Abstract Regularization of certain linear discrete ill-posed problems, as well regression can be formulated large-scale, possibly nonconvex, minimization whose objective function is the sum p th power ℓ -norm a fidelity term and q regularization term, with 0 < , ≤ 2. We describe new restarted iterative solution methods that require less computer storage execution time than described by Huang...

The three-term Lanczos process for a symmetric matrix leads to bases for Krylov subspaces of increasing dimension. The Lanczos basis, together with the recurrence coefficients, can be used for the solution of symmetric indefinite linear systems, by solving a reduced system in one way or another. This leads to well-known methods: MINRES (minimal residual), GMRES (generalized minimal residual), a...

and Kk(BA,Br) = span{Br, (BA)Br, . . . , (BA)k−1Br}, (3) where B ∈ Rn×m is the mapping and preconditioning matrix, and apply Krylov subspace iteration methods on these subspaces. For overdetermined problems, applying the standard CG method to Kk(BA,Br) leads to the preconditioned CGLS [3] or CGNR [9] method while for underdetermined problems it leads to preconditioned CGNE [9] method. The GMRES...

This paper introduces two new algorithms, belonging to the class of Arnoldi-Tikhonov regularization methods, which are particularly appropriate for sparse reconstruction. The main idea is to consider suitable adaptively-defined regularization matrices that allow the usual 2-norm regularization term to approximate a more general regularization term expressed in the p-norm, p ≥ 1. The regularizat...

A new Newton-Raphson method based preconditioner for Krylov type linear equation solvers for GPGPU is developed, and the performance is investigated. Conventional preconditioners improve the convergence of Krylov type solvers, and perform well on CPUs. However, they do not perform well on GPGPUs, because of the complexity of implementing powerful preconditioners. The developed preconditioner is...

‎the global fom and gmres algorithms are among the effective‎ ‎methods to solve sylvester matrix equations‎. ‎in this paper‎, ‎we‎ ‎study these algorithms in the case that the coefficient matrices‎ ‎are real symmetric (real symmetric positive definite) and extract‎ ‎two cg-type algorithms for solving generalized sylvester matrix‎ ‎equations‎. ‎the proposed methods are iterative projection metho...

Quadratic eigenvalue problems involving large matrices arise frequently in areas such as the vibration analysis of structures, MEMS simulation, and the solution of quadratically constrained least squares problems. The typical approach is to solve the quadratic eigenvalue problem using a mathematically equivalent linearized formulation, resulting in a doubled dimension and a lack of backward sta...

Hydrodynamic interactions play an important role in the dynamics of macromolecules. The most common way to take into account hydrodynamic effects in molecular simulations is in the context of a brownian dynamics simulation. However, the calculation of correlated brownian noise vectors in these simulations is computationally very demanding and alternative methods are desirable. This paper studie...

Generalized rational Krylov decompositions are matrix relations which, under certain conditions, are associated with rational Krylov spaces. We study the algebraic properties of such decompositions and present an implicit Q theorem for rational Krylov spaces. Transformations on rational Krylov decompositions allow for changing the poles of a rational Krylov space without recomputation, and two ...

This note explores sparse matrix dense matrix (SMDM) multiplications, useful in block Krylov or block Lanczos methods. SMDM computations are AU , and V A, multiplication of a large sparse matrix m × n matrix A by a matrix V of k rows of length m or a matrix U of k columns of length k, k << m, k << n . In a block Lanczos or Krylov algorithm, matrix matrix multiplications with the ”tall” U and ”w...