A finite element cochain complex on Cartesian meshes of any dimension based on the \(H^1\)-inner product is introduced. It yields \(H^1\)-conforming finite element spaces with exterior derivatives in \(H^1\). We use a tensor product construction to obtain \(L^2\)-stable projectors into these spaces which commute with the exterior derivative. The finite element complex is generalized to a family of

We use the scalar auxiliary variable (SAV) reformulation of the nonlinear Schrödinger (NLS) equation to construct structure-preserving SAV–Gauss methods for the NLS equation, namely \(L^2\)-conservative methods satisfying a discrete analogue of the energy (the Hamiltonian) conservation of the equation. This is in contrast to Gauss methods for the standard form of the NLS equation that are \(L^2\)-conservative

In this study, we discuss the superconvergence of the space-time discontinuous Galerkin method for the first-order linear nonhomogeneous hyperbolic equation. By using the local differential projection method to construct comparison function, we prove that the numerical solution is \((2n+1)\)-th order superconvergent at the downwind-biased Radau points in the discrete \(L^2\)-norm. As a by-product,

The velocity solution of the incompressible Stokes equations is not affected by changes of the right hand side data in form of gradient fields. Most mixed methods do not replicate this property in the discrete formulation due to a relaxation of the divergence constraint which means that they are not pressure-robust. A recent reconstruction approach for classical methods recovers this invariance property

The randomized Kaczmarz algorithm has received considerable attention recently because of its simplicity, speed, and the ability to approximately solve large-scale linear systems of equations. In this paper we propose randomized double and triple Kaczmarz algorithms to solve extended normal equations of the form \(\mathbf{A}^\top \mathbf{Ax}=\mathbf{A}^\top \mathbf{b}-\mathbf{c}\). The proposed algorithms

The block Gauss–Seidel algorithm can significantly outperform the simple randomized Gauss–Seidel algorithm for solving overdetermined least-squares problems since it moves a large block of columns rather than a single column into working memory. Here, with the help of the maximum residual rule, we construct a two-step Gauss–Seidel (2SGS) algorithm, which selects two different columns simultaneously

The spectral collocation method is investigated for the system of nonlinear Riemann–Liouville fractional differential equations (FDEs). The main idea of the presented method is to solve the corresponding system of nonlinear weakly singular Volterra integral equations obtained from the system of FDEs. In order to carry out convergence analysis for the presented method, we investigate the regularity

We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the

The Peaceman–Rachford splitting method (PRSM) is a preferred method for solving the two-block separable convex minimization problems with linear constraints at present. In this paper, we propose an inertial generalized proximal PRSM (abbreviated as IGPRSM) to improve computing efficiency, which unify the ideas of inertial proximal point and linearization technique. Both subproblems are linearized by

This paper is concerned with C0 (non-Lagrange) finite element approximations of the linear elliptic equations in non-divergence form and the Hamilton–Jacobi–Bellman (HJB) equations with Cordes coefficients. Motivated by the Miranda–Talenti estimate, a discrete analog is proved once the finite element space is C0 on the \((n-1)\)-dimensional subsimplex (face) and \(C^1\) on \((n-2)\)-dimensional subsimplex

The fractal interpolation functions with appropriate iterated function systems provide a method to perturb and approximate a continuous function on a compact interval I. This method produces a class of functions \(f^{\varvec{\alpha }}\in {\mathcal {C}}(I)\), where \(\varvec{\alpha }\) is a vector with functional components. The presence of scaling function in these fractal functions helps to get a

The Mittag-Leffler function is computed via a quadrature approximation of a contour integral representation. We compare results for parabolic and hyperbolic contours, and give special attention to evaluation on the real line. The main point of difference with respect to similar approaches from the literature is the way that poles in the integrand are handled. Rational approximation of the Mittag-Leffler

In this paper, based on the continuous collocation polynomial approximations, we derive and analyse a class of trigonometric collocation integrators for solving the highly oscillatory hyperbolic system. The symmetry, convergence and energy conservation of the continuous collocation polynomial approximations are rigorously analysed in details. Moreover, we also proved that the continuous collocation

We are interested in finding a solution to the tensor complementarity problem with a strong M-tensor, which we call the M-tensor complementarity problem. We propose a lower dimensional linear equation approach to solve that problem. At each iteration, only a lower dimensional system of linear equation needs to be solved. The coefficient matrices of the lower dimensional linear systems are independent

The need to estimate upper and lower bounds for matrix functions of the form \({\mathrm{trace}}(W^Tf(A)V)\), where the matrix \(A\in {{\mathbb {R}}}^{n\times n}\) is large and sparse, \(V,W\in {{\mathbb {R}}}^{n\times s}\) are block vectors with \(1\le s\ll n\) columns, and f is a function arises in many applications, including network analysis and machine learning. This paper describes the shifted

In this paper, we introduce a \(C^{0}\) virtual element method for the Helmholtz transmission eigenvalue problem, which is a fourth-order non-selfadjoint eigenvalue problem. We consider the mixed formulation of the eigenvalue problem discretized by the lowest-order virtual elements. This discrete scheme is based on a conforming \(H^{1}(\varOmega )\times H^{1}(\varOmega )\) discrete formulation, which

For backward differentiation formulae (BDF) applied to gradient flows of semiconvex functions, quadratic stability implies the existence of a Lyapunov functional. We compute the maximum time step which can be derived from quadratic stability for the 3-step BDF method (BDF3). Applications to the asymptotic behaviour of sequences generated by the BDF3 scheme are given.

In this paper we provide algorithms for computing the bidiagonal decomposition of the Wronskian matrices of the monomial basis of polynomials and of the basis of exponential polynomials. It is also shown that these algorithms can be used to perform accurately some algebraic computations with these Wronskian matrices, such as the calculation of their inverses, their eigenvalues or their singular values

The \(H^m\)-nonconforming virtual elements of any order k on any shape of polytope in \({\mathbb {R}}^n\) with constraints \(m> n\) and \(k\ge m\) are constructed in a universal way. A generalized Green’s identity for \(H^m\) inner product with \(m>n\) is derived, which is essential to devise the \(H^m\)-nonconforming virtual elements. By means of the local \(H^m\) projection and a stabilization term

We discuss the solution of eigenvalue problems associated with partial differential equations that can be written in the generalized form \({\mathsf {A}}x=\lambda {\mathsf {B}}x\), where the matrices \({\mathsf {A}}\) and/or \({\mathsf {B}}\) may depend on a scalar parameter. Parameter dependent matrices occur frequently when stabilized formulations are used for the numerical approximation of partial

To locate the eigenvalues of a given tensor, we present two classes of new Ky Fan type eigenvalue inclusion sets for tensors, which are tighter than those in Yang et al. (SIAM J Matrix Anal Appl 31:2517–2530, 2010) and He et al. (J Inequal Appl 114:1-9, 2014), respectively. Under certain conditions, the theoretical comparisons of the new proposed Ky Fan type eigenvalue inclusion sets for tensors are

In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive an optimal error estimate and present several numerical tests assessing the validity of the theoretical results.

The proximal term plays a significant role in the literature of proximal Alternating Direction Method of Multipliers (ADMM), since (positive-definite or indefinite) proximal terms can promote convergence of ADMM and further simplify the involved subproblems. However, an overlarge proximal parameter decelerates the convergence numerically, though the convergence can be established with it. In this paper

In this paper, we study discrete projection methods for solving the Hammerstein integral equations on the half-line with a smooth kernel using piecewise polynomial basis functions. We show that discrete Galerkin/discrete collocation methods converge to the exact solution with order \({\mathcal {O}}(n^{-min\{r, d\}}),\) whereas iterated discrete Galerkin/iterated discrete collocation methods converge

In this work we present a new mixed finite element method for a class of steady-state natural convection models describing the behavior of non-isothermal incompressible fluids subject to a heat source. Our approach is based on the introduction of a modified pseudostress tensor depending on the pressure, and the diffusive and convective terms of the Navier–Stokes equations for the fluid and a vector

The Quasi-Newton method is one of the most effective methods using the first derivative for solving all unconstrained optimization problems. The Broyden family method plays an important role among the quasi-Newton algorithms. However, the study of the convergence of the classical Broyden family method is still not enough. While in the special case, BFGS method, there have been abundant achievements

Combining the quadratic equal-order stabilized method with the approach of local and parallel finite element computations and classical iterative methods for the discretization of the steady-state Navier–Stokes equations, three parallel iterative stabilized finite element methods based on fully overlapping domain decomposition are proposed and compared in this paper. In these methods, each processor

In this work, ideas previously introduced for a discontinuous Galerkin recovery method in one dimension, that involves a penalty stabilization term, are extended to an elliptic differential equation in several dimensions and different types of boundary conditions and meshes. Using standard arguments for other existing discontinuous Galerkin methods, we show results of existence and uniqueness of the

New H(div) conforming finite element methods for incompressible flows are designed that involve the rotation form of the equations of motion and the Bernoulli function. With a specific choice of numerical fluxes, we recover the same velocity field as in Guzmán et al. (IMA J Numer Anal 37(4):1733–1771, 2016) for the incompressible Euler equation in the convection form. Error estimates are presented

Image restoration problem is often solved by minimizing a cost function which consists of data-fidelity terms and regularization terms. Half-quadratic regularization has the advantage that it can preserve image details well in the recovered images. In this paper, we consider solving the image restoration model which involves multiplicative half-quadratic regularization term. Newton method is employed

In this paper, a small and necessary revision on an assumption condition of Aminifard and Babaie-Kafaki (Calcolo, 2019. https://doi.org/10.1007/s10092-019-0312-9) is made. By a little modification, a new conjugate gradient method is proposed, in which the search directions satisfy the sufficient descent condition with the strong Wolfe line search. The main difference between two algorithms is that

The study of the long time conservation for numerical methods poses interesting and challenging questions from the point of view of geometric integration. In this paper, we analyze the long time energy and magnetic moment conservations of two-step symmetric methods for charged-particle dynamics. A two-step symmetric method is proposed and its long time behaviour is shown not only in a normal magnetic

In this paper, we study the Poisson noise removal problem with total variation regularization term. Using the dual formulation of total variation and Lagrange dual, we formulate the problem as a new constrained minimax problem. Then, a modified Chambolle-Pock first-order primal-dual algorithm is developed to compute the saddle point of the minimax problem. The main idea of this paper is using different

In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we

In this paper, we propose an alternative version of the randomized Kaczmarz method, which chooses each row of the coefficient matrix A with probability proportional to the square of the Euclidean norm of the residual of each corresponding equation. We prove that it converges with expected linear rate and the convergence rate of this method is better than the Strohmer and Vershynin’s RK method. Numerical

In this paper a new modified Nyström method is proposed to solve linear integral equations of the second kind with fixed singularities of Mellin convolution type. It is based on the Gauss–Radau quadrature formula with a suitable Jacobi weight. The stability and convergence of the method is proved in weighted spaces with uniform norm. Moreover, an error estimate of the numerical solution is given under

A wide variety of different (fixed-point) iterative methods for the solution of nonlinear equations exists. In this work we will revisit a unified iteration scheme in Hilbert spaces from our previous work [16] that covers some prominent procedures (including the Zarantonello, Kačanov and Newton iteration methods). In combination with appropriate discretization methods so-called (adaptive) iterative

In this article, we study the convergence properties of the streamline-diffusion finite element method (SDFEM) for singularly perturbed 1D parabolic convection–diffusion initial-boundary-value problems. To discretize the spatial domain, we use a layer-adaptive nonuniform grids obtained through the equidistribution principle, whereas uniform grid is used in the time direction. Here, we use the backward-Euler

For any real numbers \(a,\ b\), and c, we form the sequence of polynomials \(\{P_n(z)\}_{n=0}^\infty\) satisfying the four-term recurrence$$\begin{aligned} P_n(z)+azP_{n-1}(z)+bP_{n-2}(z)+czP_{n-3}(z)=0,\ n\in {\mathbb {N}}, \end{aligned}$$with the initial conditions \(P_0(z)=1\) and \(P_{-n}(z)=0\). We find necessary and sufficient conditions on \(a,\ b\), and c under which the zeros of \(P_n(z)\)

In this article, a new \(\alpha\)-fractal rational cubic spline is introduced with the help of the iterated function system (IFS) that contains rational functions. The numerator of the rational function contains a cubic polynomial and the denominator of the rational function contains a quadratic polynomial with three shape parameters. The convergence analysis of the \(\alpha\)-fractal rational cubic

In this paper, the implementation of second derivative general linear methods (SGLMs) in a variable stepsize environment using Nordsieck technique is discussed and various implementation issues are investigated. All coefficients of a method of order four together with its error estimate are obtained. The method is derived with the aim of good zero-stability properties for a large range of ratios of

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using

Recently, some matrix exponential-based discriminant analysis methods were proposed for high dimensionality reduction. It has been shown that they often have more discriminant power than the corresponding discriminant analysis methods. However, one has to solve some large-scale matrix exponential eigenvalue problems which constitutes the bottleneck in this type of methods. The main contribution of

Three new procedures for computation the scaling parameter in the self-scaling memoryless Broyden–Fletcher–Goldfarb–Shanno search direction with a parameter are presented. The first two are based on clustering the eigenvalues of the self-scaling memoryless Broyden–Fletcher–Goldfarb–Shanno iteration matrix with a parameter by using the determinant or the trace of this matrix. The third one is based

Computing more than one eigenvalue for (large sparse) one-parameter polynomial and general nonlinear eigenproblems, as well as for multiparameter linear and nonlinear eigenproblems, is a much harder task than for standard eigenvalue problems. We present simple but efficient selection methods based on divided differences to do this. Selection means that the approximate eigenpair is picked from candidate

In this paper, we propose a new preconditioned SOR method for solving the multi-linear systems whose coefficient tensor is an \({\mathcal{M}}\)-tensor. The corresponding comparison for spectral radii of iterative tensors is given. Numerical examples demonstrate the efficiency of the proposed preconditioned methods.

In this paper, we modify the accelerated generalized successive overrelaxation (AGSOR) method for block two-by-two complex linear systems, and use the AGSOR method as an inner iteration for the modified Newton equations to solve the nonlinear system whose Jacobian matrix is a block two-by-two complex symmetric matrix. Our new method is named modified Newton AGSOR (MN-AGSOR) method. Because generalized

We construct smooth finite elements spaces on Powell–Sabin triangulations that form an exact sequence. The first space of the sequence coincides with the classical \(C^1\) Powell–Sabin space, while the others form stable and divergence-free yielding pairs for the Stokes problem. We develop degrees of freedom for these spaces that induce projections that commute with the differential operators.

This paper concerns the approximation of bivariate functions by using the well-known filtered back projection (FBP) formula from computerized tomography. We prove error estimates and convergence rates for the FBP approximation of target functions from Sobolev spaces \(\mathrm H^\alpha ({\mathbb {R}}^2)\) of fractional order \(\alpha >0\), where we bound the FBP approximation error, which is incurred

When a tensor is partitioned into subtensors, some tensor norms of these subtensors form a tensor called a norm compression tensor. Norm compression inequalities for tensors focus on the relation of the norm of this compressed tensor to the norm of the original tensor. We prove that for the tensor spectral norm, the norm of the compressed tensor is an upper bound of the norm of the original tensor

It is shown that the problem of balancing a nonnegative matrix by positive diagonal matrices can be recast as a nonlinear eigenvalue problem with eigenvector nonlinearity. Based on this equivalent formulation some adaptations of the power method and Arnoldi process are proposed for computing the dominant eigenvector which defines the structure of the diagonal transformations. Numerical results illustrate

We explore the potential of Tensor-Train (TT) decompositions in the context of multi-feature face or object recognition strategies. We devise a new recognition algorithm that can handle three or more way tensors in the TT format, and propose a truncation strategy to limit memory usage. Numerical comparisons with other related methods—including the well established recognition algorithm based on high-order

In this paper we analyze the coupling of the Stokes equations with a transport problem modelled by a scalar nonlinear convection–diffusion problem, where the viscosity of the fluid and the diffusion coefficient depend on the solution to the transport problem and its gradient, respectively. An augmented mixed variational formulation for both the fluid flow and the transport model is proposed. As a consequence

An algorithm is proposed for computing intervals containing the Moore–Penrose inverses. For developing this algorithm, we analyze the Ben-Israel iteration. We particularly emphasize that the algorithm is applicable even for rank deficient matrices. Numerical results show that the algorithm is more successful than previous algorithms in the rank deficient cases.

In this paper we consider some applications of Odd and Even Lidstone-type polynomial sequences. In particular we deal with the Odd and Even Lidstone-type and the Generalized Lidstone interpolatory problems with respect to a linear functional \(L_1\) and, respectively, \(L_2\). Estimations of the remainder for the related interpolation polynomials are given. Numerical examples are provided. Some possible

A distributed optimal control problem with the constraint of a parabolic partial differential equation is considered. Boundary value methods are used to solve the coupled initial/final value problems arising from the first order optimality conditions for this problem. We use a block triangular preconditioning strategy for solving the resulting two-by-two linear system. By making use of a matching strategy

In this article, we study the time dependent convection–diffusion–reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical

This paper is concerned with the efficient numerical treatment of 1D stationary Schrödinger equations in the semi-classical limit when including a turning point of first order. As such it is an extension of the paper [3], where turning points still had to be excluded. For the considered scattering problems we show that the wave function asymptotically blows up at the turning point as the scaled Planck

In this paper we propose a novel way to prescribe weakly the symmetry of stress tensors in weak formulations amenable to the construction of mixed finite element schemes. The approach is first motivated in the context of solid mechanics (using, for illustrative purposes, the linear problem of linear elasticity), and then we apply this technique to reduce the computational cost of augmented fully-mixed

Real matrices with non-negative off-diagonal entries play a crucial role for modelling the positive linear dynamical systems. In the literature, these matrices are referred to as Metzler matrices or negated Z-matrices. Finding the closest stable Metzler matrix to an unstable one (and vice versa) is an important issue with many applications. The stability considered here is in the sense of Hurwitz,