Optimal sampling design for global approximation of jump diffusion stochastic differential equations

The paper deals with strong global approximation of stochastic differential equations (SDEs) driven by two independent processes: a nonhomogeneous Poisson process and a Wiener process. We assume that the jump and diffusion coefficients of the underlying SDE satisfy jump commutativity condition (see Chapter 6.3 in [21]). We establish the exact convergence rate of minimal errors that can be achieved by arbitrary algorithms based on a finite number of observations of the Poisson and Wiener processes. We consider classes of methods that use equidistant or nonequidistant sampling of the Poisson and Wiener processes. We provide a construction of optimal methods, based on the classical Milstein scheme, which asymptotically attain the established minimal errors. The analysis implies that methods based on nonequidistant mesh are more efficient, with respect to asymptotic constants, than those based on the equidistant mesh.     

Adaptive mesh point selection for the efficient solution of scalar IVPs

We discuss adaptive mesh point selection for the solution of scalar initial value problems. We consider a method that is optimal in the sense of the speed of convergence, and we aim at minimizing the local errors. Although the speed of convergence cannot be improved by using the adaptive mesh points compared to the equidistant points, we show that the factor in the error expression can be significantly reduced. We obtain formulas specifying the gain achieved in terms of the number of discretization subintervals, as well as in terms of the prescribed level of the local error. Both nonconstructive and constructive versions of the adaptive mesh selection are shown, and a numerical example is given.     

Estimate of the Constant in Two Strengthened C.B.S. Inequalities for F.E.M. Systems of 2D Elasticity: Application to Multilevel Methods and a Posteriori Error Estimators

The constant γ in the strengthened Cauchy-Buniakowski-Schwarz (C.B.S.) inequality plays a crucial role in the convergence rate of multilevel iterative methods as well as in the efficiency of a posteriori error estimators, that is in the framework of finite element approximations of SPD problems. We consider the approximation of the 2D elasticity problem by the Courant element. Concerning multilevel convergence rate, that is the γ corresponding to nested general triangular meshes of size h and 2h, we have proved that γ²≤ 3/4$ uniformly on the mesh and the Poisson ratio. Concerning error estimator, that is the γ corresponding to quadratic and linear approximations on the same mesh, numerical computations have shown that the exact γ for a reference element deteriorates that is goes to one, when the Poisson ratio tends to 1/2     

Form-preserving exponential approximation

In this paper we continue our research in constructing linear methods for approximating functions by parabolic splines with equidistant knots, inheriting such properties of approximated functions as monotony and convexity. The obtained results are generalized to the case of nonequidistant knots and to some kinds of exponential splines of the third order.     

Analysis of optimal preconditioners for CutFEM

In this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so-called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem.     

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R₊ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.     

Finite-element methods for singularly perturbed high-order elliptic two-point boundary value problems. I: reaction-diffusion-type problems

We consider singularly perturbed high-order elliptic two-pointboundary value problems of reaction-diffusion type. It is shownthat, on an equidistant mesh, polynomial schemes cannot achievea high order of convergence that is uniform in the perturbationparameter. Piecewise polynomial Galerkin finite-element methodsare then constructed on a Shishkin mesh. Almost optimal convergenceresults, which are uniform in the perturbation parameter, areobtained in various norms. Numerical results are presented fora fourth-order problem. e-mail address: stynes{at}bureau.ucc.ie.     

Automatic Smoothing With Wavelets for a Wide Class of Distributions

Wavelet-based denoising techniques are well suited to estimate spatially inhomogeneous signals. Waveshrink (Donoho and Johnstone) assumes independent Gaussian errors and equispaced sampling of the signal. Various articles have relaxed some of these assumptions, but a systematic generalization to distributions such as Poisson, binomial, or Bernoulli is missing. We consider a unifying l₁-penalized likelihood approach to regularize the maximum likelihood estimation by adding an l₁ penalty of the wavelet coefficients. Our approach works for all types of wavelets and for a range of noise distributions. We develop both an algorithm to solve the estimation problem and rules to select the smoothing parameter automatically. In particular, using results from Poisson processes, we give an explicit formula for the universal smoothing parameter to denoise Poisson measurements. Simulations show that the procedure is an improvement over other methods. An astronomy example is given.     

Preventive maintenance for homogeneous and heterogeneous systems

We consider optimal preventive maintenance for homogeneous and heterogeneous systems with major (critical) and minor (noncritical) failures. A major failure results in a replacement of a failed system, whereas minor failures can be minimally instantaneously repaired. Distinct from the homogeneous case, where the process of minimal repairs is the Poisson process, the process of minimal repairs in the heterogeneous case is the mixed Poisson process that does not possess the memoryless property. This enables considering the number of minimal repairs as the decision parameter for the corresponding optimal preventive maintenance policy. The proposed approach is theoretically justified, and the detailed illustrative numerical examples are presented.     

Convergence of Nonlocal Finite Element Energies for Fracture Mechanics

Usually, smeared crack techniques are based on the following features: the fracture is represented by means of a band of finite elements and by a softening constitutive law of damage type. Often, these methods are implemented with nonlocal operators that control the localization effects and reduce the mesh bias. We consider a nonlocal smeared crack energy defined for a finite element space on a structured grid. We characterize the limit energy as the mesh size h tends to zero and we establish a precise link between the discrete and continuum formulations of the fracture energies, showing the correct scaling and the explicit form of the mesh bias.     

Modified Defect Correction Algorithms for ODEs. Part I: General Theory

The well-known method of Iterated Defect Correction (IDeC) is based on the following idea: Compute a simple, basic approximation and form its defect w.r.t. the given ODE via a piecewise interpolant. This defect is used to define an auxiliary, neighboring problem whose exact solution is known. Solving the neighboring problem with the basic discretization scheme yields a global error estimate. This can be used to construct an improved approximation, and the procedure can be iterated. The fixed point of such an iterative process corresponds to a certain collocation solution. We present a variety of modifications to this algorithm. Some of these have been proposed only recently, and together they form a family of iterative techniques, each with its particular advantages. These modifications are based on techniques like defect quadrature (IQDeC), defect interpolation (IPDeC), and combinations thereof. We investigate the convergence on locally equidistant and nonequidistant grids and show how superconvergent approximations can be obtained. Numerical examples illustrate our considerations. The application to stiff initial value problems will be discussed in Part II of this paper.     

A numerical approach for the Poisson equation in a planar domain with a small inclusion

We consider the Poisson equation in a domain with a small inclusion. We present a simple numerical method, based on asymptotic analysis, which allows to approximate robustly the far field of the solution as the size of the inclusion goes to zero without any mesh adaptation procedure. The discretization is based on a fully standard Galerkin approach such as finite elements. We prove stability and consistency of the numerical method and provide error estimates. We end the paper with numerical experiments illustrating the efficiency of the technique.     

A remark on the optimality of adaptive finite element methods

We show that the standard assumption on the smallness of the marking parameter θ in adaptive finite element methods can be avoided for the proof of the optimality of the algorithm. To this end we propose a new technique based on comparison of the solutions of different finite element spaces obtained by different refinements of a given mesh. We consider conforming and nonconforming low-order finite elements on triangular and tetrahedral meshes.     

Estimates of the age of a heat distribution

The paper deals with the possibility to solve the heat equation backwards in time. More specifically, we treat the following problem. Given the temperature at a finite number of points of a homogeneous bar, how old can the heat distribution be? In the case that the temperature is given at equidistant points x₁, the problem is completely solved. In the case of nonequidistant x_i we find an upper bound for the age. Such a bound is also obtained when the information about the heat distribution is given by the value of a finite number of linear functionals.     

Power utility maximization in exponential Lévy models: convergence of discrete-time to continuous-time maximizers

We consider power utility maximization of terminal wealth in a 1-dimensional continuous-time exponential Lévy model with finite time horizon. We discretize the model by restricting portfolio adjustments to an equidistant discrete time grid. Under minimal assumptions we prove convergence of the optimal discrete-time strategies to the continuous-time counterpart. In addition, we provide and compare qualitative properties of the discrete-time and continuous-time optimizers.     

Rescaled Lévy-Loewner hulls and random growth

We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and SLE(κ) as limits.     

Prediction of Components in Random Sums

We consider predictions of the random number and the magnitude of each iid component in a random sum based on its distributional structure, where only a total value of the sum is available and where iid random components are non-negative. The problem is motivated by prediction problems in a Poisson shot noise process. In the context, although conditional moments are best possible predictors under the mean square error, only a few special cases have been investigated because of numerical difficulties. We replace the prediction problem of the process with that of a random sum, which is more general, and establish effective numerical procedures. The methods are based on conditional technique together with the Panjer recursion and the Fourier transform. In view of numerical experiments, procedures work reasonably. An application in the compound mixed Poisson process is also suggested.     

Modelling and analysis of a software system with improvements in the testing phase

In this paper, we consider the error detection phenomena in the testing phase when modifications or improvements can be made to the software in the testing phase. The occurrence of improvements is described by a homogeneous Poisson process with intensity rate denoted by λ. The error detection phenomena is assumed to follow a nonhomogeneous Poisson process (NHPP) with the mean value function being denoted by m(t). Two models are presented and in one of the models, we have discussed an optimal release policy for the software taking into account the occurrences of errors and improvements. Finally, we discuss the possibility of an improvement removing k errors with probability p_k, k ≥ 0 in the software and develop a NHPP model for the error detection phenomena in this situation.     

Stability for the Gravitational Vlasov–Poisson System in Dimension Two

We consider the two dimensional gravitational Vlasov–Poisson system. Using variational methods, we prove the existence of stationary solutions of minimal energy under a Casimir type constraint. The method also provides a stability criterion of these solutions for the evolution problem.     

On stochastic models of teletraffic with heavy-tailed distributions

Following the approach suggested by I. Kaj and M. Taqqu, we consider a stochastic model of teletraffic based on Poisson random measure. We show that under appropriate assumptions, the finite-dimensional distributions for the scaled workload process converge to those of a stable Lévy process. Bibliography: 10 titles.