on a model of nonlocal continuum mechanics part i existence and regularity in this work we assume that the total stored energy functional for a body b depends not only on the local strain field but also on the spatial average of the strain field over the body weighted with an influence kernel we investigate the problem of minimizing the total stored energy subject to a given bulk displacement delta geq 0 after the general setup for this problem is reviewed we give sufficient conditions for an energy minimizing strain field e cdot to satisfy an integro differential euler lagrange equation the result is general and applies to material energies that display a wide variety of singular behavior through analysis of this euler lagrange equation for a special class of influence kernels we arrive at a regularity theorem which ensures that energy minimizing strain fields must be periodic piecewise smooth and possess a finite number of simple discontinuities we then combine this with a well known existence result for relaxed minimization problems to arrive at a genera

interface problems in nonlocal diffusion and sharp transitions between local and nonlocal domains abstract we investigate interface problems in nonlocal diffusion and demonstrate how to reformulate and generalize the classical treatment of interface problems in the presence of nonlocal interactions through formal derivations we show that nonlocal diffusion interface problems converge to their classical local counterparts in the limit of vanishing nonlocality a central focus of this paper is local nonlocal interface problems or interface problems with sharp transitions between local and nonlocal domains such problems can be cast as instances of a nonlocal interface problem with a finite horizon in certain regions and a vanishing horizon in other regions we derive a local nonlocal interface problem and utilize conservation principles to obtain local nonlocal interface conditions comparisons between nonlocal local and local nonlocal interface problems are presented analytically and numerically with a focus on multiscale aspects of nonlocal models induced by their inherent length scales

homoclinic solutions of an integral equation existence and stability abstract we study existence and stability of homoclinic type solutions of a bistable integral equation these are stationary solutions of an integrodifferential equation which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity

propagation of uncertainty in test analysis correlation of substructured spacecraft determining a finite element model s fem fidelity to data or the process of test analysis correlation is a vital component of the overall verification and validation procedure that is applied to analytical models of spacecraft prior to flight the goal of this work is to develop a methodology for studying the effects of uncertainty on metrics used for test analysis correlation of complex spacecraft that are validated on a substructure by substructure basis uncertainty will be propagated from substructures to system for low frequency applications

existence and uniqueness of travelling waves for a neural network a one dimensional scalar neural network with two stable steady states is analysed it is shown that there exists a unique monotone travelling wave front which joins the two stable states some additional properties of the wave such as the direction of its velocity are discussed

motion by curvature by scaling nonlocal evolution equations we prove convergence to a motion by mean curvature by scaling diffusively a nonlinear nonlocal evolution equation this equation was introduced earlier to describe the macroscopic behavior of a ferromagnetic spin system with kac interaction which evolves with glauber dynamics the convergence is proven in any time interval in which the limiting motion is regular

analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit we present a mathematical analysis of the asymptotic preserving scheme proposed in m lemou and l mieussens siam j sci comput 31 2008 pp 334 368 for linear transport equations in kinetic and diffusive regimes we prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles this property is satisfied under an explicitly given cfl condition this condition tends to a parabolic cfl condition for small mean free paths and is close to a convection cfl condition for large mean free paths our analysis is based on very simple energy estimates

an automatic frequency domain modal parameter estimation algorithm in this article a frequency domain modal parameter estimation method is proposed the algorithm automatically separates physical poles from mathematical ones an important issue in the automatization of the algorithm is the inclusion of noise information to estimate the standard deviations of the poles these standard deviations are used together with other features as the inputs of a fuzzy clustering algorithm the clustering algorithm then classifies the poles into the mathematical and physical ones the method requires no user interaction and a parameter is available quantifying the success of the classification

a posteriori error analysis of finite element method for linear nonlocal diffusion and peridynamic models abstract in this paper we present some results on a posteriori error analysis of finite element methods for solving linear nonlocal diffusion and bond based peridynamic models in particular we aim to propose a general abstract frame work for a posteriori error analysis of the peridynamic problems a posteriori error estimators are consequently prompted the reliability and efficiency of the estimators are proved connections between nonlocal a posteriori error estimation and classical local estimation are studied within continuous finite element space numerical experiments 1d are also given to test the theoretical conclusions

asymptotically compatible fourier spectral approximations of nonlocal allen cahn equations we study fourier spectral approximations of a nonlocal allen cahn nac equation that reduces to the conventional allen cahn equation in the local limit we show that the fourier spectral methods are asymptotically compatible in the sense that they provide convergent approximations to both nonlocal and local models furthermore we provide various error estimates in particular it is shown that the numerical solutions of nonlocal models converge to those of the corresponding local models uniformly at a rate of o delta 2 this is achieved by first establishing a similar result for linear nonlocal diffusion equations a careful investigation both analytically and computationally is made of the steady state solutions of nac equations demonstrating how discontinuities may appear in solutions and how they are related to model parameters

a fast and faithful collocation method with efficient matrix assembly for a two dimensional nonlocal diffusion model abstract the peridynamic theory provides an appropriate description of the deformation of a continuous body involving discontinuities or other singularities which cannot be described properly by the classical theory of solid mechanics however the operator in the peridynamic theory is nonlocal so the resulting numerical methods generate dense or full coefficient matrices which require o n 2 of memory where n is the number of unknowns in the discretized system gaussian types of direct solvers which were traditionally used to solve these problems require o n 3 of operations furthermore due to the singularity of the kernel in the peridynamic model the evaluation and assembly of the coefficient matrix can be very expensive numerous numerical experiments have shown that in many practical simulations the evaluation and assembly of the coefficient matrix often constitute the main computational cost the significantly increased computational work and memory requirement of the peridynamic model over those for the classical partial differential equation models severely limit their applications especially in multiple space dimensions we develop a fast and faithful collocation method for a two dimensional nonlocal diffusion model which can be viewed as a scalar valued version of a peridynamic model without using any lossy compression but rather by exploiting the structure of the coefficient matrix the new method reduces the evaluation and assembly of the coefficient matrix by o n reduces the computational work from o n 3 required by the traditional methods to o n log 2 n and the memory requirement from o n 2 to o n numerical results are presented to show the utility of the fast method

a conservative nonlocal convection diffusion model and asymptotically compatible finite difference discretization in this paper we first propose a nonlocal convection diffusion model in which the convection term is constructed in a special upwind manner so that mass conservation and maximum principle are maintained in any space dimension the well posedness of the proposed nonlocal model and its convergence to the classical local convection diffusion model are established a quadrature based finite difference discretization is then developed to numerically solve the nonlocal problem and it is shown to be consistent and unconditionally stable we further demonstrate that the numerical scheme is asymptotically compatible that is the approximate solutions converge to the exact solution of the corresponding local problem when 0 0 and h 0h 0 numerical experiments are also performed to complement the theoretical analysis

cauchy problem of a nonlocal p laplacian evolution equation with nonlocal convection abstract this paper is concerned with the cauchy problem of a nonlocal equation that takes into account convective and p laplacian diffusive effects u t x t r n j x y u y t u x t p 2 u y t u x t d y g f u f u x t with j radially symmetric and g not necessarily symmetric first we prove the existence and uniqueness of solutions and if the convolution kernels j and g are rescaled appropriately we show that solutions of the nonlocal problem converge to the solution of the usual p laplacian diffusion equation with convection finally as a supplementary result we study the asymptotic behavior of solutions as t and give the decay estimate

fast and accurate implementation of fourier spectral approximations of nonlocal diffusion operators and its applications abstract this work is concerned with the fourier spectral approximation of various integral differential equations associated with some linear nonlocal diffusion and peridynamic operators under periodic boundary conditions for radially symmetric kernels the nonlocal operators under consideration are diagonalizable in the fourier space so that the main computational challenge is on the accurate and fast evaluation of their eigenvalues or fourier symbols consisting of possibly singular and highly oscillatory integrals for a large class of fractional power like kernels we propose a new approach based on reformulating the fourier symbols both as coefficients of a series expansion and solutions of some simple ode models we then propose a hybrid algorithm that utilizes both truncated series expansions and high order runge kutta ode solvers to provide fast evaluation of fourier symbols in both one and higher dimensional spaces it is shown that this hybrid algorithm is robust efficient and accurate as applications we combine this hybrid spectral discretization in the spatial variables and the fourth order exponential time differencing runge kutta for temporal discretization to offer high order approximations of some nonlocal gradient dynamics including nonlocal allen cahn equations nonlocal cahn hilliard equations and nonlocal phase field crystal models numerical results show the accuracy and effectiveness of the fully discrete scheme and illustrate some interesting phenomena associated with the nonlocal models

an integrodifferential model for phase transitions stationary solutions in higher space dimensions we study the existence and stability of stationary solutions of an integrodifferential model for phase transitions which is a gradient flow for a free energy functional with general nonlocal integrals penalizing spatial nonuniformity as such this model is a nonlocal extension of the allencahn equation which incorporates long range interactions we find that the set of stationary solutions for this model is much larger than that of the allencahn equation

some spectral properties of the canonical solution operator to on weighted fock spaces abstract we characterize the schatten class membership of the canonical solution operator to acting on l 2 e 2 where is a subharmonic function with a doubling measure the obtained characterization is in terms of as part of our approach we study hankel operators with anti analytic symbols acting on the corresponding fock space of entire functions in l 2 e 2

nonlocal convection diffusion problems and finite element approximations abstract nonlocal models are becoming increasingly popular in numerical simulations of important application problems ranging from anomalous diffusion in heterogeneous environment to crack and fracture in composites in this paper we propose and compare some nonlocal models for convection diffusion problems and study their finite element approximations by piecewise constant and piecewise bilinear basis functions our focus is on the convection dominated case through analysis and numerical experiments we demonstrate the advantage of the upwind models which are more consistent with the physical nature and allow more stable and robust numerical approximations on the other hand the central model may produce unphysical oscillations

spectral collocation method for the time fractional diffusion wave equation and convergence analysis in this paper we consider the numerical solution of the time fractional diffusion wave equation essentially the time fractional diffusion wave equation differs from the standard diffusion wave equation in the time derivative term we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

a mean field equation of motion for the dynamic ising model a mean field type of approximation is used to derive two differential equations one approximately representing the average behavior of the ising model with glauber spin flip stochastic dynamics and the other doing the same for kawasaki spin exchange dynamics the proposed new equations are compared with the cahn allen and cahn hilliard equations representing the same systems and with information about the exact behavior of the microscopic models

a nonlocal nonlinear diffusion equation with blowing up boundary conditions we deal with boundary value problems prescribing dirichlet or neumann boundary conditions for a nonlocal nonlinear diffusion operator which is analogous to the porous medium equation first we prove existence uniqueness and the validity of a comparison principle for these problems next we impose boundary data that blow up in finite time and study the behavior of the solutions

phase segregation dynamics in particle systems with long range interactions i macroscopic limits we present and discuss the derivation of a nonlinear nonlocal integrodifferential equation for the macroscopic time evolution of the conserved order parameter r t of a binary alloy undergoing phase segregation our model is ad dimensional lattice gas evolving via kawasaki exchange with respect to the gibbs measure for a hamiltonian which includes both short range local and long range nonlocal interactions the nonlocal part is given by a pair potential dj x y 0 x and y in d in the limit 0 the macroscopic evolution is observed on the spatial scale 1 and time scale 2 i e the density r t is the empirical average of the occupation numbers over a small macroscopic volume element centered atr x a rigorous derivation is presented in the case in which there is no local interaction in a subsequent paper part ii we discuss the phase segregation phenomena in the model in particular we argue that the phase boundary evolutions arising as sharp interface limits of the family of equations derived in this paper are the same as the ones obtained from the corresponding limits for the cahn hilliard equation

mathematical and numerical analysis of linear peridynamic models with nonlocal boundary conditions in this paper we study the linear bond based nonlocal peridynamic models with a particular focus on problems associated with nonstandard nonlocal displacement loading conditions both stationary and time dependent problems are considered for a one dimensional scalar equation defined on a finite bar and for a two dimensional system defined on a square the related peridynamic operators and associated functional spaces are analyzed applications to the numerical analysis of the finite dimensional approximations to peridynamic models are also illustrated

probabilistic investigation of sensitivities of advanced test analysis model correlation methods the industry standard method used to validate finite element models involves correlation of test and analysis mode shapes using reduced test analysis models tams some organizations even require this model validation approach considerable effort is required to choose sensor locations and to create a suitable tam so that the test and analysis mode shapes will be orthogonal to within the required tolerance this work uses a probabilistic framework to understand and quantify the effect of small errors in the test mode shapes on test analysis orthogonality using the proposed framework test orthogonality is a probabilistic metric and the problem becomes one of choosing sensor placement and tam generation techniques that assure that the orthogonality has a high probability of being within an acceptable range if the model is correct even though the test measurements are contaminated with random errors a simple analytical metric is derived that is shown to give a good estimate of the sensitivity of a tam to errors in the test mode shapes for a certain noise model these ideas are then applied to a generic satellite system using tams generated by the static modal and improved reduced system irs reduction methods experimental errors are simulated for a set of mode shapes and monte carlo simulation is used to estimate the probability that the orthogonality metric exceeds a threshold due to experimental error alone for the satellite system considered here the orthogonality calculation is highly sensitive to experimental errors so a set of noisy mode shapes has a small probability of passing the orthogonality criteria for some of the tams a number of sensor placement techniques are used in this study and the comparison reveals that for this system the modal tam is twice as sensitive to errors on the test mode shapes when it is created on a sensor set optimized for the static tam rather than one that was optimized specifically for the modal tam these findings are evaluated in light of previously published studies of tam sensitivity and special attention is given to gordis s theory which suggest that tam sensitivity is related to the natural frequencies of the structure when all measurement points are fixed some aspects of tam sensitivity are problem dependent so this one work cannot achieve a conclusive ranking of all of the available methodologies instead this work focuses on presenting a set of tools and a probabilistic framework that can be used to correctly quantify tam sensitivity and demonstrating the approach for one dynamic system and for a particular probabilistic model for the errors contaminating the test mode shapes

intermediate hankel operators on the fock space abstract we construct a natural sequence of middle hankel operators on the fock space i e operators which are intermediate between the small and big hankel operators these operators are connected with the minimal l 2 norm solution operator to n as well as to the polyanalytic fock spaces

a nonlocal convection diffusion equation in this paper we study a nonlocal equation that takes into account convective and diffusive effects ut j u u g f u f u in rd with j radially symmetric and g not necessarily symmetric first we prove existence uniqueness and continuous dependence with respect to the initial condition of solutions this problem is the nonlocal analogous to the usual local convection diffusion equation ut u b f u in fact we prove that solutions of the nonlocal equation converge to the solution of the usual convection diffusion equation when we rescale the convolution kernels j and g appropriately finally we study the asymptotic behaviour of solutions as t when f u u q 1u with q 1 we find the decay rate and the first order term in the asymptotic regime

generalized motion by mean curvature as a macroscopic limit of stochastic ising models with long range interactions and glauber dynamics we study themacroscopic limit of an appropriately rescaledstochastic ising model withlong range interactions evolving withglauber dynamics as well as the correspondingmean field equation which is nonlinear and nonlocal in the limit we obtain an interface evolving with normal velocity k wherek isthe mean curvature and thetransport coefficient is identified by aneffective green kubo type formula the above assertions are valid for all positive times the motion of the interface being interpreted in theviscosity sense after the onset of the geometric singularities

the polyanalytic ginibre ensembles for integers n q 1 2 3 let pol n q denote the mathbb c linear space of polynomials in z and bar z of degree n 1 in z and of degree q 1 in bar z we supply pol n q with the inner product structure of begin aligned l 2 bigl mathbb c mathrm e m z 2 mathrm d a bigr quad mbox where mathrm d a z pi 1 mathrm d x mathrm d y z x mathrm i y end aligned the resulting hilbert space is denoted by pol m n q here it is assumed that m is a positive real we let k m n q denote the reproducing kernel of pol m n q and study the associated determinantal process in the limit as m n while n m o 1 the number q the degree of polyanalyticity is kept fixed we call these processes polyanalytic ginibre ensembles because they generalize the ginibre ensemble the eigenvalue process of random normal matrices with gaussian weight there is a physical interpretation in terms of a system of free fermions in a uniform magnetic field so that a fixed number of the first landau levels have been filled we consider local blow ups of the polyanalytic ginibre ensembles around points in the spectral droplet which is here the closed unit disk bar mathbb d z in mathbb c z le1 we obtain asymptotics for the blow up process using a blow up to characteristic distance m 1 2 the typical distance is the same both for interior and for boundary points of bar mathbb d this amounts to obtaining the asymptotical behavior of the generating kernel k m n q following ameur et al in commun pure appl math 63 12 1533 1584 2010 the asymptotics of the k m n q are rather conveniently expressed in terms of the berezin measure and density open image in new window for interior points z 1 in contrast for exterior points z 1 we have instead that mathrm d b langle z rangle m n q w to mathrm d omega w z mathbb d e where mathrm d omega w z mathbb d e is the harmonic measure at z with respect to the exterior disk mathbb d e w in mathbb c w 1 for boundary points z 1 the berezin measure mathrm d b langle z rangle m n q converges to the unit point mass at z as with interior points but the blow up to the scale m 1 2 exhibits quite different behavior at boundary points compared with interior points we also obtain the asymptotic boundary behavior of the 1 point function at the coarser local scale q 1 2 m 1 2

a spectral collocation method for multidimensional nonlinear weakly singular volterra integral equation abstract this paper is concerned with the convergence properties of chebyshev spectral collocation method when used to approximate the solution of multidimensional nonlinear volterra integral equation of the second kind with a weakly singular kernel we consider the case that the underlying solution is sufficiently smooth the chebyshev collocation discretization is proposed for this equation in the present paper we provide a rigorous error analysis which justifies that the errors of approximate solution decay exponentially in weighted l 2 norm and l norm numerical results are presented to demonstrate the effectiveness of the spectral method

a convergent adaptive finite element algorithm for nonlocal diffusion and peridynamic models in this paper we propose an adaptive finite element algorithm for the numerical solution of a class of nonlocal models which correspond to nonlocal diffusion equations and linear scalar peridynamic models with certain nonintegrable kernel functions the convergence of the adaptive finite element algorithm is rigorously derived with the help of several basic ingredients such as the upper bound of the estimator the estimator reduction and the orthogonality property we also consider how the results are affected by the horizon parameter delta which characterizes the range of nonlocality numerical experiments are performed to verify our theoretical findings

analysis and approximation of nonlocal diffusion problems with volume constraints a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in mbrn the nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion including a notion of nonlocal flux the ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory in particular it is shown that fractional laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider the numerous applications elucidate different interpretations of the operator and the associated governing equations for example a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process sufficie

a triangular spectral element method using fully tensorial rational basis functions a rational approximation in a triangle is proposed and analyzed in this paper the rational basis functions in the triangle are obtained from the polynomials in the reference square through a collapsed coordinate transform optimal error estimates for the l 2 and h 1 0 orthogonal projections are derived with upper bounds expressed in the original coordinates in the triangle it is shown that the rational approximation is as accurate as the polynomial approximation in the triangle illustrative numerical results which are in agreement with the theoretical estimates are presented

robustness of system equivalent reduction expansion process on spacecraft structure model validation test analysis models are used in the validation of the nite element models of spacecraft structures here a probabilistic approach is used to assess the robustness of a system equivalent reduction expansion process based testanalysis model when experimental and analytical modes contain different levels of inaccuracy the approach is applied to three spacecraft models and monte carlo simulations were used to determine the sensitivity of the normalized cross orthogonality check to the system equivalent reduction expansion process reduced matrix the effect of parameters used in this reduction and the amount of inaccuracies that can be tolerated in the modes before failing the normalized cross orthogonality check were also determined the results show that the probability to pass the normalized cross orthogonality check is highly determined by the number of modes used in the reduction the relation between capability of the nite element models to predict the frequency response function and the quality of the model validation determined using normalized cross orthogonality check is also investigated and it is observed that the quantities are not always correlated this study also shows that the sensor locations can be optimally chosen using the system equivalent reduction expansion process reduced mass matrix and this can increase the probability to pass the normalized cross orthogonality check

analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations we consider the numerical solution of nonlocal constrained value problems associated with linear nonlocal diffusion and nonlocal peridynamic models two classes of discretization methods are presen

a spectral collocation method for nonlocal diffusion equations with volume constrained boundary conditions abstract the nonlocal diffusion model describes the diffusion process of solutes in complex media properly while the classical theory of partial differential equations can not provide an appropriate description the purpose of this paper is to provide illustrations from both theoretical and numerical perspectives of the computation of nonlocal diffusion models by legendre collocation methods compared to local numerical methods legendre collocation methods can achieve a fixed accuracy with much fewer unknowns whenever the computational domain is regular and the solutions are sufficiently smooth this paper is a groundwork towards efficient high order methods including spectral and spectral element methods for nonlocal diffusion equations with volume constrained boundary conditions

numerical solutions for solving time fractional fokker planck equations based on spectral collocation methods abstract in this paper we consider the numerical solution of the time fractional fokker planck equation we propose a spectral collocation method in both temporal and spatial discretizations with a spectral expansion of jacobi interpolation polynomial for this equation the convergence of the method is rigorously established numerical tests are carried out to confirm the theoretical results

a coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions we develop and analyze an optimization based method for the coupling of nonlocal and local diffusion problems with mixed volume constraints and boundary conditions the approach formulates the coupling as a control problem where the states are the solutions of the nonlocal and local equations the objective is to minimize their mismatch on the overlap of the nonlocal and local domains and the controls are virtual volume constraints and boundary conditions when some assumptions on the kernel functions hold we prove that the resulting optimization problem is well posed and discuss its implementation using sandia s agile software components toolkit the latter provides the groundwork for the development of engineering analysis tools while numerical results for nonlocal diffusion in three dimensions illustrate key properties of the optimization based coupling method