, and a matrix multiplication subsystem based upon a sum of products CTD is presented. An evaluation process for sum of products devices (particularly analog-analog correlators) is developed, and the feasibility of employing a particular device in a pattern classifier is determined. Finally, the possible impact of future trends in technology is considered.