Iterative Algorithms For Solving Double Saddle Point Problems

As a class of linear systems with special structures,saddle point problems have always been a research hotspot in the field of scientific computing.In fact,there are many research on traditional saddle point problems.Based on them,we mainly study the efficient method to solve double saddle point systems with special structure in this paper.First of all,by dividing the coefficient matrix of double saddle point system(i.e.double saddle point matrix)reasonably,we establish a block upper triangular preconditioner and a block diagonal preconditioner respectively.We analyze the eigenvalue properties of the preconditioned matrices in detail and give inexact forms for practical application.In numerical experiment part,we use PGMRES to solve the double saddle point problem.At the same time,we compare the numerical results with the performance of the preconditioners in the existing researches,which show the effectiveness of these new preconditioners.Secondly,we split the original double saddle point matrix with nonzero parameters and symmetric positive definite submatrices to obtain a lower triangular preconditioning matrix.And then we analyze the eigenvalue distribution of the preconditioned matrix.For two symmetric positive definite submatrices in the preconditioner,we give several different choices and record their performance in the numerical examples.The numerical results illustrate the competitiveness of this preconditioner comparing with the previous ones.Finally,we design a new algorithm of symmetric triangular decomposition,which decomposes the original double saddle point matrix into the product of a block triangular matrix and a symmetric positive definite matrix.The core of this algorithm is to continuously reduce the dimension of the matrix to one based on block recursion.Compared with the existing symmetric triangular decomposition,our new algorithm has an advantage in the decomposition time.With the help of this new decomposition algorithm,we successfully convert the nonsymmetric indefinite double saddle point system into a symmetric positive definite system,in that case,we can directly use the conjugate gradient method to solve the symmetric positive definite system.The numerical results obtained in this way are comparable to those obtained directly by PGMRES.