Matrix preconditioning: a robust operation for optical linear algebra processors

Accuracy requirements of optical linear algebra processors in adaptive optics imaging systems

A ground-based adaptive optics imaging telescope system attempts to improve image quality by measuring and correcting for atmospherically induced wavefront aberrations. The necessary control computations during each cycle will take a finite amount of time, which adds to the residual error variance since the atmosphere continues to change during that time. Thus an optical processor may be well-suited for this task. This paper investigates this possibility by studying the accuracy requirements in a general optical processor that will make it competitive with, or superior to, a conventional digital computer for adaptive optics use.

Adaptive optical processors

There are two different approaches for improving the accuracy of analog optical associative processors: postprocessing with a bimodal system and preprocessing with a preconditioner. These two approaches can be combined to develop an adaptive optical multiprocessor that can adjust the computational steps depending on the data and produce solutions of linear algebra problems with a specified accuracy in a given amount of time.

Realization of preconditioned Lanczos and conjugate gradient algorithms on optical linear algebra processors

Lanczos and conjugate gradient algorithms are important in computational linear algebra. In this paper, a parallel pipelined realization of these algorithms on a ring of optical linear algebra processors is described. The flow of data is designed to minimize the idle times of the optical multiprocessor and the redundancy of computations. The effects of optical round-off errors on the solutions obtained by the optical Lanczos and conjugate gradient algorithms are analyzed, and it is shown that optical preconditioning can improve the accuracy of these algorithms substantially. Algorithms for optical preconditioning and results of numerical experiments on solving linear systems of equations arising from partial differential equations are discussed. Since the Lanczos algorithm is used mostly with sparse matrices, a folded storage scheme to represent sparse matrices on spatial light modulators is also described.