Application Research On Preconditioning Of GMRES Method For Large-Scale Sparse Systems In Circuit Simulation

The design and analysis of large-scale integrated circuits rely on Electronic Design Automation(EDA)technology,and electronic circuits need to be simulated using numerical simulation methods.Numerical simulation of circuit systems essentially involves solving a system of linear equations Ax=b,where the coefficient matrix A is a large-scale sparse matrix that is non-symmetric,b is a known vector,and x is the vector to be solved.In today’s scientific and engineering computing tasks,the number of unknowns in linear equation systems is increasing,leading to greater computational demands.Therefore,fast solving of linear equation systems(especially those with large-scale sparse coefficient matrices)is crucial.Generally,the number of arithmetic operations required to solve an equation system using direct methods is O(n3),where n is the dimension of the coefficient matrix A.However,this method inevitably requires a huge amount of storage and computing resources,and may also lead to the destruction of the structure of the coefficient matrix.In addition,if the value of n is too large,the computational efficiency of the direct method will significantly decrease.Unlike the direct method,the iterative method gradually approximates the solution of a linear equation system,and the computational complexity required to solve a sparse linear equation system is relatively small.In addition,iterative method has a simple program design and can effectively utilize the sparsity of matrices to save memory.Therefore,when solving large-scale problems,the iterative method is usually used.Compared with the direct method,a suitable iterative solution requires less computation and has more controllable errors.They have great potential to improve the computational efficiency of solving large-scale sparse matrix equation systems,and are of significant engineering value in accelerating the simulation of complex circuits.The GMRES(Generalized Minimal Residuals)method is one of the most effective iterative methods for solving large sparse nonsymmetric systems of equations.This method uses Krylov subspace to approximate the solution of the equation system,that is,by finding a vector that can minimize the residual to approximate the actual solution.However,if the GMRES method is used for calculation,as the number of iterations increases,the computational and storage requirements also increase,and sometimes the GMRES method may not converge.In order to solve this problem,researchers have made multiple improvements and optimizations in algorithm,and deeply explored the preprocessing technology of GMRES method.Preprocessing technology can alter the structural features of the coefficient matrix of the equation system,thereby further improving computational efficiency.Therefore,the key research direction of this article is to find appropriate preprocessing methods.Based on the Krylov subspace method,this paper attempts to develop efficient acceleration strategies to improve the speed of iterative solution in the preconditioned GMRES iterative method for circuit simulation.Due to the drawbacks of the existing preconditioned GMRES iterative method used in circuit simulation to solve large-scale sparse systems,i.e,it cannot well meet the requirement in engineering computing,this paper proposes a precondition reuse LU generalized minimum residual method(RPLUGMRES),which combines the characteristics of circuit matrices.Based on the preconditioned GMRES iterative method,LU decomposition is reused as the new precondition.First,the preconditioned GMRES iterative method is used to solve linear system problems.Then,to solve the undesirable problems in the preconditioned GMRES iterative method solution process,the precondition reuse is used to change the coefficient matrix of spectral analysis,which effectively improves the convergence of the algorithm,while reducing the number of matrix vector multipliers and CPU time.The numerical stability of ILU(Incomplete LU)decomposition for precondition reuse is analyzed.Finally,the effectiveness of RPLUGMRES is verified in numerical experiment.On the one hand,due to the simple and arbitrary problem of setting the initial value of the GMRES iteration method,RPLUGMRES sets the initial value as the value solved by the pre matrix.In particular,at moment t0,LU decomposition is performed on A0,and substitutes the value of L、U into the set of linear equation A1x=b1 at the next time to obtain LUx=b1 and solve.The solution vector obtained is used as the initial value of the preconditioned GMRES iteration at the next time,and a comparative evaluation is made with the strategy of setting the initial value to zero,The results show that the initial value setting strategy of the RPLUGMRES method is effective.The RPLUGMRES algorithm proposed in this paper mainly focuses on two aspects:precondition reuse and initial value setting,and has significant improvements over traditional iterative methods and preconditioned GMRES iterative methods.The experimental results show that the RPLUGMRES method in this paper has strong potential in engineering applications,and can greatly improve computational efficiency and reduce computational cost in solving realistic circuit problems.