Exponential Time Integration Research Articles

Krylov subspace approximations for the exponential Euler method: error estimates and the harmonic Ritz approximant

We study Krylov subspace methods for approximating the matrix-function vector product $\varphi(tA)b$ where $\varphi(z) = [\exp(z)-1]/z$. This product arises in the numerical integration of large stiff systems of differential equations by the Exponential Euler Method, where $A$~is the Jacobian matrix of the system. Recently, this method has found application in the simulation of transport phenomena in porous media within mathematical models of wood drying and groundwater flow. We develop an a posteriori upper bound on the Krylov subspace approximation error and provide a new interpretation of a previously published error estimate. This leads to an alternative Krylov approximation to $\varphi(tA)b$, the so-called Harmonic Ritz approximant, which we find does not exhibit oscillatory behaviour of the residual error. References E. J. Carr, T. J. Moroney and I. W. Turner. Efficient simulation of unsaturated flow using exponential time integration. Appl. Math. Comput., 217(14):6587--6596, 2011. doi:10.1016/j.amc.2011.01.041 E. J. Carr, I. W. Turner and P. Perre. A Jacobian-free exponential integrator for simulating transport in heterogeneous porous media: application to the drying of softwood, submitted for publication. E. J. Carr, I. W. Turner and P. Perre. A new control-volume finite-element scheme for heterogeneous porous media:application to the drying of softwood. Chem. Eng. Technol., 34(7):1143--1150, 2011. doi:10.1002/ceat.201100060 E. Celledoni and I. Moret. A Krylov projection method for systems of ODEs. Appl. Numer. Math., 24(2--3):365--378, 1997. doi:10.1016/S0168-9274(97)00033-0 N. J. Higham. Functions of matrices: theory and computation. SIAM, Philadelphia, PA, USA, 2008. M. Hochbruck, M. E. Hochstenbach. Subspace extraction for matrix functions, submitted for publication. http://www.win.tue.nl/ hochsten/pdf/funext.pdf M. Hochbruck, C. Lubich and H. Selhofer. Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput., 19(5):1552--1574, 1998. doi:10.1137/S1064827595295337 B. V. Minchev and W. M. Wright. A review of exponential integrators for first order semi-linear problems. Numerics No. 2/05, Norwegian University of Science and Technology, Trondheim, Norway, 2005. http://www.math.ntnu.no/preprint/numerics/2005/N2-2005.ps P. Perre and I. Turner. A heterogeneous wood drying computational model that accounts for material property variation across growth rings. Chem. Eng. J., 86(1--2):117--131, 2002. doi:10.1016/S1385-8947(01)00270-4

Fast simplified approaches to Asian option pricing

In this paper a general and simple framework is proposed for pricing Asian options under a variety of pricing models, including the Merton model and the Carr- Geman-Madan-Yor model. The approach uses the exponential time integration scheme in combination with a dimensional splitting technique. For discretely observed Asian options, a simple splitting technique is applied, while a Strang splitting strategy is employed to improve the temporal accuracy for continuously observed averages. For European Asian options it is shown that spectral accuracy can be achieved using a Chebyshev discretization. Barrier features are easily enforced and an operator-splitting technique is used to price Amerasians. For the Black-Scholes model we show that employing best rational approximation via Carathéodory-Fejér approximation leads to algorithms with linear complexity in each spatial dimension.

Fast approximations of bond option prices under CKLS models

A new computational method for approximating prices of zero-coupon bonds and bond option prices under general Chan–Karolyi–Longstaff–Schwartz models is proposed. The pricing partial differential equations are discretized using second-order finite difference approximations and an exponential time integration scheme combined with best rational approximations based on the Carathéodory–Fejér procedure is employed for solving the resulting semi-discrete equations. The algorithm has a linear computational complexity and provides accurate bond and European bond option prices. We give several numerical results which illustrate the computational efficiency of the algorithm and uniform second-order convergence rates for the computed bond and bond option prices.

Efficient simulation of unsaturated flow using exponential time integration

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ( A) = A −1( e A − I) on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ( A) v can be approximated by Krylov subspace methods that require only matrix–vector products with A. We prove that despite the use of this approximation the scheme remains second order. Furthermore, we provide a practical variable-stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation. Numerical experiments performed on two-dimensional test problems demonstrate that this implementation outperforms second-order, variable-stepsize implementations of the backward differentiation formulae.

Numerical pricing of American options under infinite activity Lévy processes

AbstractUnder infinite activity Lévy models, American option prices can be obtained by solving a partial integro‐differential equation (PIDE), which has a singular kernel. With increasing degree of singularity, standard time‐stepping techniques may encounter difficulties. This study examines exponential time integration (ETI) for solving this problem and the performance of this scheme is compared with the Crank–Nicolson (CN) method and an implicit–explicit method in conjunction with an extrapolation (IMEX‐Extrap), in terms of computational speed and convergence orders. These findings indicate that ETI is faster and more accurate among PIDE‐based methods for solving the system of ordinary differential equations resulting from spatial discretization of the PIDE. For very singular problems, it is shown that the IMEX‐Extrap scheme becomes unfavorable compared with the other schemes as it is relatively more time consuming and the global convergence deteriorates from quadratic to linear, whereas the ETI scheme yields both point‐wise and global quadratic convergence. For illustration, under the infinite variation process, the IMEX‐Extrap achieves a precision of the order of 10−4 in 663.016 s, whereas for the same set of parameters, the CN method and the ETI scheme reach an accuracy of the order of 10−5 in 237.891 s and 22.772 s, respectively. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark 31:809–829, 2011

Fast exponential time integration scheme for option pricing with jumps

SUMMARYA fast exponential time integration scheme is considered for pricing European and double barrier options in jump‐diffusion models. After spatial discretization, the option pricing problem is transformed into the product of a matrix exponential and a vector, while the matrix bears a Toeplitz structure. The shift‐and‐invert Arnoldi method is then employed for fast approximation to such operation. Owing to the Toeplitz structure, the computational cost can be reduced by the fast Fourier transform. Furthermore, the discretized form of option pricing problem satisfies a given condition such that the error bound of the shift‐and‐invert Arnoldi approximation is unrelated to the norm of the matrix. Numerical tests are carried out to compare the proposed method with other option pricing methods. Copyright © 2010 John Wiley & Sons, Ltd.

An exponential integrator for advection-dominated reactive transport in heterogeneous porous media

We present an exponential time integrator in conjunction with a finite volume discretisation in space for simulating transport by advection and diffusion including chemical reactions in highly heterogeneous porous media representative of geological reservoirs. These numerical integrators are based on the variation of constants solution and solving the linear system exactly. This is at the expense of computing the exponential of the stiff matrix comprising the finite volume discretisation. Using real Léja points or a Krylov subspace technique compared to standard finite difference-based time integrators. We observe for a variety of example applications that numerical solutions with exponential methods are generally more accurate and require less computational cost. They hence comprise an efficient and accurate method for simulating non-linear advection-dominated transport in geological formations.

Exponential time integration using Krylov subspaces

AbstractThe application of exponential integrators based on Krylov techniques to large‐scale simulations of complex fluid flows with multiple time‐scales demonstrates the efficiency of these schemes in reducing the associated time‐step restrictions due to numerical stiffness. Savings of approximately 50% can be achieved for simulations of the three‐dimensional compressible Navier–Stokes equations while still maintaining a truncation error typical of explicit time‐stepping schemes. Exponential time integration techniques of this type are particularly advantageous for fluid flows with a wide range of temporal scales such as low‐Mach number, reactive or acoustically dominated flows. Copyright © 2008 John Wiley & Sons, Ltd.

Exponential splitting time integration for pseudospectral methods on moving meshes

Moving meshes are successfully used in many fields. Here we investigate how a recently proposed approach to combine the Strang splitting method for time integration with pseudospectral spatial discretization by orthogonal polynomials can be extended to include moving meshes. A double representation of a function (by coefficients of polynomial expansion and by values at the mesh nodes associated with a suitable quadrature formula) is an essential part of the numerical integration. Before numerical implementation the original PDE is transformed into a suitable form. The approach is illustrated on the linear heat transfer equation.

Exponential time integration for fast finite element solutions of some financial engineering problems

We consider exponential time integration schemes for fast numerical pricing of European, American, barrier and butterfly options when the stock price follows a dynamics described by a jump-diffusion process. The resulting pricing equation which is in the form of a partial integro-differential equation is approximated in space using finite elements. Our methods require the computation of a single matrix exponential and we demonstrate using a wide range of numerical tests that the combination of exponential integrators and finite element discretisations with quadratic basis functions leads to highly accurate algorithms for cases when the jump magnitude is Gaussian. Comparison with other time-stepping methods are also carried out to illustrate the effectiveness of our methods.

Exponential operator splitting time integration for spectral methods

Pseudospectral spatial discretization by orthogonal polynomials and Strang splitting method for time integration are applied to second-order linear evolutionary PDEs. Before such a numerical integration can be used the original PDE is transformed into a suitable form. Trigonometric, Jacobi (and some of their special cases), generalized Laguerre and Hermite polynomials are considered. A double representation of a function (by coefficients of a polynomial expansion and by values at the nodes associated with a suitable quadrature formula) is used for numerical implementation so that it is possible to avoid calculations of matrix exponentials.

Exponential time integration and Chebychev discretisation schemes for fast pricing of options

We consider exponential time differencing (ETD) schemes for the numerical pricing of options. Special treatments for the implementation of the boundary conditions that arise in finance are described. We show that only one explicit time step computation gives unconditional second order accuracy for European, Barrier and Butterfly spread options under both Black–Scholes geometric Brownian motion model and Merton's jump diffusion model with constant coefficients. In comparison, the commonly used Crank–Nicolson scheme is shown to be only conditionally stable due to lack of L 0 -stability. Finally, we describe how the use of spectral spatial discretisation based on a Chebychev grid point concentration strategy gives fourth order accurate option prices for both the Black–Scholes and Merton's jump–diffusion model.

Effective Antenna Pattern of Scanning Radars

It is shown that weather radar signal averaging causes a broadened apparent pattern of the scanning radar antenna. The apparent pattern is obtained by convolving the original pattern with the averaging circuit impulse response. Echo displacement and pattern broadness are examined for square law, linear, and logarithmic detectors having exponential and finite time integrators.