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Topic: 4D Visualization
Replies: 9   Last Post: Apr 8, 1994 10:36 AM

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Evelyn Sander

Posts: 187
Registered: 12/3/04
4D Part 5: Grids in PDEs
Posted: Jun 28, 1993 11:26 AM
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"Instead of looking at 4-D as being very close to 3-D, you
can often get a better feel for 4-D by thinking of it as
quite a bit along the way to infinite-D. Understanding
what happens in extremely high dimensions can clarify
the emerging trends that make 4-D a _different_ place to
live than 2- and 3-D," says Paul Burchard of the Geometry
Center. Burchard is developing software to study
Partial Differential Equations (PDEs) which are
related to differential geometry.

This article consists of observations on the problems encountered in
higher dimensions when using one of the standard methods of PDEs,
namely grids. Numerical calculations of solutions to differential
equations often involve a rectangular or triangular grid, using only
values of derivatives on the grid points to get an approximation to
the function at those grid points. I will describe some of the
problems with rectangular grids in higher dimensions. These same
problem occur in all shapes of grid, not just rectangular ones.

In a rectangular grid with k equally spaced grid points
per unit length in each spatial direction, an n
dimensional cube requires k^n grid points. In other
words, the number of grid points grows exponentially
with dimension. Thus using schemes with grid points very
quickly becomes prohibitively costly in terms of
storage and time as the dimension increases.

Aside from the large number of points in a rectangular
grid, Burchard observes another problem with this
approach of having the grid points be the vertices of a
hypercube. Namely, as n gets large, the corners of a n
dimensional hypercube stick out more and more, making it
a pointy and strangely shaped object.

In terms of distance, the following shows that corners
get further away: The length of a diagonal of a unit nD
hypercube is the square root of n (sqrt(n)). (This can be
deduced from the Pythagorean theorem and the fact that
the length of each side is one.) The distance from the
center to each corner is sqrt(n)/2. As dimension
increases, the unit cube has corners which stick out more
in linear distance.

Another measurement of the oddness of the shape of the
cube is to look at how much of the volume is in the corners.
Compare the volume of the nD hypercube to the inscribed n
dimensional ball. In some sense, the inscribed ball cuts
out the corners of the cube, leaving only the middle. If
the volume of this ball gets smaller, since the hypercube
is always unit volume, the volume contained in the
corners of the cube must be increasing as dimension
increases. Through some calculations which I will
describe separately, the inscribed nD ball has volume
V(n)=(V(n-2)*pi)/(2*n). Note that the recursive
formula contains an n in the denominator, so there is some
kind of factorial decrease in volume. Thus as n
increases, the volume of the inscribed ball quickly
decreases. This means that more and more of the volume of
the unit cube is contained away from the center, another
indication that the corners of the high dimensional cube
become more pointy.

To give a more detailed picture, it is not only the corners
which stick out in the hypercube; all of the "edges" stick
out to progressively larger extent as the dimension
increases. For example, in the three dimensional cube,
the faces (two dimensional edges) are distance 1/2 from
the center, the edges (one dimensional edges) are
distance sqrt(2)/2 from the center, and the corners
(zero dimensional edges) are furthest away, namely
sqrt(3)/2 from the center. In general, the k dimensional
edges of the n dimensional cube stick out distance
sqrt(n-k)/3 from the center.

By the above, perhaps grids are not the best method to
solve high dimensional PDEs. In fact, even in relatively
low dimensional cases, it is necessary to resort to other
techniques. According to Burchard, "From the
perspective of trying to write fast accurate code in
differential equations, five is in a practical sense
most of the way to infinity; in other words, it is
practically impossible to write the code I would like and
have it run in any reasonable amount of time. I think 4-D
will still be feasible but slow; 5- or 6-D is where it
starts to become practically impossible with
grid-based techniques."






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