Studies On Two Classes Of Tensor Equations And Associated Problems

Tensor equations play important roles in computational science and engineering applications.For instance,isotropic and anisotropic elastic-ity in continuum mechanics and engineering can be described by tensor linear systems via Einstein product.The interaction of multiple genes in biological science can be transformed into finding the sparse solution of some tensor linear systems.Besides,some high-dimensional PDEs can be discretized as multilinear systems formulated by tensor-vector product.Therefore,how to solve tensor equations effectively is quite significant from the viewpoints of theory and practical applications.This disser-tation concerns the solutions of those two classes of tensor equations,which contains the following contents:In the first chapter,we briefly introduce the background and sig-nificance of this dissertation,and present its research contents,charac-teristics and innovations.In the second chapter,we recall the basic concepts,operations and properties of tensors,and introduce in detail the tensor-train decompo-sition,the classical ADMM method as well as the Levenberg-Marquardt method and their variants.In the third chapter,the invertibility criteria of tensors based on Einstein product,and its applications in solving tensor linear systems and tensor eigenvalue problems,are studied.Specifically,the elementary transformations of tensors,the unfolding rank of tensors as well as the unfolding determinants of tensors are defined.And then some neces-sary and sufficient conditions for judging the invertibility of a tensor are proposed.Meanwhile,the elementary transformation method for com-puting the inverse tensor is derived,and the corresponding expression described by the new tensor determinant is obtained as well.Furthermore,the necessary and sufficient conditions for determin-ing the solvability of tensor linear systems are obtained by using the un-folding rank of tensors,and then the elimination method for solving this kind of tensor equations is established by using the elementary transfor-mations of tensors.Moreover,the Cramer's rule for solving tensor linear systems is proposed turning to the new tensor determinant.Particularly,together with the above elimination method,the tensor eigenvalue prob-lem proposed by Liqun Qi working at Hong Kong Polytechnic University is addressed theoretically.These results can be regarded as generaliza-tions of the corresponding results in numerical linear algebra.In the fourth chapter,we further study the Moore-Penrose gen-eralized inverses of tensors,which is a generalization of the inverse of a tensor.By using the unfolding rank,the full rank decomposition of a general tensor is given,which leads to a new representation for the Moore-Penrose generalized inverse.These results affirmatively answer an open question raised in[56].In a.ddition,by using the relevant con-clusions of Moore-Penrose generalized inverses of tensors,we study the tensor nearness problem constrained by a tensor linear system,which is a generalization of the matrix nearness problem,the tensor comple-tion problem and so on,obtain the necessary and sufficient condition for the unique existence of the solution to this problem,and represent it by using the Moore-Penrose generalized inverses of the known tensors.On the other hand,we develop a gradient-based iterative algorithm for solving the tensor nearness problem constrained by the Sylvester tensor equation which contains the tensor linear system as a special case.Chapters 5 and 6 mainly address multilinear systems with general coefficient tensors.Firstly,by using the definition of the tensor-vector product,we convert this kind of tensor equations into the optimiza-tion problem with consensus constrained conditions,and then propose a multi-block ADMM method(G-ADMM)for it.We also analyze its convergency under certain hypotheses.Secondly,following the classi-cal Levenberg-Marqudt(LM)method and its variants,we establish the two-step accelerated LM method(TALM)for the above multilinear sys--tems and prove that this algorithm is convergent cubically.However,the G-ADMM and TALM methods still suffer the so-called "curse-of-dimensionality" when the coefficient tensor is a dense one.To overcome this problem,we apply the tensor-train decomposition of the coefficient tensor to G-ADMM and TALM,and obtain their useful modifications,which avoid the curse.The performed numerical results illustrate the feasibility,efficiency and superiority of our methods.In the seventh chapter,we summarize the whole contents,and point out the future research focuses and directions.