Convergence Analysis of Iterative Laplace Transform Methods for the Coupled PDEs from Regime-Switching Option Pricing

This paper aims to analyze the convergence rates of the iterative Laplace transform methods for solving the coupled PDEs arising in the regime-switching option pricing. The so-called iterative Laplace transform methods are described as follows. The semi-discretization of the coupled PDEs with respect to the space variable using the finite difference methods (FDMs) gives the coupled ODE systems. The coupled ODE systems are solved by the Laplace transform methods among which an iteration algorithm is used in the computational process. Finally, the numerical contour integral method is used as the Laplace inversion to restore the solutions to the original coupled PDEs from the Laplace space. This Laplace approach is regarded as a better alternative to the traditional time-stepping method. The errors of the approach are caused by the FDM semi-discretization, the iteration algorithm and the Laplace inversion using the numerical contour integral. This paper provides the rigorous error analysis for the iterative Laplace transform methods by proving that the method has a second-order convergence rate in space and exponential-order convergence rate with respect to the number of the quadrature nodes for the Laplace inversion.     

Laplace transform analysis of the carbon cycle

A Laplace transform representation is used to describe the changes in atmospheric CO₂ in response to emissions. The formalism gives an explicit representation of generic relations that are less clear when model results are presented as numerical integrations with particular parameter values. In particular, the Laplace transform formalism clarifies some issues involved in inversion of ice-core data and analysis of geosequestration. The airborne fraction is expressed as the emission growth rate multiplied by the Laplace transform of the atmospheric response function, evaluated at the growth rate. This representation emphasises that historical data only capture carbon cycle dynamics over a limited range of time-scales. The Laplace transform formalism provides a basis for expressing uncertainties in the response function in terms of the Padé–Laplace transformation used for fitting sums of exponentials.     

A Flexible Inverse Laplace Transform Algorithm and its Application

A flexible efficient and accurate inverse Laplace transform algorithm is developed. Based on the quotient-difference methods the algorithm computes the coefficients of the continued fractions needed for the inversion process. By combining diagonalwise operations and the recursion relations in the quotient-difference schemes, the algorithm controls the dimension of the inverse Laplace transform approximation automatically. Application of the algorithm to the solute transport equations in porous media is explained in a general setting. Also, a numerical simulation is performed to show the accuracy and efficiency of the developed algorithm.     

遗传算法在三维网格模型数字水印中的应用

在数字水印的研究中,为保护版权信息特性,提高精确性和抗噪能力,提出了一种遗传算法的三维网格模型变换域盲水印算法。先将三维模型转化为二维数值矩阵,然后进行二维分块DCT变换生成系数矩阵,利用遗传算法寻找满足适应度函数的最佳嵌入系数,适应度函数使嵌入水印后的模型变形最小。并要嵌入矩阵的位置信息及相应的DCT系数,进行仿真。结果表明,算法可抵抗平移、旋转、各向一致缩放攻击、顶点重排序攻击、噪声攻击。证明对简化方法具有一定的鲁棒性,为设计提供参考。     

Numerical inversion of multidimensional Laplace transforms by the Laguerre method

Numerical transform inversion can be useful to solve stochastic models arising in the performance evaluation of telecommunications and computer systems. We contribute to this technique in this paper by extending our recently developed variant of the Laguerre method for numerically inverting Laplace transforms to multidimensional Laplace transforms. An important application of multidimensional inversion is to calculate time-dependent performance measures of stochastic systems. Key features of our new algorithm are: (1) an efficient FFT-based extension of our previously developed variant of the Fourierseries method to calculate the coefficients of the multidimensional Laguerre generating function, and (2) systematic methods for scaling to accelerate convergence of infinite series, using Wynn's ε-algorithm and exploiting geometric decay rates of Laguerre coefficients. These features greatly speed up the algorithm while controlling errors. We illustrate the effectiveness of our algorithm through numerical examples. For many problems, hundreds of function evaluations can be computed in just a few seconds.     

On a functional approximation for inversion of Laplace transforms via shifted Chebyshev series

A functional representation for inversion of the Laplace transform of a function is considered. The function is given as a shifted Chebyshev series expansion. Using special operational properties, each Laplace transform is converted into a set of simultaneous linear algebraic equations that are then easily solved to give the coefficients of the Chebyshev series. The method is simple and very suitable for computer programming. Applications to rational and irrational Laplace transforms are presented to demonstrate the satisfactory results that the method provides.     

Symbolic-numerical solution of systems of linear ordinary differential equations with required accuracy

In the paper, a symbolic-numerical algorithm for solving systems of ordinary linear differential equations with constant coefficients and compound right-hand sides. The algorithm is based on the Laplace transform. A part of the algorithm determines the error of calculation that is sufficient for the required accuracy of the solution of the system. The algorithm is efficient in solving systems of differential equations of large size and is capable of choosing methods for solving the algebraic system (the image of the Laplace transform) depending on its type; by doing so the efficiency of the solution of the original system is optimized. The algorithm is a part of the library of algorithms of the Mathpar system [15].     

Hybrid laplace transform/weighting function scheme for transient multidimensional conservation equations

A hybrid Laplace transform/weighting function scheme is developed for solving time-dependent multidimensional conservation equations. The new method removes the time derivatives from the governing differential equations using the Laplace transform and solves the associated equation with the weighting function scheme. The similarity transform method is used to treat the complex coefficient system of the equations, which allows the simplest form of complex number functions to be obtained, and then to use the partial fractions method or a numerical method to invert the Laplace transform and transform the functions to the physical plane. Three different examples have been analyzed by the present method. The present method solutions are compared in tables with the exact solutions and those obtained by the other numerical methods. It is found that the present method is a reliable and efficient numerical tool.     

基于DCT的2D实值离散Gabor变换

摘 要 本文提出了一种在临界抽样条件下基于2D DCT的二维实值离散Gabor变换(2D RDGT),介绍了其快速算法。并比较了该变换与二维复值离散Gabor变换(2D CDGT)的算法复杂性。     

Functional approximation for inversion of Laplace transforms via polynomial series

A method for finding the inverse of Laplace transforms using polynomial series is discussed. It is known that any polynomial series basis vector can be transformed into Taylor polynomials by use of a suitable transformation. In this paper, the cross product of a polynomial series basis vector is derived in terms of Taylor polynomials, and as a result the inverse of the Laplace transform is obtained, using the most commonly used polynomial series such as Legendre, Chebyshev, and Laguerre. Properties of Taylor series are first briefly presented and the required function is given as a Taylor series with unknown coefficients. Each Laplace transform is converted into a set of simultaneous linear algebraic equations that can be solved to evaluate Taylor series coefficients. The inverse Laplace transform using other polynomial series is then obtained by transforming the properties of the Taylor series to other polynomial series. The method is simple and convenient for digital computation. Illustrative examples are also given,     

One-way hash function construction based on 2D coupled map lattices

An algorithm for constructing one-way hash function based on spatiotemporal chaos is proposed. A two-dimensional coupled map lattices (2D CML) with parameters leading to the largest Lyapunov exponent is employed. The state of the 2D CML is dynamically determined by its previous state and the message bit at the corresponding positions. The hash value is obtained by a linear transform on the final state of the 2D CML. Theoretical analysis and computer simulation indicate that our algorithm has good statistical properties, strong collision resistance and high flexibility. It is practical and reliable, with high potential to be adopted as a strong hash function for providing data integrity.     

基于图拉普拉斯变换和极限学习机的时间序列预测算法

由于时间效率的约束,多元时间序列预测算法往往存在预测准确率不足的问题。对此,提出基于图拉普拉斯变换和极限学习机的时间序列预测算法。基于图拉普拉斯变换对时间序列进行半监督的特征提取,通过散布矩阵将监督特征和无监督特征进行融合。设计在线的极限学习机学习算法,仅需要在线更新网络的输出权重矩阵即可完成神经网络的学习。利用提取的特征在线训练极限学习机,实现对多元时间序列的实时预测。基于多个数据集进行仿真实验,结果表明该算法有效地提高了预测准确率。     

A distributed algorithm for European options with nonlinear volatility

A distributed algorithm is developed to solve nonlinear Black-Scholes equations in the hedging of portfolios. The algorithm is based on an approximate inverse Laplace transform and is particularly suitable for problems that do not require detailed knowledge of each intermediate time steps.     

Linear system identification using numerical computation of Laplace transforms

A method for system identification using sampled values of the initial transient step or impulse response is described. A polynomial fit of the sampled values is made using Lagrange interpolation and the Laplace transform of the output observed is determined. Then the coefficients of the numerator and denominator polynomials of the system transfer function are determined by minimizing the square of the difference between the observed and calculated values of the Laplace transform of the output variable at a number of discrete points. This process is considerably simplified by the use of tables of coefficients for the numerical calculation of Laplace transforms.     

A semi-infinite programming approach to identifying matrix-exponential distributions

The Laplace–Stieltjes transform of a matrix-exponential (ME) distribution is a rational function where at least one of its poles of maximal real part is real and negative. The coefficients of the numerator polynomial, however, are more difficult to characterise. It is known that they are contained in a bounded convex set that is the intersection of an uncountably infinite number of linear half-spaces. In order to determine whether a given vector of numerator coefficients is contained in this set (i.e. the vector corresponds to an ME distribution) we present a semi-infinite programming algorithm that minimises a convex distance function over the set. In addition, in the event that the given vector does not correspond to an ME distribution, the algorithm returns a closest vector which does correspond to one.     

基于功率谱的H.264/AVC快速帧内预测算法

针对H.264/AVC帧内预测复杂度高的问题,结合H.264帧内预测的特点,提出一种基于功率谱的快速帧内预测算法。该算法在帧内预测之前,首先对宏块进行离散快速傅里叶变换,将频域内的三维功率谱函数转为化二维功率谱函数,然后利用二维功率谱函数的自相关性进行阈值判决,最后根据判决结果从两种预测模式中选择一种,从而减少算法的复杂度。实验结果表明,该算法在保证图像质量和编码码率较少增加的前提下,有效提高了编码速度;并且,该算法有利于在视频编码芯片上的实现,可用于实际的视频通信产品。     

On the numerical approximation of the Laplace transform

A method is developed for the numerical approximation of the Laplace transform of a given function of time. This method is an extension of the method proposed in [2] An example is included which compares the accuracy of the original method and the extension proposed.     

Modelling and simulation of nabla fractional dynamic systems with nonzero initial conditions

The paper focuses on the numerical approximation of discrete fractional order systems with the conditions of nonzero initial instant and nonzero initial state. First, the inverse nabla Laplace transform is developed and the equivalent infinite dimensional frequency distributed model of discrete fractional order system is introduced. Then, resorting the nabla discrete Laplace transform, the rationality of the finite dimensional frequency distributed model approaching the infinite one is illuminated. Based on this, an original algorithm to estimate the parameters of the approximate model is proposed with the help of vector fitting method. Additionally, the applicable object is extended from a sum operator to a general system. Three numerical examples are performed to illustrate the applicability and flexibility of the introduced methodology.     

结合谱聚类的标记分布学习

标记分布是一种新的学习范式,现有算法大多数直接使用条件概率建立参数模型,未充分考虑样本之间的相关性,导致计算复杂度增大。基于此,引入谱聚类算法,通过样本之间相似性关系将聚类问题转化为图的全局最优划分问题,进而提出一种结合谱聚类的标记分布学习算法（label distribution learning with spectral clustering,SC-LDL）。首先,计算样本相似度矩阵;然后,对矩阵进行拉普拉斯变换,构造特征向量空间;最后,通过K-means算法对数据进行聚类建立参数模型,预测未知样本的标记分布。与现有算法在多个数据集上的实验表明,本算法优于多个对比算法,统计假设检验进一步说明算法的有效性和优越性。