UNIVERSITY OF COLORADO AT DENVER

PLACE: Mathematics Conference Room 626 UCD Building, 1250 14th St., Denver

TIME: noon (Refreshments served at 11:45 am)

DATE: February 1, 1999

SPEAKER:  Ben Fox

TITLE:    Quasi-Monte Carlo for PDEs


ABSTRACT:


     Like ordinary standard Monte Carlo (MC),
     quasi-Monte Carlo (QMC) estimates an integral
     by averaging the integrand at certain points.
     The latter, however, uses points that are
     "better spaced".  In traditional QMC, these
     are selected deterministically.
     Perhaps contrary to intuition,
     in more than one dimension, grids are not
     "well spaced".  We take the domain of the integral
     as the unit cube, if necessary mapping invertibly
     from another domain.

     How does this connect to PDEs ?
     An obvious good case for MC or QMC occurs when we want
     the solution only at a few selected points.
     More generally, we may want to estimate functionals of
     the solution.
     For example, the integral can be an L_p-norm of the
     solution or of the "error". 
     In the latter case, the error can be measured relative
     deviations from
     a known solution (for checking a more general code)
     or from an estimated solution (to assess its "reliability").

     For stochastic PDEs (with noisy coefficents and/or
     right side generated by a random field), the integral
     can be the expectation of the error.  For each input
     realization of the field, a deterministic PDE-solver
     (or, for the cases above, QMC)
     produces an estimated solution.  Average these estimates
     to get the overall estimate.

     In the talk, I deal with the Gaussian case.
     Amazing to me (but not due to me),
     the pointwise solution to
     both generalized heat equations
     of a certain form and Poisson's equation with enough
     smoothness can be recast in terms of Brownian motion.
     For stochastic PDEs, I take the random field to
     be Gaussian.

     I show how to structure the simulation to increase
     efficiency of QMC.  A "randomized" version of QMC
     will be indicated.  This gives a way to estimate error,
     via iid copies of QMC "blocks",
     and also takes greater advantage of smoothness,
     increasing simulation efficiency by another order
     of magnitude.