One question in the context of immersed boundary or fictitious domain methods is how to compute discontinuous integrands in cut elements accurately. A frequently used method is to apply a composed Gaussian **quadrature** based on a spacetree subdivision. Although this approach works robustly on any geometry, the resulting integration mesh yields a low order representation of the boundary. If high order shape functions are employed to approximate the solution, this lack of geometric approxima- tion power prevents exponential convergence in the asymptotic range. In this paper we present an algorithmic subdivision approach that aims to be as robust as the spa- cetree decomposition even for close-to-degenerate cases—but remains geometrically accurate at the same time. Based on 2D **numerical** examples, we will show that optimal convergence rates can be obtained with a nearly optimal number of integration points. Keywords: Immersed boundary methods, Finite Cell Method, **Numerical** **quadrature**

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We present a method for the **numerical** **quadrature** of an integral with a logarithmic singularity and a cosine oscillator: a modified Filon-Lobatto **quadrature** for the oscillating parts and an integral transformation based on the error function for the singularity. Since this integral can be solved analytically, we are in a position to verify the results of our investigations, with a focus on precision and computation time.

In this thesis, we examine the main types of **numerical** **quadrature** methods for a special subclass of one-dimensional highly oscillatory integrals. Along with a presentation of the methods themselves and the error bounds, the thesis contains implementations of the methods in Maple and Python. The implementations take advantage of the symbolic computational abilities of Maple and allow for a larger class of problems to be solved with greater ease to the user. We also present a new variation on Levin integration which uses di ff erentiation matrices in various interpolation bases.

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Abstract. Two conceptual developments in the Bayesian automatic adaptive **quadrature** approach to the **numerical** solution of one-dimensional Riemann integrals [Gh. Adam, S. Adam, Springer LNCS 7125, 1–16 (2012)] are reported. First, it is shown that the **numerical** **quadrature** which avoids the overcomputing and minimizes the hidden ﬂoating point loss of precision asks for the consideration of three classes of integration domain lengths endowed with speciﬁc **quadrature** sums: microscopic (trapezoidal rule), meso- scopic (Simpson rule), and macroscopic (**quadrature** sums of high algebraic degrees of precision). Second, sensitive diagnostic tools for the Bayesian inference on macroscopic ranges, coming from the use of Clenshaw-Curtis **quadrature**, are derived.

Using the Picard iteration method and treating the involved integration by **numerical** **quadrature** formulas, we propose a **numerical** scheme for the second kind nonlinear Volterra integral equa- tions. For enlarging the convergence region of the Picard iteration method, multistage algorithm is devised. We also introduce an algorithm for problems with some singularities at the limits of in- tegration including fractional integral equations. **Numerical** tests verify the validity of the pro- posed schemes.

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This paper has examined spectral risk measures based on an exponential utility function. We find that the exponential utility function leads to risk-aversion functions and spectral risk measures with intuitive and nicely behaved properties. These exponential SRMs are easy to estimate using **numerical** **quadrature** methods and accurate estimates can be obtained very quickly in real time. It is also easy to estimate confidence intervals for these SRMs using a parametric bootstrap. Illustrative results suggest that these confidence intervals are surprisingly narrow, and this indicates that SRM estimates are quite precise. Of course, the results presented here are based on an assumed normal distribution, and further work is needed to establish results for other distributions. 10

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In this paper we introduce a new fast and accurate **numerical** method for pricing exotic derivatives when discrete monitoring occurs, and the under- lying evolves according to a Markov one-dimensional stochastic processes. The approach exploits the structure of the matrix arising from the **numerical** **quadrature** of the pricing backward formulas to devise a convenient factor- ization that helps greatly in the speed-up of the recursion. The algorithm is general and is examined in detail with reference to the CEV (Constant Elasticity of Variance) process for pricing different exotic derivatives, such as Asian, barrier, Bermudan, lookback and step options for which up to date no efficient procedures are available. Extensive **numerical** experiments confirm the theoretical results. The MATLAB code used to perform the computation is available online at http://www1.mate.polimi.it/~marazzina/BP.htm. Keywords: CEV Process, Discrete Monitoring, Exotic Derivatives, Matrix Factorization, **Numerical** **Quadrature**, Option Pricing, Recursive Pricing Formula

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approximation theory [5, 15] (see also [8]), and has been applied with huge success in the context of FEMs for the **numerical** approximation of differential equations. Indeed, under certain conditions, the judicious combination of subinterval refine- ments (h-refinement) and selection of local approximation orders (p-refinement), which results in the class of so-called hp-FEMs, is able to achieve high-order alge- braic or exponential rates of convergence, even for solutions with local singularities; see, e.g. [18]. In an effort to automate the combined h- and p-refinement process, a number of hp-adaptive FEM approaches have been proposed in the literature; see, e.g, [13] and the references cited therein. In the current article, we pursue the smoothness estimation approach developed in [6, 21] (cf. also [11]), and translate the idea into the context of adaptive variable order **numerical** **quadrature**.

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We give the **numerical** results of the splitting extrapolation of types 1 and 2 and Gauss **quadrature** methods. Table 1 gives the relative error (RE) and CPU time for different dimension ( s ) and splitting times ( m ). From the Table 1, we can find that the splitting extrapolation method is suit for solving high dimen- sional integrals, and Gauss **quadrature** rule is difficult for solving more than five dimensional problems.

to approximate the solution of (1). In this method he apply the classical trape- zoidal **quadrature** rule ( for functions with continuous second order derivatives ) in [22], to compute the integrals from (4) on the knots of the uniform partition. In [14] is used the sequence of successive approximations and the classical **quadrature** trapezoidal rule for functions with continuous second order deriva- tives to approximate the solutions of Fredholm and Volterra integral equations.

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We have described a special finite difference method for solving a singularly perturbed differential difference equation with layer behaviour at one end point. In the special second order method, we have used a second order finite difference approximation for second order derivative, a modified second order upwind finite difference approximation for first order derivative and a second order average difference approximation for y. This method controls the rapid changes that occur in the boundary layer region and it gives good results. To discuss the applicability of the method we have solved some model examples by taking different values of , and . We have presented maximum absolute errors for the standard examples chosen from the literature. The **numerical** solution is compared with the exact solution. It is observed from results that the present technique approximates the exact solution very well.

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It is well known that a Hilbert space can be given a Gaussian measure. Let H be a Hilbert space equipped with Gaussian measure and L a continuous linear functional acting on H. Smale in 6 a pioneering work on continuous complexity theory defined an average with respect to the Gaussian measure error for **quadrature** rules. A result of Smale 6 says that the average error is proportional to L. More precisely,

A class of singular integro-differential equations in Lebesgue spaces are stud- ied. There are many applications of the singular integro-differential equations discussed in this paper. An example in modeling the stress distribution of an elas- tic medium with holes is discussed in the paper. Direct **numerical** schemes using a collocation method and a mechanical **quadrature** rule designed for the singular integro-differential equations are proposed for arbitrary smooth closed contours. Convergence analysis of these methods are given. **Numerical** examples are also provided.

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In this paper, we present a **numerical** method for the simulation of a cantilever beam expressed as a boundary value problem with mixed conditions. We first introduce a basis by Bernstein polynomials satisfying homogeneous mixed boundary conditions. Then, a Bernstein–spectral method is presented for the **numerical** simulation of the problem. Also, a convolution **quadrature** method combined with a second order backward difference for the handling of the Neumann condition at the free end of the cantilever is presented for the discretization of the problem. It is discussed that the resulting nonlinear system has a special structure that makes it possible to be approximated by a linear system. This paper is organized as follows. Section 2 describes the physical aspects and modeling of a cantilever beam with regard to the static governing equations of the Euler–Bernoulli beam. Section 3 introduces a basis by Bernstein polynomials in order to use with the spectral method for the discretization of the problem. In section 4, we we the transformation to a Volterra integral equation and the convolution **quadrature** method is presented. The **numerical** experiments are provided in Section 5. The paper ends with some concluding remarks.

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There are several analytical and **numerical** methods use to solve nonlinear FIE as mentioned in the previous section but these analytical solution methods are not easy to use and require tedious calculation. Also when applying these methods to solve linear and nonlinear Fredholm integral equations many definite integrals need to be computed.

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All the inequalities in (1.8) are sharp and the constant 1 2 is the best possible. In this paper we establish an Ostrowski type inequality for convex functions. Applications for **quadrature** rules, for integral means, for probability distribution functions, and for HH−divergences in Information Theory are also considered.

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to make comparison between two methods and detect which of them is better. The **numerical** results obtained in this paper ensure that the problems have small desired time to reach it. Thus they have very small step size which is preferred and use RK4 to solve the system of ordinary differential equations in order to decrease the computational time. On the other hand, the problems which have high desired time to reach it, thus have large incremental time (stiff problems) which are preferred and use implicit Euler with perturbation method to solve the system of ordinary differential equations.

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Since the kernel and the deriving term of the integral equation (1.7) are expressed by weakly singular integrals, we must use a **numerical** method which is able to compute these integrals with weak singularity at the end points. For this purpose, Navot's **quadrature** rule is used. This special **quadrature** is applied for functions having a singularity of any type on or near the integration interval.

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In classical analysis, integration is a well-posed problem. In the theory of functions, Lebesgue integration may be logically interpreted as being the inverse to the differentiation operator. The set of absolutely continuous functions in a given interval coincides with the set of functions represented in the form of an indefinite Lebesgue integral with a variable upper bound of some integrable function plus a constant. However, the indicated property of being the inverse to differentiation is not longer valid a discrete set, since the operation of **numerical** differentiation is ill-posed.

In this paper, new **quadrature** rules to approx- imate double and triple integrals with variables limits are presented. For this purpose, the CAS wavelets have been used. Presented **quadrature** rules in this paper can approximate some im- proper integrals. To show the efficiency of pre- sented methods, some test problems are consid- ered.