Nonlocal Integral Operator Research Articles

Local inverse estimates for non-local boundary integral operators

We prove local inverse-type estimates for the four non-local boundary integral operators associated with the Laplace operator on a bounded d-dimensional Lipschitz domain Omega for d >= 2 with piecewise smooth boundary. For piecewise polynomial ansatz spaces and d = 2 or 3, the inverse estimates are explicit in both the local mesh width and the approximation order. An application to efficiency estimates in a posteriori error estimation in boundary element methods is given.

Modeling of nonlocal damage-plasticity in beams using isogeometric analysis

A nonlocal damage-plastic Euler–Bernoulli beam model is proposed based on resultant damage-plasticity together with implicit approximations of the nonlocal integral operators. Isogeometric concept is used to approximate the higher order gradient terms, and numerical algorithms and consistent tangents are derived. Test cases are presented to show the important features of the proposed nonlocal beam element. Numerical results show that the proposed nonlocal beam is successfully able to preserve the objectivity of the results and a meaningful convergence is achieved with decreasing mesh size. In addition, various softening responses associated with localized damage mechanisms – including snap-back and snap-through responses – are obtained.

Numerical solution of Navier–Stokes–Korteweg systems by Local Discontinuous Galerkin methods in multiple space dimensions

Compressible liquid–vapor flow with phase transitions can be described by systems of Navier–Stokes–Korteweg type. They extend the Navier–Stokes equations by nonlinear higher-grade terms which take the form of either differential or nonlocal integral operators. A numerical approximation method on the basis of the Local Discontinuous Galerkin method in multiple space dimensions is suggested for isothermal flows. It relies on a specific discretization of a non-conservative formulation. To enhance the performance of the overall scheme two techniques are used: (i) local spatial adaptivity based on gradient indicators for the density and (ii) parallelism based on domain decomposition.The paper concludes with numerical experiments in two and three space dimensions. They show the reliability and efficiency of the proposed approach as well as they demonstrate the applicability of the models for several important phase transition phenomena.

KdV waves in atomic chains with nonlocal interactions

We consider atomic chains with nonlocal particle interactions and prove the existence of near-sonic solitary waves. Both our result and the general proof strategy are reminiscent of the seminal paper by Friesecke and Pego on the KdV limit of chains with nearest neighbor interactions but differ in the following two aspects: First, we allow for a wider class of atomic systems and must hence replace the distance profile by the velocity profile. Second, in the asymptotic analysis we avoid a detailed Fourier pole characterization of the nonlocal integral operators and employ the contraction mapping principle to solve the final fixed point problem.

Peridynamics simulations of geomaterial fragmentation by impulse loads

In this work, we have developed nonlinear peridynamics models of drained and saturated geomaterials, and applied them to simulations of dynamic fragmentation and ejecta formation due to impulse loads. First, we have re-phrased and re-interpreted the non-local state-based peridynamics formulation to connect the non-local integral operator with the local differential operator. Second, we have implemented the Drucker–Prager plasticity model in state-based peridynamics at finite strain. A peridynamics version of the Hughes and Winget algorithm is derived for the constitutive update. Third, we have developed a peridynamics U-p formulation for saturated geomaterials. Fourth, numerical simulations have been carried out to verify the proposed peridynamics formulations in the simulation of geomaterial fragmentation induced by impulse loads. Comparisons of analytical and numerical results indicate that the peridynamics model has the ability to both match traditional continuum compression examples, as well as simulate complex geomaterial fragmentation processes resulting from impulse loads. Copyright © 2015 John Wiley & Sons, Ltd.

On the numerical solution of a nonlocal boundary value problem

We study a nonlinear boundary value problem involving a nonlocal (integral) operator in the coefficients of the unknown function. Provided sufficient conditions for the existence and uniqueness of the solution, for its approximation, we propose a numerical method consisting of a classical discretization of the problem and an algorithm to solve the resulting nonlocal and nonlinear algebraic system by means of some iterative procedures. The second order of convergence is assured by different sufficient conditions, which can be alternatively used in dependence on the given data. The theoretical results are confirmed by several numerical tests.

Conditioning Analysis of Nonlocal Integral Operators in Fractional Sobolev Spaces

We study the conditioning of nonlocal integral operators with singular and integrable kernels in fractional Sobolev spaces. These operators are used, for instance, in peridynamics formulation and nonlocal diffusion. In one dimension (1D), we present sharp quantification of the extremal eigenvalues in all three parameters: size of nonlocality, mesh size, and regularity of the fractional Sobolev space. We accomplish sharpness both rigorously and numerically. For the minimal eigenvalue, we obtain sharpness analytically by using a nonlocal characterization of Sobolev spaces. We verify this estimate by exploiting the Toeplitz property of the stiffness matrix. However, the analytical approach fails to give sharp quantification of the maximal eigenvalue. Hence, in 1D, we take an algebraic approach by directly working with the stiffness matrix entries, which have complicated expressions due to all three parameters. We systematically characterize the nonzero entries and dramatically simplify their expressions by u...

Trace asymptotics for fractional Schrödinger operators

This paper proves an analogue of a result of Bañuelos and Sá Barreto [5] on the asymptotic expansion for the trace of Schrödinger operators on Rd when the Laplacian −Δ, which is the generator of the Brownian motion, is replaced by the non-local integral operator (−Δ)α/2, 0<α<2, which is the generator of the symmetric stable process of order α. These results also extend recent results of Bañuelos and Yildirim [6] where the first two coefficients for (−Δ)α/2 are computed. Some extensions to Schrödinger operators arising from relativistic stable and mixed-stable processes are obtained.

Heat trace of non-local operators

This paper extends results of M. van den Berg on two-term asymptotics for the trace of Schrödinger operators when the Laplacian is replaced by non-local (integral) operators corresponding to rotationally symmetric stable processes and other closely related Lévy processes.

Numerical and perturbative computations of solitary waves of the Benjamin–Ono equation with higher order nonlinearity using Christov rational basis functions

Computation of solitons of the cubically-nonlinear Benjamin–Ono equation is challenging. First, the equation contains the Hilbert transform, a nonlocal integral operator. Second, its solitary waves decay only as O(1/∣ x∣ 2). To solve the integro-differential equation for waves traveling at a phase speed c, we introduced the artificial homotopy H( u XX ) − c u + (1 − δ) u 2 + δu 3 = 0, δ ∈ [0, 1] and solved it in two ways. The first was continuation in the homotopy parameter δ, marching from the known Benjamin–Ono soliton for δ = 0 to the cubically-nonlinear soliton at δ = 1. The second strategy was to bypass continuation by numerically computing perturbation series in δ and forming Padé approximants to obtain a very accurate approximation at δ = 1. To further minimize computations, we derived an elementary theorem to reduce the two-parameter soliton family to a parameter-free function, the soliton symmetric about the origin with unit phase speed. Solitons for higher order Benjamin–Ono equations are also computed and compared to their Korteweg–deVries counterparts. All computations applied the pseudospectral method with a basis of rational orthogonal functions invented by Christov, which are eigenfunctions of the Hilbert transform.

A finite element discretization method for option pricing with the Bates model

In the present paper we present a finite element approach to option pricing in the framework of a well-known stochastic volatility model with jumps, the Bates model. In this model the asset log-returns are assumed to follow a jump-diffusion model where the jump component consists of a Levy process of compound Poisson type, while the volatility dynamics is described by a stochastic differential equation of CIR type, with a mean-reverting drift term and a diffusion component correlated with that of the log-returns. Like in all the Levy models, the option pricing problem can be formulated in terms of an integro-differential equation: for the Bates model the unknown C(S, V, t) (the option price) of the pricing equation depends on three independent variables and the differential operator part turns out to be of parabolic kind, while the nonlocal integral operator is calculated with respect to the Levy measure of the jumps. In this paper we will present a variational formulation of the problem suitable for a finite element approach. After evaluating European options as a benchmark, the method will be applied to Barrier (Down and Out) options.

Comparison of three spectral methods for the Benjamin–Ono equation: Fourier pseudospectral, rational Christov functions and Gaussian radial basis functions

The Benjamin–Ono equation is especially challenging for numerical methods because (i) it contains the Hilbert transform, a nonlocal integral operator, and (ii) its solitary waves decay only as O(1/| x| 2). We compare three different spectral methods for solving this one-space-dimensional equation. The Fourier pseudospectral method is very fast through use of the Fast Fourier Transform (FFT), but requires domain truncation: replacement of the infinite interval by a large but finite domain. Such truncation is unnecessary for a rational basis, but it is simple to evaluate the Hilbert Transform only when the usual rational Chebyshev functions TB n ( x) are replaced by their cousins, the Christov functions; the FFT still applies. Radial basis functions (RBFs) are slow for a given number of grid points N because of the absence of a summation algorithm as fast as the FFT; because RBFs are meshless, however, very flexible grid adaptation is possible.

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

One considers a linear thermoelastic composite medium, which consists of a homogeneous matrix containing a statistically homogeneous random set of ellipsoidal uncoated or coated heterogeneities. It is assumed that the stress–strain constitutive relations of constituents are described by the nonlocal integral operators, whereas the equilibrium and compatibility equations remain unaltered as in classical local elasticity. The general integral equations connecting the stress and strain fields in the point being considered and the surrounding points are obtained. The method is based on a centering procedure of subtraction from both sides of a known initial integral equation their statistical averages obtained without any auxiliary assumptions such as, e.g., effective field hypothesis implicitly exploited in the known centering methods. In a simplified case of using of the effective field hypothesis for analyzing composites with one sort of heterogeneities, one proves that the effective moduli explicitly depend on both the strain and stress concentrator factor for one heterogeneity inside the infinite matrix and does not directly depend on the elastic properties (local or nonlocal) of heterogeneities. In such a case, the Levin’s (1967) formula in micromechanics of composites with locally elastic constituents is generalized to their nonlocal counterpart. A solution of a volume integral equation for one heterogeneity subjected to inhomogeneous remote loading inside an infinite matrix is proposed by the iteration method. The operator representation of this solution is incorporated into the new general integral equation of micromechanics without exploiting of basic hypotheses of classical micromechanics such as both the effective field hypothesis and “ ellipsoidal symmetry” assumption. Quantitative estimations of results obtained by the abandonment of the effective field hypothesis are presented.

Nonlocal description of evaporating drops

We present a theoretical study of the evolution of a drop of pure liquid on a solid substrate, which it wets completely. In a situation where evaporation is significant, the drop does not spread, but instead the drop radius goes to zero in finite time. Our description couples the viscous flow problem to a self-consistent thermodynamic description of evaporation from the drop and its precursor film. The evaporation rate is limited by the diffusion of vapor into the surrounding atmosphere. For flat drops, we compute the evaporation rate as a nonlocal integral operator of the drop shape. Together with a lubrication description of the flow, this permits an efficient numerical description of the final stages of the evaporation problem. We find that the drop radius goes to zero like R∝(t0−t)α, where α has value close to 1/2, in agreement with experiment.

Computation of the Hartree-Fock Exchange by the Tensor-Structured Methods

AbstractWe propose a novel numerical method for fast and accurate evaluation of the exchange part of the Fock operator in the Hartree-Fock equation which is a (nonlocal) integral operator. Usually, this challenging computational problem is solved by analytical evaluation of two-electron integrals using the “analytically separable” Galerkin basis functions, like Gaussians. Instead, we employ the agglomerated “grey-box” numerical computation of the corresponding six-dimensional integrals in the tensor-structured format which does not require analytical separability of the basis set. The point of our method is a low-rank tensor representation of arising functions and operators on an n×n×n Cartesian grid and the implementation of the corresponding multi-linear algebraic operations in the tensor product format. Linear scaling of the tensor operations, including the 3D convolution product, with respect to the one-dimension grid size n enables computations on huge 3D Cartesian grids thus providing the required high accuracy. The presented algorithm for evaluation of the exchange operator and a recent tensor method for the computation of the Coulomb matrix are the main building blocks in the numerical solution of the Hartree-Fock equation by the tensor-structured methods. These methods provide a new tool for algebraic optimization of the Galerkin basis in the case of large molecules.

A note on interface dynamics for convection in porous media

We studied the fluid interface problem through porous media given by two incompressible 2-D fluids of different densities. This problem is mathematically analogous to the dynamics interface for convection in porous media, where the free boundary evolves between fluids with different temperatures. We found a new formula for the evolution equation of the free boundary parameterized as a function in the periodic case. In this formula there is not a principal value in the non-local integral operator involved in the equation, giving a simpler system. Using this formulation, we perform numerical simulations in the stable case (denser fluid below) which shows a strong regularity effect in the periodic interface.

On stabilized models in micromagnetics

The effective behaviour of stationary micromagnetic phenomena is modelled by a convexified Landau–Lifshitz minimization problem for the limit of large and soft magnets Ω with no exchange energy. The numerical simulation of the resulting minimization problem has to overcome difficulties caused by the pointwise side-constraint \(|\bf{m}|\leq 1\) and the stray-field energy on the unbounded domain \(\mathbb{R}^{d}\). A penalty method models the side-constraint and the exterior Maxwell equation is recast via a nonlocal integral operator \(\mathcal{L}\). This paper gives an overview of the available results and implementational details.

Averaging Techniques for the Effective Numerical Solution of Symm's Integral Equation of the First Kind

Averaging techniques for finite element error control, occasionally called ZZ estimators for the gradient recovery, enjoy a high popularity in engineering because of their striking simplicity and universality: One does not even require a PDE to apply the nonexpensive post-processing routines. Recently, averaging techniques have been mathematically proved to be reliable and efficient for various applications of the finite element method. This paper establishes a class of averaging error estimators for boundary integral methods. Symm's integral equation of the first kind with a nonlocal single-layer integral operator serves as a model equation studied both theoretically and numerically. We introduce four new error estimators which are proven to be reliable and efficient up to terms of higher order. The higher-order terms depend on the regularity of the exact solution. Several numerical experiments illustrate the theoretical results and show that the [normally unknown] error is sharply estimated by the proposed estimators, i.e., error and estimators almost coincide.

Numerical Analysis for a Macroscopic Model in Micromagnetics

The macroscopic behavior of stationary micromagnetic phenomena can be modeled by a relaxed version of the Landau--Lifshitz minimization problem. In the limit of large and soft magnets $\Omega$, it is reasonable to exclude the exchange energy and convexify the remaining energy densities. The numerical analysis of the resulting minimization problem, \begin{align*} \min E_0^{**}({\bf m})\text{ amongst }{\bf m}:\Omega\to\mathbb{R}^d\text{ with } |{\bf m}(x)|\le1\text{ for almost every }x\in\Omega, \end{align*} for $d=2,3$, faces difficulties caused by the pointwise side-constraint $|{\bf m}|\le1$ and an integral over the whole space $\mathbb{R}^d$ for the stray field energy. This paper involves a penalty method to model the side-constraint and reformulates the exterior Maxwell equation via a nonlocal integral operator $\mathcal{P}$ acting on functions exclusively defined on $\Omega$. The discretization with piecewise constant discrete magnetizations leads to edge-oriented boundary integrals, the implementation of which and related numerical quadrature are discussed, as are adaptive algorithms for automatic mesh-refinement. A priori and a posteriori error estimates provide a thorough rigorous error control of certain quantities. Three classes of numerical experiments study the penalization, empirical convergence rates, and performance of the uniform and adaptive mesh-refining algorithms.

Solvability of nonlinear boundary-value problems arising in modeling plasma diffusion across a magnetic field and its equilibrium configurations

We study the simplest one-dimensional model of plasma density balance in a tokamak type system, which can be reduced to an initial boundary-value problem for a second-order parabolic equation with implicit degeneration containing nonlocal (integral) operators. The problem of stabilizing nonstationary solutions to stationary ones is reduced to studying the solvability of a nonlinear integro-differential boundary-value problem. We obtain sufficient conditions for the parameters of this boundary-value problem to provide the existence and the uniqueness of a classical stationary solution, and for this solution we obtain the attraction domain by a constructive method.