270 research outputs found
Zooming from Global to Local: A Multiscale RBF Approach
Because physical phenomena on Earth's surface occur on many different length
scales, it makes sense when seeking an efficient approximation to start with a
crude global approximation, and then make a sequence of corrections on finer
and finer scales. It also makes sense eventually to seek fine scale features
locally, rather than globally. In the present work, we start with a global
multiscale radial basis function (RBF) approximation, based on a sequence of
point sets with decreasing mesh norm, and a sequence of (spherical) radial
basis functions with proportionally decreasing scale centered at the points. We
then prove that we can "zoom in" on a region of particular interest, by
carrying out further stages of multiscale refinement on a local region. The
proof combines multiscale techniques for the sphere from Le Gia, Sloan and
Wendland, SIAM J. Numer. Anal. 48 (2010) and Applied Comp. Harm. Anal. 32
(2012), with those for a bounded region in from Wendland, Numer.
Math. 116 (2012). The zooming in process can be continued indefinitely, since
the condition numbers of matrices at the different scales remain bounded. A
numerical example illustrates the process
On the expected uniform error of geometric Brownian motion approximated by the L\'evy-Ciesielski construction
It is known that the Brownian bridge or L\'evy-Ciesielski construction of
Brownian paths almost surely converges uniformly to the true Brownian path. In
the present article the focus is on the error. In particular, we show for
geometric Brownian motion that at level , at which there are points
evaluated on the Brownian path, the expected uniform error has an upper bound
of order , or equivalently, . This upper bound matches the known order for the expected uniform error
of the standard Brownian motion. We apply the result to an option pricing
example
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