David A. Voss
Western Illinois University
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Publication
Featured researches published by David A. Voss.
Journal of The Chinese Institute of Engineers | 2004
Gregory E. Fasshauer; Abdul Qayyum Masud Khaliq; David A. Voss
Abstract We study the applicability of meshfree approximation schemes for the solution of multi‐asset American option problems. In particular, we consider a penalty method which allows us to remove the free and moving boundary by adding a small and continuous penalty term to the Black‐Scholes equation. Time discretization is achieved by a linearly implicit θ method. A comparison with results obtained recently by two of the authors using a linearly implicit finite difference method is included.
Journal of Risk | 2007
Abdul Q.M. Khaliq; David A. Voss; Greg Fasshauer
In this paper we consider a meshfree radial basis function approach for the valuation of pricing options with non-smooth payoffs. By taking advantage of parallel architecture, a strongly stable and highly accurate time stepping method is developed with computational complexity comparable to the implicit Euler method implemented concurrently on each processor. This, in collusion with the radial basis function approach, provides an efficient and reliable valuation of exotic options, such as American digital options.
SIAM Journal on Numerical Analysis | 1988
David A. Voss
A fifth-order second derivative formula is developed for stiff systems of ordinary differential equations. The exponentially fitted formulas possess a large region of absolute stability, and numerical results demonstrate increased accuracy with the same computational effort when compared with similar fourth-order formulas.
International Journal of Computer Mathematics | 2015
David A. Voss; A. Q. M. Khaliq
In this paper we discuss new split-step methods for solving systems of Itô stochastic differential equations (SDEs). The methods are based on a L-stable (strongly stable) second-order split Adams–Moulton Formula for stiff ordinary differential equations in collusion with Milstein methods for use on SDEs which are stiff in both the deterministic and stochastic components. The L-stability property is particularly useful when the drift components are stiff and contain widely varying decay constants. For SDEs wherein the diffusion is especially stiff, we consider balanced and modified balanced split-step methods which posses larger regions of mean-square stability. Strong order convergence one is established and stability regions are displayed. The methods are tested on problems with one and two noise channels. Numerical results show the effectiveness of the methods in the pathwise approximation of stiff SDEs when compared to some existing split-step methods.
Applied Numerical Mathematics | 2015
V. Reshniak; Abdul Q.M. Khaliq; David A. Voss; G. Zhang
Abstract We consider split-step Milstein methods for the solution of stiff stochastic differential equations with an emphasis on systems driven by multi-channel noise. We show their strong order of convergence and investigate mean-square stability properties for different noise and drift structures. The stability matrices are established in a form convenient for analyzing their impact arising from different deterministic drift integrators. Numerical examples are provided to illustrate the effectiveness and reliability of these methods.
Mathematical and Computer Modelling | 2004
David A. Voss
Rosenbrock methods are frequently used for the numerical solution of stiff initial value problems. Such linearly implicit methods are characterized by a relatively easy implementation together with excellent linear stability properties. In this paper, we consider modified Rosenbrock methods with s external linearly implicit stages each of which contains p additional linearly implicit internal stages. The internal stages are already parallel so that they can be solved for independently of each other and, consequently, the processors need to exchange their results only after the completion of each of the s external stages. We focus on the design of fourth-order methods with three external stages. Using embedded third-order methods, a variable step size implementation is compared with well-known Rosenbrock codes for performance on the Robertson problem.
Journal of Computational and Applied Mathematics | 1999
David A. Voss; Paul H. Muir
Among the numerical techniques commonly considered for the efficient solution of stiff initial value ordinary differential equations are the implicit Runge-Kutta (IRK) schemes. The calculation of the stages of the IRK method involves the solution of a nonlinear system of equations usually employing some variant of Newtons method. Since the costs of the linear algebra associated with the implementation of Newtons method generally dominate the overall cost of the computation, many subclasses of IRK schemes, such as diagonally implicit Runge-Kutta schemes, singly implicit Runge-Kutta schemes, and mono-implicit (MIRK) schemes, have been developed to attempt to reduce these costs. In this paper we are concerned with the design of MIRK schemes that are inherently parallel in that smaller systems of equations are apportioned to concurrent processors. This work builds on that of an earlier investigation in which a special subclass of the MIRK formulas were implemented in parallel. While suitable parallelism was achieved, the formulas were limited to some extent because they all had only stage order 1. This is of some concern since in the application of a Runge-Kutta method to a system of stiff ODEs the phenomenon of order reduction can arise; the IRK method can behave as if its order were only its stage order (or its stage order plus one), regardless of its classical order. The formulas derived in the current paper represent an improvement over the previous investigation in that the full class of MIRK formulas is considered and therefore it is possible to derive efficient, parallel formulas of orders 2, 3, and 4, having stage orders 2 or 3.
Communications in Numerical Methods in Engineering | 1999
Theresa Kohler; David A. Voss
The method of lines provides a flexible and general approach for solving time-dependent PDEs. However, the numerical solution of the resulting ODE system can present certain difficulties depending on the method used. In particular, oscillations may appear in the solution when standard methods are applied to the ODE system arising from the semi-discretization of the diffusion–convection equation ∂u/∂t=α(∂2u/∂x2)−β(∂u/∂x). We examine second-order methods for such systems and present economical L-stable predictor–corrector schemes which are oscillation-free. Copyright
Applied Mathematics and Computation | 1989
David A. Voss
A class of two-step fourth order implicit Runge-Kutta Methods designed for stiff systems of ODEs has recently been investigated by the author and others. In this paper certain computational aspects of this class of methods are discussed. Similar algorithms which, in particular, exploit any sparseness properties inherent in the Jacobian matrix are considered along with an efficient way of estimating the local truncation error using embedded methods.
International Journal of Engineering Science | 1980
Rama Subba Reddy Gorla; David A. Voss
Abstract The steady and transient heat transfer characteristics of a second order viscoelastic boundary layer flow at a stagnation point have been studied in this paper. The implicit cubic spline numerical procedure is used to solve the governing boundary layer equations. The details of the temperature profiles and wall heat flux rates have been graphically illustrated. The range of values of the Prandtl number was from 5 to 1000 while the Weissenberg number was varied from 0.1 to 0.3.