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In this work, the Fourier-cosine series (COS) method has been combined with the Boundary Element Method (BEM) for a fast evaluation of barrier option prices. After a description of its use in the Black and Scholes (BS) model, the focus of the paper is on the application of the proposed methodology to the barrier option evaluation in the Heston model, where its contribution is fundamental to improve computational efficiency and to make BEM appealing among finance practitioners as a valid alternative to Monte Carlo (MC) or other more traditional approaches. An error analysis is provided on the number of terms used in the Fourier-cosine series expansion, where the error bound estimation is based on the characteristic function of the log-asset price process. A Matlab code implementing this technique is attached at the end of the paper. [ABSTRACT FROM AUTHOR]

The interaction between laser beams and backscattering object surfaces lies at the fundamental working principle of any Terrestrial Laser Scanning (TLS) system. Optical properties of surfaces such as concrete, metals, wood, etc., which are commonly encountered in structural health monitoring of buildings and structures, constitute an important category of systematic and random TLS errors. This paper presents an approach for considering the random errors caused by object surfaces. Two surface properties are considered: roughness and reflectance. The effects on TLS measurements are modeled stepwise in form of a so-called synthetic variance-covariance matrix (SVCM) based on the elementary error theory. A line of work is continued for the TLS stochastic model by introducing a new approach for determining variances and covariances in the SVCM. Real measurements of cast stone façade elements of a tall building are used to validate this approach and show that the quality of the estimation can be improved with the appropriate SVCM. [ABSTRACT FROM AUTHOR]

In this paper we develop efficient numerical schemes to solve ordinary differential equations. Our methods are of the Nordsieck type, adaptive and capable of automatically controlling the global error of a numerical solution. A special feature of the new stepsize selection algorithms introduced here is the global error estimation quality control. Two different ways of attaining the preassigned accuracy of computation are examined in the paper. Namely, we implement the global error control mechanism based on reducing the maximum stepsize bound and the other one is based on reducing the local error tolerance. An accurate starting procedure for the adaptive Nordsieck methods is presented in full detail. Our intention here is to find the most effective strategy of stepsize selection. Theoretical investigation is supplied with numerical tests. [ABSTRACT FROM AUTHOR]

In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method was introduced which achieves a high order approximation of the geometry for domains which are implicitly described by smooth level set functions. This method is based on a parametric mapping which transforms a piecewise planar interface (or surface) reconstruction to a high order approximation. In the paper [C. Lehrenfeld and A. Reusken, IMA J. Numer. Anal. 38 (2018), No. 3, 1351–1387] an a priori error analysis of the method applied to an interface problem is presented. The analysis reveals optimal order discretization error bounds in the H1-norm. In this paper we extend this analysis and derive optimal L2-error bounds. [ABSTRACT FROM AUTHOR]

In this paper, we consider the modified inhomogeneous Helmholtz equation Δ u( x, y) u( x, y) f( x, y), x , 0 y 1, with inhomogeneous Cauchy data being given at y 0. The problem is known to be ill-posed, as the solution (if exists) does not depend continuously on the given data. We propose a regularization method to obtain a stable approximate solution of the problem and get some error estimates. Finally, a numerical example shows the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper we present an algorithm to bound parameters estimated from very small data sets and use this method to suggest optimal times for data collection. The method is deterministic in that we assume that one has error bars on the data. The basic idea is based on bounded error parameter identification methods and we use this framework to assign a measure of quality to the estimated parameter. Exploiting the dynamics of the underlying mathematical model allows us to describe the quality of the estimated parameter in a way that, due to the small number of data points, is not appropriate for traditional statistical techniques. The algorithm in this paper differs from traditional bounded error parameter identification techniques in that one computes the membership set by solving an inverse problem rather than a forward interval problem. [ABSTRACT FROM AUTHOR]

Recently, Schmitt, Weiner, and Erdmann have proposed an efficient family of numerical methods termed Implicit Parallel Peer (IPP) methods. They are a subclass of s-stage general linear methods of order s – 1. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff differential equations. The special construction of IPP methods allows for a parallel implementation, which is advantageous in modern high-performance computation environment. In this paper we add one more useful functionality to IPP methods, i.e. automatic global error control. We show that the global error estimation developed by Kulikov and Shindin in multistep formulas is suitable for the methods of Schmitt, Weiner and Erdmann. Moreover, that global error estimation can be done in parallel. An algorithm of efficient stepsize selection is also discussed here. [ABSTRACT FROM AUTHOR]

This paper derives a posteriori error estimates for the mixed numerical approximation of the Laplace eigenvalue problem. We discuss a reconstruction in the standard H 0 1 {H_{0}^{1}} -conforming space for the primal variable of the mixed Laplace eigenvalue problem and compare it with analogous approaches present in the literature for the corresponding source problem. In the case of Raviart–Thomas finite elements of arbitrary polynomial degree, the resulting error estimator constitutes a guaranteed upper bound for the error and is shown to be local efficient. Our reconstruction is performed locally on a set of vertex patches. [ABSTRACT FROM AUTHOR]

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the L 2 {L^{2}} norm are derived. Numerical results are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a high-order, fully-mixed finite element method for the free convection of n-dimensional fluids, n ∈ { 2 , 3 } {n\in\{2,3\}} , with temperature-dependent viscosity and thermal conductivity. The mathematical model is given by the coupling of the equations of continuity, momentum (Navier–Stokes) and energy by means of the Boussinesq approximation, as well as mixed thermal boundary conditions and a Dirichlet condition on the velocity. Because of the dependence on the temperature of the fluid properties, several additional variables are defined, thus resulting in an augmented formulation that seeks the rate of strain, pseudostress and vorticity tensors, velocity, temperature gradient and pseudoheat vectors, and temperature of the fluid. Using a fixed-point approach, smallness-of-data assumptions and a slight higher-regularity assumption for the exact solution provide the necessary well-posedness results at both continuous and discrete levels. In addition, and as a result of the augmentation, no discrete inf-sup conditions are needed for the well-posedness of the Galerkin scheme, which provides freedom of choice with respect to the finite element spaces. In particular, we suggest a combination based on Raviart–Thomas, Lagrange and discontinuous elements for which we derive optimal a priori error estimates. Finally, several numerical examples illustrating the performance of the method and confirming the theoretical rates of convergence are reported. [ABSTRACT FROM AUTHOR]

A decoupled finite element method with different time steps for the nonstationary Darcy--Brinkman problem is considered in this paper. Moreover, for the presented method, the stability analysis and error estimates are deduced. Finally, numerical tests are provided that demonstrate the efficiency of the method. It is found the presented method can save lots of computational time compared with standard coupled method. [ABSTRACT FROM AUTHOR]

Functional a posteriori estimates have been derived for elliptic and linear parabolic problems in [S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals, 2000. S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, 2003. S. Repin, A posteriori error estimates for partial differential equations, 2008. S. Repin, A posteriori error estimation for nonlinear variational problems by duality theory, 1997] and some other publications. They provide computable upper bounds of the difference between the exact solution u and any approximation v lying in the admissible (energy) class. This paper is concerned with advanced forms of these estimates, which are discussed within the paradigm of the reaction-diffusion problem. In the first part of the paper, we derive guaranteed and computable upper bounds for problems which admit the decomposition of the domain into a set of simple (e.g., simplicial or polyhedral) subdomains. For this case an upper bound which involves constants in the Poincaré inequality for the corresponding subdomains is obtained. Estimates of this type can be helpful if computations are performed by the domain decomposition method. The second part of the paper is devoted to the derivation of two-sided error bounds in terms of weighted norms. Our analysis shows that the approach based on certain transformations of the basic integral identity earlier developed for energy error norms (see [S. Repin, Two-sided estimates of deviation from exact solutions of uniformly elliptic equations, 2003. S. Repin, A posteriori error estimates for partial differential equations, 2008]) can be successfully applied to error estimation in weighted norms. [ABSTRACT FROM AUTHOR]

In this paper an application of the homotopy analysis method for some type of nonlinear and linear integral equations of the second kind is presented. A special case of considered equation is the Volterra- Fredholm integral equation. In homotopy analysis method a series is created. It has shown that if the series is convergent, its sum is the solution of the considered equation. It has been also shown that under proper assumptions the considered equation possesses a unique solution and the series obtained in homotopy analysis method is convergent. The error of the approximate solution was estimated. This approximate solution is obtained when we limit to the partial sum of the series.Application of the method is illustrated with examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider the iteratively regularized Gauss–Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton/discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., L ∞ {L^{\infty}} as a preimage space. The theoretical findings are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

This paper deals with the homogenization problem of one-dimensional pseudo-elliptic equations with a rapidly varying random potential.The main purpose is to characterize the homogenization error (random fluctuations), i.e., the difference between the random solution and the homogenized solution, which strongly depends on the autocovariance property of the underlying random potential.It is well known that when the random potential has short-range dependence, the rescaled homogenization error converges in distribution to a stochastic integral with respect to standard Brownian motion.Here, we are interested in potentials with long-range dependence and we prove convergence to stochastic integrals with respect to the Hermite process. [ABSTRACT FROM AUTHOR]

It is still an open problem to prove a priori error estimates for finite volume schemes of higher order MUSCL type, including limiters, on unstructured meshes, which show some improvement compared to first order schemes. In this paper we use these higher order schemes for the discretization of convection dominated elliptic problems in a convex bounded domain Ω in ℝ2 and we can prove such kind of an a priori error estimate. In the part of the estimate, which refers to the discretization of the convective term, we gain h1/2. Although the original problem is linear, the numerical problem becomes nonlinear, due to MUSCL type reconstruction/limiter technique. [ABSTRACT FROM AUTHOR]

It is well known that the theory of asymptotic expansion of the global error of one-step methods is an important but complicated fact in the realm of numerical analysis of differential equations. It can hardly be confirmed for majority of numerical schemes and problems by practical computations. On the other hand, this theory is a fundamental tool to justify the extrapolation technique, which is one of the most efficient means to solve ordinary differential equations. Therefore, in the recent paper Kulikov [[Numer. Algorithms 53: 321--342, 2010]] presented his local theory of extrapolation methods, which is based on the Richardson technique only and does not require any asymptotic global error expansion. Here, we concentrate on quadratic extrapolation. We explain which property of symmetric one-step methods provides two-order growth of accuracy of the underlying method after each extrapolation step and arrive at the notion of proportional extrapolation. We also learn more about adjoint and symmetric one-step methods. In addition, we prove that the modified Aitken--Neville algorithm works for any symmetric one-step method of an arbitrary order 2 s. [ABSTRACT FROM AUTHOR]

Misspecifications of econometric models can lead to biased coefficients and incorrect interpretations of error terms, which in turn can lead to incorrectly estimated models and incorrect inference. There are specific techniques such as instrumental variables, which are used in the economics literature to deal with some individual forms of model misspecification, only addressing one problem at a time. The joint and separate solutions to the problems of unknown functional forms, omitted variables and measurement errors, discussed in this paper, prove that instrumental variables do not exist. Therefore, the specific techniques used in the literature are not feasible. This paper proposes a general method for estimating underlying parameters in the presence of a range of model misspecifications. It is argued that this method can consistently estimate the direct effect of an independent variable on a dependent variable with all of its other determinants held constant even in the presence of an unknown functional form, measurement error and omitted variables. [ABSTRACT FROM AUTHOR]

We introduce and investigate a probabilistic model of a coherent system which consists of m elements and is intended for fulfilment of homogeneous tasks. On the first stage, one task out of the total number n of tasks enters each element of the system. The result of work of an element is either a fulfilment of the task or a failure of the element. In the case of the failure of an element, it is excluded from the system, and the task which is not fulfilled returns to the queue of those waiting for fulfilment. On the second stage, m1 tasks are sent to the system, where m1 is the number of elements of the system remained operable after the first stage, and so on. In the paper, a detailed analysis of the suggested model is realised in the case where each element of the system fulfils tasks with the same probability independently of the rest of the elements. [ABSTRACT FROM AUTHOR]

The Fourier pseudo-spectral method has been studied for a one- dimensional coupled system of viscous Burgers equations. Two test problems with known exact solutions have been selected for this study. In this paper, the rate of con- vergence in time and error analysis of the solution of the first problem has been studied, while the numerical results of the second problem obtained by the present method are compared to those obtained by using the Chebyshev spectral collocation method. The numerical results show that the proposed method outperforms the conventional one in terms of accuracy and convergence rate. [ABSTRACT FROM AUTHOR]

Five different error estimation strategies suitable for the Gauss-type Nested Implicit Runge–Kutta method of order 4 have been presented and tested numerically in [Kulikov and Shindin, Lecture Notes in Computer Science: 136–143, 2007, Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. The nested Implicit Runge–Kutta schemes introduced recently are an efficient class of Implicit Runge–Kutta formulas. In this paper we deal with the methods of order 6. One scheme of such sort has been constructed in [Kulikov and Shindin, Appl. Numer. Math. 59: 707–722, 2009]. Now we present a one-parametric family of the above-mentioned formulas of order 6 by relaxing the accuracy requirement for some stage values. This allows the error estimation strategies designed for the method of order 4 to be extended to the higher-order Gauss-type Nested Implicit Runge–Kutta method. We also present the particulars of the efficient implementation of this method, which is stable and accurate. The numerical examples confirm the efficiency of the numerical scheme under consideration for both ordinary differential equations and partial differential equations. [ABSTRACT FROM AUTHOR]

This paper addresses the regularization of linear inverse problems with so-called sparsity constraints in terms of ℓ P-penalty terms. Error estimates and convergence rates are derived. The application of the generalized gradient projection method and the semismooth Newton method to the according minimization problem is shown. Numerical experiments show the efficiency of the semismooth Newton method. [ABSTRACT FROM AUTHOR]

This paper deals with asymptotically correct methods to evaluate the local and global errors of Nordsieck formulas applied to ordinary differential equations. It extends naturally the results developed by Kulikov and Shindin [ Comp. Math. Math. Phys. (2000) 40, 1255–1275] in local and global error computation of multistep methods, but shows that Kulikov and Shindin's technique becomes more complicated when implemented in numerical methods, for which the concepts of consistency and quasi-consistency are not equivalent (see Skeel [ SIAM J. Numer. Anal. (1976) 13, 664–685]). A new property termed super quasi-consistency is introduced and special cases of Nordsieck formulas with cheaper error estimation are found. Numerical examples are included to confirm practically the theory presented in this paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with a posteriori error estimates in terms of goal-oriented quantities for linear elliptic problems. The quantities are presented by locally supported linear functionals that serve as measures of the difference between the exact solution and its approximation in subdomains that can be selected using certain a priori knowledge on the expected properties of the solution. Typically, the evaluation of such type of error indicators is performed with the help of the so-called adjoint problem (see, e.g., [W. Bangerth and R. Rannacher, Adaptive Finite Element Methods for Differential Equations. Birkhäuser, Berlin, 2003; R. Becker and R. Rannacher, Weighted a posteriori error control in finite element methods. In: Proc. of ENUMATH'97, 1998, pp. 621–637]). In [S. Korotov, P. Neittaanmäki, and S. Repin, A posteriori error estimation of goal-oriented quantities by the superconvergent patch recovery. J. Numer. Math. (2003) 11, No. 1, 33–59] an approach was suggested, in which the primal and adjoint problems are solved on non-coinciding meshes, and averaging of gradients is used to evaluate the term in the estimator that cannot be computed directly. However, superconvergence requires higher differentiability of the solutions and also regularity (quasi-uniformity) of both primal and dual meshes. In the present paper we propose a new method for computation of the estimator, where we use only superconvergence of the post-processed finite-element solution of the primal problem. At this point our analysis is based on the properties of the least square surface fitting operator investigated by J. Wang [J. Wang, A Superconvergence analysis for finite element solutions by the least-squares surface fitting on irregular meshes for smooth problems. J. Math. Study (2000) 33, No. 3, 229–243]. We establish asymptotical properties of the new error indicator and verify its efficiency in a series of numerical tests. [ABSTRACT FROM AUTHOR]

Tikhonov regularization is one of the widely used procedures for the regularization of nonlinear as well as linear ill-posed problems. The error analysis carried out in most of the works that appeared in last few years on Tikhonov regularization of nonlinear ill-posed problems are under Hölder type source conditions on the unknown solution which is known to be applicable only for mildly ill-posed problems. In this paper we consider Tikhonov regularization of nonlinear ill-posed problems and derive order optimal error estimate under a general source condition together with an a posteriori parameter rule proposed by Scherzer et al., which is applicable for severely ill-posed problems as well. [ABSTRACT FROM AUTHOR]

In this paper we report on a new iterative method for regularizing a nonlinear Hammerstein type operator equation in Hilbert spaces. The proposed Newton–Lavrentiev method is a combination of Lavrentiev regularization and a Newton's iteration. Under the assumptions that the operator F is continuous Fréchet differentiable with a Lipschitz-continuous first derivative and that the solution of (1.1) fulfills a smoothness condition, we will give a convergence rate result. [ABSTRACT FROM AUTHOR]

This paper investigates an error estimate proposed by Warnock and studied by Halton (2005). That error estimate is simply the sample standard error applied to certain non-randomized quasi-Monte Carlo points. This quasi-standard error (QSE) closely tracks the actual error in an example, and looks to be at least as accurate as a standard error based on random replication. We also show that the quasi-standard error is not unreasonably large in its intended use. But there are quasi-Monte Carlo (QMC) constructions for which the QSE severely underestimates the true error. Moreover, discrepancy considerations do not separate these counter-examples from other cases where the method might be reliable. We conclude that the QSE is not yet ready to be trusted in applications. [ABSTRACT FROM AUTHOR]

This paper concerns a posteriori error estimation for the primal and dual mixed finite element methods applied to the diffusion problem. The problem is considered in a general setting with inhomogeneous mixed Dirichlet-Neumann boundary conditions. New functional-type a posteriori error estimators are proposed that exhibit the ability both to indicate the local error distribution and to ensure upper bounds for discretization errors in primal and dual (flux) variables. The latter property is a direct consequence of the absence in the estimators of any mesh-dependent constants; the only constants present in the estimates stem from the Friedrichs and trace inequalities and, thus, are global and dependent solely on the domain geometry and the bounds of the diffusion matrix. The estimators are computationally cheap and require only the projections of piecewise constant functions onto the spaces of the lowest-order Raviart-Thomas or continuous piecewise linear elements. It is shown how these projections can be easily realized by simple local averaging. [ABSTRACT FROM AUTHOR]

This paper derives some interior superconvergence estimates for finite element solutions of the Stokes problem by a local L2 projection method. The results depend only on local properties of the Stokes problem and the finite element approximations. [ABSTRACT FROM AUTHOR]

The efficient market hypothesis implies that (risk-adjusted) asset prices cannot be cointegrated. On the other hand, arbitrage processes prevent prices of fundamentally related assets from drifting too far away. An attractive model that reconciles these two conflicting facts is the non-linear error correction mechanism (ECM). Such a process tolerates small deviations from the long-run relationship. For more substantial deviations, an effective adjustment process pushes the diverging prices towards their fundamental relationship. In this paper parametric and non-parametric techniques are employed to investigate the ECM between prices of voting and non-voting stocks. Despite its intuitive appeal, we find little evidence for a non-linear relationship between German dual-class shares. Only in four out of 12 cases does the threshold ECM yield a substantial improvement of fit. In other cases, the evidence for non-linearity is rather weak and the threshold ECM fails to outperform the linear model. [ABSTRACT FROM AUTHOR]

In this paper, we consider two-parameter singularly perturbed boundary value problems via fitted splines. We propose an exponentially fitting factor to the highest order derivative of the problem and discretize by parametric cubic spline functions on a uniform mesh. An error analysis of the method is presented. The efficiency of the method is demonstrated by test examples which support the theoretical results. [ABSTRACT FROM AUTHOR]

This paper addresses the solution of one- and two-dimensional Volterra integral equations (VIEs) by means of the spectral collocation method. The novel technique takes advantage of the properties of shifted Jacobi polynomials and is applied for solving multi-dimensional VIEs. Several numerical examples demonstrate the efficiency of the method and an error analysis verifies the correctness and feasibility of the proposed method when solving VIE. [ABSTRACT FROM AUTHOR]

In this paper we give a new proof of the -error estimate for the finite element approximation of the Laplace-Beltrami equation with an additional lower order term on a surface. While the proof available in the literature uses the method of perturbed bilinear forms from Schatz and Wahlbin, we adapt Scott's proof from an Euclidean setting to the surface case. Furthermore, in contrast to the literature we use an approximative Green's function on the surface instead of an exact Green's function which is obtained by lifting an Euclidean Green's function locally from the tangent plane to the surface. [ABSTRACT FROM AUTHOR]

Online dynamic security assessment (DSA) is a computationally intensive task. In order to reduce the amount of computation, screening of contingencies is performed. Screening involves analyzing the contingencies with the system described by a simpler model so that computation requirement is reduced. Screening identifies those contingencies which are sure to not cause instability and hence can be eliminated from further scrutiny. The numerical method and the step size used for screening should be chosen with a compromise between speed and accuracy. This paper proposes use of energy function as a measure of error in the numerical solution used for screening contingencies. The proposed measure of error can be used to determine the most accurate numerical method satisfying the time constraint of online DSA. Case studies on 17 generator system are reported. [ABSTRACT FROM AUTHOR]

In the current paper, we derive the a priori error analysis ( hp version) of the streamline diffusion DG finite element approximation for optimal distributed control problem governed by the first order linear hyperbolic equation. We present the stability of such method, obtain the a priori error upper bound for the state and the control approximation, and prove the convergence of numerical method. For the optimal control problem, these estimates are apparently not available in the literature. [ABSTRACT FROM AUTHOR]

The problem of variational data assimilation for a nonlinear evolution model is formulated as an optimal control problem to nd the initial condition. The optimal solution (analysis) error arises due to the errors in the input data (background and observation errors). Under the Gaussian assumption the optimal solution error covariance can be constructed using the Hessian of the auxiliary data assimilation problem. The aim of this paper is to study the evolution of model errors via data assimilation. The optimal solution error covariances are derived in the case of imperfect model and for the weak constraint formulation, when the model equations determine the cost functional. [ABSTRACT FROM AUTHOR]

In this paper we study the convergence of Halley's method as a method for finding all zeros of a polynomial simultaneously. We present two types of local convergence theorems as well as a semilocal convergence theorem for Halley's method for simultaneous computation of polynomial zeros. [ABSTRACT FROM AUTHOR]

It is well-known that the remaining term of a classical n-point Newton-Cotes quadrature depends on at least an n-order derivative of the integrand function. Discounting the fact that computing an n-order derivative requires a lot of differentiation for large n, the main problem is that an error bound for an n-point Newton-Cotes quadrature is only relevant for a function that is n times differentiable, a rather stringent condition. In this paper, by defining two specific linear kernels, we resolve this problem and obtain new error bounds for all closed and open types of Newton-Cotes quadrature rules. The advantage of the obtained bounds is that they do not depend on the norms of the integrand function and are very general such that they cover almost all existing results in the literature. Some illustrative examples are given in this direction. [ABSTRACT FROM AUTHOR]

-The main purpose of this paper is to propose the Chebyshev spectral-collocation method for a class of the weakly singular Volterra integral equations (VIEs) with proportional delay. The proposed method also are applicable to a class of the weakly singular VIEs with proportional delay possessing unsmooth solution. To provide a rigorous error analysis for the proposed method, we prove the the uniqueness and smoothness of the solution. The error analysis shows that the numerical errors decay exponentially in the infinity norm and the Chebyshev weighted Hilbert space norms. Numerical results are presented to confirm the theoretical prediction of the exponential rate of convergence. [ABSTRACT FROM AUTHOR]

Numerical errors for models of the sea surface undulation are studied in the paper based on spectral decomposition of the stochastic field of water level. Such errors depend on the number of random harmonics in the spectral model and on the size of the domain for which the spectral model is constructed. Numerical errors are studied for temporal and spatial spectral models. [ABSTRACT FROM AUTHOR]

In this paper, we make an offer of the lattice approximate method for solving a class of multi-dimensional integro-differential equations with the initial conditions. Also, we analyze the worst case error measured in weighted Korobov spaces for these equations. Finally, numerical examples complete this work. [ABSTRACT FROM AUTHOR]

A problem of an approximate solving of Symm's integral equation for an infinitely smooth closed boundary is considered. For a fully discrete projection method, error estimates are found in the metric of Sobolev spaces. A discretization parameter is chosen by balancing principle. [ABSTRACT FROM AUTHOR]

- This paper is concerned with the derivation of conforming and non-conforming functional a posteriori error estimates for elliptic boundary value problems in exterior domains. These estimates provide computable and guaranteed upper and lower bounds for the difference between the exact and the approximate solution of the respective problem. We extend the results from [5] to non-conforming approximations, which might not belong to the energy space and are just considered to be square integrable. Moreover, we present some numerical tests. [ABSTRACT FROM AUTHOR]

- This paper presents the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting. We prove the existence and uniqueness of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem numerically, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust convergence rates and optimal complexity for the preconditioned minimal residual method. All systems can be solved totally in parallel. Furthermore, we present a complete analysis for the error introduced by the multiharmonic finite element discretization as well as some numerical results confirming our theoretical findings. [ABSTRACT FROM AUTHOR]

In this paper a general parabolic problem is considered and discretized by the discontinuous Galerkin (DG) method in time and, in general, in space. Optimal a priori error estimates in space as well as in time are derived and applied to the heat equation and to a nonlinear convection-diffusion equation. [ABSTRACT FROM AUTHOR]

In this paper we consider the problem of convergence of a three-step random fixed point iteration with errors for the case of three random asymptotically quasi-nonxpansive type operators in a real separable Banach space. The stability of the iterative process has also been discussed. We define almost stability and show that the random iteration is almost stable. The present work also generalizes several established results in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider the possibility of a posteriori error estimates for linear and nonlinear ill-posed operator equations. Given an auxiliary finite-dimensional problem , that approximates the original infinite model , with a certain level of accuracy, we try to estimate the distance between z, an approximate solution to , and , the exact solution to . The problem is assumed to accumulate different sources of error (discretization, measurements, etc.), and the computed solution z is assumed to satisfy the equation within a nonzero tolerance . We conduct both a theoretical and numerical study of a posteriori error analysis. [ABSTRACT FROM AUTHOR]

In many applications Monte Carlo (MC) sequences or Quasi-Monte Carlo (QMC) sequences are used for numerical integration. In moderate dimensions the QMC method typically yield better results, but its performance significantly falls off in quality if the dimension increases. One class of randomized QMC sequences, which try to combine the advantages of MC and QMC, are so-called mixed sequences, which are constructed by concatenating a d-dimensional QMC sequence and an ()-dimensional MC sequence to obtain a sequence in dimension s. Ökten, Tuffin and Burago proved probabilistic asymptotic bounds for the discrepancy of mixed sequences, which were refined by Gnewuch. In this paper we use an interval partitioning technique to obtain improved probabilistic bounds for the discrepancy of mixed sequences. By comparing them with lower bounds we show that our results are almost optimal. [ABSTRACT FROM AUTHOR]