Research On Theories And Algorithms For Solving Two Classes Of Tensor Equations

Tensor equation problems are generalizations of matrix equation problems in high dimensional space.It has a central role to play in tensor complementarity,finite difference,signal processing,structural design,stability and control theory,which has attracted much attention.According to the matrix algebra theory,many researchers have discussed the solvability conditions of tensor equations over the various products,and their expressions of solutions were also established.From numerical optimization point of view,this dissertation,which is based on Krylov subspace methods and splitting techniques,is committed to proposing some efficient methods for solving two classes of tensor equations,the convergence analysis of which is also established.The main contents of this dissertation are as follows:In Chapter 2,from the hierarchical identification principle and relaxation parameters,we propose a relaxed gradient based iterative method based on tensor format(RGI-BTF)for solving a third order Sylvester tensor equation with respect to n-mode product.According to the information given by the previous steps,we then develop the tensor form of modified relaxed gradient based iterative(MRGI-BTF)algorithm that can greatly improve its convergence rate,and the convergence analysis of which is also established.Compared with some existing methods,the obtained numerical results show the effectiveness of the introduced algorithms.In Chapter 3,we establish the tensor forms of the bi-conjugate gradient(BiCG_BTF)and bi-conjugate residual(BiCR_BTF)methods to solve the high order Sylvester tensor equations under the n-mode product,respectively.To improve their performance,combining with the nearest Kronecker product,two preconditioned bi-conjugate gradient(PBiCG_BTF)and bi-conjugate residual(PBiCR_BTF)methods are derived.Convergence analysis shows that the proposed algorithms are convergent to an exact solution within finite steps in the absence round-off errors for any initial tensor.Finally,we give some numerical examples to confirm the effectiveness of the algorithms proposed in here.In Chapter 4,the tensor forms of modified bi-conjugate gradient(MBiCG_BTF)and bi-conjugate residual(MBiCR_BTF)methods,which are based on bi-conjugate gradient and bi-conjugate residual methods and its own specific structure,are proposed for solving generalized coupled Sylvester equations over the n-mode product,and the convergence analysis of which is also given.From the nearest Kronecker product preconditioner generated by the block matrix,we present their preconditioned versions to accelerate the convergence of rate.Numerical examples are provided to show that the proposed algorithms are effective where the preconditioned modified bi-conjugate gradient method is superior to the other methods.In Chapter 5,we first extend the classical accelerated overrelaxation(AOR)splitting method to solve a nonlinear symmetric tensor equation.To improve its convergence rate,we then develop a Newton-AOR(NAOR)method that hybridizes the Newton method and the accelerated overrelaxation scheme,and the convergence analysis of which is also presented.The obtained numerical results show that our methods outperform some existing methods in terms of both the number of iteration steps and CPU time.