Search

Accelerated regularization algorithms for ill-posed problems have received much attention from researchers of inverse problems since the 1980s. The current optimal theoretical results indicate that some regularization algorithms, e.g. the ν -method and the Nesterov method, are such that under conventional source conditions the optimal convergence rates can be obtained with approximately the square root of the iterations of those needed for the benchmark (i.e. the Landweber iteration). In this paper, we propose a new class of regularization algorithms with parameter n, called the Acceleration Regularization of order n (AR n ). Theoretically, we prove that, for an arbitrary number n > - 1 , AR n can attach the optimal convergence rates with approximately the n + 1 root of the iterations needed for the benchmark method. Moreover, unlike the existing accelerated regularization algorithms, AR n s have no saturation restriction. Some symplectic iterative regularizing algorithms are developed for the numerical realization of AR n . Finally, numerical experiments with integral equations and inverse problems in partial differential equations demonstrate that, at least for n ≤ 2 , the numerical behavior of AR n matches our theoretical findings, also breaking the practical acceleration capability of all existing regularization algorithms. [ABSTRACT FROM AUTHOR]

For a general third-order tensor A ∈ R n × n × n the paper studies two closely related problems, an SVD-like tensor decomposition and an (approximate) tensor diagonalization. We develop a Jacobi-type algorithm that works on 2 × 2 × 2 subtensors and, in each iteration, maximizes the sum of squares of its diagonal entries. We show how the rotation angles are calculated and prove convergence of the algorithm. Different initializations of the algorithm are discussed, as well as the special cases of symmetric and antisymmetric tensors. The algorithm can be generalized to work on higher-order tensors. [ABSTRACT FROM AUTHOR]

In this paper, we first define the weakly essentially irreducible nonnegative tensors and unify the definitions of essentially positive tensors, weakly positive tensors and generalized weakly positive tensors. Then an algorithm to find the spectral radius of weakly essentially irreducible nonnegative tensors is given based on the implicit translational transformation, and the linear convergence condition of the algorithm is analyzed using the directed graph of the matrix, and finally the computational efficiency of the related algorithms is compared using numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose two kinds of Riemannian conjugate gradient methods for computing the extreme eigenvalues of even order symmetric tensors. One is a sufficient descent Dai–Yuan type conjugate gradient method, and another is Zhu's Riemannian nonmonotone conjugate gradient method. The global convergence of two proposed algorithms can be guaranteed, respectively. Numerical results are reported to demonstrate the feasibility and efficiency of the proposed methods. [ABSTRACT FROM AUTHOR]

The tensor t-function, a formalism that generalizes the well-known concept of matrix functions to third-order tensors, is introduced in Lund (Numer Linear Algebra Appl 27(3):e2288). In this work, we investigate properties of the Fréchet derivative of the tensor t-function and derive algorithms for its efficient numerical computation. Applications in condition number estimation and nuclear norm minimization are explored. Numerical experiments implemented by the t-Frechet toolbox hosted at https://gitlab.com/katlund/t-frechet illustrate properties of the t-function Fréchet derivative, as well as the efficiency and accuracy of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this work, we propose a preconditioned tensor splitting iteration method for solving the multilinear system with Einstein product and give the corresponding theoretical analysis. Then we give a global correction method and theoretically prove that the proposed global correction method accelerates the convergence of the existing algorithm. These studies give some new accelerated iterative techniques which are not appeared in the previously published works. Some numerical results are disclosed to experimentally illustrate the effectiveness of the preconditioned method and the global correction technique. [ABSTRACT FROM AUTHOR]

A fast algorithm is proposed for numerically computing an interval matrix containing the minimal nonnegative solution to the nonsymmetric T-Riccati equation. The cost of this algorithm is cubic plus that for numerically solving the equation. The algorithm proves that the solution contained in the interval matrix is unique and equal to the minimal nonnegative solution. Numerical results show the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

We propose and analyse an augmented mixed finite element method for the Oseen equations written in terms of velocity, vorticity, and pressure with non-constant viscosity and homogeneous Dirichlet boundary condition for the velocity. The weak formulation includes least-squares terms arising from the constitutive equation and from the incompressibility condition, and we show that it satisfies the hypotheses of the Babuška-Brezzi theory. Repeating the arguments of the continuous analysis, the stability and solvability of the discrete problem are established. The method is suited for any Stokes inf-sup stable finite element pair for velocity and pressure, while for vorticity any generic discrete space (of arbitrary order) can be used. A priori and a posteriori error estimates are derived using two specific families of discrete subspaces. Finally, we provide a set of numerical tests illustrating the behaviour of the scheme, verifying the theoretical convergence rates, and showing the performance of the adaptive algorithm guided by residual a posteriori error estimation. [ABSTRACT FROM AUTHOR]