@Article{ Holtz.Kunoth:2004,
author = {Markus Holtz and Angela Kunoth},
title = {B-spline-based Monotone Multigrid Methods},
note = {Also as SFB 611 preprint No. 0252, 2005},
journal = {SIAM J. Numer. Anal.},
volume = {45},
number = {3},
pages = {1175-1199},
year = {2007},
abstract = {For the efficient numerical solution of elliptic
variational inequalities on closed convex sets, multigrid
methods based on piecewise linear finite elements have been
investigated over the past decades. Essential for their
success is the appropriate approximation of the constraint
set on coarser grids which is based on function values for
piecewise linear finite elements. On the other hand, there
are a number of problems which profit from higher order
approximations. Among these are the problem of prizing
American options, formulated as a parabolic boundary value
problem involving Black-Scholes' equation with a free
boundary. In addition to computing the free boundary, the
optimal exercise prize of the option, of particular
importance are accurate pointwise derivatives of the value
of the stock option up to order two, the so-called Greek
letters.
In this paper, we propose a monotone multigrid method for
discretizations in terms of B-splines of arbitrary order to
solve elliptic variational inequalities on a closed convex
set. In order to maintain monotonicity (upper bound) and
quasi-optimality (lower bound) of the coarse grid
corrections, we propose an optimized coarse grid correction
(OCGC) algorithm which is based on B-spline expansion
coefficients. We prove that the OCGC algorithm is of
optimal complexity of the degrees of freedom of the coarse
grid and, therefore, the resulting monotone multigrid
method is of asymptotically optimal multigrid complexity.
Finally, the method is applied to a standard model for the
valuation of American options. In particular, it is shown
that a discretization based on B-splines of order four
enables us to compute the second derivative of the value of
the stock option to high precision.},
pdf = {http://wissrech.ins.uni-bonn.de/research/pub/holtz/holtzkunoth_rev.pdf}
,
annote = {article,ALM}
}