Full Index for Slitex Foilset CPS615 Numerical Integration Module

Slitex Foilset CPS615 Numerical Integration Module

This module describes numerical integration covering first the simplest trapezoidal and Simpson's rule. This is followed by their general Newton-Cotes extensions and adaptive Simpson's Rule. Gaussian integration is briefly described. Most of the presentation is devoted to Monte Carlo methods including Importance Sampling and Metropolis Method. Examples are given from financial modelling, experimental and theoretical high energy physics.

1:Introduction to Numerical Integration
2:NI.1: Newton-Cotes Formulae
3:Linear Equations for Coefficients
4:Examples of Newton Cotes Rules I
5:Examples of Newton Cotes Rules II
6:Formulae for Examples: Trapezoidal and Simpson's Rules
7:Summary of Newton Cotes Rules
8:Errors in Newton Cotes Formulae
9:Use of High Order Newton Cotes
10:Strategies for Manipulating Integrals before using Standard Numerical Integration
11:Criteria for Choosing Manipulation
12:Dangers in Integrating Non-Analytic Functions
13:Why Do We Have Problems with Non-Analytic Functions?
14:Combining Analytic and Numerical Integration
15:Cunning Choice of Weight Functions
16:Change of Variable Before Numerical Integration---I
17:Change of Variable Before Numerical Integration---II
18:How Do You Choose Correct Strategy?
19:Hybrid Strategy Example (continued)
20:NI.2: Romberg or Adaptive Integration
21:NI.3: Gaussian Integration
22:Some Mathematics Underlying Gaussian Integration
23:More Mathematics Deriving Gaussian Integration Formulae
24:Basic Gaussian Integration
25:When Should You Use Gaussian?
26:Three Choices of in Numerical Integration
27:NI.4: Parallel Computing and Integration
28:Issues for Parallel Computing
29:Two-Dimensional Integrals
30:NI.5: Monte Carlo Integration
31:Basic Formulation of Integral as a Mean (Expectation Value)
32:Basic Monte Carlo Approach to Integration
33:Why Monte Carlo Methods Are Best in Multidimensional Integrals
34:Best Multidimensional Integration Formulae
35:Distribution of Points in Two-dimensional Integral Done by Newton-Cotes Style Formulae
36:Distribution of Points in Two-dimensional Integral Done by Monte Carlo
37:Use of Bounding Boxes to Calculate --- I
38:Use of Bounding Boxes to Calculate --- II
39:Use of Bounding Boxes in Complicated Geometries --- I
40:Use of Bounding Boxes in Complicated Geometries --- II
41:IMPORTANCE Sampling Basic Theory
42:Choice of Importance Sampling Weight Function --- I
43:Choice of Importance Sampling Weight Function --- II
44:Monte Carlo Approach to Discrete Integrals
45:Why Use Monte Carlo for Summations?
46:Example of using Monte Carlo for Summations
47:The Wrong Way to Perform Multiple Monte Carlo
48:Stock Market Example of Multiple Monte Carlos --- I
49:Stock Market Example of Multiple Monte Carlos --- II
50:Number of Points Needed in Joint Monte Carlo
51:Accept/Reject Method for Generating General Probability Distributions
52:Estimate of Maximum in Accept/Reject Method
53:Introduction to Metropolis Method
54:The Metropolis Procedure
55:Why Metropolis Method Works
56:Monte Carlo Examples Example 1: An Experimental Physics Application
57:A High Energy Experiment Scenario
58:An Experimental Physics Monte Carlo
59:Double Monte Carlo's Again --- I
60:Double Monte Carlo's Again --- II
61:A Monte Carlo Event
62:Uniform Weight Events
63:What Happens If You Miscalculate Maximum Weight in Accept/Reject Approach
64:Example 2: Parallel Computing for ``Event'' Monte Carlos
65:Example 3: Lattice Monte Carlo Theoretical Physics
66:Choice of Points in Lattice Monte Carlo
67:Pictorial View of Lattice Monte Carlo Integrands
68:Metropolis and Heat Bath Methods
69:Calculation of Observables
70:Example 4: Parallel Computing for Lattice Theory
71:A Problem Lattice Decomposed Onto a 64-node Machine Arranged as a Machine Lattice
72:The 16 Time and Eight Internal Gluon Degrees of Freedom

Geoffrey Fox, 73:Northeast Parallel Architectures Center at Syracuse University, gcf@npac.syr.edu