Acceleration of LU Decomposition on FPGAs

Team Number: xohw22-006

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Background

Simulation Program with Integrated Circuit Emphasis or SPICE has now been widely used in the IC design and verification. Solving of sparse matrices often takes up most of the SPICE simulation time. Lower–upper (LU) decomposition is the most commonly used method to solve matrices. It factorizes a matrix into two factors – a lower triangular matrix L and an upper triangular matrix U. In this way, we only need to solve triangular systems to get results. However, the sparse-matrix computation is hard to parallelize on regular processors due to the irregular structure of the matrices. Modern FPGAs, however, have the potential to compute these hard-to-parallelise problems more efficiently due to its flexible reconfigurability.

Team and Project Information

• University name: the University of Edinburgh
• Supervisor: Dr Danial Chitnis (email: [email protected])
• Student: Yichen Zhang (email: [email protected])
• Board: Alveo U280 Data Center Accelerator Card
• Software version: v2021.2
• Video link: https://youtu.be/b9Mz6KtWnw8
• Report: report_xohw22-006.pdf

Experimental Results

Matrices used for test

Matrix	Order	NNZ	Sparsity	Pattern Symmetry	Numeric Symmetry
rajat11	135	665	3.65%	89.10%	63%
rajat14	180	1475	4.55%	100%	2.50%
rajat05	301	1250	1.38%	77%	70.60%
oscil_dcop_01	430	1544	0.84%	97.60%	69.80%
fpga_dcop_01	1220	5892	0.40%	81.80%	27.30%

CPU

FPGA

For smaller matrices, FPGA tends to take longer time to solve per right hand side vectors than CPU. However, when the matrices becomes larger, FPGA tends to be faster than CPU, with a speedup of about 1.2.

File Organisation

• Matrix_Sample/ Some matrices used for test.
• myKLU/ Software version for KLU decomposition
• Vitis/myKLU/host Host code for FPGA implementation
• Vitis/myKLU/host Kernel code for FPGA implementation

Build

Board used: Alveo U280 Data Center Accelerator Card

Vitis Version: v2021.2

Instructions to build and test project

For the CPU version:

• Step 1: Go to directory ./myKLU
• Step 2: Check if clang is installed. If not, modify Makefile to use appropriate compiler. Typically GCC is acceptable.
• Step 3: To make static library only, run make library
• Step 4: To run software version of LU decomposition, run make klu_kernel. The default matrix file should be put in the same directory as the executable program.
• Step 5: To benchmark it with matrices in directory ../Matrix_Sample, run make klu_bench

For the FPGA version:

• Step 1: Import the project to Vitis 2021.2.
• Step 2: Compile the CPU version above first, FPGA version requires the results from CPU version to verify that the results are correct.
• Step 3: Check the host C/C++ build settings. Ensure the include and lib settings are correct. The include path should include the directory of ./myKLU/include and ./myKLU/myKLU of the Software version. The lib should include the path of ./myKLU/lib of the Software version and require the library libklu.a, libamd.a, libbtf.a, and libsuitesparseconfig.a.
• Step 4: Build the project.