Some equivalences in linear estimation (in Russian)

Under normality, the Bayesian estimation problem, the best linear unbiased estimation problem, and the restricted least-squares problem are all equivalent. As a result we need not compute pseudo-inverses and other complicated functions, which will be impossible for large sparse systems. Instead, by reorganizing the inputs, we can rewrite the system as a new but equivalent system which can be solved by ordinary least-squares methods.Linear Bayes estimation, best linear unbiased, least squares, sparse problems, large-scale optimization

Approximate inverse based multigrid solution of large sparse linear systems

In this thesis we study the approximate inverse based multigrid algorithm FAPIN for the solution of large sparse linear systems of equations. This algorithm, which is closely related to the well known multigrid V-cycle, has proven successful in the numerical solution of several second order boundary value problems. Here we are mainly concerned with its application to fourth order problems. In particular, we demonstrate good multigrid performance with discrete problems arising from the beam equation and the biharmonic (plate) equation. The work presented also represents new experience with FAPIN using cubic B-spline, bicubic B-spline and piecewise bicubic Hermite basis functions. We recast a convergence proof in matrix notation for the nonsingular case. Central to our development are the concepts of an approximate inverse and an approximate pseudo-inverse of a matrix. In particular, we use least squares approximate inverses (and related approximate pseudo-inverses) found by solving a Frobenius matrix norm minimization problem. These approximate inverses are used in the multigrid smoothers of our FAPIN algorithms

Tensor and Matrix Inversions with Applications

Higher order tensor inversion is possible for even order. We have shown that a tensor group endowed with the Einstein (contracted) product is isomorphic to the general linear group of degree $n$. With the isomorphic group structures, we derived new tensor decompositions which we have shown to be related to the well-known canonical polyadic decomposition and multilinear SVD. Moreover, within this group structure framework, multilinear systems are derived, specifically, for solving high dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense, that is, when the tensor has an odd number of modes or when the tensor has distinct dimensions in each modes. With the notion of tensor inversion, multilinear systems are solvable. Numerically we solve multilinear systems using iterative techniques, namely biconjugate gradient and Jacobi methods in tensor format