In this article, we address the efficient numerical solution of linear and quadratic programming problems, often of large scale. With this aim, we devise an infeasible interior point method, blended with the proximal method of multipliers, which in turn results in a primal‐dual regularized interior point method. Application of this method gives rise to a sequence of increasingly ill‐conditioned linear

Image denoising is a typical inverse problem and is hard to be solved. Fortunately, a powerful two‐phase method for restoring images corrupted by high‐level impulse noise has been proposed. The key point of the method is the computational efficiency of the second phase which requires the minimization of a smooth objective function defined on the terms of edge‐preserving potential function. In this

In this article, we present a reliable hybrid algorithm for solving convex quadratic minimization problems. At the kth iteration, two points are computed: first, an auxiliary point x ˙ k is generated by performing a gradient step using an optimal steplength, and second, the next iterate xk + 1 is obtained by means of weighted sum of x ˙ k with the penultimate iterate xk − 1. The coefficient of the

Scalable approaches for uncertainty quantification are necessary for characterizing prediction confidence in large‐scale subsurface flow simulations with uncertain permeability. To this end we explore a multilevel Monte Carlo approach for estimating posterior moments of a particular quantity of interest, where we employ an element‐agglomerated algebraic multigrid (AMG) technique to generate the hierarchy

In the past decades, multigrid methods for linear systems having multilevel Toeplitz coefficient matrices with scalar entries have been widely studied. On the other hand, only few papers have investigated the case of block entries, where the entries are small generic matrices of fixed size instead of scalars. In that case the efforts of the researchers have been mainly devoted to specific applications

Time–space fractional Fokker–Planck equations (TSFFPEs) are a class of very useful models for describing some practical phenomena in statistical physics. In the present article, we focus on the fast computation for semilinear two‐dimensional TSFFPEs. A stable implicit difference scheme with second‐order accuracy in time and space is derived. In order to accelerate the convergence rate of the scheme

In [1], the following error was published on page 1. The affiliation of the second author, Pei Yuan, read: 2Institute of Computing Technology, Beijing, 100190, China. The affiliation of the second author, Pei Yuan, should read: 2Institute of Computing Technology, Chinese Academy of Sciences, Beijing, 100190, China. 3University of Chinese Academy of Sciences, Beijing, 100049, China.

Discrete ill‐posed inverse problems arise in many areas of science and engineering. Their solutions are very sensitive to perturbations in the data. Regularization methods aim at reducing this sensitivity. This article considers an iterative regularization method, based on iterated Tikhonov regularization, that was proposed in M. Donatelli and M. Hanke, Fast nonstationary preconditioned iterative methods

The row iterative method is popular in solving the large‐scale ill‐posed problems due to its simplicity and efficiency. In this work we consider the randomized row iterative (RRI) method to tackle this issue. First, we present the semiconvergence analysis of RRI method for the overdetermined and inconsistent system, and derive upper bounds for the noise error propagation in the iteration vectors. To

We propose to reduce the (spectral) condition number of a given linear system by adding a suitable diagonal matrix to the system matrix, in particular by shifting its spectrum. Iterative procedures are then adopted to recover the solution of the original system. The case of real symmetric positive definite matrices is considered in particular, and several numerical examples are given. This approach

In this article, we consider the stochastic inverse singular value problem (ISVP) of constructing a stochastic matrix from the prescribed realizable singular values. We propose a Riemannian inexact Newton‐CG method with various choices of forcing terms for solving the stochastic ISVP. We show the proposed method converges linearly or superlinearly for different forcing terms under some assumptions

We investigate the method of conjugate gradients, exploiting inaccurate matrix‐vector products, for the solution of convex quadratic optimization problems. Theoretical performance bounds are derived, and the necessary quantities occurring in the theoretical bounds estimated, leading to a practical algorithm. Numerical experiments suggest that this approach has significant potential, including in the

High‐dimensionality reduction techniques are very important tools in machine learning and data mining. The method of generalized low rank approximations of matrices (GLRAM) is a popular technique for dimensionality reduction and image compression. However, it suffers from heavily computational overhead in practice, especially for data with high dimension. In order to reduce the cost of this algorithm

Block Krylov subspace methods (KSMs) comprise building blocks in many state‐of‐the‐art solvers for large‐scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient

Following the Perron theorem, the spectral radius of a primitive matrix is a simple eigenvalue. It is shown that for a primitive matrix A, there is a positive rank one matrix X such that B = A ∘ X, where ∘ denotes the Hadamard product of matrices, and such that the row (column) sums of matrix B are the same and equal to the Perron root. An iterative algorithm is presented to obtain matrix B without

During the past few decades, uncertainty quantification (UQ) techniques have been developed and applied to many applications. The majority of these techniques have been applied directly to specifically defined problems, that is, problems described by a mathematical operator and a specific source term, both which may be endowed with uncertainty. In this article, we take an alternative approach: applying

In this article, we propose some subspace methods such as the conjugate residual, generalized conjugate residual, biconjugate gradient, conjugate gradient squared and biconjugate gradient stabilized methods based on the tensor forms for solving the tensor equation involving the Einstein product. These proposed algorithms keep the tensor structure. The convergence analysis shows that the proposed methods

In this article, we study preconditioning techniques for the control of the Navier–Stokes equation, where the control only acts on a few parts of the domain. Optimization, discretization, and linearization of the control problem results in a generalized linear saddle‐point system. The Schur complement for the generalized saddle‐point system is very difficult or even impossible to approximate, which

The greater arithmetic intensity of high‐order finite element discretizations makes them attractive for implementation on next‐generation hardware, but assembly of high‐order finite element operators as matrices is prohibitively expensive. As a result, the development of general algebraic solvers for such operators has been an open research challenge. Fast matrix‐free application of high‐order operators

In a recent article, one of the authors developed a multigrid technique for coarse‐graining dynamic powergrid models. A key component in this technique is a relaxation‐based coarsening of the graph Laplacian given by the powergrid network and its weighted graph, which is represented by the admittance matrix. In this article, we use this coarsening strategy to develop a multigrid method for solving

Nonuniform rational B‐splines (NURBS) are the most common representation form in isogeometric analysis. In this article, we study the spectral behavior of discretization matrices arising from isogeometric Galerkin and collocation methods based on d‐variate NURBS of degrees (p1,…,pd), and applied to general second‐order partial differential equations defined on a d‐dimensional domain. The spectrum of

In this article, we study the numerical technique for variable‐order fractional reaction‐diffusion and subdiffusion equations that the fractional derivative is described in Caputo's sense. The discrete scheme is developed based on Lucas multiwavelet functions and also modified and pseudo‐operational matrices. Under suitable properties of these matrices, we present the computational algorithm with high

Bounds on the spectrum of Schur complements of subdomain stiffness matrices of the discretized Laplacian with respect to interior variables are important in the convergence analysis of finite element tearing and interconnecting (FETI)‐based domain decomposition methods. Here, we are interested in bounds on the regular condition number of Schur complements of “floating” clusters, that is, of matrices

In this paper, we propose a numerical scheme based on the iterative method for solving the fractional diffusion‐wave equation involving the Caputo–Weyl fractional derivative of order 0 < α ≤ 2 . The convergence of the approximate solutions of the fractional diffusion‐wave equation is rigourously established. Moreover, the approximate solutions of the fractional diffusion‐wave equation have been presented

The Gram–Schmidt process uses orthogonal projection to construct the A = QR factorization of a matrix. When Q has linearly independent columns, the operator P = I − Q(QTQ)−1QT defines an orthogonal projection onto Q⊥. In finite precision, Q loses orthogonality as the factorization progresses. A family of approximate projections is derived with the form P = I − QTQT, with correction matrix T. When T = (QTQ)−1

Algebraic multigrid (AMG) is a widely used scalable solver and preconditioner for large‐scale linear systems resulting from the discretization of a wide class of elliptic PDEs. While AMG has optimal computational complexity, the cost of communication has become a significant bottleneck that limits its scalability as processor counts continue to grow on modern machines. This article examines the design

We present a greedy algorithm for computing selected eigenpairs of a large sparse matrix H that can exploit localization features of the eigenvector. When the eigenvector to be computed is localized, meaning only a small number of its components have large magnitudes, the proposed algorithm identifies the location of these components in a greedy manner, and obtains approximations to the desired eigenpairs

The goal of the present paper is the design of embeddings of a general sparse graph into a set of points in ℝ d for appropriate d ≥ 2. The embeddings that we are looking at aim to keep vertices that are grouped in communities together and keep the rest apart. To achieve this property, we utilize coarsening that respects possible community structures of the given graph. We employ a hierarchical multilevel

We construct an algebraic multigrid (AMG) based preconditioner for the reduced Hessian of a linear‐quadratic optimization problem constrained by an elliptic partial differential equation. While the preconditioner generalizes a geometric multigrid preconditioner introduced in earlier works, its construction relies entirely on a standard AMG infrastructure built for solving the forward elliptic equation

The fast Fourier transform (FFT) is one of the most successful numerical algorithms of the 20th century and has found numerous applications in many branches of computational science and engineering. The FFT algorithm can be derived from a particular matrix decomposition of the discrete Fourier transform (DFT) matrix. In this paper, we show that the quantum Fourier transform (QFT) can be derived by

The singular value distribution of the matrix‐sequence {YnTn[f]}n, with Tn[f] generated by f ∈ L 1 ( [ − π , π ] ) , was shown in [J. Pestana and A.J. Wathen, SIAM J Matrix Anal Appl. 2015;36(1):273‐288]. The results on the spectral distribution of {YnTn[f]}n were obtained independently in [M. Mazza and J. Pestana, BIT, 59(2):463‐482, 2019] and [P. Ferrari, I. Furci, S. Hon, M.A. Mursaleen, and S.

Based on the weak regular splitting of tensors, preconditioned technology has shown great advantages in solving tensor equations with nonsingular (strong) ℳ ‐tensors. In this article, we introduce H‐splitting and H‐compatible splitting and investigate the relationship between these different splittings and their convergence. Meanwhile, we also consider a preconditioned AOR iterative method for solving

Multigrid solvers face multiple challenges on parallel computers. Two fundamental ones read as follows: Multiplicative solvers issue coarse grid solves which exhibit low concurrency and many multigrid implementations suffer from an expensive coarse grid identification phase plus adaptive mesh refinement overhead. We propose a new additive multigrid variant for spacetrees, that is, meshes as they are

Let P(t) be an n × n (complex) exponentially parameterized pentadiagonal matrix. In this article, using a theorem of Akian, Bapat, and Gaubert, we present explicit formulas for asymptotics of the moduli of the eigenvalues of P(t) as t → ∞. Our approach is based on exploiting the relation with tropical algebra and the weighted digraphs of matrices. We prove that this asymptotics tends to a unique limit

Since recent studies have shown that the Cayley transform method can be an effective iterative method for solving the inverse eigenvalue problem, in this work, we consider using an extension of it for solving a type of parameterized generalized inverse eigenvalue problem and prove its locally quadratic convergence. This type of inverse eigenvalue problem, which includes multiplicative and additive

In this paper, preconditioners for the conjugate gradient method are studied to solve the Newton system with symmetric positive definite Jacobian. In particular, we define a sequence of preconditioners built by means of Symmetric Rank one (SR1) and Broyden‐Fletcher‐Goldfarb‐Shanno (BFGS) low‐rank updates. We develop conditions under which the SR1 update maintains the preconditioner symmetric positive

Given a nonsymmetric matrix A ∈ ℝ n × n and two unit norm vectors, the two‐sided Krylov subspace methods construct a pair of bases for two Krylov subspaces with respect to A and AT, respectively. In practical calculations, however, the two subspaces spanned by the computed bases may not be Krylov subspaces. Given two subspaces 𝒦 and ℒ , in [G. Wu et al, Toward backward perturbation bounds for approximate

The notion of the Drazin inverse of an even‐order tensors with the Einstein product was introduced, very recently [J. Ji and Y. Wei. Comput. Math. Appl., 75(9), (2018), pp. 3402‐3413]. In this article, we further elaborate this theory by establishing a few characterizations of the Drazin inverse and 𝒲 ‐weighted Drazin inverse of tensors. In addition to these, we compute the Drazin inverse of tensors

In this article, we generalize modulus‐based synchronous multisplitting methods to horizontal linear complementarity problems. In particular, first we define the methods of our concern and prove their convergence under suitable smoothness assumptions. Particular attention is devoted also to modulus‐based multisplitting accelerated overrelaxation methods. Then, as multisplitting methods are well‐suited

Uncertainty quantification for linear inverse problems remains a challenging task, especially for problems with a very large number of unknown parameters (e.g., dynamic inverse problems) and for problems where computation of the square root and inverse of the prior covariance matrix are not feasible. This work exploits Krylov subspace methods to develop and analyze new techniques for large‐scale uncertainty

This paper addresses the problem of computing the Riemannian center of mass of a collection of symmetric positive definite matrices. We show in detail that the condition number of the Riemannian Hessian of the underlying optimization problem is never very ill conditioned in practice, which explains why the Riemannian steepest descent approach has been observed to perform well. We also show theoretically

The computation of Gauss quadrature rules for arbitrary weight functions using the Stieltjes algorithm is a purely sequential process, and the computational cost significantly increases when high accuracy is required. ParaStieltjes is a new algorithm to compute the recurrence coefficients of the associated orthogonal polynomials in parallel, from which the nodes and weights of the quadrature rule can

We consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block

This article develops the preconditioning technique as a method to address the accuracy issue caused by ill‐conditioning. Given a preconditioner M for an ill‐conditioned linear system Ax=b, we show that, if the inverse of the preconditioner M−1 can be applied to vectors accurately, then the linear system can be solved accurately. A stability concept called inverse‐equivalent accuracy is introduced

This article presents a multilevel parallel preconditioning technique for solving general large sparse linear systems of equations. Subdomain coloring is invoked to reorder the coefficient matrix by multicoloring the adjacency graph of the subdomains, resulting in a two‐level block diagonal structure. A full binary tree structure 𝒯 is then built to facilitate the construction of the preconditioner

Multigrid methods are popular solution algorithms for many discretized PDEs, either as standalone iterative solvers or as preconditioners, due to their high efficiency. However, the choice and optimization of multigrid components such as relaxation schemes and grid‐transfer operators is crucial to the design of optimally efficient algorithms. It is well known that local Fourier analysis (LFA) is a

This article presents enhancement strategies for the Hermitian and skew‐Hermitian splitting method based on gradient iterations. The spectral properties are exploited for the parameter estimation, often resulting in a better convergence. In particular, steepest descent with early stopping can generate a rough estimate of the optimal upper bound. This is better than an arbitrary choice since the latter

In the present article, we consider a class of elliptic partial differential equations with Dirichlet boundary conditions and where the operator is div(−a(x)∇·), with a continuous and positive over Ω ‾ , Ω being an open and bounded subset of R d , d≥1. For the numerical approximation, we consider the classical P k Finite Elements, in the case of Friedrichs–Keller triangulations, leading, as usual,

This article develops an algebraic multigrid (AMG) method for solving systems of elliptic boundary‐value problems. It is well known that multigrid for systems of elliptic equations faces many challenges that do not arise for most scalar equations. These challenges include strong intervariable couplings, multidimensional and possibly large near‐nullspaces, analytically unknown near‐nullspaces, delicate

Although convergence of the Parareal and multigrid‐reduction‐in‐time (MGRIT) parallel‐in‐time algorithms is well studied, results on their optimality is limited. Appealing to recently derived tight bounds of two‐level Parareal and MGRIT convergence, this article proves (or disproves) hx‐ and ht‐independent convergence of two‐level Parareal and MGRIT, for linear problems of the form u ′ ( t ) + ℒ u

We present Nesterov‐type acceleration techniques for alternating least squares (ALS) methods applied to canonical tensor decomposition. While Nesterov acceleration turns gradient descent into an optimal first‐order method for convex problems by adding a momentum term with a specific weight sequence, a direct application of this method and weight sequence to ALS results in erratic convergence behavior

In this work we consider the problem of semi‐active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the ℋ ∞ ‐norm of the transfer function from the exogenous inputs to the performance outputs. We make use of a new greedy method for computing the ℋ ∞ ‐norm of a transfer function based on rational interpolation

Lanczos‐type product methods (LTPMs), in which the residuals are defined by the product of stabilizing polynomials and the Bi‐CG residuals, are effective iterative solvers for large sparse nonsymmetric linear systems. Bi‐CGstab(L) and GPBi‐CG are popular LTPMs and can be viewed as two different generalizations of other typical methods, such as CGS, Bi‐CGSTAB, and Bi‐CGStab2. Bi‐CGstab(L) uses stabilizing

In this article, we study robust tensor completion by using transformed tensor singular value decomposition (SVD), which employs unitary transform matrices instead of discrete Fourier transform matrix that is used in the traditional tensor SVD. The main motivation is that a lower tubal rank tensor can be obtained by using other unitary transform matrices than that by using discrete Fourier transform

In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved