Extended Krylov subspaces approximations of matrix functions. Application to computational electromagnetics.

Vladimir Druskin,* Leonid Knizhnerman + and  Ping Lee*.

There is now a growing interest in the area of using Krylov subspace
approximations to compute the actions of matrix functions. The main
application of this approach is the solution of ODE systems, obtained
after discretization of partial differential equations by method
of lines.  In the event that the cost of computing the matrix inverse
is relatively inexpensive, it is sometimes attractive to solve the ODE
using the extended Krylov subspaces, originated by actions of both
positive and negative matrix powers. Examples of such problems can be
found frequently in computational electromagnetics.

In this presentation, we introduce an economical Gram-Schmidt
orthogonalization on the extended Krylov subspaces of a symmetric matrix.
An error bound for a family of problems arising from the elliptic
method of lines (i.e. the matrix square root and its stable exponentials)
is derived. The bound shows that, for the same approximation quality, the
diagonal variant of the extended subspace requires about the square
root of the dimension of the standard Krylov subspaces using only
positive or negative matrix powers.

Two applications arising from geophysical electromagnetics, one to the
solution of a 2.5-D elliptic problem for direct current potential and
another for 3-D eddy-current problem in conductive media attest to a
computational efficiency of the method.

* Schlumberger-Doll Research, Old Quarry Road,
            Ridgefield, CT 06877-4108,
+ Central Geophysical Expedition, Narodnogo Opolcheniya St.,
            House 40, Bldg. 3, Moscow 123298, Russia