The parallelization of helmholtz equation related to breast cancer growth

Abstract

Detecting breast cancer at an early stage will decrease the mortality rate and improve the cancer treatment successfully. This research focuses on the parallelization of the mathematical modeling on breast cancer growth using one and two dimensional Helmholtz equations. Finite difference method (FDM) is chosen to discretize the Helmholtz equation in order to generate a large sparse grid solution. Some numerical iterative methods are used to simulate the grid solution. The numerical methods under consideration are alternating group explicit (AGE), Red Black Gauss Seidel (RBGS), Gauss Seidel (GS) and Jacobi (JB) method. The alternative numerical method can be detected and quantified by comparing and analyzing the numerical methods under consideration in the aspect of run time, number of iterations, maximum error, root mean square error and computational complexity. Domain decomposition technique of the parallel AGE, RBGS and JB can be applied to decompose the full domain solution into subdomains. The message passing among the neighbourhood of subdomain can be done efficiently using MATLAB Distributed Computing Software. This technique is a straight forward implementation on a distributed parallel computer system (DPCS) because of the non-overlapping subdomain feature. The computer system architecture of DPCS is a single instruction multiple data stream (SIMD) and well suited to support the high computational complexity of a large sparse matrix. The development of DPCS is based on the Linux platform with eight processors of Intel® Core™ Duo Processor architecture and MATLAB Distributed Computing Software version R2011a. The visualization of one and two dimensional of breast cancer growth are captured using Comsol Multiphysic version 4.3a. The parallel performance evaluations of parallel AGE, RBGS and JB are measured in terms of run time, speedup, efficiency, effectiveness and temporal performance. As a conclusion, the parallel algorithm of AGE is superior than RBGS, GS and JB for solving one and two dimensional Helmholtz equations for breast cancer growth early detection.