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This paper presents the parallel implementations of two iterative gradient based algorithms to solve the unconstrained linear quadratic regulator optimal control problem. It is shown that parallel evaluation of the step length and gradient of the quadratic cost function can be efficiently performed as a function of the number of processors. We then embed our parallel step length and gradient procedures to produce parallel implementations of the gradient and conjugate gradient methods that may be executed on an SIMD machine.

Principal component analysis (PCA) is a key statistical technique for multivariate data analysis. For large data sets the common approach to PCA computation is based on the standard NIPALS-PCA algorithm, which unfortunately suffers from loss of orthogonality, and therefore its applicability is usually limited to the estimation of the first few components. Here we present an algorithm based on Gram-Schmidt orthogonalization (called GS-PCA), which eliminates this shortcoming of NIPALS-PCA. Also, we discuss the GPU (Graphics Processing Unit) parallel implementation of both NIPALS-PCA and GS-PCA algorithms. The numerical results show that the GPU parallel optimized versions, based on CUBLAS (NVIDIA) are substantially faster (up to 12 times) than the CPU optimized versions based on CBLAS (GNU Scientific Library).

Parallel least mean square-partial parallel interference cancelation (PLMS-PPIC) is a partial interference cancelation which employs adaptive multistage structure. In this algorithm the channel phases for all users are assumed to be known. Having only their quarters in (0,2\pi), a modified version of PLMS-PPIC is proposed in this paper to simultaneously estimate the channel phases and the cancelation weights. Simulation examples are given in the cases of balanced, unbalanced and time varying channels to show the performance of the modified PLMS-PPIC method.

The successive over-relaxation (SOR) iterative method is an important solver for linear systems. In this paper, a parallel algorithm for the red-black SOR method with domain decomposition is investigated. The parallel SOR algorithm is designed by combining the traditional red-black SOR and row block domain decomposition technique, which reduces the communication cost and simplifies the parallel implementation. Two other iterative methods, Jacobi and Gauss-Seidel (G-S), ate also implemented in parallel for comparison. The three parallel iterative algorithms are implemented in C and MPI (Message Passing Interface) for solving the Dirichlet problem on a Linux cluster with eight dual processor 2.6ghz 32 bit Intel Xeons, totaling 16 processors. The performances of the three algorithms are evaluated in terms of speedup and efficiency.

This annual report briefly describes progress on research in algorithms for optimal control problems. The principal research focus has been on a new approach to the parallel implementation of iterative algorithms for optimal control based on a two level parametrization of optimally conditions, and a secondary research focus has been the investigation of fault detection in the type of computational networks used for optimal control computations. Publications describing the results in detail are listed.

Least mean square-partial parallel interference cancelation (LMS-PPIC) is a partial interference cancelation using adaptive multistage structure in which the normalized least mean square (NLMS) adaptive algorithm is engaged to obtain the cancelation weights. The performance of the NLMS algorithm is mostly dependent to its step-size. A fixed and non-optimized step-size causes the propagation of error from one stage to the next one. When all user channels are balanced, the unit magnitude is the principal property of the cancelation weight elements. Based on this fact and using a set of NLMS algorithms with different step-sizes, the parallel LMS-PPIC (PLMS-PPIC) method is proposed. In each iteration of the algorithm, the parameter estimate of the NLMS algorithm is chosen to match the elements' magnitudes of the cancelation weight estimate with unity. Simulation results are given to compare the performance of our method with the LMS-PPIC algorithm in three cases: balanced channel, unbalanced channel and time varying channel.

Algorithms for assembling in parallel the sparse system of linear equations that result from finite difference or finite element discretizations of elliptic partial differential equations, such as those that arise in structural engineering are developed. Parallel linear stationary iterative algorithms and parallel preconditioned conjugate gradient algorithms are developed for solving these systems. In addition, a model for comparing parallel algorithms on array architectures is developed and results of this model for the algorithms are given.

The principal focus of our research is a new systematic approach to design optimal control algorithms that may be implemented on parallel machines. This approach is based on a two-level parametrization of first-order optimality conditions. The first level of parametrization is concerned with the decrease of the overall amount of operations, and the second level is concerned with parallelism. By introducing parametrization matrices in the first level and then factoring those matrices to exhibit the amount of parallelism desired in the second level as a function of the number of processing elements to be used, the resulting optimality conditions may be tailored to the computing network on which the computations are to be performed. Algorithm, Parallelism, Optimal control, Computer systems, Fault tolerant. (jes)