Anjan Ghosh

Optical linear algebra processors: noise and error-source modeling.

The modeling of system and component noise and error sources in optical linear algebra processors (OLAPs) are considered, with attention to the frequency-multiplexed OLAP. General expressions are obtained for the output produced as a function of various component errors and noise. A digital simulator for this model is discussed.

Alignability of optical interconnects

Successful development, fabrication, and deployment of optical interconnects and computing systems depend on how easily the optical components and light beams can be aligned. An interconnection system difficult or time-consuming to align is costly to develop and may be unreliable. In this paper a probability theoretical framework is developed for analyzing the alignability, that is, the degree of difficulty of aligning the devices and the light beams, of a given optical interconnection system. The alignability measure is related to the other performance measures of an interconnect such as the power transfer efficiency, SNR, and the spatial and temporal bandwidths. The cost of the development and deployment of an optical processor or interconnection system can be estimated from knowledge of the alignability.

Performance of direct and iterative algorithms on an optical systolic processor

The frequency-multiplexed optical linear algebra processor (OLAP) is treated in detail with attention to its performance in the solution of systems of linear algebraic equations (LAEs). General guidelines suitable for most OLAPs, including digital–optical processors, are advanced concerning system and component error source models, guidelines for appropriate use of direct and iterative algorithms, the dominant error sources, and the effect of multiple simultaneous error sources. Specific results are advanced on the quantitative performance of both direct and iterative algorithms in the solution of systems of LAEs and in the solution of nonlinear matrix equations. Acoustic attenuation is found to dominate iterative algorithms and detector noise to dominate direct algorithms. The effect of multiple spatial errors is found to be additive. A theoretical expression for the amount of acoustic attenuation allowed is advanced and verified. Simulations and experimental data are included.

Direct and implicit optical matrix-vector algorithms.

New direct and implicit algorithms for optical matrix-vector and systolic array processors are considered. Direct rather than indirect algorithms to solve linear systems and implicit rather than explicit solutions to solve second-order partial differential equations are discussed. In many cases, such approaches more properly utilize the advantageous features of optical systolic array processors. The matrix-decomposition operation (rather than solution of the simplified matrix-vector equation that results) is recognized as the computationally burdensome aspect of such problems that should be computed on an optical system. The Householder QR matrix-decomposition algorithm is considered as a specific example of a direct solution. Extensions to eigenvalue computation and formation of matrices of special structure are also noted.

Lu and cholesky decomposition on an optical systolic array processor

Abstract Direct rather than indirect solutions to matrix-vector equations on an optical systolic array processor are considered. A frequency-multiplexed optical systolic array processor for matrix-decomposition is described. The data flow and ordering of operations for LU decomposition or gaussian elimination and LL T or Cholesky decomposition on this system are detailed using an algorithm that utilizes the parallel processing ability of the optical systolic array processor. The time required for this optical algorithm is found to be much less than for the digital equivalent. The data flow in the optical system is seen to be most excellent.

Reduced sensitivity algorithm for optical processors using constraints and ridge regression

Optical linear algebra processors that involve solutions of linear algebraic equations have significant potential in adaptive and inference machines. We present an algorithm that includes constraints on the accuracy of the processor and improves the accuracy of the results obtained from such analog processors. The constraint algorithm matches the problem to the accuracy of the processor. Calculation of the adaptive weights in a phased array radar is used as a case study. Simulation results prove the benefits advertised. The desensitization of the calculated weights to computational errors in the processor is quantified. Ridge regression isused to determine the parameter needed in the algorithm.

Triangular system solutions on an optical systolic processor

It is noted that the simplified system of equations presented by Casasent and Ghosh (1983) can also be solved optically. One realization of this is described, with attention given to data flow and speed for the case of a lower triangular system of equations. The solution of an upper triangular system of equations can be handled in analogous fashion. The problem requires a matrix-vector processor. Only one carrier frequency input to the acoustooptic cell and only one output detector are used. To multiply a matrix by a vector in this system, the time-multiplexed vector elements are fed into the acoustooptic cell.

Optical Linear Algebra

Many of the linear algebra operations and algorithms possible on optical matrix-vector processors are reviewed. Emphasis is given to the use of direct solutions and their realization on systolic optical processors. As an example, implicit and explicit solutions to partial differential equations are considered. The matrix-decomposition required is found to be the major operation recommended for optical realization (since digital systems can easily solve the simplified matrix-vector problem that results). The pipelining and flow of data and operations are noted to be key issues in the realization of any algorithm on an optical systolic array processor. A realization of the direct solution by Householder OR decomposition is provided as a specific case study.

Direct And Indirect Optical Solutions To Linear Algebraic Equations: Error Source Modeling

Direct and indirect solutions to linear algebraic equations (LAEs) are considered with attention to the use of optical acousto-optic (AO) systolic array processors. Specific attention is given to error sources in one AO systolic processor. A case study of an LAE solution is conducted. The first error source model for an optical systolic array processor is advanced. Using this and digital computer modeling, a direct solution is found to be less sensitive to various optical system error sources than is an indirect solution. Acoustic attenuation is found to be the dominant error source in the AO systolic array processor considered. Related error source remarks on different bipolar data representation schemes and on optical versus digital solutions to a triangular system of equations are also advanced.