Incomplete QR Research Articles

A note on parallel approximate pseudoinverse matrix techniques for solving linear least squares problems

A new parallel generic approximate sparse pseudoinverse matrix technique using a decoupled column-wise approach, based on modified row-threshold incomplete QR factorization techniques, is proposed. The explicit preconditioned conjugate gradient least squares method in conjunction with the new parallel generic approximate sparse pseudoinverse matrix technique is used for solving linear least square problems. Numerical results indicating the applicability and effectiveness of the proposed parallel generic approximate sparse pseudoinverse matrix techniques for solving various model problems are given.

On the Application of the Incomplete QR Algorithm to the Analysis of Microstrip Antennas

In this paper, we provide some insight into the us- age of fast, iterative, method-of-moments (MoM) solution of integral equations (IE) describing antennas and other metal- lic structures immersed in a planar multilayered environ- ment. Based on the form of multilayered media Green's functions, we extract free-space terms, associated with direct rays within the analyzed structure, reducing the number of significant interactions required to describe the rest of MoM matrix. Next, we show that it is possible to construct a hybrid algorithm, where the fast multipole method (FMM) is used to thefree-spacematrixpart,whilethereducedrankincomplete QR (iQR) decomposition is applied to the remaining portion of the MoM matrix. This HM-iQR (hybrid multipole - incomplete QR) method is applied to a relatively large (in terms o f the number of un- knowns) problem of plane wave scattering by a finite array of rectangular microstrip patches printed on a grounded di- electric slab. Computation results from the new algorithm are compared to literature data and to the results of the pure low rank IE-QR method.

Preconditioned GMRES methods for least squares problems

For least squares problems of minimizing ∥b −Ax∥2 whereA is a large sparsem ×n (m ≥n) matrix, the common method is to apply the conjugate gradient method to the normal equationA T Ax =A T b. However, the condition number ofA T A is square of that ofA, and convergence becomes problematic for severely ill-conditioned problems even with preconditioning. In this paper, we propose two methods for applying the GMRES method to the least squares problem by using ann ×m matrixB. We give the necessary and sufficient condition thatB should satisfy in order that the proposed methods give a least squares solution. Then, for implementations forB, we propose an incomplete QR decomposition IMGS(l). Numerical experiments showed that the simplest casel = 0 gives the best results, and converges faster than previous methods for severely ill-conditioned problems. The preconditioner IMGS(0) is equivalent to the caseB = (diag(A T A))−1 A T, so (diag(A T A))−1 A T was the best preconditioner among IMGS(l) and Jennings’ IMGS(τ). On the other hand, CG-IMGS(0) was the fastest for well-conditioned problems.

Preconditioned Krylov subspace method for the solution of least‐squares problems

AbstractWe consider preconditioned Krylov subspace iteration methods, e.g., CG, LSQR and GMRES, for the solution of large sparse least‐squares problems min ∥Ax – b ∥2, with A ∈ Rm×n, based on the Krylov subspaces Kk (BA, Br) and Kk (AB, r), respectively, where B ∈ Rn×m is the preconditioning matrix. More concretely, we propose and implement a class of incomplete QR factorization preconditioners based on the Givens rotations and analyze in detail the efficiency and robustness of the correspondingly preconditioned Krylov subspace iteration methods. A number of numerical experiments are used to further examine their numerical behaviour. It is shown that for both overdetermined and underdetermined least‐squares problems, the preconditioned GMRES methods are the best for large, sparse and ill‐conditioned matrices in terms of both CPU time and iteration step. Also, comparisons with the diagonal scaling and the RIF preconditioners are given to show the superiority of the newly‐proposed GMRES‐type methods. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

MIQR: A Multilevel Incomplete QR Preconditioner for Large Sparse Least‐Squares Problems

This paper describes a multilevel incomplete QR factorization for solving large sparse least‐squares problems. The algorithm builds the factorization by exploiting structural orthogonality in general sparse matrices. At any given step, the algorithm finds an independent set of columns, i.e., a set of columns that have orthogonal patterns. The other columns are then block orthogonalized against columns of the independent set, and the process is repeated recursively for a certain number of levels on these remaining columns. The final level matrix is processed with a standard QR or incomplete QR factorization. Dropping strategies are employed throughout the levels in order to maintain a good level of sparsity. A few improvements to this basic scheme are explored. Among these is the relaxation of the requirement of independent sets of columns. Numerical tests are proposed which compare this scheme with the standard incomplete QR preconditioner, the robust incomplete factorization preconditioner, and the algebraic recursive multilevel solver (on normal equations).

A Class of Incomplete Orthogonal Factorization Methods. II: Implementation and Results

We present, implement and test several incomplete QR factorization methods based on Givens rotations for sparse square and rectangular matrices. For square systems, the approximate QR factors are used as right-preconditioners for GMRES, and their performance is compared to standard ILU techniques. For rectangular matrices corresponding to linear least-squares problems, the approximate R factor is used as a right-preconditioner for CGLS. A comprehensive discussion is given about the uses, advantages and shortcomings of the preconditioners.

A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems

This paper describes a technique for constructing robust preconditioners for the CGLS method applied to the solution of large and sparse least squares problems. The algorithm computes an incomplete LDLT factorization of the normal equations matrix without the need to form the normal matrix itself. The preconditioner is reliable (pivot breakdowns cannot occur) and has low intermediate storage requirements. Numerical experiments illustrating the performance of the preconditioner are presented. A comparison with incomplete QR preconditioners is also included.