This paper describes solution methods for linear discrete ill-posed problems defined by third order tensors and the t-product formalism introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp. 641--658]. A t-product Arnoldi (t-Arnoldi) process is defined and applied to reduce a large-scale Tikhonov regularization problem for third order tensors to a problem of small size. The data may be represented by a laterally oriented matrix or a third order tensor, and the regularization operator is a third order tensor. The discrepancy principle is used to determine the regularization parameter and the number of steps of the t-Arnoldi process. Numerical examples compare results for several solution methods, and illustrate the potential superiority of solution methods that tensorize over solution methods that matricize linear discrete ill-posed problems for third order tensors.
翻译:本文介绍了由三阶高压下界定的线性离散问题的解决办法,以及[M.E.Kilmer和C.D.Martin,第三阶高压的量化战略,Linear Algebra Appl.,435(2011),pp.641-658]中引入的t-product Arnoldi(t-Arnoldi)进程的定义和应用,目的是将三阶高的大型Tikhonov正规化问题降为小问题。数据可以由横向方向矩阵或第三阶高压代表,正规化操作员是第三阶高压。差异原则用于确定三阶高压进程正规化参数和步骤数量。数字示例比较了几种解决方案方法的结果,并说明了溶解方法的优越性,这些方法在将第三阶高压下线离散问题的溶解方法上压化。