Abstract The study of non-convex penalties has recently received considerable attentions in sparse approximation. The existing non-convex penalties are proposed on the principle of seeking for a continuous alternative to the ℓ0-norm penalty. In this paper, we come up with a cubic spline penalty (CSP) which is also continuous but closer to ℓ0-norm penalty compared to the existing ones. As a result,

Abstract Recently, the structured backward errors for the generalized saddle point problems with some different structures have been studied by some authors, but their results involve some Kronecker products, the vec-permutation matrices, and the orthogonal projection of a large block matrix which make them very expensive to compute when utilized for testing the stability of a practical algorithm or

Abstract A second-order surface reconstruction (SR) method for the shallow water equations with a discontinuous bottom topography and a Manning friction source term is presented. We redefine the water surface level at the cell interface by using the minimum difference between the bottom level and the original water surface level. The reconstructed water surface level is used to define the intermediate

Abstract In general, it needs to take about nearly 10 grid points inside a wall boundary layer for low accuracy order methods to get satisfactory wall-normal gradient related results, such as friction coefficient and wall heat flux. If there exist extreme points inside the boundary layer, this situation becomes even worse. In this work, with the help of the analytic solution of the one-dimensional

Abstract In this article, we propose a new method for low-rank completion of a large sparse matrix, subject to non-negativity constraint. As a challenging prototype of this problem, we have in mind the well-known Netflix problem. Our method is based on the derivation of a constrained gradient system and its numerical integration. The methods we propose are based on the constrained minimization of a

Abstract Quantifying the error that is induced by numerical approximation techniques is an important task in many fields of applied mathematics. Two characteristic properties of error bounds that are desirable are reliability and efficiency. In this article, we present an error estimation procedure for general nonlinear problems and, in particular, for parameter-dependent problems. With the presented

Abstract Given the scalar sequence \(\{f_{m}\}^{\infty }_{m=0}\) that satisfies $$ f_{m} = \sum\limits_{i=1}^{k} {a_{i}}{\zeta}_{i}^{m},\quad m=0,1,\ldots, $$ where \(a_{i}, \zeta _{i}\in \mathbb {C}\) and ζi are distinct, the algorithm of Prony concerns the determination of the ai and the ζi from a finite number of the fm. This algorithm is also related to Padé approximants from the infinite power

Abstract This paper considers Lévy noise driven nonlinear stochastic Volterra integral equations with doubly weakly singular kernels, whose singular points include both s = 0 and s = t. The existence and uniqueness theorem of the true solution as well as the strong convergence rate of the Euler–Maruyama (EM) method are developed via establishing some fine estimates. Compared with the corresponding

Abstract This paper deals with Pythagorean hodograph curves of the spaces of algebraic trigonometric functions and trigonometric polynomials, \(\text {span}\left \{1,t,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right \} \) and \( \text {span}\left \{1,\left \{{\cos \limits } \left ({kt}\right ), {\sin \limits } \left ({kt}\right )\right \}_{k=1}^{m}\right

Abstract The paper deals with numerical discretizations of separable nonlinear Hamiltonian systems with additive noise. For such problems, the expected value of the total energy, along the exact solution, drifts linearly with time. We present and analyze a time integrator having the same property for all times. Furthermore, strong and weak convergence of the numerical scheme along with efficient multilevel

Abstract In this paper, we propose a discontinuous finite volume element method to solve a phase field model for two immiscible incompressible fluids. In this finite volume element scheme, discontinuous linear finite element basis functions are used to approximate the velocity, phase function, and chemical potential while piecewise constants are used to approximate the pressure. This numerical method

Abstract In this paper, a sine pseudo-spectral-difference scheme that preserves the discrete mass and energy is presented and analyzed for the coupled Gross–Pitaevskii equations with Dirichlet boundary conditions in several spatial dimensions. The Crank–Nicolson finite difference method is employed for approximating the time derivative, and the second-order sine spectral differentiation matrix is deduced

Abstract In this paper, we discuss a priori error estimates of two-grid mixed finite element methods for a class of nonlinear parabolic equations. The lowest order Raviart-Thomas mixed finite element and Crank-Nicolson scheme are used for the spatial and temporal discretization. First, we derive the optimal a priori error estimates for all variables. Second, we present a two-grid scheme and analyze

Abstract In this paper, we propose a linearly implicit Fourier pseudo-spectral scheme, which preserves the total mass and energy conservation laws for the damped nonlinear Schrödinger equation in three dimensions. With the aid of the semi-norm equivalence between the Fourier pseudo-spectral method and the finite difference method, an optimal L2-error estimate for the proposed method without any restriction

Abstract Let \(\mathbf d=(d_{j})_{j\in \mathbb {I}_{m}}\in \mathbb {N}^{m}\) be a finite sequence (of dimensions) and \(\alpha =(\alpha _{i})_{i\in \mathbb {I}_{n}}\) be a sequence of positive numbers (of weights), where \(\mathbb {I}_{k}=\{1,\ldots ,k\}\) for \(k\in \mathbb {N}\). We introduce the (α, d)-designs, i.e., m-tuples \({\Phi }=(\mathcal F_{j})_{j\in \mathbb {I}_{m}}\) such that \(\mathcal

Abstract We study worst-case optimal approximation of positive linear functionals in reproducing kernel Hilbert spaces induced by increasingly flat Gaussian kernels. This provides a new perspective and some generalisations to the problem of interpolation with increasingly flat radial basis functions. When the evaluation points are fixed and unisolvent, we show that the worst-case optimal method converges

Abstract The eigenvalues and eigenfunctions of the Stokes operator have been the subject of intense analytical investigation and have applications in the study and simulation of the Navier–Stokes equations. As the Stokes operator is second order and has the divergence-free constraint, computing these eigenvalues and the corresponding eigenfunctions is a challenging task, particularly in complex geometries

Abstract We introduce stochastic variants of the classical Bernstein polynomials associated with a continuous function f, built up from a general triangular array of random variables. We discuss the uniform convergence in probability of the approximation process that they represent, providing at the same time rates of convergence. In the particular case in which the triangular array of random variables

Abstract We present the nonconforming virtual element method for the fourth-order singular perturbation problem. The virtual element proposed in this paper is a variant of the C0-continuous nonconforming virtual element presented in our previous work and allows to compute two different projection operators that are used for the construction of the discrete scheme. We show the optimal convergence in

Abstract Two fundamental difficulties are encountered in the numerical evaluation of time-dependent layer potentials. One is the quadratic cost of history dependence, which has been successfully addressed by splitting the potentials into two parts—a local part that contains the most recent contributions and a history part that contains the contributions from all earlier times. The history part is smooth

Abstract This article presents a new boundary integral approach for finding optimal shapes of peristaltic pumps that transport a viscous fluid. Formulas for computing the shape derivatives of the standard cost functionals and constraints are derived. They involve evaluating physical variables (traction, pressure, etc.) on the boundary only. By employing these formulas in conjunction with a boundary

Abstract A nonlinear parabolic system is derived to describe compressible miscible displacement in a porous medium. A mixed finite element method is employed to approximate the pressure and the Darcy velocity, and a characteristic finite element method is used to approximate the concentration. Twice Newton iteration is applied on the fine grid to linearize the fully discrete problem using the coarse-grid

Abstract In this paper, we provide the rate of convergence for approximation of an eigenvalue problem describing vibrations of axisymmetric revolution elastic shells. The eigenvalue problem approximation is constructed by the method of finite elements. We study a system of differential equations with variable coefficients, some of which are non-integrable improperly near 0. To this aim, certain special

Abstract In this paper, we introduce the multi-region discontinuous Galerkin composite finite element method (MRDGCFEM) with hp-adaptivity for the discretization of second-order elliptic partial differential equations with discontinuous coefficients. This method allows for the approximation of problems posed on computational domains where the jumps in the diffusion coefficient form a micro-structure

Abstract We propose and analyze a family of decoupled and positivity-preserving discrete duality finite volume (DDFV) schemes for diffusion problems on general polygonal meshes. These schemes are further extensions of the scheme in Su et al. (J. Comput. Phys.372, 773–798, 2018) and share some benefits. First, the two sets of finite volume (FV) equations are constructed for the vertex-centered and cell-centered

Abstract Model-order reduction methods tackle the following general approximation problem: find an “easily-computable” but accurate approximation \(\hat {\boldsymbol {h}}\) of some target solution h⋆. In order to achieve this goal, standard methodologies combine two main ingredients: (i) a set of partial observations of h⋆ and (ii) some “simple” prior model on the set of target solutions. The most

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in [Gillette et al., AiCM, to appear], we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying

We prove the optimal convergence estimate for first order interpolants used in finite element methods based on three major approaches for generalizing barycentric interpolation functions to convex planar polygonal domains. The Wachspress approach explicitly constructs rational functions, the Sibson approach uses Voronoi diagrams on the vertices of the polygon to define the functions, and the Harmonic