CS205 – Class 15
Covered in class: 3, 4, 5
Readings: 8.7, 9.1, 9.2
1. Can extend quadrature to
higher dimensions
a. One dimension
( )
b
a
f x dx
 subdivide [ , ]
a b
into smaller intervals
b. Two dimensions
( , )
A
f x y dA
 subdivide A into rectangles or triangles
c. Three dimensions
( , , )
V
f x y z dV
 subdivide V into boxes or tetrahedral
d.
Monte Carlo methods
– usually used in higher dimensions
i. Random or pseudo random numbers are used to generate sample
points that are averaged and multiplied by the element “size” (e.g.
length, area, volume)
ii. Error decreases like
1/2
n
where n is the number of sample points
1. 100 times more points are needed to gain one more digit of
accuracy
2. Slow convergence, but independent of the number of
dimensions
3. Not competitive for lower dimensional problems, but the only
alternative for higher dimensional problems
2.
Richardson extrapolation
eliminate the leading order error term using 2
calculations.
a. Start an integration scheme with some step size
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 Fall '07
 Fedkiw
 Higher Dimensions, Partial differential equation, step size, order error term

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