Talk
Abstract:
Seminar
on Industrial Problems
Gaussian Spectral Rules for Second Order Finite-difference
Schemes
December 3, 1999
Presented
by:
Vladimir Druskin
Schlumberger-Doll Research
druskin@ridgefield.sdr.slb.com
570
Vincent Hall
10:10 am
The subject of this talk is targeted grid optimization for elliptic
and time-domain problems arising in remote sensing (geophysics,
computed topography, 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 quadratures 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, linear algebra
and inverse problem theory. 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.
Collaborators: Sergey Asvadurov (SLB), David Ingerman (Princeton-MIT),
Shari Moskow (UFL) and Leonid Knizhnerman (CGE).
Materials
Used for the Talk
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