UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 641 UCD Building, 1250 14th St., Denver

TIME: 1 pm

DATE: Friday, March 31, 2000

Gaussian spectral rules for second order finite-difference schemes

Vladimir Druskin
Schlumberger-Doll Research, Old Quarry Rd., Ridgefield CT 06877-4108

Leonid Knizhnerman 
Central Geophysical Expedition, Russia, 123298, Moscow, Narodnogo Opolcheniya St., house 40, building 3

Abstract

     The subject of this talk is targeted grid optimization for second order FD approximations of elliptic and hyperbolic problems arising in
remote sensing, control theory, etc., where the solution is needed only at few receiver points. The optimization can be viewed as an
extension of the conception of the Gaussian quadrature rules to the second order finite-difference schemes. A standard Gaussian k-point
quadrature for numerical integration is chosen to be exact for 2k polynomials, and an optimal grid with k nodes is chosen to match the
impedance at the receiver points for some 2k frequencies. To solve this problem we employ methods of rational approximation and linear
algebra traditionally used for optimization of iterative methods. The optimization yields exponential convergence of the impedance, i.e., the
standard second order scheme with the three-point stencil exhibits spectral superconvergence. The optimized scheme is applied to two- and
three- dimensional problems in electromagnetic and acoustic well logging. Our numerical experiments exhibit exponential superconvergence
at prescribed points (receivers), where the cost per grid node is close to that of the standard second order finite-difference scheme. We
observe more than one order speedup for practically important problems. 

     The problem of rational approximation of Markov (impedance) functions on a bounded interval of the real axis arises when
constructing optimal finite-difference grids for solving differential equations. Padé-Chebyshev approximation is a popular sort of rational
approximation due to its simplicity, though it is not optimal in general. 

     We show how to calculate a [(k-1)/k] Padé-Chebyshev approximant of a Markov function by means of a Lanczos process. We
formulate an error estimate rendering concrete Gonchar-Rakhmanov-Suetin asymptotical one. We also demonstrate that such an
approximation on a changed interval can produce a better approximant for a given interval. 

     Implementation aspects are considered. 

     Collaborators: Sergey Asvadurov (SLB), David Ingerman (Princeton-MIT) and Shari Moskow (UFL).