Justin W. L. Wan

Robust numerical valuation of European and American options under the CGMY process

We develop an implicit discretization method for pricing European and American options when the underlying asset is driven by an infinite activity Levy process. For processes of finite variation, quadratic convergence is obtained as the mesh and time step are refined. For infinite variation processes, better than first order accuracy is achieved. The jump component in the neighborhood of log jump size zero is specially treated by using a Taylor expansion approximation and the drift term is dealt with using a semi-Lagrangian scheme. The resulting Partial Integro-Differential Equation (PIDE) is then solved using a preconditioned BiCGSTAB method coupled with a fast Fourier transform. Proofs of fully implicit timestepping stability and monotonicity are provided. The convergence properties of the BiCGSTAB scheme are discussed and compared with a fixed point iteration. Numerical tests showing the convergence and performance of this method for European and American options under processes of finite and infinite variation are presented.

A Boundary Condition--Capturing Multigrid Approach to Irregular Boundary Problems

We propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions. In this method, we adopt a matrix-free approach; i.e., we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices, as many other robust multigrid methods do. The main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function. Numerical results are given to compare our multigrid method with black box and algebraic multigrid methods.

Physically-based simulation of plant leaf growth

A mathematical model is presented to simulate the growth of a plant leaf. The tissue in the leaf has been regarded as a viscous, incompressible fluid whose 2D expansion comes from the non‐zero specific growth rate in area. The resulting system of equations are composed of the modified Navier‐Stokes equations. The level set method is used to capture the expanding leaf front. Numerical simulations indicate that different portions of the leaf expand at different rates, which is consistent with the biological observations in the growth of a plant leaf. Numerical results for the case of the Xanthium leaf growth are also presented. A standard ray tracing technique is applied to produce an animation simulating the leaf growth process of three days. The key results with their physical and practical implications are discussed. Copyright

A spectral k-means approach to bright-field cell image segmentation

Automatic segmentation of bright-field cell images is important to cell biologists, but difficult to complete due to the complex nature of the cells in bright-field images (poor contrast, broken halo, missing boundaries). Standard approaches such as level set segmentation and active contours work well for fluorescent images where cells appear as round shape, but become less effective when optical artifacts such as halo exist in bright-field images. In this paper, we present a robust segmentation method which combines the spectral and k-means clustering techniques to locate cells in bright-field images. This approach models an image as a matrix graph and segment different regions of the image by computing the appropriate eigenvectors of the matrix graph and using the k-means algorithm. We illustrate the effectiveness of the method by segmentation results of C2C12 (muscle) cells in bright-field images.

Simulating the dynamics of auroral phenomena

Simulating natural phenomena has always been a focal point for computer graphics research. Its importance goes beyond the production of appealing presentations, since research in this area can contribute to the scientific understanding of complex natural processes. The natural phenomena, known as the Aurora Borealis and Aurora Australis, are geomagnetic phenomena of impressive visual characteristics and remarkable scientific interest. Aurorae present a complex behavior that arises from interactions between plasma (hot, ionized gases composed of ions, electrons, and neutral atoms) and Earths electromagnetic fields. Previous work on the visual simulation of auroral phenomena have focused on static physical models of their shape, modeled from primitives, like sine waves. In this article, we focus on the dynamic behavior of the aurora, and we present a physically-based model to perform 3D visual simulations. The model takes into account the physical parameters and processes directly associated with plasma flow, and can be extended to simulate the dynamics of other plasma phenomena as well as astrophysical phenomena. The partial differential equations associated with these processes are solved using a complete multigrid implementation of the electromagnetic interactions, leading to a simulation of the shape and motion of the auroral displays. In order to illustrate the applicability of our model, we provide simulation sequences rendered using a distributed forward mapping approach.

Real-time intensity-based rigid 2d-3d medical image registration using RapidMind Multi-core Development Platform

In this paper, we present an efficient intensity-based rigid 2D-3D image registration method. We implement the algorithm using the RapidMind Multi-core Development Platform1 to exploit the highly parallel multi-core architecture of graphics processing units (GPUs). We use a ray casting algorithm to generate the digitally reconstructed radiographs (DRRs) on GPUs and efficiently reduce the complexity of DRR construction. The registration optimization problem is solved by the Gauss-Newton method. To fully exploit the multi-core parallelism, we implement almost the entire registration process in parallel by RapidMind. We also discuss the RapidMind implementation of the major computation steps. Numerical results are presented to demonstrate the efficiency of our method.

A parallel quasi-Monte Carlo approach to pricing multidimensional American options

In this paper, we develop parallel algorithms for pricing American options on multiple assets. Our parallel methods are based on the Low Discrepancy (LD) mesh method which combines the quasi-Monte Carlo technique with the stochastic mesh method. We present two approaches to parallelise the backward recursion step, which is the most computational intensive part of the LD mesh method. We perform parallel run time analysis of the proposed methods and prove that both parallel approaches are scalable. The algorithms are implemented using MPI. The parallel efficiency of the methods are demonstrated by pricing several American options, and near optimal speedup results are presented.

Kernel Preserving Multigrid Methods for Convection-Diffusion Equations

We propose a kernel preserving multigrid approach for solving convection-diffusion equations. The multigrid methods use Petrov-Galerkin coarse grid correction and linear interpolation. The restriction operator is constructed by preserving the kernel of the convection-diffusion operator. The construction considers constant and variable coefficient problems as well as cases where the convection term is not known explicitly. For constant convection-diffusion problems, we prove that the resulting Petrov-Galerkin coarse grid correction has small phase errors and the coarse grid matrix is almost an M-matrix. We demonstrate numerically the effectiveness of the multigrid methods by solving a constant convection problem, a recirculating flow problem, and a real application problem for pricing Asian options.

Low-bias simulation scheme for the Heston model by Inverse Gaussian approximation

Fast and accurate sampling of conditional time-integrated variance in the Heston model is an important and challenging problem. We proved that this very complicated distribution converges to the moment-matched Inverse Gaussian distribution as the time interval goes to infinity. Leveraging on this theoretical result, we develop an efficient and accurate Inverse Gaussian approximation to sample conditional time-integrated variance. Numerical results demonstrate that our scheme compares favourably with state-of-the-art methods in accuracy given the same computational time for moderately path-dependent options.

Simulating Mechanism of Brain Injury During Closed Head Impact

In this paper, we study the mechanics of the brain during closed head impact via numerical simulation. We propose a mathematical model of the human head, which consists of three layers: the rigid skull, the cerebrospinal fluid and the solid brain. The fluid behavior is governed by the Navier-Stokes equations, and the fluid and solid interact together according to the laws of mechanics. Numerical simulations are then performed on this model to simulate accident scenarios. Several theories have been proposed to explain whether the ensuing brain injury is dominantly located at the site of impact (coup injury) or at the site opposite to it (contrecoup injury). In particular, we investigate the positive pressure theory, the negative pressure theory, and the cerebrospinal fluid theory. The results of our numerical simulations together with pathological findings show that no one theory can explain the mechanics of the brain during the different types of accidents. We therefore highlight the accident scenarios under which each theory presents a consistent explanation of brain mechanics.