Analysis and Optimization of Block LU Decomposition for Execution on Tightly Coupled Processor Arrays

LU decomposition is a widely used application for solving systems of linear equations. It involves decomposing a given matrix into a lower and upper triangular matrix. But if the matrix size is large, using a block-based LU decomposition on smaller submatrices can be advantageous. This approach allows for an adaptation to a target architecture's memory and computing resources. In this paper, we analyze different strategies for mapping block LU decompositions onto Tightly Coupled Processor Arrays (TCPAs). Each decomposition introduces a dependence graph of matrix operations of smaller size: an unblocked LU decomposition, a triangular matrix solver, and a general matrix-matrix multiplication. First, we propose one piecewise regular algorithm for each corresponding loop nest, analyze its complexity, and then explore various reuse schemes for configuration, data, and synchronization. It will be shown that these schemes have a significant impact on the execution time of the entire algorithm and must be considered by the scheduling approach that governs the individual loop program invocations. Our performance analysis of the mapped block LU decomposition shows a maximal speedup of 12 on a TCPA of size
$4\times 4$
compared to a CPU, and a measured speedup of 10 measured on an FPGA-based SoC.