An adaptive moving mesh method for a time-fractional Black–Scholes equation

In this paper we study the numerical method for a time-fractional Black–Scholes equation, which is used for option pricing. The solution of the fractional-order differential equation may be singular near certain domain boundaries, which leads to numerical difficulty. In order to capture the singular phenomena, a numerical method based on an adaptive moving mesh is developed. A finite difference method is used to discretize the time-fractional Black–Scholes equation and error analysis for the discretization scheme is derived. Then, an adaptive moving mesh based on an a priori error analysis is established by equidistributing monitor function. Numerical experiments support these theoretical results.

Numerical and Experimental Study of Bubble Impact on a Solid Wall

The dynamic characteristics of a bubble initially very close to a rigid wall, or with a very narrow gap, are different from those of a bubble away from the wall. Especially at the contraction stage, a high-speed jet pointing toward the wall will be generated and will impact the rigid surface directly, which could cause more severe damage to the structure. Based on the velocity potential theory and boundary element method (BEM), the present paper aims to overcome the numerical difficulty and simulate the bubble impact on a solid wall for the axisymmetric case. The convergence study has been undertaken to verify the developed numerical method and the computation code. Extensive experiments are conducted. Case studies are made using both experimental data and numerical results. The effects of dimensionless distance on the bubble dynamics are investigated.

An Algorithm for the Proximity Operator in Hybrid TV-Wavelet Regularization, with Application to MR Image Reconstruction

Total variation (TV) and waveletL1norms have often been used as regularization terms in image restoration and reconstruction problems. However, TV regularization can introduce staircase effects and wavelet regularization some ringing artifacts, but hybrid TV and wavelet regularization can reduce or remove these drawbacks in the reconstructed images. We need to compute the proximal operator of hybrid regularizations, which is generally a sub-problem in the optimization procedure. Both TV and waveletL1regularisers are nonlinear and non-smooth, causing numerical difficulty. We propose a dual iterative approach to solve the minimization problem for hybrid regularizations and also prove the convergence of our proposed method, which we then apply to the problem of MR image reconstruction from highly random under-sampled k-space data. Numerical results show the efficiency and effectiveness of this proposed method.

GENERALIZED EIGENPROBLEM FOR ACOUSTIC WAVE PROPAGATION IN PERIODICALLY LAYERED ANISOTROPIC MEDIA

A stable matrix method is presented for studying acoustic wave propagation in thick periodically layered anisotropic media at high frequencies. The method enables Floquet waves to be determined reliably based on the solutions to a generalized eigenproblem involving scattering matrix. The method thus overcomes the numerical difficulty in the standard eigenproblem involving cell transfer matrix, which occurs when the unit cell is thick or the frequency is high. With its numerical stability and reliability, the method is useful for analysis of periodic media with wide range of thickness at high frequencies.

A method to find the best bounds in a multivariate discrete moment problem if the basis structure is given

The multivariate discrete moment problem (MDMP) is to find the minimum and/or maximum of the expected value of a function of a random vector which has a discrete finite support. The probability distribution is unknown, but some of the moments are given. The MDMP has been initiated by Prékopa who developed a linear rogramming methodology to solve it. The central results in this respect concern the structure of the dual feasible bases. These bases provide us with bounds without any numerical difficulty (which is arising in the usual solution methods). In this paper we shortly summarize the properties of the above mentioned basis structures, and then we show a new method which allows us to get the basis corresponding to the best bound out of the known structures by optimizing independently on each variable. We illustrate the efficiency of this method by numerical examples.

A Computational Issue and Modified Formulas for Nonlinear Dissipative Controllers

Ball, Helton, and Walker (BHW) derived the nonlinear dissipative controller formulas with the assumption implying that no stable mode uncontrollable from the exogenous input. The assumption is more restrictive than that considered in DGKF. In this paper, we address the numerical difficulty encountered by BHW’s controller formulas when the assumption is not satisfied. Next, we propose a modified nonlinear dissipative controller and successfully remove the numerical difficulty. We also show that the linear version of the proposed controller formulas is identical to the DGKF H∞ controller. An example is given to demonstrate constructing the proposed controller and simulating the closed-loop pulse responses.

Wave Propagation Behaviour of Spatial Structures: Impact Response Analysis of Single Layer Lattice Domes

Many researchers have investigated the wave propagation behaviors of continuum spatial structures such as plate and cylindrical shell. But due to the numerical difficulty, there is very little research related to discrete spatial structures. The objective of this paper is to examine wave propagation behavior of single layer lattice domes subject to vertical impact load at the centre node. In the first part, an analytical method based on the Fourier transform and a matrix method is presented. In the second part, an impacted two dimensional lattice beam is analyzed in order to estimate the validity of the analytical program. In the last part, nine types of single layer lattice domes are analyzed to estimate the effect of member distribution density, member arrangement pattern and the jointing system, etc., upon the wave propagation behavior.