The aim of this paper is the analysis of exponential mean-square stability properties of nonlinear stochastic linear multistep methods. In particular it is known that, under certain hypothesis on the drift and diffusion terms of the equation, exponential mean-square contractivity is visible: the qualitative feature of the exact problem is here analysed under the numerical perspective, to understand

We consider a general regularised interpolation problem for learning a parameter vector from data. The well-known representer theorem says that under certain conditions on the regulariser there exists a solution in the linear span of the data points. This is at the core of kernel methods in machine learning as it makes the problem computationally tractable. Most literature deals only with sufficient

In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. Here, we consider the continuous Kaiser–Bessel, continuous exp-type, sinh-type, and continuous cosh-type window functions with the same support and same shape parameter. We present novel explicit

In this paper, an improved superconvergence analysis is presented for both the Crouzeix-Raviart element and the Morley element. The main idea of the analysis is to employ a discrete Helmholtz decomposition of the difference between the canonical interpolation and the finite element solution for the first-order mixed Raviart–Thomas element and the mixed Hellan–Herrmann–Johnson element, respectively

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive

We investigate the use of the so-called variably scaled kernels (VSKs) for learning tasks, with a particular focus on support vector machine (SVM) classifiers and kernel regression networks (KRNs). Concerning the kernels used to train the models, under appropriate assumptions, the VSKs turn out to be more expressive and more stable than the standard ones. Numerical experiments and applications to breast

We consider the general question of constructing a partition of unity formed by translates of a compactly supported function g : ℝd → ℂ. In particular, we prove that such functions have a special structure that simplifies the construction of partitions of unity with specific properties. We also prove that it is possible to modify the function g in such a way that it becomes symmetric with respect to

A new sampling technique for the application of proper orthogonal decomposition to a set of snapshots has been recently developed by the authors to facilitate a variety of data processing tasks (J. Comput. Phys. 335, 2017). According to it, robust modal expansions result from performing the decomposition on a limited number of relevant snapshots and a limited number of discretization mesh points, which

The design of globally Cs-smooth (s ≥ 1) isogeometric spline spaces over multi-patch geometries with possibly extraordinary vertices, i.e. vertices with valencies different from four, is a current and challenging topic of research in the framework of isogeometric analysis. In this work, we extend the recent methods Kapl et al. Comput. Aided Geom. Des. 52–53:75–89, 2017, Kapl et al. Comput. Aided Geom

We consider a fractional order viscoelasticity problem modelled by a power-law type stress relaxation function. This viscoelastic problem is a Volterra integral equation of the second kind with a weakly singular kernel where the convolution integral corresponds to fractional order differentiation/integration. We use a spatial finite element method and a finite difference scheme in time. Due to the

A high-order accurate quadrature rule for the discretization of boundary integral equations (BIEs) on closed smooth contours in the plane is introduced. This quadrature can be viewed as a hybrid of the spectral quadrature of Kress (Math. Comput. Model. 15(3-5), 229–243 1991) and the locally corrected trapezoidal quadrature of Kapur and Rokhlin (SIAM J. Numer. Anal. 34(4), 1331–1356, 1997). The new

Consider an integral with a point singularity in its integrand, such as ρ−α or \(\log \rho \). We introduceand discuss two methods for approximating such integrals, in both two and three dimensions. The methods are first introduced using the unit disk as the quadrature region, and then, they are extended to other regions and to three dimensions. The error behavior of the numerical integration for singular

In this paper, we extendthe structure-preserving interpolatory model reduction framework, originally developed for linear systems, to structured bilinear control systems. Specifically, we give explicit construction formulae for the model reduction bases to satisfy different types of interpolation conditions. First, we establish the analysis for transfer function interpolation for single-input single-output

We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator’s banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form K(x, y) = K(x − y) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical

In this paper, we study the solute transport in heterogeneous media described by the time-fractional mobile/immobile advection diffusion model, where the integer and the fractional time derivatives are employed to characterize the movement of the particles in the mobile and immobile zone, respectively. We propose a fully discrete characteristic finite element scheme for the model, in which the modified

The dynamics of particle processes can be described by population balance equations which are governed by phenomena including growth, nucleation, breakage and aggregation. Estimating the kinetics of the aggregation phenomena from measured density data constitutes an ill-conditioned inverse problem. In this work, we focus on the aggregation problem and present an approach to estimate the aggregation

Consider a linear elliptic PDE defined over a stochastic stochastic geometry a function of N random variables. In many application, quantify the uncertainty propagated to a quantity of interest (QoI) is an important problem. The random domain is split into large and small variations contributions. The large variations are approximated by applying a sparse grid stochastic collocation method. The small

In this paper, we consider the multi-order fractional differential equation and recast it into an integral equation. Based on the integral equation, we develop an hp-version Legendre spectral collocation method and the integral terms with the weakly singular kernels are calculated precisely according to the properties of Legendre and Jacobi polynomials. The hp-version error bounds under the L2-norm

In this paper, we propose a certified reduced basis (RB) method for quasilinear parabolic problems with strongly monotone spatial differential operator. We provide a residual-based a posteriori error estimate for a space-time formulation and the corresponding efficiently computable bound for the certification of the method. We introduce a Petrov-Galerkin finite element discretization of the continuous

We consider in this paper numerical approximations of the immiscible binary mixture of nematic liquid crystals (LCs) and viscous fluids. We develop a second-order time marching scheme by adopting the recently developed stabilized-SAV (scalar auxiliary variable) approach where several critical stabilization terms are added to enhance the stability; thus, large time steps are allowed in computations

In this paper, we propose a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation. We prove that the difference scheme is stable and convergent under mild conditions. The optimal convergence order \(\mathcal {O}(\tau ^{2}+{h_{x}^{2}}+{h_{y}^{2}})\) is obtained in the pointwise sense by developing a new two-dimensional

The scaled boundary finite element method (SBFEM) is a relatively recent boundary element method that allows the approximation of solutions to partial differential equations (PDEs) without the need of a fundamental solution. A theoretical framework for the convergence analysis of SBFEM is proposed here. This is achieved by defining a space of semi-discrete functions and constructing an interpolation

We present the convergence analysis of convex combination of the alternating projection and Douglas–Rachford operators for solving the phase retrieval problem. New convergence criteria for iterations generated by the algorithm are established by applying various schemes of numerical analysis and exploring both physical and mathematical characteristics of the phase retrieval problem. Numerical results

In the paper, a variational Bayesian method is used to identify the reaction coefficient for space-time nonlocal diffusion equations using nonlocal averaged flux data. To show the posterior measure to be well-defined, we rigorously prove that the forward operator is continuous with respect to the unknown reaction field. Then, gradient-based prior information is proposed to explore oscillation features

We study a posteriori error analysis of linear-quadratic boundary control problems under bilateral box constraints on the control which acts through a Neumann-type boundary condition. We adopt the hybridizable discontinuous Galerkin method as the discretization technique, and the flux variables, the scalar variables, and the boundary trace variables are all approximated by polynomials of degree k.

In this paper, we consider a quadratic finite volume method (FVM) for solving second-order nonlinear elliptic problems. Under reasonable assumptions, we shall establish the existence and uniqueness of the quadratic FVM approximation and develop the error analysis of the approximation solution. To be specific, without any additional requirements on the underlying triangular meshes, we derive the optimal

The l parameterized generalized eigenvalue problems for the nonsquare matrix pencils, proposed by Chu et al.in 2006, can be formulated as an optimization problem on a corresponding complex product Stiefel manifold. In this paper, an effective and efficient algorithm based on the Riemannian Newton’s method is established to solve the underlying problem. Under our proposed framework, to solve the corresponding

In this paper, we introduce a new finite element method for solving the Stokes equations in the primary velocity-pressure formulation using H(div) finite elements to approximate velocity. Like other finite element methods with velocity discretized by H(div) conforming elements, our method has the advantages of an exact divergence-free velocity field and pressure-robustness. However, most of H(div)

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel

A methodology for using random sketching in the context of model order reduction for high-dimensional parameter-dependent systems of equations was introduced in Balabanov and Nouy (Part I; Advances in Computational Mathematics 45:2969–3019, 2019). Following this framework, we here construct a reduced model from a small, efficiently computable random object called a sketch of a reduced model, using

We analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside

In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and \({\mathscr{H}}\mathcal {C}^{2}\)-smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose \({\mathscr{H}}\mathcal

A novel fourth-order three-point compact operator for the nonlinear convection term uux is provided in this paper. The operator makes the numerical analysis of higher-order difference schemes become possible for a wide class of nonlinear evolutionary equations under the unified framework. We take the classical viscous Burgers’ equation as an example and establish a new conservative fourth-order implicit

Fully adaptive computations of the resistive magnetohydrodynamic (MHD) equations are presented in two and three space dimensions using a finite volume discretization on locally refined dyadic grids. Divergence cleaning is used to control the incompressibility constraint of the magnetic field. For automatic grid adaptation a cell-averaged multiresolution analysis is applied which guarantees the precision

This paper develops a new recovery-based linear C0 finite element method for approximating the weak solution of a fourth-order singularly perturbed Monge-Ampère equation, which is known as the vanishing moment approximation of the Monge-Ampère equation. The proposed method uses a gradient recovery technique to define a discrete Laplacian for a given linear C0 finite element function (offline), the

We present a new method for the stable reconstruction of a class of binary images from a small number of measurements. The images we consider are characteristic functions of algebraic domains, that is, domains defined as zero loci of bivariate polynomials, and we assume to know only a finite set of uniform samples for each image. The solution to such a problem can be set up in terms of linear equations

In this paper, we consider separating the discretisation of the diffusive and advective fluxes in the complete flux scheme. This allows the combination of several discretisation methods for the homogeneous flux with the complete flux (CF) method. In particular, we explore the combination of the hybrid mimetic mixed (HMM) method and the CF method, in order to utilise the advantages of each of these

This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific

In this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products

A novel residual a posteriori error estimator for the Oseen equations achieves efficiency and reliability by including multilevel contributions in its construction. Originates from the Multiscale Hybrid Mixed (MHM) method, the estimator combines residuals from the skeleton of the first-level partition of the domain, along with the contributions from element-wise approximations. The second-level estimator

Polynomial splines are ubiquitous in the fields of computer-aided geometric design and computational analysis. Splines on T-meshes, especially, have the potential to be incredibly versatile since local mesh adaptivity enables efficient modeling and approximation of local features. Meaningful use of such splines for modeling and approximation requires the construction of a suitable spanning set of linearly

In this article, we propose an adaptive spectral graph wavelet method to solve partial differential equations on network-like structures using so-called spectral graph wavelets. The concept of spectral graph wavelets is based on the discrete graph Laplacian. The beauty of the method lies in the fact that the same operator is used for the approximation of differential operators and for the construction

This paper considers multivariate deconvolution density estimations under the local Hölder condition by wavelet methods. A pointwise lower bound of the deconvolution model is first investigated; then we provide a linear wavelet estimate to obtain the optimal convergence rate. The nonlinear wavelet estimator is introduced for adaptivity, which attains a nearly optimal rate (optimal up to a logarithmic

A well-balanced high-order scheme for shallow water equations with variable topography and temperature gradient is constructed. This scheme is of van Leer-type and is based on exact Riemann solvers. The scheme is shown to be able to capture almost exactly the stationary smooth solutions as well as stationary elementary discontinuities. Numerical tests show that the scheme gives a much better accuracy

We demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality

In this work, we consider the tensor completion problem of an incomplete and noisy observation. We introduce a novel completion model using bilevel minimization. Therefore, bilevel model-based denoising for the tensor completion problem is proposed. The denoising and completion tasks are fully separated. The upper-level directly addresses the completion problem with the truncated nuclear norm, while

We consider a primal-dual algorithm for minimizing \(f({\mathbf {x}})+h\square l({\mathbf {A}}{\mathbf {x}})\) with Fréchet differentiable f and l∗. This primal-dual algorithm has two names in literature: Primal-Dual Fixed-Point algorithm based on the Proximity Operator (PDFP2O) and Proximal Alternating Predictor-Corrector (PAPC). In this paper, we prove its convergence under a weaker condition on

In this paper, stability and error estimates for time discretizations of linear and semilinear parabolic equations by the two-step backward differentiation formula (BDF2) method with variable step-sizes are derived. An affirmative answer is provided to the question: whether the upper bound of step-size ratios for the \(l^{\infty }(0,T;H)\)-stability of the BDF2 method for linear and semilinear parabolic

The recently developed dual-porosity-Stokes model describes a complicated dual-porosity-conduit system which uses a dual-porosity/permeability model to govern the flow in porous media coupled with free flow via four physical interface conditions. This system has important applications in unconventional reservoirs especially the multistage fractured horizontal wellbore problems. In this paper, we propose

In this paper, we study the dimension of bivariate polynomial splines of mixed smoothness on polygonal meshes. Here, “mixed smoothness” refers to the choice of different orders of smoothness across different edges of the mesh. To study the dimension of spaces of such splines, we use tools from homological algebra. These tools were first applied to the study of splines by Billera (Trans. Am. Math. Soc

We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and

Frames provide redundant, stable representations of data which have important applications in signal processing. We introduce a connection between symplectic geometry and frame theory and show that many important classes of frames have natural symplectic descriptions. Symplectic tools seem well-adapted to addressing a number of important questions about frames; in this paper, we focus on the frame

Due to their flexibility, frames of Hilbert spaces are attractive alternatives to bases in approximation schemes for problems where identifying a basis is not straightforward or even feasible. Computing a best approximation using frames, however, can be challenging since it requires solving an ill-conditioned linear system. One consequence of this ill-conditioning is that the coefficients of such a

A virtual element discretisation for the numerical approximation of the three-field formulation of linear poroelasticity introduced in R. Oyarzúa and R. Ruiz-Baier, (SIAM J. Numer. Anal. 54 2951–2973, 2016) is proposed. The treatment is extended to include also the transient case. Appropriate poroelasticity projector operators are introduced and they assist in deriving energy bounds for the time-dependent

The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even

We consider the reduction of parametric families of linear dynamical systems having an affine parameter dependence that allow for low-rank variation in the state matrix. Usual approaches for parametric model reduction typically involve exploring the parameter space to identify representative parameter values and the associated models become the principal focus of model reduction methodology. These

This paper discusses model order reduction of linear time-invariant (LTI) systems over limited frequency intervals within the framework of balanced truncation. Two new frequency-dependent balanced truncation methods are developed, one is single-frequency (SF)-type frequency-dependent balanced truncation to cope with the cases that only a single dominating point of the operating frequency interval is

In this paper, we use the non-conforming Crouzeix-Raviart element method to solve a Stekloff eigenvalue problem arising in inverse scattering. The weak formulation corresponding to this problem is non-self-adjoint and indefinite, and its Crouzeix-Raviart element discretization does not meet the condition of the Strang lemma. We use the standard duality technique to prove an extension of the Strang

In Bayesian analysis, the well-known beta–binomial model is largely used as a conjugate structure, and the beta prior distribution is a natural choice to model parameters defined in the (0,1) range. The Kumaraswamy distribution has been used as a natural alternative to the beta distribution and has received great attention in statistics in the past few years, mainly due to the simplicity and the great

To simulate the two-phase flow of conducting fluids, we propose a coupled model of the Cahn-Hilliard equations and the inductionless and incompressible magnetohydrodynamic (MHD) equations. The model describes the dynamic behavior of conducting fluid under the influence of magnetic field. Based on the “invariant energy quadratization” method, we propose two fully discrete time-marching schemes which