Pseudospectra of the convection-diffusion operator
SIAM Journal on Applied Mathematics
The Solution of Parametrized Symmetric Linear Systems
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
A column pre-ordering strategy for the unsymmetric-pattern multifrontal method
ACM Transactions on Mathematical Software (TOMS)
Algorithm 832: UMFPACK V4.3---an unsymmetric-pattern multifrontal method
ACM Transactions on Mathematical Software (TOMS)
On the numerical inversion of the Laplace transform of certain holomorphic mappings
Applied Numerical Mathematics
SIAM Journal on Numerical Analysis
A Spectral Order Method for Inverting Sectorial Laplace Transforms
SIAM Journal on Numerical Analysis
Stability of ADI schemes applied to convection-diffusion equations with mixed derivative terms
Applied Numerical Mathematics
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Evaluation of generalized Mittag---Leffler functions on the real line
Advances in Computational Mathematics
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A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., 30 (2010), pp. 334-350] for integrating semidiscrete advection-diffusion PDEs is improved and extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parameters are derived theoretically and tested numerically. An improvement on the existing method is the use of Krylov methods for the shifted linear systems, the solution of which represents the major computational cost of the algorithm. A parallel implementation is also considered. Test examples presented are the Black-Scholes PDE in one space dimension and the Heston PDE in two dimensions, for both vanilla and barrier options. In the Heston case efficiency is compared to ADI splitting schemes, and experiments show that the contour integral method is superior for the range of medium to high accuracy requirements.