Convergence analysis of a discontinuous Galerkin method for a sub-diffusion equation

We employ a piecewise-constant, discontinuous Galerkin method for the time discretization of a sub-diffusion equation. Denoting the maximum time step by k, we prove an a priori error bound of order k under realistic assumptions on the regularity of the solution. We also show that a spatial discretization using continuous, piecewise-linear finite elements leads to an additional error term of order h ² max (1,logk ^− 1). Some simple numerical examples illustrate this convergence behaviour in practice. We thank the University of New South Wales for financial support provided by a Faculty Research Grant.     

A second-order accurate numerical method for a fractional wave equation

We study a generalized Crank–Nicolson scheme for the time discretization of a fractional wave equation, in combination with a space discretization by linear finite elements. The scheme uses a non-uniform grid in time to compensate for the singular behaviour of the exact solution at t = 0. With appropriate assumptions on the data and assuming that the spatial domain is convex or smooth, we show that the error is of order k ² + h ², where k and h are the parameters for the time and space meshes, respectively.     

Existence and Attractivity of <Emphasis Type="Italic">k</Emphasis>-Pseudo Almost Automorphic Sequence Solution of a Model of Bidirectional Neural Networks

In this paper we discuss the existence and global attractivity of k-pseudo almost automorphic sequence solution of a model of bidirectional cellular neural networks. We consider the corresponding difference equation analogue of the model system using suitable discretization method and obtain certain conditions for the existence of solution. The k-pseudo almost automorphic sequence solutions generalize the results of pseudo almost periodic, almost periodic and almost automorphic sequences solutions. Moreover the results proved in this paper are new and compliment the existing one.     

On solving the planar k-centrum problem with Euclidean distances

This paper presents a solution procedure based on a gradient descent method for the k-centrum problem in the plane. The particular framework of this problem for the Euclidean norm leads to bisector lines whose analytical expressions are easy to handle. This allows us to develop different solution procedures which are tested on different problems and compared with existing procedures in the literature of Location Analysis. The computational analysis reports that our procedures provide better results than the existing ones for the k-centrum problem.     

A theoretical analysis of a new finite volume scheme for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

We consider the wave equation, on a multidimensional spatial domain. The discretization of the spatial domain is performed using a general class of nonconforming meshes which has been recently studied for stationary anisotropic heterogeneous diffusion problems, see Eymard et al. (IMAJ Numer Anal 30 (2010), 1009–1043). The discretization in time is performed using a uniform mesh. We derive a new implicit finite volume scheme approximating the wave equation and we prove error estimates of the finite volume approximate solution in several norms which allow us to derive error estimates for the approximations of the exact solution and its first derivatives. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, the convergence order is \begin{align*} h_\mathcal{D}\end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. This estimate is valid under the regularity assumption \begin{align*}u\in C^3(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*} for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Mixed finite element methods for incompressible flow: Stationary Stokes equations

In this article, we develop and analyze a mixed finite element method for the Stokes equations. Our mixed method is based on the pseudostress‐velocity formulation. The pseudostress is approximated by the Raviart‐Thomas (RT) element of index k ≥ 0 and the velocity by piecewise discontinuous polynomials of degree k. It is shown that this pair of finite elements is stable and yields quasi‐optimal accuracy. The indefinite system of linear equations resulting from the discretization is decoupled by the penalty method. The penalized pseudostress system is solved by the H(div) type of multigrid method and the velocity is then calculated explicitly. Alternative preconditioning approaches that do not involve penalizing the system are also discussed. Finally, numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Non-horizontal submanifolds and coarea formula

Let k be a positive integer and let m be the dimension of the horizontal subspace of a stratified group. Under the condition k ≤ m, we show that all submanifolds of codimension k are generically non-horizontal. For these submanifolds, we prove an area-type formula that allows us to compute their Q − k dimensional spherical Hausdorff measure. Finally, we observe that a.e. level set of a sufficiently regular vector-valued mapping on a stratified group is a non-horizontal submanifold. This allows us to establish a sub-Riemannian coarea formula for vector-valued Riemannian Lipschitz mappings on stratified groups.     

Robust multigrid preconditioners for the high‐contrast biharmonic plate equation

We study the high‐contrast biharmonic plate equation with Hsieh–Clough–Tocher discretization. We construct a preconditioner that is robust with respect to contrast size and mesh size simultaneously based on the preconditioner proposed by Aksoylu et al. (Comput. Vis. Sci. 2008; 11 :319–331). By extending the devised singular perturbation analysis from linear finite element discretization to the above discretization, we prove and numerically demonstrate the robustness of the preconditioner. Therefore, we accomplish a desirable preconditioning design goal by using the same family of preconditioners to solve the elliptic family of PDEs with varying discretizations. We also present a strategy on how to generalize the proposed preconditioner to cover high‐contrast elliptic PDEs of order 2k, k>2. Moreover, we prove a fundamental qualitative property of the solution to the high‐contrast biharmonic plate equation. Namely, the solution over the highly bending island becomes a linear polynomial asymptotically. The effectiveness of our preconditioner is largely due to the integration of this qualitative understanding of the underlying PDE into its construction. Copyright © 2010 John Wiley & Sons, Ltd.     

Recursive Marginal Quantization of the Euler Scheme of a Diffusion Process

Abstract

We propose a new approach to quantize the marginals of the discrete Euler diffusion process. The method is built recursively and involves the conditional distribution of the marginals of the discrete Euler process. Analytically, the method raises several questions like the analysis of the induced quadratic quantization error between the marginals of the Euler process and the proposed quantizations. We show in particular that at every discretization step t_k of the Euler scheme, this error is bounded by the cumulative quantization errors induced by the Euler operator, from times t₀ = 0 to time t_k. For numerics, we restrict our analysis to the one-dimensional setting and show how to compute the optimal grids using a Newton–Raphson algorithm. We then propose a closed formula for the companion weights and the transition probabilities associated to the proposed quantizations. This allows us to quantize in particular diffusion processes in local volatility models by reducing dramatically the computational complexity of the search of optimal quantizers while increasing their computational precision with respect to the algorithms commonly proposed in this framework. Numerical tests are carried out for the Brownian motion and for the pricing of European options in a local volatility model. A comparison with the Monte Carlo simulations shows that the proposed method may sometimes be more efficient (w.r.t. both computational precision and time complexity) than the Monte Carlo method.     

Nonparametric Bayesian Assessment of the Order of Dependence for Binary Sequences

This article discusses inference on the order of dependence in binary sequences. The proposed approach is based on the notion of partial exchangeability of order k. A partially exchangeable binary sequence of order k can be represented as a mixture of Markov chains. The mixture is with respect to the unknown transition probability matrix θ. We use this defining property to construct a semiparametric model for binary sequences by assuming a nonparametric prior on the transition matrix θ. This enables us to consider inference on the order of dependence without constraint to a particular parametric model. Implementing posterior simulation in the proposed model is complicated by the fact that the dimension of θ changes with the order of dependence k. We discuss appropriate posterior simulation schemes based on a pseudo prior approach. We extend the model to include covariates by considering an alternative parameterization as an autologistic regression which allows for a straightforward introduction of covariates. The regression on covariates raises the additional inference problem of variable selection. We discuss appropriate posterior simulation schemes, focusing on inference about the order of dependence. We discuss and develop the model with covariates only to the extent needed for such inference.     

A new coupled approach high accuracy numerical method for the solution of 3D non-linear biharmonic equations

In this paper, we derive a new fourth order finite difference approximation based on arithmetic average discretization for the solution of three-dimensional non-linear biharmonic partial differential equations on a 19-point compact stencil using coupled approach. The numerical solutions of unknown variable u(x,y,z) and its Laplacian ∇²u are obtained at each internal grid point. The resulting stencil algorithm is presented which can be used to solve many physical problems. The proposed method allows us to use the Dirichlet boundary conditions directly and there is no need to discretize the derivative boundary conditions near the boundary. We also show that special treatment is required to handle the boundary conditions. The new method is tested on three problems and the results are compared with the corresponding second order approximation, which we also discuss using coupled approach.     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016     

Finite element method for Grwünwald–Letnikov time-fractional partial differential equation

In this article, we consider the finite element methods (FEM) for Grwünwald–Letnikov time-fractional diffusion equation, which is obtained from the standard two-dimensional diffusion equation by replacing the first-order time derivative with a fractional derivative (of order α, with 0?h ^r+1?+?τ^2-α), where h, τ and r are the space step size, time step size and polynomial degree, respectively. A numerical example is presented to verify the order of convergence.     

A comparison of asymptotic analytical formulae with finite-difference approximations for pricing zero coupon bond

In this paper we solve numerically a degenerate parabolic equation with dynamical boundary condition for pricing zero coupon bond and compare numerical solution with asymptotic analytical solution. First, we discuss an approximate analytical solution of the model and its order of accuracy. Then, starting from the divergent form of the equation we implement the finite-volume method of Song Wang (IMA J Numer Anal 24:699–720, 2004) to discretize the differential problem. We show that the system matrix of the discretization scheme is a M-matrix, so that the discretization is monotone. This provides the non-negativity of the price with respect to time if the initial distribution is nonnegative. Numerical experiments demonstrate second order of convergence for difference scheme when the node is not too close to the point of degeneration.     

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.     

On a new method for the study of the spatial behavior in a homogeneous elastic arch-like region

We consider a two-dimensional homogeneous elastic state in the arch-like region a?≤?r?≤?b, 0?≤?θ?≤?α, where (r,θ) denotes plane polar coordinates. We assume that three of the edges are traction-free, while the fourth edge is subjected to a (in plane) self-equilibrated load. The Airy stress function ‘?’ satisfies a fourth-order differential equation in the plane polar coordinates with appropriate boundary conditions. We develop a method which allows us to treat in a unitary way the two problems corresponding to the self-equilibrated loads distributed on the straight and curved edges of the region. In fact, we introduce an appropriate change for the variable r and for the Airy stress functions to reduce the corresponding boundary value problem to a simpler one which allows us to indicate an appropriate measure of the solution valuable for both the types of boundary value problems. In terms of such measures we are able to establish some spatial estimates describing the spatial behavior of the Airy stress function. In particular, our spatial decay estimates prove a clear relationship with the Saint-Venant's principle on such regions.     

Asymptotic Integration of Parabolic Problems with Large High-Frequency Summands

We develop the averaging method theory for parabolic problems with rapidly oscillating summands some of which are large, i.e., proportional to the square root of the frequency of oscillations. In this case the corresponding averaged problems do not coincide in general with those obtained by the traditional averaging, i.e., by formally averaging the summands of the initial problem (since the principal term of the asymptotic expansion of a solution to the latter problem is not in general a solution to the so-obtained problem). In this article we consider the question of time periodic solutions to the first boundary value problem for a semilinear parabolic equation of an arbitrary order 2k whose nonlinear terms, including the large, depend on the derivatives of the unknown up to the order k-1. We construct the averaged problem and the formal asymptotic expansion of a solution. When the large summands depend on the unknown rather than its derivatives we justify the averaging method and the complete asymptotic expansion of a solution.Original Russian Text Copyright © 2005 Levenshtam V. B.The author was supported by the Russian Foundation for Basic Research (Grant 01-01-00678) and the Program “ Universities of Russia” (UR.04.01.029).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 805–821, July–August, 2005.     

The effect of numerical integration on the finite element approximation of linear functionals

In this paper, we have studied the effect of numerical integration on the finite element method based on piecewise polynomials of degree k, in the context of approximating linear functionals, which are also known as “quantities of interest”. We have obtained the optimal order of convergence, O(h^2k){\mathcal{O}(h^{2k})}, of the error in the computed functional, when the integrals in the stiffness matrix and the load vector are computed with a quadrature rule of algebraic precision 2k − 1. However, this result was obtained under an increased regularity assumption on the data, which is more than required to obtain the optimal order of convergence of the energy norm of the error in the finite element solution with quadrature. We have obtained a lower bound of the error in the computed functional for a particular problem, which indicates the necessity of the increased regularity requirement of the data. Numerical experiments have been presented indicating that over-integration may be necessary to accurately approximate the functional, when the data lack the increased regularity.     

Elliptic Regularity and Quantitative Homogenization on Percolation Clusters

We establish quantitative homogenization, large‐scale regularity, and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense, made precise) like that of euclidean space. Then, following the work of Barlow [8], we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart [7] to estimate the yet larger scale on which solutions on the cluster can be well‐approximated by harmonic functions on ℝ^d. This is the first quantitative homogenization result in a porous medium, and the harmonic approximation allows us to estimate the scale on which a higher‐order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville‐type result that states that, for each k ∊ ℕ, the vector space of solutions growing at most like o(|x|^k+1) as |x| → ∞ has the same dimension as the set of harmonic polynomials of degree at most k, generalizing a result of Benjamini, Duminil‐Copin, Kozma, and Yadin from k ≤ 1 to k ∊ ℕ. © 2018 Wiley Periodicals, Inc.     

Numerical Solution of the Bagley-Torvik Equation

We consider the numerical solution of the Bagley-Torvik equation Ay(t) + BD _* ^3/2 y(t) + Cy(t) = f(t), as a prototype fractional differential equation with two derivatives. Approximate solutions have recently been proposed in the book and papers of Podlubny in which the solution obtained with approximate methods is compared to the exact solution. In this paper we consider the reformulation of the Bagley-Torvik equation as a system of fractional differential equations of order 1/2. This allows us to propose numerical methods for its solution which are consistent and stable and have arbitrarily high order. In this context we specifically look at fractional linear multistep methods and a predictor-corrector method of Adams type.