Solutions Of Large-scale Linear Systems And Fractional Differential Equations With Applications To Electromagnetic Computations

There is a strong need for fast solutions of large linear systems in many application fields,such as electromagnetic scattering,peridynamics simulation,numerical methods of fractional differential equations,etc.The development of robust linear systems solvers is one of hottest topics in scientific computing.In most cases,the coefficient matrices from practical simulations have the special structures,so how to employ these properties to establish the efficient solution techniques should be highly preferable.Thus,the current dissertation focus on fast(preconditioned)iterative solvers of several linear systems which have special coefficient matrices,including complex symmetric linear systems,Toeplitz-like linear systems and shifted linear systems.The main contents are as follows:1.Two Krylov subspace methods,i.e.COCG and COCR,are built by the PetrovGalerkin principle for complex symmetric linear systems.However,these two methods often exhibit the irregular convergence behaviors(curves),even stagnation.We use the quasi-minimal residual strategy to remedy this difficulty,then we derive two new iterative solvers,namely QMRCOCG and QMRCOCR.The proposed methods can improve COCG and COCR,respectively,in terms of smoother convergence behavior rather than irregular convergence behaviors.Finally,numerical experiments involving electromagnetic simulations are reported to support our theoretical and numerical findings.2.We revisited the derivation of SCBiCG class of iterative solvers for complex symmetric linear systems.In fact,it is useful to note that the SCBiCG(?,n)contains the early established methods,such as COCG,BiCGCR and COCR.Especially,we discuss the framework about deriving the SCBiCG(?,n)methods.Moreover,we find and prove that BiCGCR is mathematical equivalent to COCR,but the latter often is preferable due to the less of inner product per iteration.Meanwhile,we construct a two-step preconditioning for handling electro-quasistatic frequency domain simulation.At last,numerical examples are reported to compare the performance among COCG,BiCGCR,COCR and SQMR,and also verify the efficiency of the proposed two-step preconditioning.3.In peridynamics simulations,(one-dimensional)pseudo-differential equations are needed to solve numerically.Due to the nonlocal operator,numerical methods for pseudo-differential equations result in the dense matrices.Fortunately,earlier work had found that the coefficient matrix can be written into the sum of a Toeplitz matrix and a tridiagonal matrix.Then we modified the conventional circulant preconditioners to accelerate the convergence of PCG via the fast Toeplitz matrix-vector product.The proposed numerical approach can reduce the complexity and storage from (N3)and (N2)to(N log N)and (N),where N is the grid nodes,respectively.Theocratical and numerical results are carried out to show the effectiveness of our proposed preconditioners.4.The shift-invariance property of Krylov subspace is used to generalize the BiCRtype methods for shifted linear systems.These methods employ the biconjugate Aorthonormalization process to generate one Krylov subspace for all shifted systems,so that the computational costs(i.e.,the number of matrix–vector products and the number of inner products)for such shifted systems are equal to those for a single system.We established the shifted BiCR method and the shifted BiCRStab method.Then numerical experiments involving numerical solutions of time-dependent PDEs are reported to illustrate the feasibility of our proposed solvers in comparison with the existing methods,i.e.,the shifted BiCG and the shifted BiCGStab,respectively.5.We construct a novel class of numerical methods,which combine the spatial discretization and boundary value methods(BVMs),for solving the space fractional diffusion equations.There is no need of complicate stability analysis(which is very sensitive for conventional time-stepping schemes)in our proposed methods,who are unconditionally stable.We discrete the fractional diffusion equations into a large system of linear equations.In order to solve such system efficiently,we establish the block circulant and BCCB preconditioners for accelerating the convergence of GMRES.Numerical examples are carried out to support our theoretical and numerical finds.