Direct and iterative methods for the numerical solution of mixed integral equations

In this paper we analyze and compare two classical methods to solve Volterra–Fredholm integral equations. The first is a collocation method; the second one is a fixed point method. Both of them are proposed on a particular class of approximating functions. Precisely the first method is based on a linear spline class approximation and the second one on Schauder linear basis. We analyze some problems of convergence and we propose some remarks about the peculiarities and adaptability of both methods. Numerical results complete the work.     

Diffractive optics with harmonic radiation in 2d nonlinear photonic crystal waveguide

The propagation of modulated light in a 2d nonlinear photonic waveguide is investigated in the framework of diffractive optics. It is shown that the dynamics obeys a nonlinear Schr?dinger equation at leading order. We compute the first and second corrector and show that the latter may describe some dispersive radiation through the structure. We prove the validity of the approximation in the interval of existence of the leading term.     

Numerical differentiation with noisy signal

The aim of this paper is to present a new numerical method, which ables one to filter and compute numerical derivatives of a function whose values are known in some points from experimental measurements, inducing noisy data. We use a piecewise cubic spline interpolation to generate a function whose Fourier coefficients give an approximation of the numerical derivatives we are looking for. Error and stability analysis of this numerical algorithm are provided. Numerical results are presented for data smoothing and for the first and second derivatives computed from noisy data. They show that this method gives good numerical results. Comparison with other methods is done.     

主特征值估计的新故事

本文介绍主特征值估计的两种通用方法，着重于两个侧面：来自黎曼几何的第一种方法如何应用于概率论；来自概率论的第二种方法如何应用于黎曼几何。此外，还将概述若干基本结果。     

The iterative instrumental variables method and the full information maximum likelihood method for estimating interdependent systems

The “iterative instrumental variables” (IIV) method for estimating interdependent systems, originally referred to as a symmetric counterpart to the “fix-point” (FP) method, shares its symmetry properties with Durbin's iterative method for performing the “full information maximum likelihood” (FIML) estimation. Classical interdependent systems are considered and identities may occur among the structural equations. Alternative symmetric procedures for obtaining FIML estimates are also dealt with, including the sequential maximization of the likelihood function with respect to the coefficients of one structural equation at a time.Two recent estimation methods developed by Brundy and Jorgenson (1971, Review of Economics and Statistics53, 207–224) as well as Dhrymes (1971, Austral. J. Statist.13, 168–175) can be considered the second approximation of the IIV method and Durbin's method respectively with the first approximation obtained by the “ordinary instrumental variables” (OIV) method. In practice the second approximation depends heavily on the choice of initial instrumental variables, although the asymptotic distribution is not changed by the continued iteration.     

Monte-Carlo estimation of time-dependent statistical characteristics of random dynamical systems

The problem of estimating the time-dependent statistical characteristics of a random dynamical system is studied under two different settings. In the first, the system dynamics is governed by a differential equation parameterized by a random parameter, while in the second, this is governed by a differential equation with an underlying parameter sequence characterized by a continuous time Markov chain. We propose, for the first time in the literature, stochastic approximation algorithms for estimating various time-dependent process characteristics of the system. In particular, we provide efficient estimators for quantities such as the mean, variance and distribution of the process at any given time as well as the joint distribution and the autocorrelation coefficient at different times.     

风险值的估计及其周期分析

本文提出了两种风险值的估计方法，这两种方法均是先估计出收益的分布，然后求得分布左侧p分位点作为风险值的估计．第一种方法是用核估计方法得到收益的分布估计；第二种方法则是由分布的核估计算得收益的众数，引入所谓的广义半t分布拟合众数左侧的样本．文章以上证指数为实例验证了这两种方法的可行性与精确性．最后我们利用上述两种估计方法得到了上证指数风险值的波动主周期．     

Malliavin Greeks without Malliavin calculus

We derive and analyze Monte Carlo estimators of price sensitivities (“Greeks”) for contingent claims priced in a diffusion model. There have traditionally been two categories of methods for estimating sensitivities: methods that differentiate paths and methods that differentiate densities. A more recent line of work derives estimators through Malliavin calculus. The purpose of this article is to investigate connections between Malliavin estimators and the more traditional and elementary pathwise method and likelihood ratio method. Malliavin estimators have been derived directly for diffusion processes, but implementation typically requires simulation of a discrete-time approximation. This raises the question of whether one should discretize first and then differentiate, or differentiate first and then discretize. We show that in several important cases the first route leads to the same estimators as are found through Malliavin calculus, but using only elementary techniques. Time-averaging of multiple estimators emerges as a key feature in achieving convergence to the continuous-time limit.     

B-Spline Estimation for Varying-Coefficient Single-Index Model

The varying-coefficient single-index models (VCSIM) have been applied in many fields since they combine the advantages of single-index models and varying-coefficient models. In this paper, their estimation method is proposed based on B-spline approximation technique and two calculation methods can be used. The first one is to directly calculate the parametric and nonparametric parts simultaneously by Newton-Raphson iteration algorithm. The second one is to calculate the two parts by profile method individually. We suggest that the second method is for our preference when the large amount of parameters are involved, otherwise the first method will be more convenient. Two simulated examples are given to illustrate the performances of the proposed estimation methodologies and calculation procedures.     

A new regularized stochastic approximation framework for stochastic inverse problems

We study the nonlinear inverse problem of estimating stochastic parameters in the fourth-order partial differential equation with random data. The primary focus is on developing a novel stochastic approximation framework for inverse problems consisting of three key components. As a first step, we reformulate the inverse problem into a stochastic convex optimization problem. The second step includes developing a new regularized stochastic extragradient framework for a nonlinear variational inequality, which subsumes the optimality conditions for the optimization formulation of the inverse problem. The third step involves modeling random variables by a Karhunen–Loève type finite-dimensional noise representation, allowing the direct and the inverse problems to be conveniently discretized. We show that the regularized extragradient methods are strongly convergent in a Hilbert space setting, and we also provide several auxiliary results for the inverse problem, including Lipschitz continuity and a derivative characterization of the solution map. We provide the outcome of computational experiments to estimate stochastic and deterministic parameters. The numerical results demonstrate the feasibility and effectiveness of the developed framework and validate stochastic approximation as an effective method for stochastic inverse problems.     

Vector assignment problems: a general framework

We present a general framework for vector assignment problems. In such problems one aims at assigning n input vectors to m machines such that the value of a given target function is minimized. While previous approaches concentrated on simple target functions such as max–max, the general approach presented here enables us to design a polynomial time approximation scheme (PTAS) for a wide class of target functions. In particular, thanks to a novel technique of preprocessing the input vectors, we are able to deal with nonmonotone target functions. Such target functions arise in vector assignment problems in the context of video transmission and broadcasting.     

The ridge function representation of polynomials and an application to neural networks

The first goal of this paper is to establish some properties of the ridge function representation for multivariate polynomials, and the second one is to apply these results to the problem of approximation by neural networks. We find that for continuous functions, the rate of approximation obtained by a neural network with one hidden layer is no slower than that of an algebraic polynomial.     

Using wavelet methods to solve noisy Abel-type equations with discontinuous inputs

One way of estimating a function from indirect, noisy measurements is to regularise an inverse of its Fourier transformation, using properties of the adjoint of the transform that degraded the function in the first place. It is known that when the function is smooth, this approach can perform well and produce estimators that have optimal convergence rates. When the function is unsmooth, in particular when it suffers jump discontinuities, an analogue of this approach is to invert the wavelet transform and use thresholding to decide whether wavelet terms should be included or excluded in the final approximation. We evaluate the performance of this approach by applying it to a large class of Abel-type transforms, and show that the smoothness of the target function and the smoothness of the transform interact in a particularly subtle way to determine the overall convergence rate. The most serious difficulties arise when the target function has a jump discontinuity at the origin; this has a considerably greater, and deleterious, impact on performance than a discontinuity elsewhere. In the absence of a discontinuity at the origin, the rate of convergence is determined principally by an inequality between the smoothness of the function and the smoothness of the transform.     

Case deletion diagnostics in multilevel models

This paper studies case deletion diagnostics for multilevel models. Using subset deletion, diagnostic measures for identifying influential units at any level are developed for both fixed and random parameters. Two approximate update formulae are derived. The first formula uses one-step approximation, while the second formula also includes the impact of estimating the random parameter. Two examples are used to illustrate the methodology developed.     

Weighted BMO and discrete time hedging within the Black-Scholes model

The paper combines two objects rather different at first glance: spaces of stochastic processes having weighted bounded mean oscillation (weighted BMO) and the approximation of certain stochastic integrals, driven by the geometric Brownian motion, by integrals over piece-wise constant integrands. The consideration of the approximation error with respect to weighted BMO implies L_p and uniform distributional estimates for the approximation error by a John-Nirenberg type theorem. The general results about weighted BMO are given in the first part of the paper and applied to our approximation problem in the second one.     

A moment matching approach to log-normal portfolio optimization

We consider the problem where a manager aims to minimize the probability of his portfolio return falling below a threshold while keeping the expected return no worse than a target, under the assumption that stock returns are Log-Normally distributed. This assumption, common in the finance literature for daily and weekly returns, creates computational difficulties because the distribution of the portfolio return is difficult to estimate precisely. We approximate it with a single Log-Normal random variable using the Fenton–Wilkinson method and investigate an iterative, data-driven approximation to the problem. We propose a two-stage solution approach, where the first stage requires solving a classic mean-variance optimization model and the second step involves solving an unconstrained nonlinear problem with a smooth objective function. We suggest an iterative calibration method to improve the accuracy of the method and test its performance against a Generalized Pareto Distribution approximation. We also extend our results to the design of basket options.     

Adaptive radial basis function interpolation using an error indicator

In some approximation problems, sampling from the target function can be both expensive and time-consuming. It would be convenient to have a method for indicating where approximation quality is poor, so that generation of new data provides the user with greater accuracy where needed. In this paper, we propose a new adaptive algorithm for radial basis function (RBF) interpolation which aims to assess the local approximation quality, and add or remove points as required to improve the error in the specified region. For Gaussian and multiquadric approximation, we have the flexibility of a shape parameter which we can use to keep the condition number of interpolation matrix at a moderate size. Numerical results for test functions which appear in the literature are given for dimensions 1 and 2, to show that our method performs well. We also give a three-dimensional example from the finance world, since we would like to advertise RBF techniques as useful tools for approximation in the high-dimensional settings one often meets in finance.     

Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients

We demonstrate the construction of generalized Rough Polyharmonic Splines (GRPS) within the Bayesian framework, in particular, for multiscale PDEs with rough coefficients. The optimal coarse basis can be derived automatically by the randomization of the original PDEs with a proper prior distribution and the conditional expectation given partial information on, for example, edge or first order derivative measurements as shown in this paper. We prove the (quasi)-optimal localization and approximation properties of the obtained bases. The basis with respect to edge measurements has first order convergence rate, while the basis with respect to first order derivative measurements has second order convergence rate. Numerical experiments justify those theoretical results, and in addition, show that edge measurements provide a stabilization effect numerically.     

Approximation Methods for Solving the Cauchy Problem

In this paper we give some new results concerning solvability of the 1-dimensional differential equation y′ = f(x, y) with initial conditions. We study the basic theorem due to Picard. First we prove that the existence and uniqueness result remains true if f is a Lipschitz function with respect to the first argument. In the second part we give a contractive method for the proof of Picard theorem. These considerations allow us to develop two new methods for finding an approximation sequence for the solution. Finally, some applications are given.