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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 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]

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]

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]

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]

A robust finite element method is introduced for solving elastic vibration problems in two dimensions. The temporal discretization is carried out using the P 1 {P_{1}} -continuous discontinuous Galerkin (CDG) method, while the spatial discretization is based on the Crouziex–Raviart (CR) element. It is shown after a technical derivation that the error of the displacement (resp. velocity) in the energy norm (resp. L 2 {L^{2}} norm) is bounded by O ⁢ (h + k) {O(h+k)} (resp. O ⁢ (h 2 + k) {O(h^{2}+k)}), where h and k denote the mesh sizes of the subdivisions in space and time, respectively. Under some regularity assumptions on the exact solution, the error bound is independent of the Lamé coefficients of the elastic material under discussion. A series of numerical results are offered to illustrate numerical performance of the proposed method and some other fully discrete methods for comparison. [ABSTRACT FROM AUTHOR]

We perform the error analysis of a stabilized discontinuous Galerkin scheme for the initial boundary value problem associated with the magnetic induction equations using standard discontinuous Lagrange basis functions. In order to obtain the quasi-optimal convergence incorporating second-order Runge–Kutta schemes for time discretization, we need a strengthened 4 / 3 {4/3} -CFL condition ( Δ ⁢ t ∼ h 4 / 3 {\Delta t\sim h^{4/3}}). To overcome this unusual restriction on the CFL condition, we consider the explicit third-order Runge–Kutta scheme for time discretization. We demonstrate the error estimates in L 2 {L^{2}} -sense and obtain quasi-optimal convergence for smooth solution in space and time for piecewise polynomials with any degree l ≥ 1 {l\geq 1} under the standard CFL condition. [ABSTRACT FROM AUTHOR]

In this article, we discuss error estimates for nonlinear parabolic problems using discontinuous Galerkin methods which include HDG method in the spatial direction while keeping time variable continuous. When piecewise polynomials of degree k ⩾ 1 are used to approximate both the potential as well as the flux, it is shown that the error estimate for the semi-discrete flux in L∞(0, T; L2)-norm is of order k + 1. With the help of a suitable post-processing of the semi-discrete potential, it is proved that the resulting post-processed potential converges with order of convergence O ( log ⁡ (T / h 2) h k + 2 ) $\begin{array}{} \displaystyle O\big(\!\sqrt{{}\log(T/h^2)}\,h^{k+2}\big) \end{array}$ in L∞(0, T; L2)-norm. These results extend the HDG analysis of Chabaud and Cockburn [Math. Comp. 81 (2012), 107–129] for the heat equation to non-linear parabolic problems. [ABSTRACT FROM AUTHOR]

. By motivating the result of Ramachandran et al. [Certain results on q-starlike and q-convex error functions, Math. Slovaca, 68(2) (2018), 361{368], in this present investigation we derive the classical Fekete Szegö theorem for a close-to-convex error function of order β and the sharp estimates also obtained for real μ. [ABSTRACT FROM AUTHOR]

In this work, we develop a nonlinear explicit method suitable for both autonomous and non-autonomous type of initial value problems in Ordinary Differential Equations (ODEs). The method is found to be third order accurate having L-stability. It is shown that if a variable step-size strategy is employed then the performance of the proposed method is further improved in comparison with other methods of same nature and order. The method is shown to be working well for initial value problems having singular solutions, singularly perturbed and stiff problems, and blow-up ODE problems, which is illustrated using a few numerical experiments. [ABSTRACT FROM AUTHOR]

Under the three-treatment three-period crossover design with simple carry-over effects, we derive the least-squares estimators for period effects, treatment effects and carry-over effects, as well as their covariance matrix in closed and explicit expressions. Using Monte Carlo simulation, we compare the test procedure adjusting carry-over with that ignoring carry-over with respect to Type I error and power. We further compare interval estimators adjusting carry-over with those ignoring carry-over with respect to the coverage probability and the average length. When the variation of responses within patients is small, the test procedure and interval estimators ignoring carry-over can lose accuracy in the presence of carry-over effects. When the variation of responses within patients is large, this loss of accuracy may become small or even minimal. We note that the loss of efficiency due to the adjustment of carry-over under the simple carry-over three-period crossover design is moderate, and is much less than that found for a two-period crossover design. We use the double-blind three-period crossover trial comparing formoterol solution aerosol and salbutamol suspension aerosol with a placebo for patients suffering from exercise-induced asthma on the forced expiratory volume in one second (FEV1) to illustrate the use of test procedures and interval estimators discussed here. [ABSTRACT FROM AUTHOR]

An error estimate for the minimal error method for nonlinear ill-posed problems under general a Hölder-type source condition is not known. We consider a modified minimal error method for nonlinear ill-posed problems. Using a Hölder-type source condition, we obtain an optimal order error estimate. We also consider the modified minimal error method with noisy data and provide an error estimate. [ABSTRACT FROM AUTHOR]

In this article, we discuss a linear regression change-point model with skew normal errors. We propose a testing procedure, based on a modified version of the Schwarz information criterion, which is named the modified information criterion (MIC) to locate change points in such a linear regression model. Due to the difficulty of derivation of the asymptotic null distribution of the associated test statistic analytically, the empirical critical values at different significance levels are approximated through simulations. Simulations have also been conducted under different changes among parameters of interest with various sample sizes to investigate the performance of the proposed test. Such a procedure has been applied on a NASA data to illustrate the detecting process. [ABSTRACT FROM AUTHOR]

A method for joint measuring the power law exponent and the structure constant of atmospheric turbulence is proposed and examined. The measurements are equivalent to solve the simultaneous equations formed by the irradiance scintillation index and the angle-of-arrival fluctuations variance, where the measured parameters are regarded as the unknowns. The measured error analysis is also presented. Based on our proposed method, the measured results accord with the daily trend of atmospheric turbulence. [ABSTRACT FROM AUTHOR]

We present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions. [ABSTRACT FROM AUTHOR]

We study error minimizing relaxation (EMR) strategies for use in Landweber and Kaczmarz type iterations applied to linear systems with or without convex constraints. Convergence results based on operator theory are given, assuming exact data. The advantages and disadvantages of these relaxation strategies on a noisy and ill-posed problem are illustrated using examples taken from the field of image reconstruction from projections. We also consider combining EMR with penalization. [ABSTRACT FROM AUTHOR]

A Neumann boundary control problem for a second order elliptic state equation is considered. The problem is regularized by an energy term which is equivalent to the H- 1/2( Γ) -norm of the control. Both the unconstrained and the control constrained cases are investigated. The regularity of the state, control, and co-state variables is studied with particular focus on the singularities due to the corners of the two-dimensional domain. The state and co-state are approximated by piecewise linear finite elements. For the approximation of the control variable we use carefully designed spaces of piecewise linear or piecewise constant functions, such that an inf-sup condition is satisfied. Error estimates for the approximate solution are proved for all three variables and we show a relation between convergence rate and the opening angles at corners of the domain. As the control grows in general unboundedly near the concave corners for unconstrained problems, it becomes active and hence regular when control constraints are present. We show that in this case the convergence rates are higher than in the unbounded case. Numerical tests suggest that the estimates derived are optimal in the unconstrained case but too pessimistic in the control constrained case. [ABSTRACT FROM AUTHOR]

The relations between stability and accuracy of the operator marching method (OMM) are usually conflicting in waveguides with strong range dependence. To explain this phenomenon, this study intends to present an error estimate for the OMM in range dependent waveguides. We utilize 'approximation level' to measure truncation error for a marching method in various range step sizes. Then, the error estimate is developed to analyze the performances of the OMM. Through an error analysis, we verify the following features of the OMM: (i) it is valid to apply the OMM in slowly varying waveguides with very large range step sizes; (ii) the OMM may blow up suddenly when the range dependence is strong and the step size is extremely small in the same time. We also develop a three-number set to describe the stability and accuracy level of a general marching method for computing wave propagation in a waveguide. In the end, extensive numerical experiments are implemented to verify the correctness of the error analysis. [ABSTRACT FROM AUTHOR]

We are interested on an inverse problem of distributed parameter estimation in a partial differential equation (PDE) from measures of the PDE's solution. The considered parameter is supposed to be a piecewise constant function. Identifying the parameterization consists on identifying both values of the parameter and shapes of zones where the parameter is constant. We develop a posteriori error estimators for the considered inverse problem and we present a new algorithm, result of the combination between adaptive parameterization technique, leading to overcome the underdetermination problem and, an adaptive mesh technique, guided by a posteriori error estimators, providing more precise results. [ABSTRACT FROM AUTHOR]

We present the error analysis of a particular Oldroyd-B type model with the limiting Weissenberg number going to infinity. Assuming a suitable regularity of the exact solution we study the error estimates of a standard finite element method and of a combined finite element/finite volume method. Our theoretical result shows first order convergence of the finite element method and the error of the order 퓞( h3/4) for the finite element/finite volume method. These error estimates are compared and confirmed by the numerical experiments. [ABSTRACT FROM AUTHOR]

Due to improved laser scanning technology, laser scanner based deformation analyses are presently widespread. These deformation analyses are no longer based on individual points representing the deformation of an object at selected positions. Instead, they are based on a large number of scan points sampling the whole object. This fact either leads to challenges regarding metrological aspects as well as regarding modeling aspects: -- Estimating and quantifying spatial correlations between scan points and incorporating them into the deformation analysis -- Separating the laser scanners' internal systematic errors from areal deformations -- Minimizing the bias at areal deformation analyses due to a worse network configuration and limited object knowledge -- Developing freeform parameterizations to reproduce arbitrary areal deformations of an object by individual parameters -- Incorporating an extended uncertainty model considering also model errors due to imperfect knowledge and simplification of the sampled object. -- Only when considering all of these aspects, laser scanner based deformation analyses can benefit from the potential of the areal object sampling. This study aims at naming and reasoning these aspects. Furthermore, it introduces first methodologies and approaches for dealing with them. [ABSTRACT FROM AUTHOR]

In this article, we consider an inverse problem for the integral equation of the convolution type in a multidimensional case. This problem is severely ill-posed. To deal with this problem, using a priori information (sourcewise representation) based on optimal recovery theory we propose a new method. The regularization and optimization properties of this method are proved. An optimal minimal a priori error of the problem is found. Moreover, a so-called optimal regularized approximate solution and its corresponding error estimation are considered. Efficiency and applicability of this method are demonstrated in a numerical example of the image deblurring problem with noisy data. [ABSTRACT FROM AUTHOR]

Say that interest focuses on the relationship between an exposure variable and an outcome variable; however, the exposure variable is subject to measurement error. While exceptions have been identified, in almost all circumstances nondifferential measurement error leads to attenuated regression coefficients and lost power to detect associations. In the case of an ordinal exposure variable subject to nondifferential misclassification, we confirm that power is always lost in a general test for association between the exposure and outcome variables. Surprisingly, however, we find that for a linear test of trend, a gain in power is possible. [ABSTRACT FROM AUTHOR]

We extend our hybrid bounds for twisted automorphic L-functions on the critical line [J. reine angew. Math. 621 (2008), 53-79] to cusp forms f with arbitrary nebentypus, and correct an error in the proof. [ABSTRACT FROM AUTHOR]