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DTSTART;VALUE=DATE:20121008T171500
DTEND;VALUE=DATE:20121008T171500
UID:5655@agenda.unifr.ch
DESCRIPTION:When the Laplace transform is applied to a semi-discrete parabolic PDE, the result is a\ncontour integral of Bromwich type that has to be computed numerically. The integrand of\nthis formula involves the computation of the resolvent of a typically large matrix A, which\ncan be expensive. Therefore, for the sake of computational efficiency the number of function\nevaluations has to be limited. This can be done via a judicious choice of quadrature scheme\nand a matching good contour. The choice of the trapezoidal rule combined with a Hankel\ncontour has stood the test of time and forms the focal point of this talk.\nStarting with the original proposal of A. Talbot in the mid-1970s (which was based on\nthe work of his student J. Green in the mid-1950s), we survey a number of such contours\nthat have been proposed in the literature. These contours are typically defined by a number\nof parameters that can be tuned for optimal accuracy. Assuming some information on the\nspectrum of A, we demonstrate how to find practical estimates for these parameters. A\nconsequence of this parameter tuning is that the error in the trapezoidal rule can be shown\nto decrease exponentially with the number of nodes in the rule (also known as geometric\nor spectral convergence). Thus it often happens that convergence to ten-digit accuracy or\nbetter can be achieved with as few as a dozen nodes in the trapezoidal rule.\nAdditional issues that will be addressed in the talk are the effect of non-normality\nin the matrix A (for example when solving a convection dominated problem), numerical\ninstability and the control of roundoff error, and the efficient computation of the resolvent\nvia recent techniques for the solution of shifted linear systems. Applications will be drawn\nfrom the area of mathematical finance (notably the work on the Black-Scholes and Heston\nequations done in collaboration with K. in ’t Hout of the University of Antwerp, Belgium).\n<br> [invited by J.-P. Berrut]
SUMMARY:JAC Weideman (Stellenbosch): Efficient Contours for the Numerical Computation of the Bromwich Integral
CATEGORIES:Colloque / Congrès / Forum
LOCATION:PER 08\, Phys 2.52\, Chemin du Musée 3\, 1700 Fribourg
URL;VALUE=URI:https://agenda.unifr.ch/e/fr/5655
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