This paper provides iterative construction of a common solution associated with a class of equilibrium problems and split convex feasibility problems. In particular, we are interested in the equilibrium problems defined with respect to the pseudomonotone and Lipschitz-type continuous equilibrium problem together with the generalized split null point problems in real Hilbert spaces. We propose an iterative

The paper shows a new weak approximation for generalized expectation of composition of a Schwartz tempered distribution and a solution to stochastic differential equation. Any order discretization is provided by using stochastic weights which do not depend on the Schwartz distribution. The error bound is obtained through stochastic analysis, which is consistent with the results of numerical experiments

This paper considers the semidiscrete Galerkin finite element approximation for time fractional diffusion equations with memory in a bounded convex polygonal domain. We use novel energy arguments in conjunction with repeated applications of time integral operators to study the error analysis. Our error estimates cover both smooth and nonsmooth initial data cases. Since the continuous solution u of

In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a key contribution, the steady-state Poisson-Nernst-Planck equations are studied systematically and rigorous analysis for a residual-based a posteriori error estimate

In this paper, we develop a adaptive finite volume method with the truncation of the nonlocal boundary operators for the wave scattering by periodic structures. The related truncation parameters are chosen through sharp a posteriori error estimate of the finite volume method. The crucial part of the a posteriori error analysis is to develop a duality argument technique and use a L2-orthogonality property

In this paper, we consider the unconditionally optimal error estimates of the linearized backward Euler scheme with the weak Galerkin finite element method for semilinear parabolic equations. With the error splitting technique and elliptic projection, the optimal error estimates in L2-norm and the discrete H1-norm are derived without any restriction on the time stepsize. Numerical results on both polygonal

We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength

This paper deals with the numerical approximation of solutions of Stokes and Brinkman systems using meshless methods. The aim is to solve a problem containing a nonzero body force, starting from the well known decomposition in terms of a particular solution and the solution of a homogeneous force problem. We propose two methods for the numerical construction of a particular solution. One method is

We propose an Extended Hybrid High-Order scheme for the Poisson problem with solution possessing weak singularities. Some general assumptions are stated on the nature of this singularity and the remaining part of the solution. The method is formulated by enriching the local polynomial spaces with appropriate singular functions. Via a detailed error analysis, the method is shown to converge optimally

We analyze a new stabilized dual-mixed method applied to incompressible linear elasticity problems, considering two kinds of data on the boundary of the domain: non homogeneous Dirichlet and mixed boundary conditions. In this approach, we circumvent the standard use of the rotation to impose weakly the symmetry of stress tensor. We prove that the new variational formulation and the corresponding Galerkin

An efficient direct solver for solving the Lippmann–Schwinger integral equation modeling acoustic scattering in the plane is presented. For a problem with N degrees of freedom, the solver constructs an approximate inverse in \(\mathcal {O}(N^{3/2})\) operations and then, given an incident field, can compute the scattered field in \(\mathcal {O}(N \log N)\) operations. The solver is based on a previously

We present rigorous analysis on the error bound and conservation laws of a fourth-order compact finite difference scheme for Zakharov system (ZS) with a dimensionless parameter ε ∈ (0,1], which is inversely proportional to the acoustic speed. In the subsonic limit regime, i.e., 0 < ε ≪ 1, the solutions have highly oscillatory waves and outgoing initial layers due to the perturbation from wave operator

In this paper, we design two classes of high-accuracy conservative numerical algorithms for the nonlinear Klein-Gordon-Schrödinger system in two dimensions. By introducing the energy quadratization technique, we first transform the original system into an equivalent one, where the energy is modified as a quadratic form. The Gauss-type Runge-Kutta method and the Fourier pseudo-spectral method are then

We study the problem of finding a common element that solves the multiple-sets feasibility and equilibrium problems in real Hilbert spaces. We consider a general setting in which the involved sets are represented as level sets of given convex functions, and propose a constructible linear approximation scheme that involves the subgradient of the associated convex functions. Strong convergence of the

The total positivity of collocation, Wronskian and Gram matrices corresponding to bases of the form (eλt,teλt,…,tneλt) is analyzed. A bidiagonal decomposition providing the accurate numerical resolution of algebraic linear problems with these matrices is derived. The numerical experimentation confirms the accuracy of the proposed methods.

Based on finite element discretization and a fully overlapping domain decomposition, we propose and study some parallel iterative subgrid stabilized algorithms for the simulation of the steady Navier-Stokes equations with high Reynolds numbers, where the quadratic equal-order elements are used for the velocity and pressure approximations, and the subgrid-scale model based on an elliptic projection

Phase-field modeling is strongly influenced by the shape of a free energy functional. In the theory of thermodynamics, it is a logarithmic type potential that is legitimate for modeling and simulating binary systems. Nevertheless, a tremendous amount of works have been dedicated to phase-field equations driven by 4-th double-well potentials as a polynomial approximation to the logarithmic type, which

This paper proposes a new type of multilevel method for solving the Steklov eigenvalue problem based on Newton’s method. In this iteration method, solving the Steklov eigenvalue problem is replaced by solving a small-scale eigenvalue problem on the coarsest mesh and a sequence of augmented linear problems on refined meshes, derived by Newton step. We prove that this iteration scheme obtains the optimal

Hierarchical \({\mathscr{H}}^{2}\)-matrices are asymptotically optimal representations for the discretizations of non-local operators such as those arising in integral equations or from kernel functions. Their O(N) complexity in both memory and operator application makes them particularly suited for large-scale problems. As a result, there is a need for software that provides support for distributed

For the Oseen problem, we present a new stabilized virtual element method on polygonal meshes that allows us to employ “equal-order” virtual element pairs to approximate both velocity and pressure. By introducing the local projection type stabilization terms to the virtual element method, the method can not only circumvent the discrete Babuška-Brezzi condition, but also maintain the favorable stability

The paper introduces an one-step optimization method for solving a monotone equilibrium problem including a Lipschitz-type condition in a Hilbert space. The method uses variable stepsizes and is constructed by the proximal-like mapping associated with the cost bifunction and incorporated with regularization terms. Comparing with the extragradient-like methods, our new method has an elegant and simple

This paper is devoted to an optimal control problem of a coupled spin drift-diffusion Landau–Lifshitz–Gilbert system describing the interplay of magnetization and spin accumulation in magnetic-nonmagnetic multilayer structures, where the control is given by the electric current density. A variational approach is used to prove the existence of an optimal control. The first-order necessary optimality

In this paper, a novel three-point fourth-order compact operator is considered to construct new linearized conservative compact finite difference scheme for the symmetric regularized long wave (SRLW) equations based on the reduction order method with three-level linearized technique. The discrete conservative laws, boundedness and unique solvability are studied. The convergence order \(\mathcal {O}(\tau

This paper is devoted to the study of a time-discrete scheme and its corresponding fully discretization approximating a d-dimensional chemotaxis model describing tumor invasion, d ≤ 3. This model describes the chemotactic attraction experienced by the tumor cells and induced by a so-called active extracellular matrix, which is a chemical signal produced by a biological reaction between the extracellular

Recently, a Monte Carlo approach was proposed for processing highly redundant continuous frames. In this paper, we present and analyze applications of this new theory. The computational complexity of the Monte Carlo method relies on the continuous frame being so-called linear volume discretizable (LVD). The LVD property means that the number of samples in the coefficient space required by the Monte

In this paper, we are concerned about a time-domain carpet cloak model, which was originally derived in our previous work Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014). Some finite element schemes have been developed for this model and used to simulate the cloaking phenomenon in Li et al. (SIAM J. Appl. Math., 74(4), pp. 1136–1151, 2014) and Li et al. (Methods Appl. Math., 19(2), pp.

A priori error analysis of the finite element approximation of Stokes equations under slip boundary condition of friction type has been centered on the interpolation error on the slip zone. In this work, we propose a novel approach based on the approximation of the tangential component of traction force by a truncated (cutoff) function. More precisely, we carry out (i) a complete analysis of the truncated

We present a method to stabilize bases with local supports by means of extension. It generalizes the known approach for tensor product B-splines to a much broader class of functions, which includes hierarchical and weighted variants of polynomial, trigonometric, and exponential splines, but also box splines, T-splines, and other function spaces of interest with a local basis. Extension removes elements

Fast and high-order accurate algorithms for three-dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three-dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier

Simulating the flow of water in district heating networks requires numerical methods which are independent of the CFL condition. We develop a high order scheme for networks of advection equations allowing large time steps. With the MOOD technique, unphysical oscillations of nonsmooth solutions are avoided. In numerical tests, the applicability to real networks is shown.

The purpose of this paper is twofold. The first is to give some new structural results for the invertibility of Bessel multipliers. Secondly, as applications of these results, we provide some conditions regarding the scaling sequence c = {cn}n which can be used in the role of the scalability of a given frame, a notion which has found more and more applications in the last decade. More precisely, we

AbstractIn this work, an r-linearly converging adaptive solver is constructed for parabolic evolution equations in a simultaneous space-time variational formulation. Exploiting the product structure of the space-time cylinder, the family of trial spaces that we consider are given as the spans of wavelets-in-time and (locally refined) finite element spaces-in-space. Numerical results illustrate our

We consider approximations formed by the sum of a linear combination of given functions enhanced by ridge functions—a Linear/Ridge expansion. For an explicitly or implicitly given objective function, we reformulate finding a best Linear/Ridge expansion in terms of an optimization problem. We introduce a particle grid algorithm for its solution. Several numerical results underline the flexibility, robustness

We give a systematic self-contained exposition of how to construct geometrically decomposed bases and degrees of freedom in finite element exterior calculus. In particular, we elaborate upon a previously overlooked basis for one of the families of finite element spaces, which is of interest for implementations. Moreover, we give details for the construction of isomorphisms and duality pairings between

As an extension of the traditional principal component analysis, the multi-view canonical correlation analysis (MCCA) aims at reducing m high dimensional random variables \(\boldsymbol {s}_{i}\in \mathbb {R}^{n_{i}}~(i=1,2,\ldots ,m)\) by proper projection matrices \(X_{i}\in \mathbb {R}^{n_{i}\times \ell }\) so that the m reduced ones \(\boldsymbol {y}_{i}=X_{i}^{\mathrm {T}}\boldsymbol {s}_{i}\in

Panel-based, kernel-split quadrature is currently one of the most efficient methods available for accurate evaluation of singular and nearly singular layer potentials in two dimensions. However, it can fail completely for the layer potentials belonging to the modified Helmholtz, modified biharmonic, and modified Stokes equations. These equations depend on a parameter, denoted α, and kernel-split quadrature

Collocation boundary element methods for integral equations are easier to implement than Galerkin methods because the elements of the discretisation matrix are given by lower-dimensional integrals. For that same reason, the matrix assembly also requires fewer computations. However, collocation methods typically yield slower convergence rates and less robustness, compared to Galerkin methods. We explore

The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we propose a wavelet-based adaptive version of the POD (the

It is well known in the literature that standard hierarchical matrix (\({\mathscr{H}}\)-matrix)-based methods, although very efficient for asymptotically smooth kernels, are not optimal for oscillatory kernels. In a previous paper, we have shown that the method should nevertheless be used in the mechanical engineering community due to its still important data compression rate and its straightforward

The technique of complex scaling for time harmonic wave-type equations relies on a complex coordinate stretching to generate exponentially decaying solutions. In this work, we use a Galerkin method with ansatz functions with infinite support to discretize complex-scaled scalar Helmholtz-type resonance problems with inhomogeneous exterior domains. We show super-algebraic convergence of the method with

We consider a system of coupled equations modeling a shallow water flow with solute transport and introduce an artificial dissipation in order to improve the dissipation properties of the original cell-vertex central-upwind numerical scheme applied to these equations. Namely, a formulation is proposed which involves an artificial dissipation parameter and guarantees a consistency property between the

Persistence diagrams are one of the main tools in the field of Topological Data Analysis (TDA). They contain fruitful information about the shape of data. The use of machine learning algorithms on the space of persistence diagrams proves to be challenging as the space lacks an inner product. For that reason, transforming these diagrams in a way that is compatible with machine learning is an important

The theory of Chebyshev approximation has been extensively studied. In most cases, the optimality conditions are based on the notion of alternance or alternating sequence (that is, maximal deviation points with alternating deviation signs). There are a number of approximation methods for polynomial and polynomial spline approximation. Some of them are based on the classical de la Vallée-Poussin procedure

The algebraic characterization of dual univariate interpolating subdivision schemes is investigated. Specifically, we provide a constructive approach for finding dual univariate interpolating subdivision schemes based on the solutions of certain associated polynomial equations. The proposed approach also makes it possible to identify conditions for the existence of the sought schemes.

Since its introduction, the virtual element method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the convergence on a theoretical basis are therefore quite general. They have been deduced in analogy to the similar conditions developed in the finite element method (FEM)

In this paper, we consider the reconstruction of the shape and the impedance function of an obstacle from measurements of the scattered field at a collection of receivers outside the object. The data is assumed to be generated by plane waves impinging on the unknown obstacle from multiple directions and at multiple frequencies. This inverse problem can be reformulated as an optimization problem: that

In this paper, we introduce a family of rational approximations of the reciprocal of a ϕ-function involved in the explicit solutions of certain linear differential equations, as well as in integration schemes evolving on manifolds. The derivation and properties of this family of approximations applied to scalar and matrix arguments are presented. Moreover, we show that the matrix functions computed

We consider the following repulsive-productive chemotaxis model: find u ≥ 0, the cell density, and v ≥ 0, the chemical concentration, satisfying $$ \left\{ \begin{array}{l} \partial_t u - {\Delta} u - \nabla\cdot (u\nabla v)=0 \ \ \text{ in}\ {\Omega},\ t>0,\\ \partial_t v - {\Delta} v + v = u^p \ \ { in}\ {\Omega},\ t>0, \end{array} \right. $$(1) with p ∈ (1, 2), \({\Omega }\subseteq \mathbb {R}^{d}\)

In this paper, we consider the noisy phase retrieval problem under the measurements of Fourier transforms with complex random masks. Here two kinds of Riemannian optimization algorithms, namely, Riemannian gradient descent algorithm (RGrad) and Riemannian conjugate gradient descent algorithm (RCG), are presented to solve such problem from these special but widely used measurements in practical applications

In this paper we present the first known deterministic algorithm for the construction of multiple rank-1 lattices for the approximation of periodic functions of many variables. The algorithm works by converting a potentially large reconstructing single rank-1 lattice for some d-dimensional frequency set I ⊂{0,…,N − 1}d into a collection of much smaller rank-1 lattices which allow for accurate and efficient

The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization

In this paper, we introduce a computational framework for recovering a high-resolution approximation of an unknown function from its low-resolution indirect measurements as well as high-resolution training observations by merging the frameworks of generalized sampling and functional principal component analysis. In particular, we increase the signal resolution via a data-driven approach, which models

In this paper, we present a divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes. The velocity and the pressure are discretized by the H(div) virtual element and discontinuous piecewise polynomials, respectively. An additional polynomial space that lives on the element edges is introduced to approximate the tangential trace of the velocity. The velocity at

We present a novel family of C1 quadrilateral finite elements, which define global C1 spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the construction by Brenner and Sung (J. Sci. Comput. 22(1-3), 83-118, 2005), which is based on polynomial elements of tensor-product degree p ≥ 6, to all degrees p ≥ 3. The proposed C1 quadrilateral is based upon the