Inner-outer Iteration Method Research Articles

A Cross-Product Free Jacobi–Davidson Type Method for Computing a Partial Generalized Singular Value Decomposition of a Large Matrix Pair

The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix $A$ with the singular values in a given interval. The resulting FEAST SVDsolver is subspace iteration applied to an approximate spectral projector of $A^TA$ corresponding to the desired singular values in a given interval, and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which $A$ is projected to obtain Ritz approximations. Differently from a commonly used contour integral-based FEAST solver, we propose a robust alternative that constructs approximate spectral projectors by using the Chebyshev--Jackson polynomial series, which are symmetric positive semi-definite with the eigenvalues in $[0,1]$. We prove the pointwise convergence of this series and give compact estimates for pointwise errors of it and the step function that corresponds to the exact spectral projector. We present error bounds for the approximate spectral projector and reliable estimates for the number of desired singular triplets, establish numerous convergence results on the resulting FEAST SVDsolver, and propose practical selection strategies for determining the series degree and for reliably determining the subspace dimension. The solver and results on it are directly applicable or adaptable to the real symmetric and complex Hermitian eigenvalue problem. Numerical experiments illustrate that our FEAST SVDsolver is at least competitive with and is much more efficient than the contour integral-based FEAST SVDsolver when the desired singular values are extreme and interior ones, respectively, and it is also more robust than the latter.

Hierarchical Orthogonal Factorization: Sparse Least Squares Problems

In this work, we develop a fast hierarchical solver for solving large, sparse least squares problems. We build upon the algorithm, spaQR (sparsified QR Gnanasekaran and Darve in SIAM J Matrix Anal Appl 43(1):94–123, 2022), that was developed by the authors to solve large sparse linear systems. Our algorithm is built on top of a Nested Dissection based multifrontal QR approach. We use low-rank approximations on the frontal matrices to sparsify the vertex separators at every level in the elimination tree. Using a two-step sparsification scheme, we reduce the number of columns and maintain the ratio of rows to columns in each front without introducing any additional fill-in. With this improvised scheme, we show that the runtime of the algorithm scales as \(\mathcal {O}(M \log N)\) and uses \(\mathcal {O}(M)\) memory to store the factorization. This is achieved at the expense of a small and controllable approximation error. The end result is an approximate factorization of the matrix stored as a sequence of sparse orthogonal and upper-triangular factors and hence easy to apply/solve with a vector. Numerical experiments compare the performance of the spaQR algorithm with direct multifrontal QR, an inner-outer iterative method, and CGLS iterative method with a diagonal preconditioner and Robust Incomplete Factorization (RIF) preconditioner.

A general multi-step matrix splitting iteration method for computing PageRank

Based on the general inner-outer (GIO) iteration method [5,34] and the iteration framework [6], we present a general multi-step matrix splitting (GMMS) iteration method for computing PageRank, and analyze its overall convergence property. Moreover, the same idea can be used as a preconditioning technique for accelerating the Krylov subspace methods, such as GMRES method. Finally, several numerical examples are given to illustrate the effectiveness of the proposed algorithm.

A Note on the Inner-Outer Iterative Method for Solving the Linear Equation Ax = b

Recently, Tian et al. [Computers and Mathematics with Applications, 75(2018): 2710-2722] came up with the inner-outer iterative method to solve the linear equation Ax=b and studied the corresponding convergence of the method. In this paper, we improve the main results of the inner-outer method and get weaker convergence results. Moreover, the parameters can be adjusted suitably so that the convergence property of the method can be substantially improved.

Scenario-based Unit Commitment Optimization for Power System with Large-scale Wind Power Participating in Primary Frequency Regulation

Continuous increase of wind power penetration brings high randomness to power system, and also leads to serious shortage of primary frequency regulation (PFR) reserve for power system whose reserve capacity is typically provided by conventional units. Considering large-scale wind power participating in PFR, this paper proposes a unit commitment optimization model with respect to coordination of steady state and transient state. In addition to traditional operation costs, losses of wind farm de-loaded operation, environmental benefits and transient frequency safety costs in high-risk stochastic scenarios are also considered in the model. Besides, the model makes full use of interruptible loads on demand side as one of the PFR reserve sources. A selection method for high-risk scenarios is also proposed to improve the calculation efficiency. Finally, this paper proposes an inner-outer iterative optimization method for the model solution. The method is validated by the New England 10-machine system, and the results show that the optimization model can guarantee both the safety of transient frequency and the economy of system operation.

An inner-outer iteration method for solving convex optimization problems involving the sum of three convex functions

In this paper, we consider solving a class of convex optimization problem which minimizes the sum of three convex functions $f(x)+g(x)+h(Bx)$, where $f(x)$ is differentiable with a Lipschitz continuous gradient, $g(x)$ and $h(x)$ have a closed-form expression of their proximity operators and $B$ is a bounded linear operator. This type of optimization problem has wide application in signal recovery and image processing. To make full use of the differentiability function in the optimization problem, we take advantage of two operator splitting methods: the forward-backward splitting method and the three operator splitting method. In the iteration scheme derived from the two operator splitting methods, we need to compute the proximity operator of $g+h \circ B$ and $h \circ B$, respectively. Although these proximity operators do not have a closed-form solution in general, they can be solved very efficiently. We mainly employ two different approaches to solve these proximity operators: one is dual and the other is primal-dual. Following this way, we fortunately find that three existing iterative algorithms including Condat and Vu algorithm, primal-dual fixed point (PDFP) algorithm and primal-dual three operator (PD3O) algorithm are a special case of our proposed iterative algorithms. Moreover, we discover a new kind of iterative algorithm to solve the considered optimization problem, which is not covered by the existing ones. Under mild conditions, we prove the convergence of the proposed iterative algorithms. Numerical experiments applied on fused Lasso problem, constrained total variation regularization in computed tomography (CT) image reconstruction and low-rank total variation image super-resolution problem demonstrate the effectiveness and efficiency of the proposed iterative algorithms.

The general inner-outer iteration method based on regular splittings for the PageRank problem

In this paper, combined the regular splittings of the coefficient matrix I−αP with the inner-outer iteration framework [9], a general inner-outer (GIO) iteration method is presented for solving the PageRank problem. Firstly, the AOR and modified AOR (MAOR) methods for solving the PageRank problem are constructed, and several comparison results are also given. Next, the GIO iteration scheme is developed, and its overall convergence is analyzed in detail. Furthermore, the preconditioner derived from the GIO iteration can be used to accelerate the Krylov subspace methods, such as GMRES method. Finally, some numerical experiments on several PageRank problems are provided to illustrate the efficiency of the proposed algorithm.

On convergence of the inner-outer iteration method for computing PageRank

Without imposing any restriction on the damping factors and the stopping tolerances, we prove the overall convergence of the inner-outer iteration method for computing the PageRank vector, which was proposed by Gleich, Gray, Greif and Lau (SIAM J. Sci. Comput. 32(2010)349-371). Based on the formula of the contraction factor of the method, we discuss possible choices of the iteration parameters, which could be practically useful for accelerating the convergence rate of the inner-outer iteration method.

A Modified Nonlinear Inexact Uzawa Algorithm with a Variable Relaxation Parameter for the Stabilized Saddle Point Problem

This paper proposes a modified nonlinear inexact Uzawa (MNIU) algorithm for solving the stabilized saddle point problem, by introducing a variable overrelaxation parameter to speed up convergence. MNIU is an inner-outer iteration method with variable inner accuracy. We give a detailed error analysis for the convergence of MNIU, based upon a newly defined error norm which helps to handle variable inner accuracy for the Uzawa method. We also simply formulate the optimal overrelaxation parameter. Sufficient conditions are given for the convergence of MNIU. We show that MNIU converges in a relatively large range for the variable inner accuracy setting. For constant inner accuracy $\delta$, MNIU is convergent when $\delta<1$, too. Compared with the original nonlinear inexact Uzawa algorithm (NIU) that converges only for constant accuracy and $\delta<1/3$, this is a significant improvement. We also show a practical approach for estimating the optimal overrelaxation parameter for numerical computation. Numerical experiments of MNIU are given and compared with other methods. These results confirm the significant improvements of MNIU.

NUMERICAL SIMULATION OF LAMINAR SOLID-LIQUID TWO-PHASE FLOW IN STIRRED TANKS

Laminar flow of solid-liquid suspension has been rarely reported in the literature. In this article, the laminar solid-liquid two-phase flow in a lab-scale stirred tank is measured with digital particle image velocimetry (DPIV) and numerically simulated with the improved inner-outer iterative method. The simulation results show good agreement with the present data. In the range of low solid volume concentration under investigation, solid particles have minor impact on two-phase flow, and the simulated flow fields of both phases are similar to the counterpart of single-phase laminar liquid flow. The comparison of simulation with experiments of single-phase laminar flow shows reasonable agreement. These observations suggest that the numerical method employed in the present study is reliable for analysis of stirred tanks. The influence of impeller off-bottom clearance on velocities is found significant: the lower the impeller, the smaller the radial and axial velocities, and the higher the solid concentration.

Inner-outer iterative techniques in the explicit alternating direction (ead) method

This paper is concerned with developing an inner-outer iterative method for steady state problems by setting up an inner iterative procedure (specifically the AGE-1D algorithm, chosen due to its suitability for parallel computers) within the outer iterative procedures that are usually applied (e.g. the ADI method). The experiments are carried out on the Laplace equation in two dimensions. The study involves examining what termination criterion should be applied for the inner procedures. It examines the influence of grid refinement on this criterion, and on the choice of the acceleration parameters of the inner/outer iteration procedures and their effect on the rate of convergence.