Large Scale Circuit Analysis by Preconditioned Relaxation Methods

This paper discusses circuit simulation with preconditioned relaxation methods. Circuit simulation is one of the most important application problem of the numerical problems with discrete structures, and has been thought to be diicult to parallelize because of the high serialism of the direct linear solvers: no eecient iterative method applicable to circuit simulation is known. This paper reports that preconditioning on relaxation methods shows as good performance as the LU decomposition method in circuit simulation. The preconditioned Jacobi method possesses a high and natural parallelism, and a good parallelizability is connrmed by an implementation on AP1000. This paper also discusses two sparse explicit preconditioners, ensparsed inverse and decomposed inverse, and their computational algorithms. x1 Introduction Simulations of electronic circuits are among the most important large scale application problems. Also, they are good examples of the three modes of problems (though all of them are static structured): device simulation is a partial diierential equation (continuous for time and space), circuit simulation is an ordinary diierential equation (continuous for time and discrete for space), and logic simulation is a discrete event simulation (discrete for time and space). These simulations require diierent techniques to solve and parallelize. This paper discusses solution and parallelization of, especially the systems of linear equations of, the circuit simulation. The ordinary diierential equations can be solved by the backward Euler method or one of the Gill's methods, and the non-linear equations for time steps can be solved by the Newton method or a modiied one. They also include some diicult and challenging problems, but in general, they are very easy to parallelize when the nodes of the circuit have been appropriately distributed among processing elements. On the other hand, linear equations which appear in the Newton iterations are diicult to solve eeciently in parallel, and are the main obstacles of parallelization of circuit simulation. Usually the linear equations of circuit simulations are solved by the direct (LU decomposition) method, because of the reliability. However parallelization of the LU decomposition method is by no means an easy task: the LU decomposition method is an essentially serial algorithm. Therefore the parallelism is very low, especially for the substitution phase. Most researches on parallel circuit simulation are extracting parallelism as high as possible from given circuits. Iterative linear solvers are in general easier to parallelize than the direct method. However linear equations in circuit simulation are against most iterative linear solvers: …