Research Group of Prof. Dr. M. Griebel
Institute for Numerical Simulation

  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
		  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 = { 1},
  annote = {article,ALM}