Finite Difference

1
Introduction
2
What is Numerical Analysis

• Design and analysis of algorithms for solving
  mathematical problems in science and engineering

• Study of algorithms for solving problems of
  continuous mathematics

3
What is Scientific Computing ?
4
Classical Science vs Modern Methods
Test car models Simulation of Earthquakes and
Volcanoes Protein Folding Climate Change
Nature
Observation
Physical Experimentation
Theory
5
Evolution of Supercomputers

• 1906 Lee De Forest Electronic Valve

• 1936 Z1 Konrad Zuse calculations for Henschel
  Aircraft Company

• 1943 Church Turing Thesis Alan Turing and Alonzo
  Church

"I think there is a world market for maybe five
computers.", Thomas Watson, chairman of IBM.

• 1944 Harvard Mark 1 Howard Aiken and Grace
  Hopper gunnery and ballistic calculations

• 1946 ENIAC I John Mauchly and J Presper Eckert
  used for writing artillery-firing tables

• 1947 First Transistor (William B. Shockley, John
  Bardeen and Walter H. Brattain), Magnetic Drum
  Storage

• 1949-52 EDVAC von Neumann First Magnetic Tape

"Computers in the future may weigh no more than
1.5 tons. Popular Mechanics

• 1950 Alan Turing Test of Machine Intelligence

• 1954 IBM 650 first mass-produced computers

"I have traveled the length and breadth of this
country and talked with the best people, and I
can assure you that data processing is a fad that
won't last out the year. The editor in charge of
business books for Prentice Hall.
6

• 1958 Jack Kilby and Robert Noyce Integrated
  Circuit

• 1965 CDC 6600 Seymour Cray (first
  supercomputer)

• 1970 Unix Dennis Ritchie and Kenneth Thompson

• 1971 First Microprocessor developed by Intel

• 1976 Crayl first commercially developed
  supercomputers Seymour Cray

"There is no reason anyone would want a computer
in their home." Ken Olson, Digital Equipment Corp.

• 1978 8086 by Intel, first PC First Video Game

• 1981 Cosmic Cube Charles Seitz and Geoffery Fox

• 1984 Macintosh Released

• 1985 Microsoft Windows released

• 1986 Connection Machine, Thinking Machine
  Corporation parallel processing introduced

7

• 1989 World Wide Web Tim Berners-Lee

""Windows NT addresses 2 Gigabytes of RAM which
is more than any application will ever need".
Microsoft

• 1994 Beowulf Thomas Sterling and Don Becker
  NASAs Goddard Space Flight Center

• 1997-2000 ASCI Red, ASCI Blue Pacific, ASCI
  White IBM

• 2002 Earth Simulator NASDA, JAERI, and JAMSTEC

• 2005 Blue Gene IBM, MHD, ITER(Nuclear Fusion)

8
60 Years of Speed Increases
One Billion Times Faster!
9
Algorithms are Important

• Moore's Law(1965)--- the number of transistors
  on an integrated circuit (computing power)
  doubles every 24 months.
• Over 36 years, processor architecture goes
  through 18 doubling periods
• Algorithms produce an equal factor of speedup on
  a small problem much more on a larger problem

Speedup on a 3D Poisson problem
Many branches of computer science are being
tapped to create better algorithms
10
Steps in Computational Simulation

• Develop mathematical model (equations)
• Develop algorithms to solve equations numerically
• Implement algorithms in software
• Run software to simulate physical process
• Represent results
• Interpret and validate computed results
• Repeat .

11
Well-Posed Problem

• A Problem is well-posed if
• A solution exists
• Solution is unique
• Depends continuously on the input data
• (small perturbations to input do not cause abrupt
  change in solution)

Even if a problem is well-posed solutions can
still be sensitive to changes.
12
Solving Computational Problems

• Infinite Dimension to Finite Dimension
• Infinite Process to Finite Process
• Differential Equations to Algebraic Equations
• Nonlinear Problems to Linear Problems
• General Matrices to Matrices with Specific Form

Summary Complicated to Simple But at what cost ?
13
Approximations

• Approximations before computing
• Modeling
• Empirical measurements
• Previous Computations

• Approximations after computing
• Truncation
• Rounding

14
Example

• Let Surface area of earth be calculated as

A 4?(r2)
Earth is considered to be a sphere modeling
error
Value of r is an empirical measurement
measuring error
Value of ? is infinite has to be truncated
truncation error
Calculated in a computer rounding error
15
What is the Measurement of Error

• Absolute value is meaningful only in context
• Absolute Error approximate value-true value
• Relative ErrorAbsolute Error/True Value
• What is the catch here ???

We do not know the true value We generally
calculate bounds of error than exact value
16
Other Measurements

• If a value has relative error of 10-p, then its
  decimal representation has about p correct
  significant digits
• Precision number of digits with which a number
  is expressed
• Accuracy number of correct significant digits

17
Sources of Error

• Data Error Error due to approximation of input
  data
• Computational Error Error due to computation

g appx of function f a appx of data x
Total error g(a)-f(x)
g(a)-f(a) f(a)-f(x)
18
Example

• Calculate value of sin(?/8)

19
Computational Errors

• Truncation Error Approximations due to
  truncating infinite series
• Rounding Error Approximations due to
  finite-precision rounded arithmetic

20
Forward Error

• Compute value of a function yf(x)
• Forward error difference between true and
  calculated value of y
• ?yY-y
• Difficult to estimate forward error

21
Backward Error

• Calculate value of a function yf(x)
• Backward error difference between value of x
  that gives the computed value and actual value of
  x
• ?xX-x
• Where Yf(X)
• Why is backward error important ?